CARLEMAN ESTIMATES AND UNIQUE CONTINUATION
FOR SECOND ORDER PARABOLIC EQUATIONS WITH
NONSMOOTH COEFFICIENTS
HERBERT KOCH AND DANIEL TATARU
Abstract. In this work we obtain strong unique continuation results for
variable coefficient second order parabolic equations. The coefficients in
the principal part are assumed to satisfy a Lipschitz condition in x and a
Hölder C
3 condition in time. The coefficients in the lower order terms, i.e.
the potential and the gradient potential, are allowed to be unbounded and
required only to satisfy mixed norm bounds in scale invariant L
x spaces.
1. Introduction
The evolution of the understanding of the strong unique continuation prob-
lem for second order parabolic equations mirrors and is closely related to the
corresponding strong unique continuation problem for second order elliptic
equations. Consequently, we begin with a brief overview of the latter problem.
To a second order elliptic operator ∆g = ∂ig
ij∂j and potentials V , W in R
we associate the elliptic equation
(1.1) −∆gu = W∇u+ V u
Given a function u ∈ L2loc(Rn) and x0 ∈ Rn we say that u vanishes of infinite
order at x0 if there exists R so that for each integer N we have
(1.2)
B(x0,r)
|u|2 dx ≤ c2Nr2N , r < R
The elliptic strong unique continuation property ESUCP has the form
Let u be a solution to (1.1) which vanishes of infinite order
at x0. Then u(x) = 0 for x in a neighborhood of x0.
ESUCP
ESUCP type results go back to the pioneering work of Carleman [5] in
dimension n = 2, later extended to higher dimension in by Aronszajn and
collaborators [3], [4]. Their results apply to Lipschitz metrics g but only mildly
unbounded potentials V and W . A key ingredient in their approach was
to obtain a class of weighted L2 estimates which were later called Carleman
estimates. The simplest Carleman estimate has the form
‖|x|−τu‖L2 . ‖|x|2−τ∆u‖L2
The first author was supported in part by the DFG grant KO1307/5-3. The second
author was supported in part by the NSF grant DMS-0301122. Part of the work was done
while the first author was supported by the Miller Institute for Basic Research in Science.
http://arxiv.org/abs/0704.1349v2
and holds uniformly for τ away from ±(n−2
+ N). This restriction is related
to the spectrum of the spherical Laplacian.
Adding some extra convexity to the |x|−τ weight makes the above estimate
more robust and allows one to also use it in the variable coefficient case. The
role played by the convexity was further clarified and explained by Hörmander
[12], [13], who introduced the pseudoconvexity condition for weights as an
almost necessary and sufficient condition in order for the Carleman estimates
to hold.
The problem becomes more difficult if one seeks to work with unbounded
potentials V in or near the scale invariant L
2 space. There the L2 Carle-
man estimates are insufficient. Instead the key breakthrough was achieved
in Jerison-Kenig [15], where the L2 Carleman estimates are replaced by Lp
estimates of the form
‖|x|−τu‖
. ‖|x|−τ∆u‖
Relevant to the present paper is also the alternative proof of this result which
was given by Jerison [14], taking advantage of Sogge’s [25] spectral projection
bounds for the spherical Laplacian. In the case of operators with smooth
variable coefficients Lp Carleman estimates were first obtained by Sogge [26],
[27].
Working with gradient potentials in the scale invariant space W ∈ Ln intro-
duces an added layer of difficulty. There not even the Lp Carleman estimates
can hold. Wolff’s solution to this in [29] is a weight osculation argument, which
allows one to taylor the weight in the Carleman estimate to the solution u,
producing estimates of the form
‖e−τφ(x)u‖
+ ‖e−τφ(x)W∇u‖
. ‖eτφ(x)∆u‖
where the choice of φ depends on both u, W and τ .
Finally, the authors’s article [17] combines the ideas above into a nearly
optimal scale invariant ESUCP result for the elliptic problem, with (i) a
Lipschitz metric g, (ii) an L
2 potential V , and (iii) an almost Ln gradient
potential W . The present paper is the counterpart of [17] for the parabolic
strong unique continuation problem.
We consider the second order backwards parabolic operator
(1.3) P = ∂t + ∂kg
kl(t, x)∂l
in R × Rn and potentials V,W1,W2. To these we associate the parabolic
equation
(1.4) Pu = V u+W1∇xu+∇x(W2u)
Given a function u ∈ L2loc and (t0, x0) ∈ R × Rn we say that u vanishes of
infinite order at (t0, x0) if there exists R so that for each integer N we have
(1.5)
B(x0,r)
|u|2 dxdt ≤ c2Nr2N , r < R
Alternatively we may only require that x→ u(t0, x) vanishes of infinite order
at x0, i.e.
(1.6)
B(x0,r)
|u(t0, x)|2 dx ≤ c2Nr2N , r < R
The two conditions (1.5) and (1.6) are largely equivalent provided that the
coefficients gkl have some uniform regularity as t → t0. However, our as-
sumptions in this article are not strong enough to guarantee this, therefore we
consider the two separate cases.
Now we can define the strong unique continuation property SUCP(I) :
Let u be a solution to (1.4) which vanishes of infinite order
at (t0, x0). Then u(t0, x) = 0 for x in a neighborhood of x0.
SUCP(I)
and the slightly stronger variant
Let u be a solution to (1.4) so that x→u(t0, x) vanishes of infinite
order at x0. Then u(t0, x) = 0 for x in a neighborhood of x0.
SUCP(II)
The study of unique continuation for parabolic equations began with early
work of Mizohata [21] and Yamabe [30], followed by Saut-Scheurer [23]; Lp
Carleman estimates were first obtained by Sogge [24].
The study of the parabolic strong unique continuation problem began with
work of Lin [20] who considered SUCP(II) for the heat equation with W = 0
and V bounded and time independent. This continued with work of Chen [6]
and Poon [22]. Fernández [11], and Escauriaza, Fernández and Vessella [7] con-
sidered SUCP(II) under various assumptions on the coefficients and pointwise
bounds for W = 0 and V . It is a consequence of Alessandrini and Vessella [1]
that SUCP(I) andSUCP(II) are equivalent under weak assumptions on the
coefficients, and they derived SUCP(II) in [2] for bounded W and V .
The article of Poon [22] contributed to clarifying the correct form of the
L2 Carleman estimates for the parabolic strong unique continuation problem
in Escauriaza and Fernández’s work [9]. In the simplest form, these have the
‖t−τ−
8t u‖L2 . ‖t−τ+
8t (∂t +∆)u‖L2
and hold uniformly with respect to τ away from (2n + N)/4. This restriction
is connected with the spectral properties of the Hermite operator.
The Lp spectral projection bounds for the Hermite operator were indepen-
dently obtained by Thangavelu [28] and Kharazdhov [16]; see also the sim-
plified proof in the authors’s paper [19]. These bounds were essential in the
proof of Lp Carleman inequalities for the heat operator of Escauriaza [8] and
Escauriaza and Vega [10] which yield SUCP(I) when g = In, W = 0 and
V ∈ L1L∞ + L∞Ln/2.
Our aim is to prove that SUCP(I) respectively, SUCP(II) hold under
sharp scale invariant assumptions on the metric g and Lp conditions on the
potentials V and W1,W2. The contribution of this work is comparable to [17]
for the elliptic problem: We study almost optimal conditions on
(1) the coefficients g
(2) the potential V
(3) the gradient potentials Wj
The combination of rough variable coefficients and Lp conditions on the po-
tential seems to be new. Also, to the best of our knowledge this is the first
result on unique continuation for parabolic problems under Lp conditions on
the coefficients of the gradient term.
For simplicity we always assume that t0 = 0, x0 = 0. For SUCP(I) it is
natural1 to consider a larger class of operators P which have the form
(1.7) P =
+ ∂kg
kl∂l +
dkl∂l + ∂ld
where (gkl), (dkl) and (ekl) are real valued and (gkl) and (ekl) are symmetric.
Then simple scale invariant assumptions for the coefficients would be
(1.8) ‖d‖L∞ + ‖(t+ x2)
2∂xd‖L∞ + ‖t∂td‖L∞ ≪ 1
Here and in the sequel d stands for a generic coefficient of the form gkl − δkl,
dkl and ekl. For V and W1,2 we could consider conditions of the form
(1.9) ‖V ‖L1L∞+L∞Ln/2 ≪ 1,
(1.10) ‖W1,2‖L2L∞+L∞Ln ≪ 1.
Here and in the sequel we use the notation LpLq = L
The situation is however more complex and we may take (1.8) to (1.10)
only as guidelines. We will have to strengthen (1.8) to include some dyadic
summability. on the other hand we are able to weaken the time differentiability
to a C
3 Holder condition on small time scales.
We are also able to slightly weaken (1.9) almost to uniform bounds on
dyadic sets. However, we are unable to use mixed norms for W1 and W2, and
we restrict ourselves to a summable Ln+2 norm in dyadic sets.
To state our assumptions on g, V , W1 and W2 we consider a double infinite
dyadic partition of the space,
(1.11) R+ × Rn =
where
(1.12) Aij = {(t, x) ∈ R+ ×Rn | e−4i−4 ≤ t ≤ e−4i, ej ≤ 1+ 2|x|t−
2 ≤ ej+1}.
Consider the subset of indices
(1.13) A = {(i, j) : j ≤ 2i+ 2}
defining a partition of the cylinder
Q = [0, 1]× B(0, 1)
1This becomes clearer later on after a change of coordinates and conjugation with respect
to a Gaussian weight
Define
(1.14) A(τ) = {(i, j) ∈ A : 4i ≥ ln τ + 1, j ≤ 1
ln τ + 2}
which corresponds to a partition of the cut parabola
Qτ = {(t, x) : |x|2 ≤ τt ≤ 1}.
t = τ−1
|x| = 1
Figure 1. The cut parabola
We also consider a decomposition of Q into dyadic time slices
Ai = [e
−4i−4, e−4i]× B(0, 1)
and a similar partition of the cut parabola Qτ into the sets
Aτi = Ai ∩Qτ
Given a function space X and 1 ≤ q < ∞ we introduce the Banach spaces
lq(A, X) with norms
‖V ‖q
lq(A,X)
i,j∈A
‖V ‖q
X(Aij)
In a similar manner we define the spaces l∞(A, X).
Within the sets Aij we define the modulus of continuity (mij) in time
mij(ρ) = e
4iρ+ e
(2i−j)ρ
and denote by C
t the space of continuous functions with finite seminorm
= sup
t1,t2,x
|u(t1, x)− u(t2, x)|
mij(|t1 − t2|)
For the reader’s convenience we note that within Aij we have e
4i ≈ t−1 and
(2i−j) ≈ (t+ |x|2)− 13 .
For the coefficients of the operator P in (1.7) we change the condition (1.8)
(1.15) sup
‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖Cmijt (Aij) ≪ 1.
where we note that ej−2i ≈ (t + x2) 12 in Aij .
The pointwise bound for g − In in (1.15), namely
(1.16) sup
‖g − In‖l1(A(τ),L∞) ≪ 1
is not really needed for our results. It can be always obtained from the other
bounds after a change of coordinates. This is discussed in the appendix.
The assertion (1.15) is satisfied for g ∈ Lipx∩C
t provided that g(0, 0) = In.
Indeed by scaling we may assume that the Lipx ∩ C
t norm is small therefore
it suffices to compute
‖(t + |x|2)
3‖l1(A(τ),L∞) ≤
i≥ln τ
j≤ln(τ)/2+2
(j−2i) . 1.
For the potential V we consider:
‖V ‖l∞(A,L1L∞+L∞Ln/2) ≪ 1 for n > 2
‖V ‖l∞(A,L1L∞+LpLp′ ) ≪ 1 for n > 2, 1 ≤ p <∞,
‖V ‖l∞(A,L1L∞+L2L1) ≪ 1 for n = 1
(1.17)
where p′ in the second line is the dual exponent. In addition we require that
(1.18) sup
‖χiV ‖L1L∞+L∞Ln/2 ≪ 1 n > 2
with the obvious modifications for n = 1, 2, where χi is the characteristic
function of the set
{(t, x) : e−4i−4 ≤ t ≤ e−4i, t−1/2|x| ≤ i}.
Both (1.17) and (1.18) are fulfilled if V ∈ L1L∞ + L∞Ln/2 with small norm.
Finally for the gradient potentials W1,2 we introduce the summability con-
dition with respect to time slices
(1.19) sup
‖W1,2‖Ln+2(Aτ
) ≪ 1.
As a consequence of this we note the uniform bound
(1.20) sup
‖W1,2‖Ln+2(Ai) ≪ 1.
Now we can state our main results.
Theorem 1. Let P be as in (1.7) with coefficients satisfying (1.15). As-
sume that the potentials V and W1,2 satisfy (1.17), (1.18) and (1.19). Then
SUCP(I) holds at (0, 0) for H1 functions u satisfying (1.4).
It is part of the conclusion that the trace of u at t = 0 exists near x = 0.
The assumptions on the operator seem to be too weak to imply existence of a
trace in general. More precisely shall prove
‖u(t, .)‖L2(B(0,1/8)) . e−
for some δ > 0.
The C
3 Hölder regularity in time for the metric g seems so be new, improv-
ing the C
2 Hölder regularity in [9]. It is not clear to the authors whether this
condition is optimal or not.
We may replace the assumptions by stronger translation invariant assump-
tions,
(1.21) ‖g‖Lipx + ‖g‖
(1.22) ‖V ‖L1L∞+L∞Ln/2 ≪ 1
(1.23)
‖W1,2‖Ln+2(Ai) . 1
Then we also obtain a stronger conclusion.
Theorem 2. Let P be as in (1.3) with coefficients as in (1.21). Assume that
the potentials V and W satisfy (1.22) respectively (1.23). Then SUCP(II)
holds at (0, 0) for H1 functions u satisfying (1.4).
We remark that the condition (1.21) is really too strong, and that with some
additional work (see Remark 2.3) one can bring it almost to the level of (1.15).
Precisely, it suffices to replace (1.15) by
(1.24) sup
‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖
t (Aij)
where the slightly stronger time continuity modulus m2ij is given by
m2ij(ρ) = e
4i−2jρ+ e
(2i−j)ρ
However, we cannot keep the additional terms in (1.7), because we need to
be able to meaningfully talk about the trace of the solution at time t = 0.
Both theorems are consequences of quantitative estimates, which also imply
weak unique continuation under the assumptions of Theorem 2:
Let u be a solution to (1.4) for which u(t0, .) vanishes in
the closure of an open set. Then (u(t0, .) vanishes in a
neighborhood of the closure.
If u satisfies the assumptions and vanishes in an open set U , then it vanishes
in the time slices t = t0 in an open neighborhood of the closure of Ut0 = {(x :
(t, x) ∈ U}.
Theorems 1 and 2 are nontrivial consequences of a Carleman inequality. To
state a first version of the Carleman inequality we introduce an additional fam-
ily B(τ) of sets which is a partition of the cylinder [0, τ−1)×B(0, 1), consisting
(1.25) Aij , ln τ ≤ 4i ≤ τ 1/2, 0 ≤ j ≤ ln τ/2 + 2,
(1.26) [e−4i−4, e−4i]× B(0, e−2iτ 1/2), 4i > τ
(1.27) Aij , ln τ ≤ 4i, ln τ/2 ≤ j ≤ 2i.
This partition is coarser than the partition of the same cylinder into the sets
Aij . This is the reason why we need the assumption (1.18). More precisely
Assumptions (1.17) and (1.18) imply
(1.28) ‖V ‖l∞(B(τ),L1L∞+L∞Ln/2) ≪ 1.
Theorem 3. Let τ0 ≫ 1, ε > 0 and P as in (1.7) with coefficients satisfying
(1.15). Suppose that W1,2 satisfY (1.19) with constants depending on ε and
τ0. Then there exists C > 0 such that for all τ ≥ τ0 the following is true:
Suppose that v ∈ L2(H1) is compactly supported in [0, 8τ−1) × B(0, 2) and
that it vanishes of infinite order near (0, 0). Then we can find a function
φ ∈ C∞([0, 8τ−1]×B(0, 1)\{0, 0}) and h ∈ C∞(R+) which satisfy
(1.29) τ ≤ h′ ≤ (1 + ε)τ
(1.30)
∣∣∣∣φ(x, t)−
h(− ln t)− x
)∣∣∣∣ ≤ ε
such that
‖eφv‖
l2(B(τ),L∞L2∩L2L
n−2 ))
≤ C‖eφ(P +W1∇ +∇W2)v‖
l2(B(τ),L1L2+L2L
n+2 )
for n ≥ 3, respectively
‖eφv‖l2(B(τ),L∞L2∩Lp′Lq′ ) ≤ C‖e
φ(P +W1∇+∇W2)v‖l2(B(τ),L1L2+LpLq),
for n = 2, 1
, 2 < p, and
‖eφv‖l2(B(τ),L∞L2∩L4L∞) ≤ C‖eφ(P +W1∇ +∇W2)v‖l2(B(τ),L1L∞+L4/3L1)
for n = 1.
The statement of the Carleman inequality is involved for several reasons.
The weight t−τe−|x|
2/8t, which works for the constant coefficient case, has to be
modified so that it has more convexity in order to handle variable coefficients,
spatial localization and the gradient potential. However the polynomial growth
in time (imposed by the assumption of vanishing of infinite order) limits the
available amount of convexity; this is the origin of the l1 summability in (1.16),
(1.20), and to a lesser extend of (1.18).
If W = 0 then the Carleman inequality holds for a large explicit class
of weights eφ. This cannot be possibly true for general gradient potentials.
Instead, we are only able to prove that there exists some weight function φ,
which now depends on τ , u andW , for which the uniform Carleman inequality
holds. This strategy goes back to the seminal work of T. Wolff [29] and has
been used by the authors for the elliptic problem [17].
The partition Aij is much finer than the dyadic decomposition in t only,
which would correspond to the dyadic decomposition in the elliptic case. We
are only able to localize the estimates to the sets Aij if we make the weight
function sufficiently convex. We can do this for many Aij, but not for all of
them. The sets (1.26) correspond directly to the assumption (1.18). We need
to have control of the L1L∞+L∞Ln/2 norm of V in sets which are not smaller
than those of the partition in (1.25), (1.26) and (1.27).
We have stated Theorem 3 in a simpler form which suffices to derive Theo-
rem 1 and Theorem 2. However the full estimate we prove is stronger in that
it also contains precise L2 bounds. These are essential for the localization and
perturbations techniques we use.
The strategy of the proof is the same as in [17]:
(1) We construct families of pseudoconvex weights and derive L2 Carle-
man inequalities. The convexity of weights determines the space-time
localization scales and the admissible size of perturbations.
(2) We enhance the above L2 Carleman inequalities to include Lp esti-
mates. Due to the L2 localization it suffices to do this in small sets.
This allows us to use perturbation arguments starting from the case of
the heat equation with the weight t−τe−|x|
2/8t.
(3) Lp estimates for the spectral projections to spherical harmonics im-
ply the Lp Carleman inequalities in the elliptic case. Here spectral
projection for the Hermite operator play a similar role.
(4) Finally we include Wolff’s osculating argument into the scheme in order
to handle the gradient potentials. The efficiency of this part depends
on the flexibility in the choice of the weight functions.
The complexity of the weights and the L2 Carleman estimates comes mainly
from the geometry of the classical harmonic oscillator. Orbits are contained in
a sphere in R2n. The projection down in the x space is a ball, where frequency
variables have a different behavior in radial and angular directions and near
the boundary of the the ball. It turns out that our analytic estimates reflect
these features.
2. Proof of Theorem 1 and 2
In this section we prove Theorem 1 and 2 assuming Theorem 3. The relation
between Carleman estimates and unique continuation is fairly straightforward
in the elliptic case. In the parabolic situation the argument is less direct due
to the more complex geometry of the level sets of the weight functions.
It is a standard consequence of a localized energy inequality that for the
parabolic equation (1.4) u(t) and its gradient can be controlled by L2 norms
of u.
Proposition 2.1. Let n ≥ 3 and suppose that v solve the parabolic equation
vt + ∂ka
kl∂lv =W1∇v +∇(W2v) + V v
on the space-time cylinder Q = [0, 2]×B(0, 2) with a ∈ Lip uniformly elliptic
‖W1,2‖Ln+2(Q) + ‖V ‖L1L∞+L∞Ln/2 ≪ 1
F extτ
F intτ
1/16τ
211/42δ
Figure 2. The sets Eδ, F
τ and F
0≤t≤1
‖v(t)‖L2(B(0,1)) + ‖∇xv‖L2([0,1]×B(0,1)) . ‖v‖L2(Q).
If n = 2 then the same statement is true with L∞Ln/2 replaced by LpLq with
1 ≤ p < ∞, 1 < q ≤ ∞ and 1
= 1. Similarly, if n = 1 we have to replace
it by L4L∞.
Given our assumptions (1.15), (1.17), (1.18), (1.19) and (1.20) we can apply
Proposition 2.1 rescaled in sets of the form [t0, 2t0] × B(x0, t1/20 ), which are
subsets of the Aij. Summing up with respect to such sets contained in a
parabolic cube
Qr = [0, r
2]×B(0, r)
we obtain the following consequence.
Corollary 2.2. The following estimate holds under the assumptions of Theo-
rem 1
2u‖L∞L2(Qr) + ‖t
2∇u‖L2(Qr) . ‖u‖L2(Q2r)
Proof of Theorem 1. We choose τ ≥ τ0 and 0 < δ ≪ τ−1/2 and introduce the
[0, 2δ2]×B(0, 2δ)
[0, δ2]× B(0, δ)
F extτ =
[0, 2/τ ]×B(0, 2)
[0, 1/τ ]×B(0, 1)
F intτ =[1/32τ, 1/16τ ]× B(0, 1/4)
Our strategy will be to truncate u in Eδ and F
τ and to apply Theorem 3
to the truncated function in order to obtain a good bound on u in F intτ .
Let η be a cutoff function supported in [0, 2)× B(0, 2) and identically 1 in
[0, 1]×B(0, 1). For δ ≪ τ−1/2 we define
vδ(t, x) = (1− η(t/δ2, x/δ))η(τt/8, x)u(x, t)
which satisfies
(P +W∇)vδ = V vδ − [P +W∇, η(t/δ2, x/δ)]u+ [P +W∇, η(τt/8, x)]u.
The second term on the right hand side is supported in Eδ and the third one
in F extτ .
We apply Theorem 3 to vδ. One should keep in mind that the corresponding
weight φ depends on δ but that the bounds we prove are uniform with respect
to δ. We normalize the function h by h(0) = 0. We have to control the size of
φ in the sets Eδ, F
τ and F
τ . Due to (1.29) we have
τs ≤ h(s) ≤ 2τs, s ≥ 0
By (1.30) we obtain a rough polynomial bound in δ
(2.1) eφ ≤ t−2τe−
+ε(τ+x
) ≤ t1/2c(τ)δ−4τ−1 in Eδ.
M = sup{eh(− ln(t))e−
+ε(τ+x
) : (t, x) ∈ F extτ }
By (1.29) the supremum is attained at a point (t0, x0) with
≤ 8τt0 ≤ 1 and
|x0| = 1. A simple computation also shows that
sup{t−1/2eh(− ln(t))e−
+ε(τ+x
) : (t, x) ∈ F extτ } . τ 1/2M.
Then M dominates eφ in F extτ :
(2.2) eφ ≤M, t−1/2eφ . τ 1/2M in F extτ .
Next we need to bound eφ from below in F intτ in terms of M ,
(2.3) inf
F intτ
eφ ≥ e
To see this we compute for (t, x) ∈ F intτ and sufficiently small ε:
φ(t, x)− φ(t0, x0) ≥ h(− ln t)− h(− ln t0) +
− 2ε(τ + 1
− 20ε)τ ≥ 1
and use (2.1) and (2.2).
Theorem 3 applied to vδ yields
‖eφvδ‖
l2(B(τ),L2L
n−2 ∩L∞L2)
.‖eφV vδ‖
l2(B(τ),L2L
n+2+L1L2)
+ ‖eφ[P +W∇, η(t/δ2, x/δ)]u‖
l2(B(τ),L2L
n+2+L1L2)
+ ‖eφ[P +W∇, η(τt, x)]u‖
l2(B(τ),L2L
n+2 +L1L2)
(2.4)
By Hölder’s inequality we have
‖eφV vδ‖
l2(B(τ),L2L
n+2+L1L2)
.‖V ‖l∞(B(τ),L1L∞+L∞Ln/2)‖eφvδ‖
l2(B(τ),L2L
n−2 ∩L∞L2)
Due to the smallness in (1.28) we can absorb this term on the left hand side
of the inequality.
We calculate the first commutator
fδ = [P +W∇, η(t/δ2, x/δ)]u
+ ∂kg
kl∂l + 2
dkl∂l +W∇)η(t/δ2, x/δ)
+ 2δ−1gkl(∂kη)(t/δ
2, x/δ)∂lu
By (2.1) we have
‖eφfδ‖
l2(B(τ),L2L
n+2 +L1L2)
. c(τ)δ−4τ−1‖t
2 fδ‖
l2(B(τ),L2L
n+2 +L1L2)
For the W term we use (1.19) and Holder’s inequality. For term involving
dkl we bound the L1L2 norm in terms of an L∞L2 norm, using (1.15) which
implies that the pointwise bound for dkl is summable with respect to dyadic
time regions. For the remaining terms we simply bound the L1L2 norm in
terms of the L2 norm. This yields
‖eφfδ‖
l2(B(τ),L2L
n+2+L1L2)
.c(τ)δ−4τ−2(‖u‖L2(Eδ) + ‖(∂xη)(t/δ
2, x/δ)t
2∇u‖L2
+ ‖(∂xη)(t/δ2, x/δ)t
2u‖L∞L2)
Then we can apply a straightforward modification of Corollary 2.2 on the Eδ
scale to finally obtain
‖eφ[P +W∇, η(t/δ2, x/δ)]u‖
l2(B(τ),L2L
n+2 +L1L2)
. c(τ)δ−4τ−2‖u‖L2(Eδ).
Similarly we can estimate the second commutator
‖eφ[P +W∇, η(τt, x)]u‖
l2(B(τ),L2L
n+2 +L1L2)
.Mτ 1/2‖u‖L2(F extτ ).
Hence by inequality (2.4) we get
(2.5) ‖eφvδ‖
l2(B(τ),L2L
n−2∩L∞L2)
. Mτ 1/2‖u‖L2(F extτ ) + c(τ)δ
−4τ−2‖u‖L2(Eδ).
Within F intτ we have vδ = u. Then by (2.3) we obtain
(2.6) ‖u‖L∞L2(F intτ ) . τ
1/2e−
τ‖u‖L2(F extτ ) + c(τ)δ
−4τ−2‖u‖L2(Eδ).
Also by the vanishing of infinite order the second term tends to zero as δ → 0.
Hence as δ → 0 we obtain
(2.7) ‖u‖L∞L2(F intτ ) . τ
1/2e−
τ‖u‖L2(F extτ )
For 0 < t≪ 1 we choose τ = 1
to obtain
‖u(t, .)‖L2(B(0,1/4)) . t−1/2e−
32t .
This completes the proof of Theorem 1. �
Proof of Theorem 2. We extend the potentials V and W by zero to negative
time, and gkl(t, x) = gkl(0, x) for t < 0. By definition, possibly after rescaling,
we have u(0, .) ∈ L2(B(0, 2)). We now solve the mixed problem
(2.8) ut + ∂kg
kl∂lu = 0 for t < 0 and |x| ≤ 2
with the boundary condition
u(t, x) = 0 if |x| = 2 and t < 0
and the obvious initial condition to obtain an extension of u to negative. The
heat kernel for (2.8) satisfies Gaussian estimates. In particular we obtain from
(1.6) for all positive integers N with a constant cN possibly differing from (1.6)
(2.9)
Br(0)
|u|2dxdt . c2Nr2N .
We seek to prove that the bound (2.7) still holds in this context. The
difficulty is that we only know that u vanishes of infinite order at (0, 0) for
negative time. To account for this we shift the time up, t→ t+2δ. Arguing as
in the previous proof we obtain (2.6) with u replaced by u(t+ 2δ, x). Letting
δ → 0 by (2.9) we obtain (2.7) and conclude as above.
Remark 2.3. If one wants to prove Theorem 2 under the weaker assumptions
on g in (1.24) then the origin needs to be avoided in the above argument. Hence
the time translation needs to be accompanied by a spatial translation, namely
uδ(t, x) = u(t+ 2δ
2, x− 8τδe1)
This translation places the image of the origin, or better of the cube [0, τδ2]×
B(0, 4τδ), within the region {τt < x2}. But in this region the conjugated
operator Pψ, introduced later, is elliptic so only pointwise bounds for g are
needed for the Carleman estimates.
3. L2 bounds in the flat case and the Hermite operator
In this section we prove the simplest possible L2 Carleman estimate for the
constant coefficient backward parabolic equation
∂tu+∆xu = f
This serves as a good pretext to introduce the class of weight functions which
is later modified for the variable coefficient case.
We also describe the change of coordinates which turns the backward par-
abolic operator into a forward parabolic equation for the Hermite operator
H . In this way we are able to relate the L2 Carleman estimates for the heat
operator to spectral information for H .
Proposition 3.1. Let u ∈ L2 with compact support away from 0. Then
(3.1) ‖t−τ−
8t u‖L2 ≤ ‖t−τ+
8t (∂t +∆)u‖L2
uniformly with respect to τ away from (2n+ N)/4.
Proof. In R+ × Rn we introduce new coordinates (s, y) ∈ R× Rn defined by
(3.2)
t = e−4s
x = 1
e−2sy
= −4e−4s
− e−2sy
Hence in the new coordinates our operator becomes
4t(∂t +∆x) = −
− 2y ∂
If we conjugate it by tn/4e−
8t = e−nse−
2 we obtain
4t1+n/4e−
8t (∂t +∆x)t
−n/4e
8t = − ∂
+∆y − y2 =: −∂s −H =: −P0
where H is the Hermite operator
H = −∆y + y2
Then it is natural to define the new functions
v(s, y) = e−nse−
2 u(e4s,
e2sy), g(s, y) = e(−n−4)se−
2 f(e4s,
e2sy)
which are related by
P0v = g ⇐⇒ (∂t +∆)u = f
In the new coordinates, the bound (3.1) becomes
(3.3) ‖e4τsv‖L2 . ‖e4τsP0v‖L2.
Denoting w = e4τsv, we conjugate
e4τsP0v = e
4τsP0e
−4τsw = (−∂s −H + 4τ)w
and the above bound becomes
(3.4) ‖w‖L2 . ‖(−∂s −H + 4τ)w‖L2.
Since ∂s and H − 4τ commute we expand
‖(−∂s −H + 4τ)w‖2L2 = ‖∂sw‖2L2 + ‖(H − 4τ)w‖2L2 ≥ d(4τ, n+ N)2‖w‖2L2.
Note the spectral gap, which is essential in order to obtain strong unique
continuation results. �
For later use we also record the following slight generalization of the above
result. For expediency this is stated in the (y, s) coordinates, i.e. in the form
of an analogue of (3.3).
Proposition 3.2. Let h be an increasing, convex, twice differentiable function
so that
d(h′,N) + h′′ ≥ 1
(3.5)
‖(1 + h′′)1/2eh(s)v‖L2 +
∥∥∥∥min
(1 + h′′)1/2
1 + h′
eh(s)Hv
. ‖eh(s)(∂s −H)v‖L2
for all compactly supported v ∈ L2.
Proof. After the substitution w = eh(s)v the bound (3.5) becomes
(3.6)
‖(1 + h′′)1/2w‖2L2 +
∥∥∥∥min
(1 + h′′)1/2
1 + h′
. ‖(∂s −H + h′(s))w‖2L2.
and we obtain the L2 estimate through expanding the term on the right hand
side with respect to its selfadjoint and skewadjoint part:
‖(∂s −H + h′(s))w‖2L2 =‖∂sw‖2L2 + ‖(−H + h′)w‖2L2 + ‖(h′′)1/2w‖2L2
(d(h′,N)2 + h′′)‖w(s)‖2L2ds
To complete the proof we observe that for each s we have
‖Hw(s)‖L2 . ‖(−H + h′)w(s)‖L2 + h′(s)‖w(s)‖L2
4. Resolvent bounds for the Hermite operator
As seen in the previous section, the spectral properties of the Hermite op-
erator play an essential role even in the simplest L2 Carleman estimates for
the heat equation. In this section we take a look at L2 and Lp bounds for its
spectral projectors and its resolvent.
The spectrum of H is n+ 2N, and its eigenfunctions are the Hermite func-
tions defined by
uα = cα(∂y − y)αe−
2 , Huα = (n+ 2|α|)uα
As |α| increases, so does the multiplicity of the eigenvalues. We denote the
spectral projectors by Πλ for λ ∈ n + 2N. We consider both the spectral
projectors and the resolvent of H and obtain both Lp and localized L2 bounds.
4.1. Weighted L2 bounds. We consider two parameters
1 ≤ d, R . λ
We denote
BR = {y : |y| < R}, , Bjd = {y : |yj| < d}, j = 1, ..., n
By χR, respectively χ
d we denote bump functions in BR, respectively B
d which
are smooth on the corresponding scales.
Proposition 4.1. The spectral projectors Πλ satisfy the localized L
2 bounds
(4.1) R−
4‖χRΠλf‖L2 +R−
4‖χR∇Πλf‖L2 . ‖f‖2L2,
respectively
(4.2) d−
2‖|Dj|
dΠλf)‖L2 . ‖f‖L2
Proof. The inequality (4.1) is trivial unless R ≪ λ 12 . To prove it in dimension
n = 1 we only need to consider the case when f is a Hermite function,
f = Πλf = hλ
in which case it follows from the pointwise bound
4 |h′λ(x)|+ λ
4 |hλ(x)| . ‖hλ‖L2, |x| ≤
In dimension n = 1 (4.2) follows by interpolation from (4.1) with R = d.
This extends trivially to higher dimension by separation of variables.
It remains to prove (4.1) in higher dimensions. Summing up (4.2) with
d = R over j we obtain the bound
2‖|D|
2χRΠλf‖L2 . ‖f‖2L2
For |x| . R ≪ λ 12 we have |ξ|2 ≈ λ in the characteristic set ofH−ℜz, therefore
the above norm should essentially control the left hand side of (4.1). For later
use we prove a slightly more general result, which in particular concludes the
proof of (4.1).
(4.3) λ
4‖v‖L2+λ−
4‖∇v‖L2. ‖|D|
2 v‖L2+‖|y|
2 v‖L2+‖(H−λ)v‖
yL2+∇L2+λ
Indeed the norm on the right is equivalent to
4v‖L2 + ‖(H + λ)−
2 (H − λ)v‖L2 & λ
4‖v‖L2 + λ−
2v‖L2
In our case we apply (4.3) to v = R−
2χRΠλf . Then
2 v‖L2 . ‖Πλf‖L2
while
(H − λ)v = 2R−
2∇(∇χRΠλf) +R−
2∆χRΠλf
which yields
‖(H − λ)v‖
yL2+∇L2+λ
2‖f‖L2
To state the corresponding resolvent bounds we define the spaces X̃2(z) by
‖u‖X̃2(z) = (1+ |ℑz|)
2‖u‖L2+‖(H−z)u‖
yL2+∇L2+|z|
2‖|Dj|
du‖L2
and the corresponding dual spaces X̃∗2 (z). These spaces are larger than the
corresponding “elliptic” spaces,
(4.4) ‖v‖X2(z) . |z|
2‖v‖L2 + ‖yv‖L2 + ‖∇v‖L2
On the other hand by extending the bound (4.3) to complex λ we obtain a
counterpart of (4.1), namely
(4.5) R−
4‖χRu‖L2 +R−
4‖χR∇u‖L2 . ‖u‖X2(z)
Finally, the result of (4.2) can be written in the following dual forms
(4.6) ‖Πλf‖X2(λ) . ‖f‖L2, ‖Πλf‖L2 . ‖f‖X∗2 (λ)
The localized L2 resolvent bounds have the form
Proposition 4.2. Let n ≥ 2, z ∈ C with dist(z, n + 2N) & 1, and 1 ≤ d ≤
R ≪ ℜz. Then
(4.7) ‖u‖X̃2(z) . ‖(H − z)f‖X̃∗2 (z)
where the d component of norms is omitted in dimension n = 1.
Proof. We first note that the bounds (4.6) almost imply (4.7) up to a loga-
rithmic divergence. They do imply easily a bound for higher powers of the
resolvent for z away from the spectrum of H ,
(4.8) ‖(H − z)−1−kf‖X̃2(z) . (1 + |ℑz|)
−k‖f‖X̃∗2 (z), k ≥ 1.
as well as
(4.9) ‖u‖L2 . (1 + |ℑz|)−
2‖(H − z)u‖X̃∗2 (z).
Hence it remains to show that
(4.10) ‖u‖X̃2(z) . (1 + |ℑz|)
2‖u‖L2 + ‖(H − z)u‖X̃∗2 (z).
Using a positive commutator technique we first prove a one dimensional
estimate. For this we define the one dimensional skewadjoint pseudodifferential
operator2
Qr = iOp
w(χ(yr−1)χ(ξ|y|−1))
where χ is a mollified signum function which satisfies
χ′(x) =
, |x| ≤ 2
Its properties are summarized in the following
2As defined the symbol of Q is not smooth at 0. However, any smooth modification in
the ball {x2 + ξ2 < r2} will do.
Lemma 4.3. a) Qr is bounded in L
p for 1 ≤ p ≤ ∞ uniformly for r ≥ 1.
b) Qr is also bounded in X̃2(z) uniformly with respect to z ∈ C and r ≥ 1.
c) Qr satisfies the commutator estimate
(4.11) r−1‖|D|
2χru‖2L2 . (1 + |ℑz|)‖u‖2L2 + 〈(H − z)u,Qu〉
Proof. a) The Lp boundedness is straightforward and is left for the reader.
b) For the X̃2(z) boundedness we consider first the d terms, which without
any loss in generality we can write in the form
2‖(d2 +D2)
4χdu‖L2
Since Q is bounded in L2 it suffices to prove the commutator bound
‖[Qr, (d2 +D2)
4χd]u‖L2 . ‖u‖L2
But this is easily verified using the pdo calculus.
Next we consider the term
‖(H − z)u‖
yL2+∇L2+|z|
for which it suffices to prove the commutator bound
‖[Qr, H ]u‖
yL2+∇L2+|z|
. ‖u‖L2
or equivalently, by duality,
‖[Qr, H ]u‖L2 . ‖yu‖L2 + ‖∇u‖L2 + ‖|z|
2u‖L2
This follows again from the pseudodifferential calculus.
c) Since Qr is skewadjoint we have the identity
〈(H − z)u,Qru〉 = 〈[H,Qr]u, u〉+ ℑz〈iQru, u〉
therefore it suffices to insure that
(4.12) 〈[H,Qr]u, u〉 & ‖u‖2X̄2(r) +O(‖u‖
For this we compute the commutator [H,Q],
[H,Q] = Opw({ξ21 + y21, χ(y1r−1)χ(ξ1|y1|−1)}) +O(1)L2→L2
= Opw(2r−1χ′(y1r
−1)ξ1χ(ξ1|y1|−1)) +O(1)L2→L2
r−1(χ1r)
2(ξ21 + r
2 . r−1χ′(y1r
−1)ξ1χ(ξ1|y1|−1) + 1
and the conclusion follows by Garding’s inequality. �
We return to the proof of the proposition. By separation of variables the
bound (4.11) extends to higher dimensions and gives
r−1‖|Dj|
2χjru‖2L2 . (1 + |ℑz|)‖u‖2L2 + 2ℜ〈(H − z)u,Qjru〉
where Qjr the higher dimensional analogue of Qr with respect to the j variable.
TheX2(z) boundedness ofQr also extends easily to higher dimension. Hence
by Cauchy-Schwartz we obtain
r−1‖|Dj|
2χjru‖2L2 . (1 + |ℑz|)‖u‖2L2 + ‖(H − z)u‖X∗2 (z)‖u‖X2(z)
To conclude the proof of the estimate (4.10) it remains to show that
‖(H − z)u‖
2L2+yL2+∇L2
. ‖(H − z)u‖X∗2 (z)
which follows by duality from (4.4).
The final step in the L2 resolvent bounds is to replace the y′ derivatives by
angular derivatives. Let ∇⊥ = y|y| ∧∇ be the angular derivative and |D⊥|
the corresponding fractional derivative.
We split the coordinates into y = (y1, y
′) and use the notation ′ for coordi-
nates and derivatives in the obvious sense. For 1 ≤ d ≤ R ≤
λ we define the
sector
BR,d = {R < |y1| < 2R, |y′| ≤ d}
and χR,d a bump function inBR,d. Then we define the function spaceX2(λ,R, d)
‖u‖2X2(λ,R,d) = ‖u‖
L2 +R
−1/2λ−1/4‖∇(χR,du)‖2L2
+ R−1/2λ1/4‖χR,du‖2L2 + d−1/2‖|D⊥|
2χR,du‖2L2
and X∗2 (λ,R, d) as its dual.
Lemma 4.4. Suppose that n ≥ 2 and 1 ≤ d ≤ R. Then
‖u‖X2(λ,R,d) ≈ R−
4‖χR,du‖L2 +R−
4‖∇χR,du‖L2 + d−
2‖|D′|
2χR,du‖L2
Proof. Within BR,d the angular derivatives are close to the y
′ derivatives,
namely
|D⊥u| . |D′u|+
|∇u|, |D′u| . |D⊥u|+
|∇u|.
This implies the corresponding bounds for L2 norms, and the conclusion follows
by interpolation. �
¿From the above lemma we obtain
‖u‖X2(λ,R,d) . ‖u‖X̃2(λ,R,d)
Hence, we may replace X̃2 by X2 in (4.7) and (4.6):
Corollary 4.5. a) For λ in the spectrum of H we have
(4.13) ‖Πλf‖X2(λ,R,d) . ‖f‖L2, ‖Πλf‖L2 . ‖f‖X∗2 (λ,R,d)
b) For z away from the spectrum of H and 1 ≤ d ≤ R . ℜz we have
(4.14) ‖(H − z)−1−kf‖X2(λ,R,d) . (1 + |ℑz|)−k‖f‖X∗2 (λ,R,d), k ≥ 0
4.2. The Lp bounds of the resolvent. The Lp bounds for the spectral
projectors and the resolvent were proved in [16], [28] (see also [10]). For the
sake of completeness we also present them here in a simpler manner following
the approach in [19]. We refer the reader to the same paper for further results.
We consider pairs of exponents satisfying
(4.15)
where the range for p is
(4.16) p ≥ 4 for n = 1, p > 2 for n = 2, p ≥ 2 for n ≥ 3.
This leads to the following range3 for q:
(4.17)
q ∈ [2,∞] for n = 1, q ∈ [2,∞) for n = 2, q ∈ [2, 2n
] for n ≥ 3.
The dual exponents are denoted by p′ and q′ as usual.
Proposition 4.6. Let q be as in (4.17). Then
a) The spectral projectors Πλ satisfy
‖Πλ‖Lq′→L2 . 1, ‖Πλ‖L2→Lq . 1, n ≥ 2
‖Πλ‖Lq′→L2 . λ
p , ‖Πλ‖L2→Lq . λ−
p , n = 1
(4.18)
b) For z away from n+ N the resolvent (H − z)−1 satisfies
‖(H − z)−1‖Lq′→Lq . (1 + |ℑz|)
p′ , n ≥ 1(4.19)
Outline. To revisit the Lp bounds associated to the spectral projectors we recall
the approach in [19]. The first step there is to establish pointwise bounds for
the Schrödinger evolution4
(4.20) ‖eitH‖L1→L∞ . (sin t)−
This immediately (see also [18]) leads to Strichartz estimates for the solution
to the inhomogeneous equation
ivt −Hv = g, v(0) = v0
namely
(4.21) ‖v‖Lp([0,2π];Lq) . ‖v0‖L2 + ‖g‖Lp′([0,2π];Lq′)
where (p, q) are as described in (4.15), (4.16).
To obtain (4.18) we apply (4.21) to v = e−iλtΠλu, which yields L
2 → Lp
bounds, and hence by duality and selfadjointness all estimates of (4.18) for
3The exponent q = ∞ is actually allowed in the spectral projection bounds in dimension
n = 2. However, it is not allowed in any of the resolvent bounds.
4These bounds are very robust and are in effect established in [19] for a much larger class
of operators
n ≥ 2. The case n = 1 can be dealt with directly using the pointwise bounds
for the Hermite functions.
We note a consequence of the bounds (4.18), namely
(4.22) ‖(H − z)−1−k‖Lq′→Lq . (1 + |ℑz|)
, n ≥ 2, k ≥ 1
which is obtained by interpolating between q = 2 and q = 2n
Similarly we get
(4.23) ‖(H − z)−1‖Lq′→L2 . (1 + |ℑz|)
p′ , n ≥ 2.
Then we apply (4.21) to
v(x, t) = χ(t)e−iztu(x), g = χ′(t)e−iztu(x) + χ(t)e−izt(H − z)u
where χ is a unit bump function on an interval of size (1+ |ℑz|)−1. This yields
‖u‖Lq . (1 + |ℑz|)
p‖u‖L2 + (1 + |ℑz|)
p′ ‖(H − z)u‖Lq′ . ‖(H − z)u‖Lq′ .
concluding the proof of (4.19) for n ≥ 2. The case n = 1 is a variation on the
same theme. �
4.3. Combining the estimates. Here we combine the L2 and the Lp com-
ponents in the resolvent bounds:
Proposition 4.7. For z away from n+ 2N the resolvent (H − z)−1 satisfies
(4.24)
‖(H − z)−1‖Lq′→X2(ℜz,R,d) . (1 + |ℑz|)
2 , n ≥ 2, (n, q) 6= (2,∞)
with the obvious modification for n = 1.
Proof of Proposition 4.7. Taking into account the bounds (4.19) and (4.23), it
remains to prove the estimate
‖u‖X̃2(ℜz,R,d) .(1 + |ℑz|)
p′ ‖(H − z)u‖Lq′
+ (1 + |ℑz|)
p‖u‖Lq + (1 + |ℑz|)
2‖u‖L2
But this follows from (4.12) in the same way as for Proposition 4.2 since the
operator Q is bounded in Lq. �
5. Lp estimates in the flat case and parametrix bounds
In this section we begin with the mixed norm LpLq Carleman estimates
in the simplest case, i.e. with constant coefficients and a polynomial weight.
These were proved in [8] except for the endpoint which was obtained later in
[10] .
After a conformal change of coordinates and conjugation with respect to the
exponential weight the Carleman estimates reduce to proving LpLq estimates
for a parametrix K for ∂t −H + τ . In this article we need a stronger version
of these bounds, where we add in localized L2 norms.
In a simplified form, Escauriaza-Vega’s result in [10] has the form:
Theorem 4. [10] Let p and q be as above. Then
‖t−τe−
8t u‖L∞(L2)∩Lp(Lq) ≤ ‖t−τe−
8t (∂t +∆)u‖L1(L2)+Lp′(Lq′ ),
for all u with compact support in Rn × [0,∞) vanishing of infinite order at
(0, 0) uniformly with respect to 4τ with a positive distance from integers.
One can write the estimate in the (s, y) coordinates using the same trans-
formation as in Section 3:
(5.1) ‖eτsv‖L∞(L2)∩Lp(Lq) . ‖eτs(∂s +H)v‖L1(L2)+Lp′ (Lq′ )
Setting w = eτsv this becomes
(5.2) ‖w‖L∞(L2)∩Lp(Lq) . ‖(∂s +H − τ)w‖L1(L2)+Lp′ (Lq′ )
Denoting by Πλ the spectral projection onto the λ eigenspace ofH we obtain
a parametrix K for (∂t −H + τ),
K(∂t +H − τ) = I
where the s-translation invariant kernel of K is
K(s) =
s(τ−λ)1s(τ−λ)<0
Since w decays at ±∞ we have
w = K(∂s +H − τ)w
therefore (5.2) can be rewritten in the form
(5.3) ‖Kf‖L∞(L2)∩Lp(Lq) . ‖f‖L1(L2)+Lp′ (Lq′ )
The main result of this section is an improvement of (5.2), namely
Proposition 5.1. Assume that τ is away from n + N and that
1 ≤ d ≤ R . τ
(5.4) ‖Kf‖L∞(L2)∩Lp(Lq)∩L2X2(τ,R,d) . ‖f‖L1(L2)+Lp′ (Lq′ )+L2X∗2 (τ,R,d)
Proof. We work in dimension n ≥ 2; some obvious adjustments are needed in
dimension n = 1, which is slightly easier. We consider four endpoints:
A: The L1L2 → L∞L2 bound follows easily since the projectors Πλ are L2
bounded.
B: The L1L2 → LpLq bound. Here it suffices to prove
‖K(.)f‖LptLqx . ‖f‖L2
Splitting f into spectral projections and using (4.18) we obtain
‖K(t)f‖Lq .
e−|(λ−τ)t|‖Πλf‖L2
For |t| ≥ 1 we can use Cauchy-Schwartz to obtain
‖K(t)f‖Lq∩X2(R,d) . e−c|t|‖f‖L2
which suffices for all q. For |t| ≤ 1 we consider the most difficult case p = 2
and compute
‖K(t)f‖2L2([−1,1],Lq) .
e−|(λ−τ)t|‖Πλf‖L2
|λ− τ | + |µ− τ |
‖Πλf‖L2‖Πµf‖L2
0≤i,j
2−i−j
|λ−τ |≈2i
‖Πλf‖L2
|µ−τ |≈2i+j
‖Πµf‖L2
|λ−τ |≈2i
‖Πλf‖2L2
|µ−τ |≈2i+j
‖Πµf‖2L2
.‖f‖2L2
C: The L1L2 → X2(τ, R, d) bound for K follows in the same way from
‖Πu‖X2(τ,R,d) . ‖u‖L2.
D: The Lp
+ L2X∗2 (τ, R, d) → L∞L2 bound for K is equivalent to the
L1L2 → LpLq ∩X2(τ, R, d) bound for K∗. By reversing time this is seen to be
the same as the L1L2 → LpLq bound for K.
E: The Lp
+ L2(X∗2 (R, d)) → LpLq ∩ L2X2(R, d) bound. Using (4.18)
and (4.13) directly yields
‖K(s)‖
n+2+X∗2 (R,d)→L
n−2 ∩X2(R,d)
e|s(τ−λ)| . s−1e−cs
Similarly we obtain
‖K(s)‖
n−2 ∩X2(R,d)
2 e−cs, ‖K(s)‖
(R,d)
2 e−cs
Interpolation with the L2 estimate gives
‖K(s)‖Lq′→Lq . s
‖K(s)‖Lq′→X2(R,d) . s
2 , ‖K(s)‖X∗2 (R,d)→Lq . s
If p > 2 then the Hardy-Littlewood Sobolev inequality implies
‖K ∗ f‖LpLq . ‖f‖Lp′Lq′ .
‖K ∗ f‖L2X2(R,d) . ‖f‖Lp′Lq′ , ‖K ∗ f‖LpLq . ‖f‖L2X∗2 (R,d).
With obvious changes the analysis is similar if n = 1, 2.
It remains to prove the L2 → L2 type bounds, namely
‖Kf‖L2X2(R,d) . ‖f‖L2X∗2 (R,d) (n = 1, 2)
respectively
L2(X2(R,d)∩L
n−2 )
. ‖f‖
L2(X∗2 (R,d)+L
n+2 )
(n > 2)
For this, following an idea in [10], we consider a dyadic frequency decompo-
sition in time. By the Littlewood-Paley theory it suffices to prove the bound
for a single dyadic piece at frequency 2j, namely
(5.5) ‖Sj(Ds)Kf‖
L2(X2(R,d)∩L
n−2 )
. ‖f‖
L2(X∗2 (R,d)+L
n+2 )
(n > 2)
and its one and two dimensional counterpart.
Taking a time Fourier transform we can write (for f ∈ S(Rn))
ŜjK(σ)f = s(2
λ− τ − iσ
Πλf = s(2
−jσ)(H − τ − iσ)−1f
therefore by the inversion formula
(SjK)(t)f =
eitσs(2−jσ)(H − τ − iσ)−1fdσ
=− t−2
(s(2−jσ)(H − τ − iσ)−1)fdσ
Hence using the resolvent bounds (4.19) and (4.22) and the first line for |t| ≤
2−j and the second line for |t| ≥ 2−j we obtain
‖SjK(t)‖
(R,d)
n−2 ∩X2(R,d)
1 + 22jt2
and the similar estimate in one and two dimensions. The bound on the right
is integrable in t, therefore (5.5) follows. �
6. Modified weights and pseudoconvexity
The main result of this section, Theorem 5 is a considerable improvement
of Section 3. The weights t−τ in Section 3, while easy to use, satisfy merely
a degenerate pseudoconvexity condition, in the sense that the selfadjoint and
the skewadjoint parts of the operator in (3.4) commute. This is in contrast to
strong pseudoconvexity where one obtains better L2 bounds from the positivity
of the commutator. A perturbation argument easily implies an L2 Carleman
estimate for variable coefficients as soon as g = In+O(t). However, even arbi-
trarily small perturbations of g from In at t = 0 destroy the pseudoconvexity.
To obtain results for general variable coefficients we need a more robust
weight with additional convexity. A good way of doing this is by adding
convexity in t and by using a weight of the form
eh(− ln t)
with a convex function h. Then we obtain for the heat operator the strength-
ened L2 estimates of Proposition 3.2. The assumption of vanishing of infinite
order forces us to work with functions h with at most linear growth at infinity.
This in turn limits the convexity of h, and hence the gain from the convexity
in the L2 bounds.
These Carleman inequalities with the weight e−h(ln t) are more stable with
respect to perturbations. They can be obtained for coefficients satisfying
(6.1) |g − In|+ (t + |x|2)|∂tg|+ (t+ |x|2)1/2|∇g| . (t+ |x|2)ε.
with suitable functions h. It is not difficult to weaken (6.1) almost to our condi-
tions (1.15) and (1.16). This venue was pursued by Escauriaza and Fernández
In this paper we seek to obtain Lp Carleman inequalities and also to handle
Lp gradient potentials. Both require good spatial and temporal localization,
which depends on the strength of the L2 estimates. The weights eh(− ln t) seem
to be insufficient for this purpose.
Consequently we consider a larger class of weights of the form
eh(− ln t)+φ(xt
−1/2,− ln(t))
having some additional convexity in y = xt−1/2. Here we think of φ essen-
tially as a function of y with a milder dependence on s = − ln t. Obtaining
pseudoconvexity is not entirely straightforward because the Hamilton flow for
the Hermite operator H is periodic so no nonconstant function of y can be
convex along its orbits. We note that the projection of the orbits to the y
space are ellipses of size O(
τ ) where τ is the energy, centered at 0. Hence
we can choose φ to be convex in y for |y| ≪
τ . We compensate the lack of
convexity of φ when |y| ≈
τ by the s convexity of h. To elaborate this idea
we explain the precise setup.
Let δ1 be small positive constant. We begin with constants {αij}A (see (1.13)
and (1.14) for the notation) which control the regularity of the coefficients5
gkl − δkl, dkl and ekl of P given by (1.7) as in (1.15).
(6.2) δ1αij = ‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖Cmijt (Aij).
The condition (1.15) guarantee that for all τ ≥ 1
(6.3) ‖αij‖l1(A(τ)) ≤ 1. ‖αij‖l∞(A) ≤ 1.
We first adjust the αij ’s upward so that they vary slowly and do not con-
centrate in irrelevant regions. This readjustment depends on the choice of the
parameter τ .
Lemma 6.1. Let αij be a sequence satisfying (6.3). Then for each τ ≫ 1
there exists a double sequence (εij)A(τ) with the following properties:
5denoted generically by d here and later
(1) For each (i, j) ∈ A(τ)
αij ≤ εij
(2) We have εij ∈ l1(A(τ)),
‖εij‖l1(A(τ)) . 1.
(3) The sequence εij is slowly varying,
| ln εi1j1 − ln εi2j2| ≤
(|i1 − i2|+ |j1 − j2|), (i1, j1), (i2, j2) ∈ A(τ).
(4) The sequence (εi) defined by
j:(i,j)∈A(τ)
εij, i ≥ ln τ.
has the following properties
(6.4) | ln εi1 − ln εi2 | ≤
|i2 − i1|, εij . εi, εi[ln τ/2]+2 ≈ εi
(5) For each i ≥ ln τ there exists an unique 0 ≤ j(i) ≤ [ln τ/2]+2 with the
following properties:
(6.5) εij(i) ≈ εi.
εij ≤ e−jτ−1/2 if 0 ≤ j ≤ j(i), j(i) 6= 0
εij > e
−jτ−1/2 if j(i) < j ≤ [ln τ/2] + 2.
(6.6)
We shall see that j(i) is an important threshold. If j ≥ j(i) then we can
localize our estimates to the corresponding Aij and even to smaller sets. On
the other hand, we cannot localize to sets smaller than
(6.7) Bi0 =
j≤j(i)
Proof. To fulfill the conditions (1)-(4) we simply mollify the αij,
ε̃ij = max
(k,l)∈A(τ)
(|i−k|+|j−l|), ε̃i =
[ln τ/2]+2∑
ε̃ij.
For the last part of (4) we redefine
ε̃ij := ε̃ij + e
|j−([ln τ/2]+2)|ε̃i
This also increases ε̃i by a fixed factor.
For (5) we begin with a preliminary guess for j(i) which we call j0(i) ∈ R+.
We consider three cases.
j0(i) =
ln τ/2 if ε̃i < τ
− ln(ε̃iτ 1/2) if τ−1 ≤ ε̃i < τ−1/2,
0 if τ−1/2 ≤ ε̃i.
We define
(6.8) εij := max{ε̃ij, 2e−
|j−j0(i)|ε̃i},
which is still slowly varying because ε̃i is slowly varying.
We define j(i) according to (6.6). It is uniquely determined since the se-
quence εij is slowly varying compared to e
2 . Since ε̃ij is slowly varying we
must have ε̃ij ≤ ε̃i/2. This allows us to conclude that for j close to j0(i), the
second term in (6.8) is larger than the first one,
εij = 2e
|j−j0(i)|ε̃i for |j − j0(i)| ≤ 2.
If j0(i) = 0 then εi0 = 2ε̃i ≥ 2τ−1/2 and hence j(i) = 0. If 0 < j0(i) < ln τ/2
then for |j − j0(i)| ≤ 2 we have
εij = 2e
|j−j0(i)|e−j0(i)τ−1/2
therefore j0(i)− 2 < j(i) ≤ j0(i). If j0(i) = ln τ/2 then for |j − j0(i)| ≤ 2 we
εij ≤ 2e−
|j−j0(i)|e−j0(i)τ−1/2
and we arrive again at j0(i)−2 ≤ j(i). In all three cases we have |j0(i)−j(i)| ≤
2 therefore (6.5) holds. We observe that εi . ε̃i. �
The sequence (εij)A(τ) is used to describe the amount of spatial convexity
needed in the region Aij , which will be reflected in the construction of φ below.
The partial sums εi measure the amount of s-convexity needed in [i, i+1]. The
purpose of part (5) above is to correlate the two amounts in a region where
they have the same strength (where j is close to j(i)).
Our weights have the form
(6.9) ψ(s, y) = h(s) + φ(s, y)
Their choice is described in the next two lemmas:
Lemma 6.2. Let τ and (εi) be as in Lemma 6.1. Then there is a convex
function h with the following properties:
(1) h′ ∈ [τ, 2τ ].
(2) h′′(s) + dist(h′(s),N) > 1
(3) εiτ . h
′′(s) . εiτ + 1 for s ∈ [i, i+ 1].
(4) |h′′′| . h′′.
The proof of the lemma is fairly straightforward and uses only the fact that
(εi) is slowly varying and summable. The second part is needed in order to
avoid the eigenvalues of the Hermite operator.
Lemma 6.3. Let τ , (εij) and (εi) be as in Lemma 6.1. Then there exists a
smooth spherically symmetric function
φ : R× Rn → R
with the following properties:
(1) (Bounds) The function φ is supported in |y| ≤ 2τ 1/2 and satisfies
(6.10) 0 ≤ φ(s, y) . εiτ, |∂sφ(s, y))|+ |∂2sφ(s, y)| . εiτ
(6.11)
l,k=0
(1 + |y|)k|D1+ky ∂lsφ| . ǫiτ 1/2 for i ≤ s ≤ i+ 1,
(2) (Monotonicity)
(6.12) ∂rφ(s, y) ≈ εiτ
2 for (s, y) ∈ Aij , (i, j) ∈ A(τ), j ≥ j(i) + 1
(3) (Convexity)
(6.13) (1 + |y|)∂2rφ(s, y) ≈ εijτ
2 in Aij, (i, j) ∈ A(τ).
Proof. Let
φj(y) =
e2j + |y|2, j ≥ 0.
We fix a smooth partition of unity 1 =
η(s− i) and define
ln aj(s) =
η(s− i) ln εij .
These functions satisfy the bounds
(6.14) aj(s) ≈ εij, i ≤ s ≤ i+ 1, |a′j |, |a′′j |, |a′′′j | . aj.
Their sum satisfies
a(s) :=
[ln τ/2]+2∑
aj(s) ≈ εi, i ≤ s ≤ i+ 1.
We define
φ(s, y) = τ
2χ(|y|τ−1/2)
[ln τ/2]+2∑
aj(s)φj(|y|)
where χ is a smooth function supported in [−2, 2] and identically 1 in [−3
We verify the properties:
0 ≤ φ(s, y) . a(s)τ . εiτ, i ≤ s ≤ i+ 1.
The remaining part of (6.10) follows from (6.14). Estimate (6.11) is a conse-
quence of (6.14) and
|(1 + |y|)kDk+1φj(y)| . 1, 0 ≤ k ≤ 3.
The upper bound from (6.12) is covered by (6.11) and the lower one follows
∂rφ(s, y) & τ
j≤j(i)
aj ≈ τ 1/2
j(i)∑
εij ∼ εiτ 1/2
in Aij with j ≥ j(i) where we use εij(i) ≈ εi. The assertion (6.13) follows from
immediate bounds on second derivatives of the φj. �
Our aim in this section is to prove L2 Carleman estimates for the variable
coefficient operator P with the exponential weight
ψ(− ln t
ψ(s, y) = h(s) + δ2φ(s, y).
where δ2 is a small constant and h and φ are as in in Lemma 6.2 and 6.3.
The calculations are involved. For a first orientation we outline the key part
of the argument for the constant coefficient heat equation. Using the change
of coordinates of Section 3 we transform the problem to weighted estimates for
the operator P0 = ∂s +H and the exponential weight e
ψ(s,y). This translates
to obtaining bounds from below for the conjugated operator
P0,ψ = e
ψ(s,y)P0e
−ψ(s,y).
Lemma 6.4. Let τ be large enough. Let ψ be as in (6.9) with h, φ as in the
above two Lemmas 6.2,6.3 with δ2 ≪ 1 . Then the operator P0,ψ satisfies the
bound from below
‖(h′′)
2 v‖2 + δ2τ−1(‖a2int∇v‖2 + ‖a2⊥∇⊥v‖2) . ‖P0,ψv‖2L2
for all functions v supported in {|y|2 ≤ 9τ} where the weights aint, a⊥ are
defined by
a4int = εij(1 + |y|)−1τ
2 , a4⊥ = 1j≥j(i)εi(1 + |y|)−1τ
2 in Aij .
Proof. We decompose P0,ψ into its selfadjoint and its skewadjoint part
P0,ψ = L
0,ψ + L
where
(6.15) Lr0,ψ := −∆y + y2 − ψs − ψ2y , Li0,ψ := ∂s + ψy∂ + ∂ψy.
Expansion of the norm gives
(6.16) ‖(Lr0.ψ + Li0,ψ)v‖2L2 = ‖Lr0,ψv‖2L2 + ‖Li0,ψv‖2L2 + 〈[Lr0,ψ, Li0,ψ]v, v〉
The conclusion of the lemma follows from the commutator bound
(6.17) 〈[Lr0,ψ, Li0,ψ]v, v〉 & ‖(h′′)
2 v‖2 + δ2τ−1(‖a2int∇v‖2 + ‖a2⊥∇⊥v‖2)
The commutator is explicitly computed
[Lr0,ψ, L
0,ψ] = ψss + 4ψyψyyψy − 4∂ψyy∂ − 4yψy + 4ψyψsy −∆2ψ
Since δ2 ≪ 1 the first term has size h′′(s). The second one is nonnegative since
ψ is convex for |y|2 < 9τ .
The Hessian of the radial function ψ can be written in the form
(6.18) ψyy = ψrr
One can see that the radial and angular derivatives carry different weights.
Our construction of φ guarantees that
ψrr .
, ψyy & ψrrIn
hence the weight ψrr can be used for all derivatives. For the size of the two
weights we have
ψrr ≈ a4int,
& a4⊥
This gives the last two terms in (6.17).
It remains to see that the remaining terms in the commutator are negligible
compared to the first term on the right hand side of (6.17). For this we use
the bound (6.11) to conclude that in Aij we have
| − 4yψy + 4ψyψsy −∆2ψ| . δ2εiτ . δ2h′′
To switch to operators with variable coefficients it is convenient to extend
the weights to the full space and to regularize them. Precisely we shall assume
a4int(s, y) ≈ εiτ in Aij if |y|2 ≥ τ
a4int(s, y) ≈ εij(1 + |y|)−1τ
2 in Aij if |y|2 ≤ τ.
(6.19)
Observe that the two cases above match since εi ≈ εij in the region where
y2 ≈ τ . We also introduce a modification a of aint which is used to include the
effect of the spectral gap in regions where we have very little convexity:
a4(s, y) ≈ 1 + εiτ in Aij if |y|2 ≥ τ
a4(s, y) ≈ 1 + εij(1 + |y|)−1τ
2 in Aij if |y|2 ≤ τ.
(6.20)
Finally we choose a⊥ with the properties
supp a⊥ ⊂
{Aij : j(i)− 1 ≤ j ≤
ln τ}
a4⊥(s, y) .εi(1 + |y|)−1τ
2 in Aij
a4⊥(s, y) ≈εi(1 + |y|)−1τ
2 in Aij if j(i) ≤ j ≤
ln τ − 1
(6.21)
The bounds for the weights from above are assumed to remain true after
applying powers of the differential operators ∂s, ∂y and y∂y to them.
Consider now a the more general class of operators P with real variable
coefficients given by (1.7). We repeat the change of coordinates and write in
the (s, y) coordinates:
4e−4sP = − ∂
− 2y ∂
+ ∂ig
ij∂j + yid
ij∂j + ∂id
ijyj + yie
This further leads to
4e−(n+4)s−
2 Pens+
2 = −P̃
where P̃ is given by
− ∂igij∂j − yi(gij − 2δij + 2dij + eij)yj
− yi(gij − δij + dij)∂j − ∂i(gij − δij)yj
We rewrite it in the generic form
P̃ = P0 − ∂d∂ − ydy − yd∂ − ∂dy
with P0 = ∂s +H .
To simplify as much as possible the proof of the main L2 Carleman estimate
we introduce a stronger condition on the regularity of the coefficients:
|d|+ 〈y〉(|dy|+ τ−
2 |dyy|+ τ−1|dyyy|+ τ−
2 |ds|) . δ1εij in Aij
|d|+ 〈y〉(|dy|+ τ−
2 |dyy|+ τ−1|dyyy|+ τ−
2 |ds|) . δ1
(6.22)
This improved regularity will be gained later on by regularizing the coefficients.
We are now in the position to formulate the Carleman estimate.
Proposition 6.5. Let τ be large enough and δ1 ≪ δ2 ≪ 1. Let ψ be as in
(6.9) with h, φ as in Lemmas 6.2,6.3. Assume that the coefficients g − In, d
and e satisfy (6.22). Then the following L2 Carleman estimate holds for all
functions u supported in {y ≤ 9τ}:
j=0,1,2
2‖a2eψDju‖+ τ− 12‖a2⊥eψD⊥u‖
. ‖eψP̃ u‖.(6.23)
Proof. After conjugation
Pψ := e
ψ(s,y)P̃ e−ψ(s,y)
we decompose Pψ into its selfadjoint and its skewadjoint part
Pψ = L
ψ + L
which for y2 < 9τ can be expressed in the generic form (see also (6.15)):
Lrψ = L
0,ψ + ∂d∂ + τd
Liψ = L
0,ψ + τ
2 (d∂j + ∂jd)
with d satisfying (6.22). Then (6.23) follows from
(6.24)
j=0,1,2
τ−j(δ2‖a2intDjv‖2 + 〈h′′〉
2Djv‖2) + δ2τ−1‖a2⊥∇⊥v‖2 . ‖P̃ψv‖2.
The proof will consist of three steps.
Step 1: First we show that for v supported in {|y|2 ≤ 9τ} we have
(‖a2int∇v‖2+‖a2⊥∇⊥v‖2)+‖(h′′)
2v‖2+‖Lrψv‖2 . ‖P̃ψv‖2+ δ1‖a2intv‖2
(6.25)
We compute
‖Pψv‖2L2 = ‖Lrψv‖2L2 + ‖Liψv‖2L2 + 〈[Lrψ, Liψ]v, v〉
We expand the commutator
[Lrψ, L
ψ] = [L
0,ψ, L
0,ψ] + [∂d∂ + τd, L
0,ψ] + τ
2 [Lrψ, d∂j + ∂jd]
The main contribution in (6.25) comes from the first commutator, for which
we use (6.17) to obtain the terms on the left side of (6.25).
The second commutator is estimated by
|〈[M r, Li0,ψ]v, v〉| . δ1(‖a2intv‖2 + τ−1‖a2int∇v‖2)
and the second term on the right is negligible since δ1 ≪ δ2. Indeed, we write
[∂d∂ + τd, Li0,ψ] = −∂kqkl∂l + r
where the coefficients q, r have the generic form
q = ds + ψydy + ψyyd+ dψyy, r = ∂d∂∆ψ + τ(ds + ψydy)
Using the bounds (6.22) for d and (6.11) for φ we estimate
|q| . δ1τ−1a4int, |r| . δ1a4int.
Finally, the third commutator is estimated in a similar fashion. We write it
in the form
2 [Lrψ, d∂j + ∂jd] = −∂kqkl∂l + r
where the coefficients q, r have the generic form
q = τ
2 (dy + ddy), r = τ
2 (∆dy + ∂d∂dy + dψys + dψyyψy + τddy)
Using (6.22) and (6.11) we obtain the same bounds for q and r as in the
previous case. This concludes the proof of (6.25).
Step 2: We use an elliptic estimate to show that for functions v supported
in {|y|2 ≤ 9τ} we have
(6.26) δ2
τ−j‖a2intDjv‖2+τ−1‖a2⊥D⊥v‖2
+‖(h′′)
2v‖2+‖Lrψv‖2L2 . ‖P̃ψv‖2
The elliptic bound
‖D2v‖+ τ‖v‖ . τ
2‖Dv‖+ ‖(−∆− h′(s))v‖
can easily proven by a Fourier transform. It implies
‖D2v‖+ τ‖v‖ . τ
2‖Dv‖+ ‖(H − h′(s))v‖+ ‖y2v‖,
We can replace H − h′(s) by Lrψ due to the pointwise estimate
|(Lrψ − (H − h′(s)))v| . δ1(|D2v|+ τ
2 |Dv|+ τ |v|)
Then (6.26) follows from (6.25).
Step 3: Here we use the spectral gap condition to improve our bound when
h′′ ≪ 1 and show that (6.26) implies (6.23). It suffices to show that if h′′(s) < 1
‖v‖L2 + τ−1‖D2v‖ . ‖Lrψv‖L2
Indeed, let s ∈ [i, i + 1] so that h′′(s) < 1
. Then h′ has a positive distance
from the integers. Also εi . 1 which implies that at time s we must have
|g − In| . δ1τ−1, |Dg| . δ1τ−1, |ψr| . δ2τ−
Hence we may think of Lrψ as a small perturbation of H − h′(s) and compute
‖v‖+ τ−1‖D2v‖ . ‖(H − h′(s))v‖ . ‖Lrv‖+ (δ1 + δ22)‖v‖+ δ1τ−1‖D2v‖
where the last two terms on the right are negligible compared to the left hand
side. The proof of the proposition is concluded.
We want to reformulate the previous result in a more symmetric fashion.
To do this we weaken the estimates slightly by using a coarser partition of the
space. We distinguish three cases for i corresponding to the value of j(i) in
Lemma 6.1 (v).
Definition 6.6. We define the partition Bij as follows.
(1) If j(i) = 0 (which corresponds to εi & τ
2 ) we set
Bij = Aij, b ≈ a, b⊥ ≈ a⊥
(2) If 0 < j(i) < [ln τ/2+2] (which corresponds to τ−1 . εi . τ
2 ) we set
Bij = Aij, b ≈ a, b⊥ = a⊥ j ≥ j(i)
respectively
Bi0 =
j<j(i)
Aij , b ≈ a|Aij(i) b⊥ = 0 on Bi0
(3) If j(i) = [ln τ/2 + 2] (which corresponds to τ−1 . εi) we set
Bi0 =
[ln τ/2]+2⋃
Aij, b = 1, b⊥ = 0 on Bi0.
Heuristically the definition of the Bij partition is motivated by the fact that
in regions Aij with j < j(i) the weight φ is ineffective, i.e. it changes by at
most O(1). Thus the convexity there is useless, and instead we rely directly
on localized bounds for the Hermite operator.
Since the εij are slowly varying b . a and b⊥ . a⊥ and we may replace the
a’s by b’s in the above proposition.
To provide some bounds on the size of b and b⊥ we introduce a function
1 ≤ r(s) ≤ τ 12 which is smooth and slowly varying on the unit scale in s so
r(s) ≈ ej(i) s ∈ [i, i+ 1]
This describes the region where b is tapered off and b⊥ = 0. Precisely, consider
two cases corresponding to the three cases above.
(1) If r(s) ≈ 1 then we have the bounds
Mτ(1 + r)−
2 . b4(r, s) .Mτ(1 + r)−1
b4⊥(r, s) .Mτ(1 + r)
(6.27)
where the parameter M ≥ 1 is defined by M ≈ ε(s)τ 12 .
(23) If r(s) ≫ 1 then
τr(s)−
2 (r(s) + r)−
2 . b4(r, s) . τr(s)−1(r(s) + r)−1
b4⊥(r, s) . τr(s)
−1(r(s) + r)−1
(6.28)
with approximate equality when r . r(s) and approximate equality on the
right when r = τ
By slightly changing b and b⊥ we may and do assume that the functions b
and b⊥ are smooth with controlled derivatives. Thus b and b⊥ are smooth on
the unit scale in s and on the dyadic scale in y, and their derivatives satisfy
the bounds
(6.29) |bs|+ (rs + r)|br|+ (rs + r)2|brr| . b, r2 < 9τ
(6.30) |b⊥s|+ (rs + r)|b⊥r|+ (rs + r)2|b⊥rr| . b⊥ + b r2 < 9τ
In addition we have
(6.31) supp b⊥r ⊂ {r > rs}
Using the functions b and b⊥ we define the Banach space X
2 with norm
= ‖bv‖2L2 + τ−1/2‖b⊥D
Then the symmetrized version of Proposition 6.5 has the form
Proposition 6.7. Assume that the coefficients of P satisfy (6.22). Let ψ be
as in (6.9) with h, φ as in Lemmas 6.2,6.3. Then the following L2 Carleman
estimate holds for all functions u supported in {y ≤ 9τ}:
(6.32) ‖eψ(s,y)u‖X02 . ‖e
ψ(s,y)Pu‖(X02 )∗
Proof. Conjugating with respect to the exponential weight, the bound (6.32)
is rewritten in the form
(6.33) ‖v‖X02 . ‖Pψv‖(X02 )∗
Observing that
D2⊥ = −y−2∆Sn−1
we introduce the operator
Q = Q(|y|, (−∆Sn−1)
2 ), q(r, λ) = (b4(r) + r−2τ−1b4⊥(r)λ
Then the inequality (6.33) can be written as
‖Qv‖L2 . ‖Q−1Pψv‖L2
whereas inequality (6.23) implies
‖Q2w‖L2 . ‖Pψw‖L2.
Hence it is natural to apply (6.23) to the function w = Q−1v, which solves
Pψw = Q
−1Pψv +Q
−1[Q,Pψ]w
Thus (6.33) would follow provided that the commutator term is small,
‖Q−1[Q,Pψ]w‖L2 ≪ ‖b2w‖L2 + τ−1/2‖b2∇w‖L2 + τ−1‖b2D2yw‖L2
Unfortunately a direct computation shows that the smallness fails when j is
close to j(i) even in the flat case, i.e. with Pψ replaced by
P0,ψ = ∂s −∆+ y2 − ψs − ψ2y
To remedy this we introduce an additional small parameter δ and use it to
define a modification Qδ of Q. We modify r(s) to rδ(s) defined by
rδ(s)
−2 = δ8r(s)−2 + δ2τ−1
and use it to define the function
bδ(r, s)
4 = δ−12τ(r2 + rδ(s)
We can still compare it with b,
bδ(r, s)
4 . δ−4b(r, s)4
Its usefulness lies in the fact that it is larger than b exactly in the region where
the commutator term above is not small.
The modification Qδ of Q has symbol
qδ(r, s, λ) = q(r, s, λ) + bδ(r, s) = (b
4(r, s) + r−2τ−1b4⊥(r, s)λ
4 + bδ(r, s)
which satisfies
(6.34) q ≤ qδ . δ−1q
We claim that it satisfies the bound
(6.35) ‖Q−1δ [Qδ, Pψ]w‖L2 . (δ + c(δ)δ1)
j=0,1,2
2‖b2Djw‖L2
Suppose this is true. Then we fix δ sufficiently small, and for δ1 small enough
we apply (6.23) to w = Q−1δ v. By (6.35) he commutator term in the equation
for w can be neglected, and we obtain
‖Q2w‖L2 . ‖Q−1δ Pψv‖L2
which by (6.34) implies that
‖Qv‖L2 . δ−1‖Q−1Pψv‖L2
It remains to prove (6.35).
I. We first calculate the commutator in the flat case, i.e. with Pψ replaced
by P0,ψ. Due to the spherical symmetry the only contribution comes from the
radial part of the Laplacian and the s derivative. Hence using polar coordinates
we compute
Q−1δ [Qδ, P0,ψ] = Q
Qδrr +
Qδr + 2Qδr∂r −Qδs
Then is suffices to verify that on the symbol level we have
(6.36) |qδrr|+ r−1|qδr|+ |qδs|+ τ 1/2|qδr| . δqδb2(1 + τ−1r−2λ2)
I.(1). We begin with the q component of qδ. Using (6.29) and (6.30) one
obtains
|qrr|+ r−1|qr|+ |qs|+ τ 1/2|qr| . (r(s) + r)−1τ
Thus it remains to show that
(r(s) + r)−1τ
2 q . δqδb
2(1 + τ−1r−2λ2)
Optimizing with respect to λ it suffices to consider the cases λ = 0 respectively
λ = rτ
2 , where the above inequality becomes
(r(s) + r)−1τ
2 (b+ b⊥) . δ(bδ + b)b
or equivalently
(b+ b⊥)b
1 . δ(bδ + b)b
which is true since by (6.27) and (6.28) we have b⊥b1 . b
1 while b1 . δbδ.
I.(2). Next we consider the bδ component of qδ, for which it suffices to prove
(6.37) |bδrr|+ r−1|bδr|+ |bδs|+ τ 1/2|bδr| . δbδb2
I.(2).(a). For the s derivative we compute
(r2δ(s))s
r2 + r2δ(s)
therefore we want to show that
(r2δ(s))s . δ(r
2 + r2δ(s))b
We optimize the right hand side with respect to r. The minimum is attained
when r2 = min{r2δ(s), τ}. We need to consider two cases:
I.(2).(a).(i). If rδ(s) . τ
2 then r2δ(s) ≈ δ−8r2(s) and τ > δ−8r2(s). Hence
using the estimate from below in (6.27) and (6.28) we obtain b4(rδ) & δ
Then the above bound for r = rδ follows since |(rδ(s)−2)s| . rδ(s)−2.
I.(2).(a).(ii). If rδ(s) & τ
2 then by6 (6.28) we evaluate b2(τ
2 ) ≈ τ 14 r(s)− 12 .
Then the above bound becomes
δ8(r(s)−2)s . δr
4 r(s)−
Since |(r(s)−2)s| . r(s)−2 it suffices to show that
δ8r(s)−2 . δr−2δ τ
4 r(s)−
The worst case is r(s)2 = δ6τ , rδ(s) = δ
−2τ when it is verified directly.
I.(2).(b). For the r derivatives the last term is the worst. Since
r2 + r2δ(s)
we want to show that
2 r . δ(r2 + r2δ(s))b
Optimizing with respect to r the worst case is when r2 = min{r2δ(s), τ}.
I.(2).(b).(i). If r2δ(s) . τ then rδ(s) ≈ δ−4r(s) therefore for r = rδ(s) the
above relation becomes
2 . δ−3r(s)b2(rδ(s))
which follows from the bound from below in (6.27) and (6.28).
I.(2).(b).(ii). If r2δ(s) & τ then as before we evaluate b
2 ) ≈ τ 14 r(s)− 12 and
rewrite the above bound as
τ . δr2δ(s)τ
4 r(s)−
The right hand side is smallest either when rδ(s) = τ
2 and r(s) = δ4τ
when rδ(s) = δ
2 and r(s) = τ
2 . In both cases the inequality is easily
verified.
II. Now we deal with the general case, which we treat as a perturbation.
Since we do not care about the dependence of the constants on δ to keep the
notations simple we include bδ in b and work with Q instead of Qδ. Thus in
the computations below we allow the implicit constants to depend on δ.
Suppose that A is a pseudodifferential operator of order 1 and let η be any
Lipschitz function. Then
(6.38) ‖[A, η]f‖L2 . ‖f‖L2.
We write
q(λ) = b+ (b4 + r−2b4⊥λ
4 − b =: b+ q1(λ).
Even though q1 has order
, we treat it as an operator of order 1 and estimate
〈λ〉k−1|q1(k)(λ)| .
6the equality holds on the right when r2 = τ
Hence, for each r we obtain the bound on the sphere Sn−1
(6.39) ‖[Q, η]f‖L2 .
‖η‖Lip(Sn−1)‖f‖L2.
As a consequence, it also follows that
(6.40) ‖[Q, η∇jθ]f‖L2 .
‖η‖Lip(Sn−1)‖∇jθf‖L2.
where ∇θ stands for the vector fields xi∂j − xj∂i generating the tangent space
of Sn−1.
To use these bounds we write the difference Pψ − P 0ψ in polar coordinates,
Pψ − P 0ψ = P 0θ ∂2r + P 1θ ∂r + P 2θ
where P
θ are spherical differential operators of order j. Modulo zero homoge-
neous coefficients which are polynomials in xr−1 we can write
P 0θ = d, P
θ = dr
−1∇θ + τ
2d+ dy
P 2θ = dr
−2∇2θ + (τ
2d+ dy)r
−1∇θ + τd + τ
where d stands for coefficients satisfying (6.22). In the support of b⊥ we have
a ≈ b therefore our regularity assumptions on d show that for fixed r we have
‖d‖L∞ + ‖d‖Lip(Sn−1) . δb4τ−1
The coefficients involving dy satisfy better Lipschitz bounds and are neglected
in the sequel.
We expand the commutator
[Q,Pψ] =
j=0,1,2
r + P
θ (Qrr + 2Qr∂r) + P
Using the trivial b−1 bound for Q−1 and (6.17), (6.40) we estimate the first
term,
j=0,1,2
‖Q−1[Q,P jθ ]∂
r w‖L2 . b−1δb4τ−1
j=0,1,2
2‖Djw‖L2
This is bounded by the right hand side in (6.35) since
b2⊥ . rτ
The second term in the commutator is estimated by
‖Q−1P 0θ (Qrr + 2Qr∂r)w‖L2 . δ(‖Qrrw‖L2 + ‖Qr∂rw‖L2)
This is bounded by the right hand side in (6.35) provided that
|qrr|+ τ−
2 |qr| . b2(1 + τ−
2 r−1λ)
which follows from (6.36). The third term in the commutator is treated simi-
larly. This concludes the proof of the proposition. �
To conclude our study of the L2 Carleman estimates we need to also pay
some attention to elliptic estimates. The conjugated operator Pψ is elliptic in
the region {y2+ ξ2 ≥ 4τ}. Precisely, in this region we have the symbol bound
|Lrψ(s, y, ξ)| & y2 + ξ2
Consequently, we can improve our estimates in this region. We consider a
smooth symbol ae(y, ξ) with the following properties
supp ae ⊂ {y2 + ξ2 ≥ 8τ}
ae(y, ξ) = (y
2 + ξ2)
2 in {y2 + ξ2 ≥ 9τ}
We define the space X2 with norm
(6.41) ‖v‖2X2 = ‖v‖
+ ‖awe (y,D)v‖2
The dual space X∗2 has norm
(6.42) ‖f‖2X∗2 = inf{‖f1‖
(X02 )
∗ + ‖f2‖2; f = f1 + awe (y,D)f2}
We note that due to the elliptic bound for high frequencies, we also have the
dual bounds
(6.43) τ−
2‖bDv‖ . ‖v‖X2 , ‖∇f‖X∗2 . ‖b
Then our final L2 Carleman estimate is
Theorem 5. Assume that the coefficients of P satisfy (6.22). Let ψ be as
in (6.9) with h, φ as in Lemmas 6.2,6.3. Then the following L2 Carleman
estimate holds for all functions u for which the right hand side is finite:
(6.44) ‖eψ(s,y)u‖X2 . ‖eψ(s,y)Pu‖X∗2
Proof. We first prove the result using the stronger assumption (6.22) on the
coefficients. After conjugation we have to show that
(6.45) ‖v‖X02 . ‖Pψv‖X∗2
We consider two overlapping smooth cutoff symbols χi = χi(y
2 + ξ2) and
χe = χε(y
2+ξ2). The interior one χi is supported in {y2+ξ2 ≤ 7τ} and equals
1 in {y2 + ξ2 ≥ 6τ}. The exterior one χe is supported in {y2 + ξ2 ≥ 4τ} and
equals 1 in {y2 + ξ2 ≤ 5τ}. We need the following bounds for χi and χe:
Lemma 6.8. a) The operator χi(x,D) satisfies the bound
(6.46) ‖χi(x,D)f‖(X02 )∗ . ‖f‖X∗2
b) The operators χi(x,D) and χ
e (x,D) satisfy the following commutator
estimates:
(6.47) ‖b−1[χi(x,D), Pψ]v‖ . τ−
4‖bv‖+ ‖χev‖
(6.48) ‖[χwe , Pψ]v‖ . τ
8‖bv‖
Proof. a) By duality the bound (6.46) is equivalent to
‖χi(x,D)v‖X2 . ‖v‖X02
We have
‖ae(x,D)χi(x,D)v‖ . τ−N‖f‖
since the supports of the symbols (1 − χe(x, ξ)) and ae(x, ξ) are O(τ
2 ) sepa-
rated. Then it remains to show that
‖χi(x,D)v‖X02 . ‖v‖X02
which is fairly straightforward and is left for the reader.
b) We now consider the bound (6.47). Commute first χi with ∂s+H−h′(s).
We have
[χi(x,D), ∂s +H − h′(s)] = [χi(x,D), H ]
Since χe = 1 in the support of ∇x,ξχi and the Poisson bracket of χi and x2+ξ2
vanishes, by standard pdo calculus we obtain
‖[χi(x,D), ∂s +H − h′(s)]v‖ . ‖χev‖+ τ−N‖v‖
The difference Pψ − (∂s +H − h′(s)) can be expressed in the form
Pψ − (∂s +H − h′(s)) = ∂g∂ + τ
2 (g∂ + ∂g) + τg
where the function g satisfies the bounds
|g|+ 〈y〉|gy|+ 〈y〉|gyy| . εi
These lead to an estimate for fixed s ∈ [i, i+ 1],
‖b−1[χi(x,D), Pψ − (∂s +H − h′(s))]v‖ . εiτ
2‖〈y〉−1b−1v‖
Then (6.47) follows since
2 〈y〉−1 . τ−
Finally, the proof of the estimate (6.48) is similar but simpler.
We continue with the proof of the proposition. For the nonelliptic part we
apply (6.32) to the function χi(x,D)v which is supported in {y2 < 9τ}. This
gives
‖χi(x,D)v‖X02 . ‖χi(x,D)Pψv‖(X02 )∗ + ‖[χi(x,D), Pψ]v‖L2
For the first term on the right we use the bound (6.46) while for the second
we use (6.47). This yields
(6.49) ‖χi(x,D))v‖X02 . ‖Pψv‖X∗2 + τ
4‖bv‖+ ‖χev‖
On the other hand for the estimate in the elliptic region we compute
(6.50) 〈(χwe )2v, Pψv〉 = 〈χwe v, Lrψχwe v〉+ 〈χwe v, [χwe , P rψ]v〉
For the first term we split Lrψ into H−h′ plus a perturbation. Using pointwise
bounds for the coefficients of Pφ we obtain
|Lrψv − (H − h′)v| . δ1((τ + y2)|v|+ τ
2 |Dv|+ |D2v|)
which shows that
〈χwe v, Lrψχwe v〉 = 〈χwe v, (H − h′)χwe v〉+O(δ1〈χwe v, (H + τ)χwe v〉)
The symbol of H − h′ is elliptic in the support of χe, therefore a standard
elliptic argument yields
〈χwe v, (H + τ)χwe v〉 . 〈χwe v, (H − h′)χwe v〉+ Cτ−N‖v‖2
for a large constant C. This further gives
〈χwe v, (H + τ)χwe v〉 . 〈χwe v, Lrψχwe v〉+ Cτ−N‖v‖2
Returning to (6.50), we obtain
c〈χwe v, (H + τ)χwe v〉 ≤ −〈(χwe )2v, Pψv〉+ 〈χwe v, [χwe , P rψ]v〉+ Cτ−N‖v‖2
We use (6.48) and then the Cauchy-Schwartz inequality to obtain
〈χwe v, (H + τ)χwe v〉 . ‖(H + τ)−
2Pψv‖2 + τ−
4‖bv‖2
The first term on the right is properly controlled due to the straightforward
estimate
‖(H + τ)−
2f‖ . ‖f‖X∗2
Hence combining the above inequality with (6.49) we obtain
‖χ(ix,D))v‖X02 + ‖(H + τ)
2χwe (x,D)v‖ . ‖Pψv‖X∗2 + τ
4‖bv‖+ ‖χwe (x,D)v‖
The last two terms on the right are negligible compared to the left hand side,
therefore we obtain
(6.51) ‖v‖X02 + ‖(H + τ)
2χwe (x,D)v‖ . ‖Pψv‖X∗2
Then (6.45) follows since χe = 1 in the support of ae.
It remains to show that the assumption (6.22) on the coefficients for (6.7)
can be replaced by the weaker condition (1.15). This is a direct consequence
of (6.43) combined with the following regularization result:
Lemma 6.9. Let d be a function which satisfies (1.15). Then there is an
approximation g1 of it satisfying (6.22) so that
|g − g1| . b2τ−1
Proof. First we transfer (1.15) to the (s, y) coordinates. A short computation
yields the equivalent form
(6.52) ‖d‖L∞(Aij) + ej‖d‖Lipy(Aij) + ‖g‖Cmijt (Aij) . εij
where the new continuity modulus m̃ij is given by
m̃ij(ρ) = ρ+ e
Within Aij we regularize d in y on the δy = τ
2 scale and in s on the
δs = e
4 scale,
d1 = S
(Dy)S
(Ds)d
These localized regularizations are assembled together using a partition of unit
corresponding to Aij. In Aij we compute
|d− d1| . εij(e−jδy +mij(δs)) ≈ εij(e−jτ−
2 + e−
4 ) . ε
4 . b2τ−1
while
|∂sg1| . εij
mij(δs)
≈ εije−jτ
The bounds for higher order derivatives of g1 follow trivially due to the fre-
quency localization. �
7. Lp Carleman estimates for variable coefficient operators
The variable coefficient counterpart of Proposition 4 uses the more convex
weights constructed in Section 6. For convenience we write it in the (s, y)
coordinates. Let τ >> 1 and B(τ) be as in (1.25),(1.26) and (1.27).
We define the function space X through its norm
(7.1) ‖v‖X := ‖v‖X2 + ‖v‖l2(B(τ);L∞t L2x) + ‖v‖l2(B(τ);LptLqx)
where (p, q) is an arbitrary Strichartz pair, with X2 as defined in (6.41).
Its (pre)dual space has the norm
‖f‖X∗ = inf
f=f1+f2+f3
‖f1‖X∗2 + ‖f2‖L1L2 + ‖f3‖Lp′Lq′(7.2)
Then we have the following improvement of Theorem 5.
Theorem 6. There exists ψ as in (6.9) with h and φ as in Lemma 6.2 and
6.3. Then the following estimate holds for all compactly supported sufficiently
regular functions u.
(7.3) ‖eψu‖X . ‖eψP̃ u‖X∗
The relation between ψ and the partition B(τ) remains a bit mysterious at
this level. If we replace it by the empty partition then the statement remains
true for all ψ with h and φ as in Lemma 6.2. The same is true for a partition
into time slices of size 1. The convexity properties of φ allow a localization
to the finer partition Bij as in 6.6 (and, as we shall soon see, to an even finer
partition). q It is possible to choose φ and h so that the partition (Bij) is
finer than the one defined by B(τ). We assume in the sequel that ψ has been
chosen with these properties.
Proof. As usual this is equivalent to proving a bound from below for the con-
jugated operator,
(7.4) ‖v‖X . ‖Pψv‖X∗
The main step in the proof is to produce a parametrix for Pψ. The key
point is that the parametrix is allowed to have a fairly large L2 error. This
is because L2 errors can be handled by Theorem 5. The advantage in having
a large L2 error is that it permits to localize the parametrix construction to
relatively small sets, on which we can freeze the coefficients and eventually
reduce the problem to the case of the Hermite operator. The properties of the
parametrix are summarized in the following
Proposition 7.1. a) Under the assumptions of the theorem there exists a
parametrix T for Pψ with the following properties:
(7.5) ‖Tf‖X . ‖f‖X∗
(7.6) ‖PψTf − f‖X∗2 . ‖f‖X∗
b) The same result holds with Pψ replaced by P
We first use the proposition to conclude the proof of the Theorem. Let
Pψv = f + g, ‖f‖X∗2 + ‖g‖L1L2+Lp′Lq′ ≈ ‖Pψv‖X∗
With T as in part (a) of the proposition we set
w = v − Tg, Pψw = f + g − PψTg
By (7.5) we can bound Tg in X , therefore it suffices to bound w in X . On the
other hand by (7.6) we obtain
‖Pψw‖X∗2 . ‖f‖X∗2 + ‖g − PψTg‖X∗2 . ‖Pψv‖X∗
It remains to show that
‖w‖X . ‖Pψw‖X∗2
By Theorem 5 we can estimate the X2 norm of w and replace this with the
weaker bound
(7.7) ‖w‖L∞L2∩LpLq . ‖w‖X2 + ‖Pψw‖X∗2
This is proved using a duality argument and the parametrix T for P ∗ψ given
by part (b) of the proposition. For f ∈ X∗ we write
〈w, f〉 =〈w, P ∗ψTf〉+ 〈w, f − P ∗ψTf〉
=〈Pψw, Tf〉+ 〈w, f − P ∗ψTf〉
Using both (7.5) and (7.6) with Pψ replaced by P
ψ we obtain
|〈w, f〉| . (‖Pψw‖X∗2 + ‖w‖X2)‖f‖X∗
and (7.7) follows. This concludes the proof of Theorem 6. �
It remains to prove the Proposition 7.1.
Proof of Proposition 7.1. The strategy for the proof is simple: On sufficiently
small sets we can approximate the problem by one with constant coefficients
and the properties of the parametrix follow from Section 4. We use a partition
of unity to construct a global parametrix from local ones. We obtain L2 errors
(1) Commuting cutoff functions with the operator. Hence the partition
has to be sufficiently coarse.
(2) Approximating the variable coefficient operator by constant coefficient
operators. Hence the partition has to be sufficiently fine.
To elaborate on this we define the notion of a local parametrix:
Definition 7.2 (Local parametrix). Given a convex set B we call T a (B-)
local parametrix for Pψ if for all f supported in B
(7.8) ‖Tf‖X . ‖f‖X∗ ,
(7.9) ‖PψTf − f‖X∗2 . ‖f‖X∗
and Tf is supported in 2B.
If T is a parametrix and η is supported on 2B, η = 1 on B then ψT is a
local parametrix, but with constants depending on the commutator of Pψ and
η. Vice verse, if (Bj) is a covering, (ηj) a subordinate partition of 1 and Tj
are local parametrices then
is a global parametrix, because (7.8) is obtained by summation, and
Tj(ηjf)− f =
(PψTjηjf − ηjf
provided ∑
‖ηjf‖2X∗ . ‖f‖2X∗
and its adjoint
‖u‖2X .
‖ηju‖2X .
This is obvious for the L2 part and has to be checked for the other part. This
strategy of constructing local parametrices leads, if it is possible, to estimates
which are stronger than in Proposition 7.1 and Theorem 6, because we may
replace the function space X by l2X(Bj) respectively. l
2X∗(Bj).
In the first part of the proof we study the localization, and in the second
part we provide the local parametrices.
7.1. Localization scales. Here we introduce a localization scale which is finer
than the Bij partition of the space, and show that it suffices to construct the
parametrix in each of these smaller sets. Precisely, the sets Bkij introduced
below are the smallest sets to which one can localize the L2 estimates for the
operator Pψ. The choice of their size is not yet apparent at this point, but
will become clear in the very last step of the proof, where we estimate the
commutator of Pψ with cutoff functions on such sets. We consider three cases
depending on the size of εi.
(1) If εi ≤ τ−1 then we use Bi0 as it is.
(2) If τ−1 ≤ εi ≤ τ−
2 then we partition the set Bi0 into time slices B
i0 of
thickness
δs = b−2i0 .
(3) If τ−1 ≤ εi and j 6= 0 then we partition Bij into subsets Bkij which have
the time scale, radial scale and angular scale given by
δs = b−2ij , δy = τ
2 b−2ij , δy
⊥ = τ
2 b−2ij,⊥
This gives a decomposition of the space
R× Rn =
We also consider a subordinated partition of unity
Suppose that in each set Bkij we have a parametrix T
ij satisfying (7.5) and
(7.6). Then we define the global parametrix T by
T kijχ
We have by an iterated application of Minkowski’s inequality
‖χkijf‖l2(L1L2+Lp′Lq′) . ‖f‖L1L2+Lp′Lq′
and the dual bound
T kijχ
ijf‖L∞L2∩LpLq . ‖T kijχkijf‖l2(L∞L2∩LpLq).
Hence (7.5) for T would follow if we proved that
vkij‖X2 . ‖vkij‖l2X2
where vkij = T
ijf are supported in B
ij . This is trivial by orthogonality for
the L2 component of the X2 norm. It is also straightforward for the elliptic
part since the kernel of the operator in the following sense: Let χ0 ∈ C∞(R)
be supported in [−3/2, 3/2], identically 1 in [−1, 1]. We define χe(x, ξ) =
χ0(5− (x2 + ξ2)/τ 2). Then
χwe v
ij‖X2 . ‖χwe vkij‖l2X2
and the adjoint estimate holds since the kernel of χwe is rapidly decreasing
beyond the τ−
2 scale, which is much smaller than the smallest possible spatial
size for Bkij, namely δy & τ
It remains to consider the angular part of the X2 norm, which is best de-
scribed using the spherical multiplier Q appearing in Proposition 6.7. The
symbol of Q is smooth with respect to λ on the τ
2 rb2b−2⊥ scale therefore its
kernel is rapidly decaying on the angular scale δθ = τ−
2 r−1b−2b2⊥ which cor-
responds to δy⊥ = τ
2 b2⊥b
−2. But by (3) and (6.5) this can be no larger than
8 which is again much smaller than the smallest possible spatial size for Bkij .
Thus orthogonality arguments still apply.
Then we have
[P0, χ
ij]u = 2χijy uy + χ
yyu+ χ
We claim that the right hand side is negligible in the estimate. For this it
suffices to verify that
|χijy | ≪ τ−
2 b2ij , |χijyy| ≪ b2ij , |χijs | ≪ b2ij
The last relation is trivial. For the first two we consider three cases.
(1) If j = 0 and εj < Cτ
−1 then we need no spatial truncation. We are
allowed to truncate at |y| > Cτ 12 to separate the elliptic region, though.
(2) If j = 0 and εj > Cτ
−1 then
|χijy | . e−j(i), |χijyy| . e−2j(i)
while
b4i0 = a
ij(i) = εij(i)τ
2 e−j(i) ≈ Ce−2j(i)τ
(3) Otherwise,
|χijy | . e−j , |χijyy| . e−2j
while
b4ij = a
ij = εijτ
2 e−j & Ce−2j(i)τ
The results can be summarized by saying that it suffices to construct local
parametrices in the sets Bkij.
7.2. Freezing coefficients. Our first observation is that restricting the result
in the proposition to a single region Bkij allows us to freeze the weights b, b⊥
in the X2 norms.
Next we are interested in freezing the coefficients of P . We consider the
same three cases as above:
The case εi ≤ τ−1. In this case we are localized to
Bi0 = [i, i+ 1]× Rn
and we have
bi0 ≈ 1, εij . τ−1
By (6.2) the second relation leads to
|d| . τ−1
Then using also (6.43) we can estimate the terms involving d in the expression
(6.22) for Pψ,
(7.10) ‖∂d∂v‖X∗2 + τ‖dv‖X∗2 + τ
2‖(d∂ + ∂d)v‖X∗2 . ‖v‖X2
Hence without any restriction of generality we can assume that d = 0 in Pψ,
which corresponds to taking g = In.
We also observe that in this case we have
|φ| . 1, |φy| . τ−
Then we can also drop the φ component of ψ. Finally, since
|hss| . 1
we can replace h by its linearization at some point in the corresponding s
region.
Conclusion: It suffices to prove the result when d = 0, ψ(y, s) = τs.
We note that the separation of τ from integers is no longer needed due to
the localization to unit s intervals.
The case τ−1 ≤ εi. In this case we are localized to a region of the form
Bki0 = [s0, s0 + e
j(i)τ−
2 ]×B(0, ej(i))
and we have
b4i0 ≈ τe−2j(i), εij . e−j(i)τ−
The second relation leads to
|d| . e−j(i)τ−
Then (7.10) is still valid, so we can assume again that d = 0 in Pψ.
We also observe that in this case we have
|φ| . 1, |φy| . e−j(i)
Then we can also drop the φ component of ψ. Finally, since
|hss| . e−j(i)τ−
we can replace h by its linearization at some point in the corresponding s
region.
Conclusion: It suffices to prove the result when d = 0, ψ(y, s) = τs.
The case τ−1 ≤ εi. In this case we are localized to a region of the form
Bkij = [s0, s0 + τ
ij ]× B(y0, τ−
ij ), |y0| ≈ ej
and we have
b2ij ≈ ε
Using (6.2) it follows that
|g(s, y)− g(s0, y0)| . ε
Arguing as before, this allows us to freeze d within Bkij. However, we note that
we are no longer allowed to replace d by 0.
Next we turn our attention to the weight function ψ. First we have
|hss| . εjτ . εijτ
2 e−j
which allows us to replace h by its linearization in s at s0.
Secondly, we claim that we can replace φ by its linearization at y0. In the
radial direction we have weaker localization but a stronger bound
|φrr| . εijτ
2 e−j
In the transversal direction we have better localization but a weaker bound,
|φyy| . εiτ
2 e−j .
The first bound allows us to obtain the relation
|φ2y(y, s)− φ2y(y0, s0)| . ε
2 ≈ b2ij
Using also the second bound we can write
(φy(y, s)− φy(y0, s0))∂y = νr∂r + ν⊥∂⊥
where the coefficients νr and ν⊥ are smooth on the B
ij scale and satisfy the
bounds
|νr| . τ−
2 b2ij , |νr| . τ−
2 b2ij,⊥
Conclusion: It suffices to prove the result when d = g(s0, y0), ψ(y, s) =
τs + cy, |c| . εiτ
Additional simplification in the highly localized case. Given the above simpli-
fications we need to work with a constant coefficient operator Pψ which has
the form
Pψ = −∂t +H − τ + ∂d∂ + c∂, |d| . εij, |c| ≤ εi
We diagonalize the second order part with a linear change of variables to obtain
Pψ = −∂t +H − τ + c∂ +O(εij)y2
We can freeze the last term at y0 and add it into τ . To deal with c we make
the change of variable
y → y − (s− s0)c
Then our operator becomes
P̃ψ = −∂t +∆− (y − c(s− s0))2 + τ
and the s− s0 terms are negligible due to the s localization.
Conclusion: We can assume without any restriction in generality that g = In
and ψ = τs.
7.3. The localized parametrix. We begin with the global parametrix K
constructed in Section 5. Then we define the parametrix TB in B by
TB = χ2BK, B = B
and show that it satisfies (7.8) and (7.9).
The Lp part of (7.8) follows directly from (5.4). It remains to prove the X2
part,
‖TBf‖X2 . ‖f‖L1L2+Lp′Lq′
The elliptic part of the X2 bound, namely
‖awe (x,D)χ2BKf‖L2 . ‖f‖L1L2+Lp′Lq′ ,
is obtained by an argument which is similar to the one beginning with (6.50).
For the rest we consider two cases.
i) If j = 0 then B is a ball, and we can use (5.4) directly with R = d.
ii) If j > 0 then B is contained in a sector B ⊂ BR,d but may be shorter
than R. This is why we can use (5.4) for the angular part of the X2 norm, but
not for the L2 part. However, the L2 part can be always obtained by taking
advantage of the time localization,
‖bijTBf‖L2 . ‖TBf‖L∞L2 . ‖f‖L1L2+Lp′Lq′
It remains to consider the error estimate (7.9). We have
f − (∂s −H + τ)TBf = [χ2B, ∂s −H + τ ]Kf = [χ2B, ∂s −H + τ ]χ4BKf
But arguing as above χ4BK0f satisfies the same X2 bound as χ2BK0f . Hence
it suffices to show that
[χ2B , ∂s −H + τ ] : X2 → X∗2
This is where the dimensions of the set B are essential; they are chosen to be
minimal so that the above property holds. We have
[χ2B, ∂s −H + τ ] = −∂sχ2B + (∂yχ2B)∂y + ∂y(∂yχ2B)
= −∂sχ2B + (∂rχ2Br)∂r + ∂r(∂rχ2Br) + (∂⊥χ2By)∂⊥ + ∂⊥(∂⊥χ2B)
For the first factor we use the bound
|∂sχ2B| . b2ij
For the radial derivatives of χ2B we combine (6.43) with
|∂rχ2B| . τ−
2 b2ij
Finally, for the angular derivatives we use the angular H
2 norm in X2 and the
bound
|∂rχ2B| . τ−
2 b2ij⊥
8. The gradient term
In this section we consider the full problem, i.e. involving also the gradient
potential W . Ideally one might want to have a stronger version of Theorem 6
which includes additional bounds for the gradient, more precisely for
‖eψ(s,y)∇u‖L2
But such bounds cannot hold, for this would imply that one can improve the
Lp indices in a restriction type theorem. To overcome this difficulty we proceed
as in [17], using Wolff’s osculation Lemma. Wolff’s idea is that by varying the
weight one can ensure concentration in a sufficiently small set, in which the
gradient potential term is only as strong as the potential term. Thus we still
obtain a one parameter family of Carleman estimates, but with the weight
depending not only on the parameter but also on the function we apply the
estimate to.
Given a gradient potential W satisfying (1.19), we first readjust the param-
eters εij, εi constructed in Lemma 6.1 in order to insure that we have the
additional condition
‖W‖Ln+2(Aτi ) ≪ εi
Then we begin with the spherically symmetric weights ψ constructed in Sec-
tion 6 and modify them as follows:
(8.1) Ψ(s, y) = ψ(s, y) + δk(s, y)
where the perturbation k is supported in {|y| ≤ 9τ} and is subject to the
following conditions:
(8.2) |∂αs ∂βy ∂
⊥k(s, y)| . εiτ
2 s ∈ [i, i+ 1]
Here δ is a sufficiently small parameter.
In order to prove the strong unique continuation result in the presence of
the gradient potential W we need the following modification of Theorem (6):
Theorem 7. Assume that (1.15) holds. Then for each τ > 0 and W subject
‖W‖Ln+2(Aτi ) ≤ εi
and each function u vanishing of infinite order at (0, 0) and ∞ there exists a
perturbation k as in (8.2) so that
(8.3) ‖eΨu‖X+‖eΨW∇u‖X∗+‖eΨ∇(Wu)‖X∗+τ
2‖eΨWu‖X∗ . ‖eΨ(x)P̃ u‖X∗
Here and in the sequel we will omit indices for W . After returning to the
(x, t) coordinates and taking (1.19) into account this implies Theorem 3. The
reader should note that the choice of φ depends on both u and W . This is
essential since for fixed φ (8.3) cannot hold uniformly for all u and W .
Proof. Up to a point the proof follows the steps which were discussed in detail
before. We outline the main steps:
STEP 1: Show that the L2 Carleman estimate (6.23) holds with ψ replaced
by Ψ for all perturbations k as in (8.2). The new conjugated operator PΨ is
obtained from Pψ after conjugating with respect to the weight e
k(y,s). This
adds a few extra components to the selfadjoint and skewadjoint parts,
LrΨ = L
ψ + k
y + ks + 2ky(ψy + d)
LiΨ = L
ψ − ky(1 + d)∂ − ∂ky(1 + d)
Observing that we can write
k2y + ks + 2ky(ψy + d) = τd, ky(1 + d) = τ
with d as in (6.22) we conclude that the conjugated operator PΨ retains the
same form as Pψ, therefore the proof of (6.23) rests unchanged.
STEP 2: Show that the symmetric L2 Carleman estimate (6.23) holds with
ψ replaced by Ψ for all perturbations k as in (8.2). Since PΨ has the same
form as Pψ, this argument is identical.
STEP 3: Show that the symmetric mixed L2 ∩ Lp Carleman estimate in
Theorem 6 holds with ψ replaced by Ψ for all perturbations k as in (8.2).
Since PΨ has the same form as Pψ, this argument is also identical.
STEP 4: Decompose W into a low and a high Hermite-frequency part,
W =Wlow +Whigh, Wlow = χ
i (x,D)W
where the smooth symbol χ1i (x, ξ) is supported in {x2 + ξ2 ≤ 81τ} and equals
1 in the region {x2 + ξ2 ≤ 64τ}. Then we show that the high frequency part
of W satisfies the desired estimates for all perturbations k as in (8.2), namely
(8.4) ‖eΨWhigh∇v‖X∗ +‖∇Whighv‖X∗ + τ
2‖eΨWhighv‖X∗ . ‖eΨu‖X‖W‖Ln+2
After conjugation this becomes
‖Whigh∇v‖X∗ + ‖∇Whighv‖X∗ + τ
2‖Whighv‖X∗ . ‖v‖X‖W‖Ln+2
We only consider the first term on the left. The second one is equivalent
by duality, and the third one is similar but simpler. We divide v into two
components,
v = (1− χ1e)v + χ1ev
where the smooth symbol χ1e(x, ξ) is supported in {x2 + ξ2 ≥ 9τ} and equals
1 in the region {x2 + ξ2 ≥ 10τ}.
For the high frequency component of v we use the H1 part of the X2 norm
to estimate
‖Whigh∇χ1ev‖
2(n+2)
. ‖Whigh‖Ln+2‖∇χ1ev‖L2 . ‖W‖Ln+2‖v‖X
For the low frequency component of v it is still possible to estimate directly
the high frequency of the output,
‖χ1e(Whigh∇(1− χ1e)v)‖H−1 .τ−
2‖Whigh∇(1− χ1e)v‖L2
.‖Whigh‖Ln+2τ−
2‖∇(1− χ1e)v‖
2(n+2)
.‖W‖Ln+2‖v‖
2(n+2)
Finally, the last remaining part has a much better L2 estimate,
‖(1− χ1e)(Whigh∇(1− χ1e)v)‖ . τ−N‖W‖Ln+2‖v‖
which is due to the unbalanced frequency localizations of the two factors.
Due to the estimate (8.4), it suffices to prove (8.3) withW replaced byWlow.
This allows us to replace the term ∇(Wlowv) by
∇(Wlowv) =Wlow∇v + (∇Wlow)v
where we can estimate
‖∇Wlow‖Ln+2 . τ
2‖W‖Ln+2
Hence without any restriction in generality we can drop the third term in (8.3)
and show that we can choose the perturbation k so that
(8.5) ‖eΨW∇u‖X∗ + τ
2‖eΨWu‖X∗ . ‖eΨ(x)P̃ u‖X∗
STEP 5: Show that, given u and W , we can choose the perturbation k
so that (8.5) holds. At this stage we no longer need the full X∗ norm for
the W terms, it suffices instead to consider the L
2(n+2)
n+4 norm. Begin with the
unperturbed integral
Fψdxdt, Fψ = |eψW∇u|
2(n+2)
n+4 + |τ
2 eψWu|
2(n+2)
We can select a subset I of R consisting of time intervals of length 1 with unit
separation at least 8 so that
Fψdxdt .
Fψdxdt
By a small abuse of notation we label
Ii, Ii ⊂ [i− 1, i+ 1]
We define a family of perturbations k depending on parameters bi, σi by
k(y, s) =
100χ3τIi + χ2Iiχ|y|2≤τ (biy + σi(s− i))
|bi| ≤ τ
2 , |σi| ≤ τ
Due to the choice of the intervals Ii it is easy to see that after changing the
weight ψ to Ψ we retain the concentration to a dilate of I,
FΨdxdt .
FΨdxdt
The choice of the parameters bi, σi can be made independently for each i.
We consider two cases.
i) Suppose εi . τ
2 . Then the choice of the parameters is irrelevant since
in 3Ii we can estimate
‖eψW∇u‖
2(n+2)
2‖eψWu‖
2(n+2)
. ‖W‖Ln+2(‖eΨ∇u‖L2 + τ
2‖eΨu‖L2)
. (τ−1/2‖eΨ∇u‖L2 + ‖eΨu‖L2)
. ‖u‖X2
ii) Suppose εi ≫ τ−
2 . Then we need to choose the parameters bi, σi in a
favorable manner. This choice is made using Wolff’s Lemma:
Lemma 8.1 (Wolff’s Lemma [29]). Let µ be a measure in Rn and B a convex
set. Then one can find bk ∈ B and disjoint convex sets Ek ⊂ Rn so that the
measures exbkµ are concentrated in Ek,∫
exbkdµ &
exbkdµ
and ∑
|Ek|−1 & |B|
We apply the lemma for the measures
dµi = 13IiFψ
In our case we have
Bi = δεi
[−τ, τ ]×B(0, τ
, |Bi| ≈ εn+1i τ
Hence we can find parameters bki and σ
i and convex sets E
i ⊂ 3Ii×B(0, 3τ
so that the corresponding measures FΨk are concentrated in E
i with
|Eki |−1 & εn+1i τ
At the same time we have
|W |n+2dxdt . εn+2i
Hence we can choose k so that∫
|W |n+2dxdt . εiτ−
2 |Eki |−1
which by Holder’s inequality leads to
(8.6) ‖W‖
2 (Eki )
Denoting this index k by k(i) we can write
‖eΨW (∇, τ
2 )u‖
2(n+2)
. ‖eΨW (∇, τ
2 )u‖
2(n+2)
n+4 (Ii)
. ‖eΨW (∇, τ
2 )u‖
2(n+2)
n+4 (E
. ‖W (∇, τ
2 )(eΨu)‖
2(n+2)
n+4 (E
Decomposing the function v = eΨu into low and high frequencies we further
estimate
‖eΨW (∇, τ
2 )u‖
2(n+2)
. ‖W (∇, τ
2 )(1− χ1i (x,D))(eΨu)‖
2(n+2)
n+4 (Ii)
+ ‖W (∇, τ
2 )χ1i (x,D)(e
2(n+2)
n+4 (E
The first term on the right is estimated as in Step 4,
‖W (∇, τ
2 )(1− χ1i (x,D))(eΨu)‖
2(n+2)
n+4 (Ii)
. ‖W‖l∞Ln+2‖(1− χ1i (x,D))(eΨu)‖H1
. ‖W‖l∞Ln+2‖eΨu‖X
It is only for the second term on the right that we need to use (8.6):
‖W (∇, τ
2 )χ1i (x,D)(e
2(n+2)
n+4 (E
. ‖W‖
l∞i L
‖(∇, τ
2 )χ1i (x,D)(e
2(n+2)
n+4 (E
. ‖eΨu‖
2(n+2)
. ‖eΨu‖X
The proof of the Theorem is concluded. �
Appendix A. The change of coordinates
Suppose that the coefficients g satisfy (1.15). In this section we verify that
we can change coordinates so that (1.15) and (1.16) are both satisfied. Due
to the anisotropic character of the equation we must leave the time variable
unchanged and consider changes of coordinates which have the form
(t, x) → (s, y), s = t, y = χ(t, x).
The expression for the operator P in the new coordinates is
P = ∂t + ∂kg̃
kl(t, y)∂l +
d̃kl∂l
where the new coefficients g̃, d̃ are computed using the chain rule,
g̃kl =
d̃kl =
Dχ−1lm(∂
There is a price to pay for this, namely in the new coordinates we obtain
lower order terms which cannot be treated perturbatively. Instead we obtain
coefficients d̃k which have the same regularity and size as g1 − In.
The Lipschitz condition (1.15) ensures that g has a limit at (0, 0) so we
assume that g is continuous. After a linear change of coordinates we may and
do choose g with g(0, 0) = In. Again by (1.15) this implies
(A.1) |g(t, x)− In| ≪ 1.
Proposition A.1. Let g be a metric which satisfies (1.15) with g(0, 0) = In.
Then there is change of coordinates (t, y) = (t, χ(t, x)) which is close to the
identity
(A.2) ‖∂xχ− In‖L∞ ≪ 1
and has regularity
(A.3)
‖(t+ x2)−1/2(t∂t)α((t+ x2)1/2∂x)βχ‖l1(A(τ);L∞) ≪ 1, 2 ≤ 2α + |β| ≤ 4
so that in the new coordinates both functions g̃ and d̃ satisfy (1.15), while g̃−In
and d̃ satisfy (1.16).
Proof. Consider the covering of the [0, 2]× B(0, 2) = ∪Aij with an associated
smooth partition of unity ηij . We can assume that the functions ηij satisfy
(A.4) |∂αt ∂βxηij | . cαβt−α(t + x2)−
We choose the points
(ti, xij) = (e
−4i, e−2i+j) ∈ Aij.
and insure that ηij = 1 near (ti, xij). By (1.15) we have
(A.5) sup
(i,j)∈A(τ)
|g(ti, xij)−g(ti, xi,(j−1))|+ |g(ti, xij)−g(ti+1, x(i+1),j)| ≪ 1.
Within a fixed set Aij we consider the linear map defined by the matrix
χij = g
−1/2(ti, xij).
It transforms the coefficients at (ti, xij) to the identity and has the desired
properties within Aij . We assemble the maps defined by χij using the partition
of unity,
χ(t, x) =
ηij(t, x)χijx.
∇χ(t, x)− In =
(∇ηij)χijx+ ηij(χij − In),
Let (t, x) ∈ Ai0,j0. Since
∇ηij = 0 we have
∇xχ(t, x)− In =
ηij(t,x)>0
∇xηij(t, x)(χij − χi0,j0) + ηij(χij − In).
The first term on the right hand is small by (A.5) (for χij) and the second one
by (A.1) therefore the smallness of ∇χ− In follows.
For the second order spatial derivatives we write
D2xχ(t, x) =
D2ηij(t, x)χijx+ 2Dxηij(t, x)χij
D2xηij(t, x)(χij − χi0,j0)x+ 2Dxηij(χij − χi0,j0).
Hence by (A.4) and (A.5) (again for χij) we obtain
‖(|x|2 + t)1/2D2xχ‖l1(A(τ);L∞) ≪ 1.
∂tχ(t, x) =
∂tηij(χij − χi0j0)x
gives the desired bound for the time derivative. A similar computation yields
the bound for the higher order derivatives in (A.3).
Consider now the new metric g̃. Since both Dχ and (Dχ)−1 are Lipschitz on
the dyadic scale with l1(A(τ)) summability, from (1.15) for g we easily obtain
(1.15) for g̃. In addition, our construction insures that g̃(ti, xij) = In. This in
turn leads to the bound
‖g̃ − In‖L∞(Aij) . ‖g̃‖Lipx(Aij) + ‖g̃‖Cmijt (Aij)
which shows that (1.15) for g̃ implies (1.16) for g̃ − In.
It remains to consider the lower order terms. From ∂tχ we obtain coefficients
d̃ of the form
d̃ = t
∂tηijχij
Within Ai0,j0 this gives
d̃ = t
∂tηij(t, x)(χij − χi0j0)
The functions t∂tηij(t, x) are bounded and smooth on the dyadic scale, while
the l1(A(τ)) summability comes from the χij−χi0j0 factor due to (A.5). Hence
both (1.15) and (1.16) are satisfied.
The contribution of ∂2xχ to d̃ has the form
(∂xχ)
−1(∂2xχ)g
There is no singularity at x = 0 since χ is linear in x for x2 ≪ t. Then from
(A.2) and (A.3) we obtain
|d̃| .
with added l1(A(τ)) summability inherited from ∂2xχ. This is better than
(1.16), and in effect this term can be included inW and treated perturbatively.
The bound (1.16) is also easy to obtain from the similar bounds for g and
derivatives of χ.
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Mathematisches Institut der Universität Bonn, Beringstr.1, 53115 Bonn,
Germany
E-mail address : koch@math.uni-bonn.de
Department of Mathematics, University of California, Berkeley, CA 94720
E-mail address : tataru@math.berkeley.edu
	1. Introduction
	2. Proof of Theorem 1 and 2 
	3.  L2 bounds in the flat case and the Hermite operator
	4. Resolvent bounds for the Hermite operator 
	4.1. Weighted L2 bounds
	4.2. The Lp bounds of the resolvent
	4.3. Combining the estimates
	5. Lp estimates in the flat case and parametrix bounds
	6.  Modified weights and pseudoconvexity
	7. Lp Carleman estimates for variable coefficient operators
	7.1. Localization scales.
	7.2. Freezing coefficients
	7.3. The localized parametrix.
	8. The gradient term
	Appendix A. The change of coordinates
	References

  This work is devoted to the strong unique continuation problem for second
order parabolic equations with nonsmooth coefficients. Introduction and
bibliography have been revised.

Introduction
The evolution of the understanding of the strong unique continuation prob-
lem for second order parabolic equations mirrors and is closely related to the
corresponding strong unique continuation problem for second order elliptic
equations. Consequently, we begin with a brief overview of the latter problem.
To a second order elliptic operator ∆g = ∂ig
ij∂j and potentials V , W in R
we associate the elliptic equation
(1.1) −∆gu = W∇u+ V u
Given a function u ∈ L2loc(Rn) and x0 ∈ Rn we say that u vanishes of infinite
order at x0 if there exists R so that for each integer N we have
(1.2)
B(x0,r)
|u|2 dx ≤ c2Nr2N , r < R
The elliptic strong unique continuation property ESUCP has the form
Let u be a solution to (1.1) which vanishes of infinite order
at x0. Then u(x) = 0 for x in a neighborhood of x0.
ESUCP
ESUCP type results go back to the pioneering work of Carleman [5] in
dimension n = 2, later extended to higher dimension in by Aronszajn and
collaborators [3], [4]. Their results apply to Lipschitz metrics g but only mildly
unbounded potentials V and W . A key ingredient in their approach was
to obtain a class of weighted L2 estimates which were later called Carleman
estimates. The simplest Carleman estimate has the form
‖|x|−τu‖L2 . ‖|x|2−τ∆u‖L2
The first author was supported in part by the DFG grant KO1307/5-3. The second
author was supported in part by the NSF grant DMS-0301122. Part of the work was done
while the first author was supported by the Miller Institute for Basic Research in Science.
http://arxiv.org/abs/0704.1349v2
and holds uniformly for τ away from ±(n−2
+ N). This restriction is related
to the spectrum of the spherical Laplacian.
Adding some extra convexity to the |x|−τ weight makes the above estimate
more robust and allows one to also use it in the variable coefficient case. The
role played by the convexity was further clarified and explained by Hörmander
[12], [13], who introduced the pseudoconvexity condition for weights as an
almost necessary and sufficient condition in order for the Carleman estimates
to hold.
The problem becomes more difficult if one seeks to work with unbounded
potentials V in or near the scale invariant L
2 space. There the L2 Carle-
man estimates are insufficient. Instead the key breakthrough was achieved
in Jerison-Kenig [15], where the L2 Carleman estimates are replaced by Lp
estimates of the form
‖|x|−τu‖
. ‖|x|−τ∆u‖
Relevant to the present paper is also the alternative proof of this result which
was given by Jerison [14], taking advantage of Sogge’s [25] spectral projection
bounds for the spherical Laplacian. In the case of operators with smooth
variable coefficients Lp Carleman estimates were first obtained by Sogge [26],
[27].
Working with gradient potentials in the scale invariant space W ∈ Ln intro-
duces an added layer of difficulty. There not even the Lp Carleman estimates
can hold. Wolff’s solution to this in [29] is a weight osculation argument, which
allows one to taylor the weight in the Carleman estimate to the solution u,
producing estimates of the form
‖e−τφ(x)u‖
+ ‖e−τφ(x)W∇u‖
. ‖eτφ(x)∆u‖
where the choice of φ depends on both u, W and τ .
Finally, the authors’s article [17] combines the ideas above into a nearly
optimal scale invariant ESUCP result for the elliptic problem, with (i) a
Lipschitz metric g, (ii) an L
2 potential V , and (iii) an almost Ln gradient
potential W . The present paper is the counterpart of [17] for the parabolic
strong unique continuation problem.
We consider the second order backwards parabolic operator
(1.3) P = ∂t + ∂kg
kl(t, x)∂l
in R × Rn and potentials V,W1,W2. To these we associate the parabolic
equation
(1.4) Pu = V u+W1∇xu+∇x(W2u)
Given a function u ∈ L2loc and (t0, x0) ∈ R × Rn we say that u vanishes of
infinite order at (t0, x0) if there exists R so that for each integer N we have
(1.5)
B(x0,r)
|u|2 dxdt ≤ c2Nr2N , r < R
Alternatively we may only require that x→ u(t0, x) vanishes of infinite order
at x0, i.e.
(1.6)
B(x0,r)
|u(t0, x)|2 dx ≤ c2Nr2N , r < R
The two conditions (1.5) and (1.6) are largely equivalent provided that the
coefficients gkl have some uniform regularity as t → t0. However, our as-
sumptions in this article are not strong enough to guarantee this, therefore we
consider the two separate cases.
Now we can define the strong unique continuation property SUCP(I) :
Let u be a solution to (1.4) which vanishes of infinite order
at (t0, x0). Then u(t0, x) = 0 for x in a neighborhood of x0.
SUCP(I)
and the slightly stronger variant
Let u be a solution to (1.4) so that x→u(t0, x) vanishes of infinite
order at x0. Then u(t0, x) = 0 for x in a neighborhood of x0.
SUCP(II)
The study of unique continuation for parabolic equations began with early
work of Mizohata [21] and Yamabe [30], followed by Saut-Scheurer [23]; Lp
Carleman estimates were first obtained by Sogge [24].
The study of the parabolic strong unique continuation problem began with
work of Lin [20] who considered SUCP(II) for the heat equation with W = 0
and V bounded and time independent. This continued with work of Chen [6]
and Poon [22]. Fernández [11], and Escauriaza, Fernández and Vessella [7] con-
sidered SUCP(II) under various assumptions on the coefficients and pointwise
bounds for W = 0 and V . It is a consequence of Alessandrini and Vessella [1]
that SUCP(I) andSUCP(II) are equivalent under weak assumptions on the
coefficients, and they derived SUCP(II) in [2] for bounded W and V .
The article of Poon [22] contributed to clarifying the correct form of the
L2 Carleman estimates for the parabolic strong unique continuation problem
in Escauriaza and Fernández’s work [9]. In the simplest form, these have the
‖t−τ−
8t u‖L2 . ‖t−τ+
8t (∂t +∆)u‖L2
and hold uniformly with respect to τ away from (2n + N)/4. This restriction
is connected with the spectral properties of the Hermite operator.
The Lp spectral projection bounds for the Hermite operator were indepen-
dently obtained by Thangavelu [28] and Kharazdhov [16]; see also the sim-
plified proof in the authors’s paper [19]. These bounds were essential in the
proof of Lp Carleman inequalities for the heat operator of Escauriaza [8] and
Escauriaza and Vega [10] which yield SUCP(I) when g = In, W = 0 and
V ∈ L1L∞ + L∞Ln/2.
Our aim is to prove that SUCP(I) respectively, SUCP(II) hold under
sharp scale invariant assumptions on the metric g and Lp conditions on the
potentials V and W1,W2. The contribution of this work is comparable to [17]
for the elliptic problem: We study almost optimal conditions on
(1) the coefficients g
(2) the potential V
(3) the gradient potentials Wj
The combination of rough variable coefficients and Lp conditions on the po-
tential seems to be new. Also, to the best of our knowledge this is the first
result on unique continuation for parabolic problems under Lp conditions on
the coefficients of the gradient term.
For simplicity we always assume that t0 = 0, x0 = 0. For SUCP(I) it is
natural1 to consider a larger class of operators P which have the form
(1.7) P =
+ ∂kg
kl∂l +
dkl∂l + ∂ld
where (gkl), (dkl) and (ekl) are real valued and (gkl) and (ekl) are symmetric.
Then simple scale invariant assumptions for the coefficients would be
(1.8) ‖d‖L∞ + ‖(t+ x2)
2∂xd‖L∞ + ‖t∂td‖L∞ ≪ 1
Here and in the sequel d stands for a generic coefficient of the form gkl − δkl,
dkl and ekl. For V and W1,2 we could consider conditions of the form
(1.9) ‖V ‖L1L∞+L∞Ln/2 ≪ 1,
(1.10) ‖W1,2‖L2L∞+L∞Ln ≪ 1.
Here and in the sequel we use the notation LpLq = L
The situation is however more complex and we may take (1.8) to (1.10)
only as guidelines. We will have to strengthen (1.8) to include some dyadic
summability. on the other hand we are able to weaken the time differentiability
to a C
3 Holder condition on small time scales.
We are also able to slightly weaken (1.9) almost to uniform bounds on
dyadic sets. However, we are unable to use mixed norms for W1 and W2, and
we restrict ourselves to a summable Ln+2 norm in dyadic sets.
To state our assumptions on g, V , W1 and W2 we consider a double infinite
dyadic partition of the space,
(1.11) R+ × Rn =
where
(1.12) Aij = {(t, x) ∈ R+ ×Rn | e−4i−4 ≤ t ≤ e−4i, ej ≤ 1+ 2|x|t−
2 ≤ ej+1}.
Consider the subset of indices
(1.13) A = {(i, j) : j ≤ 2i+ 2}
defining a partition of the cylinder
Q = [0, 1]× B(0, 1)
1This becomes clearer later on after a change of coordinates and conjugation with respect
to a Gaussian weight
Define
(1.14) A(τ) = {(i, j) ∈ A : 4i ≥ ln τ + 1, j ≤ 1
ln τ + 2}
which corresponds to a partition of the cut parabola
Qτ = {(t, x) : |x|2 ≤ τt ≤ 1}.
t = τ−1
|x| = 1
Figure 1. The cut parabola
We also consider a decomposition of Q into dyadic time slices
Ai = [e
−4i−4, e−4i]× B(0, 1)
and a similar partition of the cut parabola Qτ into the sets
Aτi = Ai ∩Qτ
Given a function space X and 1 ≤ q < ∞ we introduce the Banach spaces
lq(A, X) with norms
‖V ‖q
lq(A,X)
i,j∈A
‖V ‖q
X(Aij)
In a similar manner we define the spaces l∞(A, X).
Within the sets Aij we define the modulus of continuity (mij) in time
mij(ρ) = e
4iρ+ e
(2i−j)ρ
and denote by C
t the space of continuous functions with finite seminorm
= sup
t1,t2,x
|u(t1, x)− u(t2, x)|
mij(|t1 − t2|)
For the reader’s convenience we note that within Aij we have e
4i ≈ t−1 and
(2i−j) ≈ (t+ |x|2)− 13 .
For the coefficients of the operator P in (1.7) we change the condition (1.8)
(1.15) sup
‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖Cmijt (Aij) ≪ 1.
where we note that ej−2i ≈ (t + x2) 12 in Aij .
The pointwise bound for g − In in (1.15), namely
(1.16) sup
‖g − In‖l1(A(τ),L∞) ≪ 1
is not really needed for our results. It can be always obtained from the other
bounds after a change of coordinates. This is discussed in the appendix.
The assertion (1.15) is satisfied for g ∈ Lipx∩C
t provided that g(0, 0) = In.
Indeed by scaling we may assume that the Lipx ∩ C
t norm is small therefore
it suffices to compute
‖(t + |x|2)
3‖l1(A(τ),L∞) ≤
i≥ln τ
j≤ln(τ)/2+2
(j−2i) . 1.
For the potential V we consider:
‖V ‖l∞(A,L1L∞+L∞Ln/2) ≪ 1 for n > 2
‖V ‖l∞(A,L1L∞+LpLp′ ) ≪ 1 for n > 2, 1 ≤ p <∞,
‖V ‖l∞(A,L1L∞+L2L1) ≪ 1 for n = 1
(1.17)
where p′ in the second line is the dual exponent. In addition we require that
(1.18) sup
‖χiV ‖L1L∞+L∞Ln/2 ≪ 1 n > 2
with the obvious modifications for n = 1, 2, where χi is the characteristic
function of the set
{(t, x) : e−4i−4 ≤ t ≤ e−4i, t−1/2|x| ≤ i}.
Both (1.17) and (1.18) are fulfilled if V ∈ L1L∞ + L∞Ln/2 with small norm.
Finally for the gradient potentials W1,2 we introduce the summability con-
dition with respect to time slices
(1.19) sup
‖W1,2‖Ln+2(Aτ
) ≪ 1.
As a consequence of this we note the uniform bound
(1.20) sup
‖W1,2‖Ln+2(Ai) ≪ 1.
Now we can state our main results.
Theorem 1. Let P be as in (1.7) with coefficients satisfying (1.15). As-
sume that the potentials V and W1,2 satisfy (1.17), (1.18) and (1.19). Then
SUCP(I) holds at (0, 0) for H1 functions u satisfying (1.4).
It is part of the conclusion that the trace of u at t = 0 exists near x = 0.
The assumptions on the operator seem to be too weak to imply existence of a
trace in general. More precisely shall prove
‖u(t, .)‖L2(B(0,1/8)) . e−
for some δ > 0.
The C
3 Hölder regularity in time for the metric g seems so be new, improv-
ing the C
2 Hölder regularity in [9]. It is not clear to the authors whether this
condition is optimal or not.
We may replace the assumptions by stronger translation invariant assump-
tions,
(1.21) ‖g‖Lipx + ‖g‖
(1.22) ‖V ‖L1L∞+L∞Ln/2 ≪ 1
(1.23)
‖W1,2‖Ln+2(Ai) . 1
Then we also obtain a stronger conclusion.
Theorem 2. Let P be as in (1.3) with coefficients as in (1.21). Assume that
the potentials V and W satisfy (1.22) respectively (1.23). Then SUCP(II)
holds at (0, 0) for H1 functions u satisfying (1.4).
We remark that the condition (1.21) is really too strong, and that with some
additional work (see Remark 2.3) one can bring it almost to the level of (1.15).
Precisely, it suffices to replace (1.15) by
(1.24) sup
‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖
t (Aij)
where the slightly stronger time continuity modulus m2ij is given by
m2ij(ρ) = e
4i−2jρ+ e
(2i−j)ρ
However, we cannot keep the additional terms in (1.7), because we need to
be able to meaningfully talk about the trace of the solution at time t = 0.
Both theorems are consequences of quantitative estimates, which also imply
weak unique continuation under the assumptions of Theorem 2:
Let u be a solution to (1.4) for which u(t0, .) vanishes in
the closure of an open set. Then (u(t0, .) vanishes in a
neighborhood of the closure.
If u satisfies the assumptions and vanishes in an open set U , then it vanishes
in the time slices t = t0 in an open neighborhood of the closure of Ut0 = {(x :
(t, x) ∈ U}.
Theorems 1 and 2 are nontrivial consequences of a Carleman inequality. To
state a first version of the Carleman inequality we introduce an additional fam-
ily B(τ) of sets which is a partition of the cylinder [0, τ−1)×B(0, 1), consisting
(1.25) Aij , ln τ ≤ 4i ≤ τ 1/2, 0 ≤ j ≤ ln τ/2 + 2,
(1.26) [e−4i−4, e−4i]× B(0, e−2iτ 1/2), 4i > τ
(1.27) Aij , ln τ ≤ 4i, ln τ/2 ≤ j ≤ 2i.
This partition is coarser than the partition of the same cylinder into the sets
Aij . This is the reason why we need the assumption (1.18). More precisely
Assumptions (1.17) and (1.18) imply
(1.28) ‖V ‖l∞(B(τ),L1L∞+L∞Ln/2) ≪ 1.
Theorem 3. Let τ0 ≫ 1, ε > 0 and P as in (1.7) with coefficients satisfying
(1.15). Suppose that W1,2 satisfY (1.19) with constants depending on ε and
τ0. Then there exists C > 0 such that for all τ ≥ τ0 the following is true:
Suppose that v ∈ L2(H1) is compactly supported in [0, 8τ−1) × B(0, 2) and
that it vanishes of infinite order near (0, 0). Then we can find a function
φ ∈ C∞([0, 8τ−1]×B(0, 1)\{0, 0}) and h ∈ C∞(R+) which satisfy
(1.29) τ ≤ h′ ≤ (1 + ε)τ
(1.30)
∣∣∣∣φ(x, t)−
h(− ln t)− x
)∣∣∣∣ ≤ ε
such that
‖eφv‖
l2(B(τ),L∞L2∩L2L
n−2 ))
≤ C‖eφ(P +W1∇ +∇W2)v‖
l2(B(τ),L1L2+L2L
n+2 )
for n ≥ 3, respectively
‖eφv‖l2(B(τ),L∞L2∩Lp′Lq′ ) ≤ C‖e
φ(P +W1∇+∇W2)v‖l2(B(τ),L1L2+LpLq),
for n = 2, 1
, 2 < p, and
‖eφv‖l2(B(τ),L∞L2∩L4L∞) ≤ C‖eφ(P +W1∇ +∇W2)v‖l2(B(τ),L1L∞+L4/3L1)
for n = 1.
The statement of the Carleman inequality is involved for several reasons.
The weight t−τe−|x|
2/8t, which works for the constant coefficient case, has to be
modified so that it has more convexity in order to handle variable coefficients,
spatial localization and the gradient potential. However the polynomial growth
in time (imposed by the assumption of vanishing of infinite order) limits the
available amount of convexity; this is the origin of the l1 summability in (1.16),
(1.20), and to a lesser extend of (1.18).
If W = 0 then the Carleman inequality holds for a large explicit class
of weights eφ. This cannot be possibly true for general gradient potentials.
Instead, we are only able to prove that there exists some weight function φ,
which now depends on τ , u andW , for which the uniform Carleman inequality
holds. This strategy goes back to the seminal work of T. Wolff [29] and has
been used by the authors for the elliptic problem [17].
The partition Aij is much finer than the dyadic decomposition in t only,
which would correspond to the dyadic decomposition in the elliptic case. We
are only able to localize the estimates to the sets Aij if we make the weight
function sufficiently convex. We can do this for many Aij, but not for all of
them. The sets (1.26) correspond directly to the assumption (1.18). We need
to have control of the L1L∞+L∞Ln/2 norm of V in sets which are not smaller
than those of the partition in (1.25), (1.26) and (1.27).
We have stated Theorem 3 in a simpler form which suffices to derive Theo-
rem 1 and Theorem 2. However the full estimate we prove is stronger in that
it also contains precise L2 bounds. These are essential for the localization and
perturbations techniques we use.
The strategy of the proof is the same as in [17]:
(1) We construct families of pseudoconvex weights and derive L2 Carle-
man inequalities. The convexity of weights determines the space-time
localization scales and the admissible size of perturbations.
(2) We enhance the above L2 Carleman inequalities to include Lp esti-
mates. Due to the L2 localization it suffices to do this in small sets.
This allows us to use perturbation arguments starting from the case of
the heat equation with the weight t−τe−|x|
2/8t.
(3) Lp estimates for the spectral projections to spherical harmonics im-
ply the Lp Carleman inequalities in the elliptic case. Here spectral
projection for the Hermite operator play a similar role.
(4) Finally we include Wolff’s osculating argument into the scheme in order
to handle the gradient potentials. The efficiency of this part depends
on the flexibility in the choice of the weight functions.
The complexity of the weights and the L2 Carleman estimates comes mainly
from the geometry of the classical harmonic oscillator. Orbits are contained in
a sphere in R2n. The projection down in the x space is a ball, where frequency
variables have a different behavior in radial and angular directions and near
the boundary of the the ball. It turns out that our analytic estimates reflect
these features.
2. Proof of Theorem 1 and 2
In this section we prove Theorem 1 and 2 assuming Theorem 3. The relation
between Carleman estimates and unique continuation is fairly straightforward
in the elliptic case. In the parabolic situation the argument is less direct due
to the more complex geometry of the level sets of the weight functions.
It is a standard consequence of a localized energy inequality that for the
parabolic equation (1.4) u(t) and its gradient can be controlled by L2 norms
of u.
Proposition 2.1. Let n ≥ 3 and suppose that v solve the parabolic equation
vt + ∂ka
kl∂lv =W1∇v +∇(W2v) + V v
on the space-time cylinder Q = [0, 2]×B(0, 2) with a ∈ Lip uniformly elliptic
‖W1,2‖Ln+2(Q) + ‖V ‖L1L∞+L∞Ln/2 ≪ 1
F extτ
F intτ
1/16τ
211/42δ
Figure 2. The sets Eδ, F
τ and F
0≤t≤1
‖v(t)‖L2(B(0,1)) + ‖∇xv‖L2([0,1]×B(0,1)) . ‖v‖L2(Q).
If n = 2 then the same statement is true with L∞Ln/2 replaced by LpLq with
1 ≤ p < ∞, 1 < q ≤ ∞ and 1
= 1. Similarly, if n = 1 we have to replace
it by L4L∞.
Given our assumptions (1.15), (1.17), (1.18), (1.19) and (1.20) we can apply
Proposition 2.1 rescaled in sets of the form [t0, 2t0] × B(x0, t1/20 ), which are
subsets of the Aij. Summing up with respect to such sets contained in a
parabolic cube
Qr = [0, r
2]×B(0, r)
we obtain the following consequence.
Corollary 2.2. The following estimate holds under the assumptions of Theo-
rem 1
2u‖L∞L2(Qr) + ‖t
2∇u‖L2(Qr) . ‖u‖L2(Q2r)
Proof of Theorem 1. We choose τ ≥ τ0 and 0 < δ ≪ τ−1/2 and introduce the
[0, 2δ2]×B(0, 2δ)
[0, δ2]× B(0, δ)
F extτ =
[0, 2/τ ]×B(0, 2)
[0, 1/τ ]×B(0, 1)
F intτ =[1/32τ, 1/16τ ]× B(0, 1/4)
Our strategy will be to truncate u in Eδ and F
τ and to apply Theorem 3
to the truncated function in order to obtain a good bound on u in F intτ .
Let η be a cutoff function supported in [0, 2)× B(0, 2) and identically 1 in
[0, 1]×B(0, 1). For δ ≪ τ−1/2 we define
vδ(t, x) = (1− η(t/δ2, x/δ))η(τt/8, x)u(x, t)
which satisfies
(P +W∇)vδ = V vδ − [P +W∇, η(t/δ2, x/δ)]u+ [P +W∇, η(τt/8, x)]u.
The second term on the right hand side is supported in Eδ and the third one
in F extτ .
We apply Theorem 3 to vδ. One should keep in mind that the corresponding
weight φ depends on δ but that the bounds we prove are uniform with respect
to δ. We normalize the function h by h(0) = 0. We have to control the size of
φ in the sets Eδ, F
τ and F
τ . Due to (1.29) we have
τs ≤ h(s) ≤ 2τs, s ≥ 0
By (1.30) we obtain a rough polynomial bound in δ
(2.1) eφ ≤ t−2τe−
+ε(τ+x
) ≤ t1/2c(τ)δ−4τ−1 in Eδ.
M = sup{eh(− ln(t))e−
+ε(τ+x
) : (t, x) ∈ F extτ }
By (1.29) the supremum is attained at a point (t0, x0) with
≤ 8τt0 ≤ 1 and
|x0| = 1. A simple computation also shows that
sup{t−1/2eh(− ln(t))e−
+ε(τ+x
) : (t, x) ∈ F extτ } . τ 1/2M.
Then M dominates eφ in F extτ :
(2.2) eφ ≤M, t−1/2eφ . τ 1/2M in F extτ .
Next we need to bound eφ from below in F intτ in terms of M ,
(2.3) inf
F intτ
eφ ≥ e
To see this we compute for (t, x) ∈ F intτ and sufficiently small ε:
φ(t, x)− φ(t0, x0) ≥ h(− ln t)− h(− ln t0) +
− 2ε(τ + 1
− 20ε)τ ≥ 1
and use (2.1) and (2.2).
Theorem 3 applied to vδ yields
‖eφvδ‖
l2(B(τ),L2L
n−2 ∩L∞L2)
.‖eφV vδ‖
l2(B(τ),L2L
n+2+L1L2)
+ ‖eφ[P +W∇, η(t/δ2, x/δ)]u‖
l2(B(τ),L2L
n+2+L1L2)
+ ‖eφ[P +W∇, η(τt, x)]u‖
l2(B(τ),L2L
n+2 +L1L2)
(2.4)
By Hölder’s inequality we have
‖eφV vδ‖
l2(B(τ),L2L
n+2+L1L2)
.‖V ‖l∞(B(τ),L1L∞+L∞Ln/2)‖eφvδ‖
l2(B(τ),L2L
n−2 ∩L∞L2)
Due to the smallness in (1.28) we can absorb this term on the left hand side
of the inequality.
We calculate the first commutator
fδ = [P +W∇, η(t/δ2, x/δ)]u
+ ∂kg
kl∂l + 2
dkl∂l +W∇)η(t/δ2, x/δ)
+ 2δ−1gkl(∂kη)(t/δ
2, x/δ)∂lu
By (2.1) we have
‖eφfδ‖
l2(B(τ),L2L
n+2 +L1L2)
. c(τ)δ−4τ−1‖t
2 fδ‖
l2(B(τ),L2L
n+2 +L1L2)
For the W term we use (1.19) and Holder’s inequality. For term involving
dkl we bound the L1L2 norm in terms of an L∞L2 norm, using (1.15) which
implies that the pointwise bound for dkl is summable with respect to dyadic
time regions. For the remaining terms we simply bound the L1L2 norm in
terms of the L2 norm. This yields
‖eφfδ‖
l2(B(τ),L2L
n+2+L1L2)
.c(τ)δ−4τ−2(‖u‖L2(Eδ) + ‖(∂xη)(t/δ
2, x/δ)t
2∇u‖L2
+ ‖(∂xη)(t/δ2, x/δ)t
2u‖L∞L2)
Then we can apply a straightforward modification of Corollary 2.2 on the Eδ
scale to finally obtain
‖eφ[P +W∇, η(t/δ2, x/δ)]u‖
l2(B(τ),L2L
n+2 +L1L2)
. c(τ)δ−4τ−2‖u‖L2(Eδ).
Similarly we can estimate the second commutator
‖eφ[P +W∇, η(τt, x)]u‖
l2(B(τ),L2L
n+2 +L1L2)
.Mτ 1/2‖u‖L2(F extτ ).
Hence by inequality (2.4) we get
(2.5) ‖eφvδ‖
l2(B(τ),L2L
n−2∩L∞L2)
. Mτ 1/2‖u‖L2(F extτ ) + c(τ)δ
−4τ−2‖u‖L2(Eδ).
Within F intτ we have vδ = u. Then by (2.3) we obtain
(2.6) ‖u‖L∞L2(F intτ ) . τ
1/2e−
τ‖u‖L2(F extτ ) + c(τ)δ
−4τ−2‖u‖L2(Eδ).
Also by the vanishing of infinite order the second term tends to zero as δ → 0.
Hence as δ → 0 we obtain
(2.7) ‖u‖L∞L2(F intτ ) . τ
1/2e−
τ‖u‖L2(F extτ )
For 0 < t≪ 1 we choose τ = 1
to obtain
‖u(t, .)‖L2(B(0,1/4)) . t−1/2e−
32t .
This completes the proof of Theorem 1. �
Proof of Theorem 2. We extend the potentials V and W by zero to negative
time, and gkl(t, x) = gkl(0, x) for t < 0. By definition, possibly after rescaling,
we have u(0, .) ∈ L2(B(0, 2)). We now solve the mixed problem
(2.8) ut + ∂kg
kl∂lu = 0 for t < 0 and |x| ≤ 2
with the boundary condition
u(t, x) = 0 if |x| = 2 and t < 0
and the obvious initial condition to obtain an extension of u to negative. The
heat kernel for (2.8) satisfies Gaussian estimates. In particular we obtain from
(1.6) for all positive integers N with a constant cN possibly differing from (1.6)
(2.9)
Br(0)
|u|2dxdt . c2Nr2N .
We seek to prove that the bound (2.7) still holds in this context. The
difficulty is that we only know that u vanishes of infinite order at (0, 0) for
negative time. To account for this we shift the time up, t→ t+2δ. Arguing as
in the previous proof we obtain (2.6) with u replaced by u(t+ 2δ, x). Letting
δ → 0 by (2.9) we obtain (2.7) and conclude as above.
Remark 2.3. If one wants to prove Theorem 2 under the weaker assumptions
on g in (1.24) then the origin needs to be avoided in the above argument. Hence
the time translation needs to be accompanied by a spatial translation, namely
uδ(t, x) = u(t+ 2δ
2, x− 8τδe1)
This translation places the image of the origin, or better of the cube [0, τδ2]×
B(0, 4τδ), within the region {τt < x2}. But in this region the conjugated
operator Pψ, introduced later, is elliptic so only pointwise bounds for g are
needed for the Carleman estimates.
3. L2 bounds in the flat case and the Hermite operator
In this section we prove the simplest possible L2 Carleman estimate for the
constant coefficient backward parabolic equation
∂tu+∆xu = f
This serves as a good pretext to introduce the class of weight functions which
is later modified for the variable coefficient case.
We also describe the change of coordinates which turns the backward par-
abolic operator into a forward parabolic equation for the Hermite operator
H . In this way we are able to relate the L2 Carleman estimates for the heat
operator to spectral information for H .
Proposition 3.1. Let u ∈ L2 with compact support away from 0. Then
(3.1) ‖t−τ−
8t u‖L2 ≤ ‖t−τ+
8t (∂t +∆)u‖L2
uniformly with respect to τ away from (2n+ N)/4.
Proof. In R+ × Rn we introduce new coordinates (s, y) ∈ R× Rn defined by
(3.2)
t = e−4s
x = 1
e−2sy
= −4e−4s
− e−2sy
Hence in the new coordinates our operator becomes
4t(∂t +∆x) = −
− 2y ∂
If we conjugate it by tn/4e−
8t = e−nse−
2 we obtain
4t1+n/4e−
8t (∂t +∆x)t
−n/4e
8t = − ∂
+∆y − y2 =: −∂s −H =: −P0
where H is the Hermite operator
H = −∆y + y2
Then it is natural to define the new functions
v(s, y) = e−nse−
2 u(e4s,
e2sy), g(s, y) = e(−n−4)se−
2 f(e4s,
e2sy)
which are related by
P0v = g ⇐⇒ (∂t +∆)u = f
In the new coordinates, the bound (3.1) becomes
(3.3) ‖e4τsv‖L2 . ‖e4τsP0v‖L2.
Denoting w = e4τsv, we conjugate
e4τsP0v = e
4τsP0e
−4τsw = (−∂s −H + 4τ)w
and the above bound becomes
(3.4) ‖w‖L2 . ‖(−∂s −H + 4τ)w‖L2.
Since ∂s and H − 4τ commute we expand
‖(−∂s −H + 4τ)w‖2L2 = ‖∂sw‖2L2 + ‖(H − 4τ)w‖2L2 ≥ d(4τ, n+ N)2‖w‖2L2.
Note the spectral gap, which is essential in order to obtain strong unique
continuation results. �
For later use we also record the following slight generalization of the above
result. For expediency this is stated in the (y, s) coordinates, i.e. in the form
of an analogue of (3.3).
Proposition 3.2. Let h be an increasing, convex, twice differentiable function
so that
d(h′,N) + h′′ ≥ 1
(3.5)
‖(1 + h′′)1/2eh(s)v‖L2 +
∥∥∥∥min
(1 + h′′)1/2
1 + h′
eh(s)Hv
. ‖eh(s)(∂s −H)v‖L2
for all compactly supported v ∈ L2.
Proof. After the substitution w = eh(s)v the bound (3.5) becomes
(3.6)
‖(1 + h′′)1/2w‖2L2 +
∥∥∥∥min
(1 + h′′)1/2
1 + h′
. ‖(∂s −H + h′(s))w‖2L2.
and we obtain the L2 estimate through expanding the term on the right hand
side with respect to its selfadjoint and skewadjoint part:
‖(∂s −H + h′(s))w‖2L2 =‖∂sw‖2L2 + ‖(−H + h′)w‖2L2 + ‖(h′′)1/2w‖2L2
(d(h′,N)2 + h′′)‖w(s)‖2L2ds
To complete the proof we observe that for each s we have
‖Hw(s)‖L2 . ‖(−H + h′)w(s)‖L2 + h′(s)‖w(s)‖L2
4. Resolvent bounds for the Hermite operator
As seen in the previous section, the spectral properties of the Hermite op-
erator play an essential role even in the simplest L2 Carleman estimates for
the heat equation. In this section we take a look at L2 and Lp bounds for its
spectral projectors and its resolvent.
The spectrum of H is n+ 2N, and its eigenfunctions are the Hermite func-
tions defined by
uα = cα(∂y − y)αe−
2 , Huα = (n+ 2|α|)uα
As |α| increases, so does the multiplicity of the eigenvalues. We denote the
spectral projectors by Πλ for λ ∈ n + 2N. We consider both the spectral
projectors and the resolvent of H and obtain both Lp and localized L2 bounds.
4.1. Weighted L2 bounds. We consider two parameters
1 ≤ d, R . λ
We denote
BR = {y : |y| < R}, , Bjd = {y : |yj| < d}, j = 1, ..., n
By χR, respectively χ
d we denote bump functions in BR, respectively B
d which
are smooth on the corresponding scales.
Proposition 4.1. The spectral projectors Πλ satisfy the localized L
2 bounds
(4.1) R−
4‖χRΠλf‖L2 +R−
4‖χR∇Πλf‖L2 . ‖f‖2L2,
respectively
(4.2) d−
2‖|Dj|
dΠλf)‖L2 . ‖f‖L2
Proof. The inequality (4.1) is trivial unless R ≪ λ 12 . To prove it in dimension
n = 1 we only need to consider the case when f is a Hermite function,
f = Πλf = hλ
in which case it follows from the pointwise bound
4 |h′λ(x)|+ λ
4 |hλ(x)| . ‖hλ‖L2, |x| ≤
In dimension n = 1 (4.2) follows by interpolation from (4.1) with R = d.
This extends trivially to higher dimension by separation of variables.
It remains to prove (4.1) in higher dimensions. Summing up (4.2) with
d = R over j we obtain the bound
2‖|D|
2χRΠλf‖L2 . ‖f‖2L2
For |x| . R ≪ λ 12 we have |ξ|2 ≈ λ in the characteristic set ofH−ℜz, therefore
the above norm should essentially control the left hand side of (4.1). For later
use we prove a slightly more general result, which in particular concludes the
proof of (4.1).
(4.3) λ
4‖v‖L2+λ−
4‖∇v‖L2. ‖|D|
2 v‖L2+‖|y|
2 v‖L2+‖(H−λ)v‖
yL2+∇L2+λ
Indeed the norm on the right is equivalent to
4v‖L2 + ‖(H + λ)−
2 (H − λ)v‖L2 & λ
4‖v‖L2 + λ−
2v‖L2
In our case we apply (4.3) to v = R−
2χRΠλf . Then
2 v‖L2 . ‖Πλf‖L2
while
(H − λ)v = 2R−
2∇(∇χRΠλf) +R−
2∆χRΠλf
which yields
‖(H − λ)v‖
yL2+∇L2+λ
2‖f‖L2
To state the corresponding resolvent bounds we define the spaces X̃2(z) by
‖u‖X̃2(z) = (1+ |ℑz|)
2‖u‖L2+‖(H−z)u‖
yL2+∇L2+|z|
2‖|Dj|
du‖L2
and the corresponding dual spaces X̃∗2 (z). These spaces are larger than the
corresponding “elliptic” spaces,
(4.4) ‖v‖X2(z) . |z|
2‖v‖L2 + ‖yv‖L2 + ‖∇v‖L2
On the other hand by extending the bound (4.3) to complex λ we obtain a
counterpart of (4.1), namely
(4.5) R−
4‖χRu‖L2 +R−
4‖χR∇u‖L2 . ‖u‖X2(z)
Finally, the result of (4.2) can be written in the following dual forms
(4.6) ‖Πλf‖X2(λ) . ‖f‖L2, ‖Πλf‖L2 . ‖f‖X∗2 (λ)
The localized L2 resolvent bounds have the form
Proposition 4.2. Let n ≥ 2, z ∈ C with dist(z, n + 2N) & 1, and 1 ≤ d ≤
R ≪ ℜz. Then
(4.7) ‖u‖X̃2(z) . ‖(H − z)f‖X̃∗2 (z)
where the d component of norms is omitted in dimension n = 1.
Proof. We first note that the bounds (4.6) almost imply (4.7) up to a loga-
rithmic divergence. They do imply easily a bound for higher powers of the
resolvent for z away from the spectrum of H ,
(4.8) ‖(H − z)−1−kf‖X̃2(z) . (1 + |ℑz|)
−k‖f‖X̃∗2 (z), k ≥ 1.
as well as
(4.9) ‖u‖L2 . (1 + |ℑz|)−
2‖(H − z)u‖X̃∗2 (z).
Hence it remains to show that
(4.10) ‖u‖X̃2(z) . (1 + |ℑz|)
2‖u‖L2 + ‖(H − z)u‖X̃∗2 (z).
Using a positive commutator technique we first prove a one dimensional
estimate. For this we define the one dimensional skewadjoint pseudodifferential
operator2
Qr = iOp
w(χ(yr−1)χ(ξ|y|−1))
where χ is a mollified signum function which satisfies
χ′(x) =
, |x| ≤ 2
Its properties are summarized in the following
2As defined the symbol of Q is not smooth at 0. However, any smooth modification in
the ball {x2 + ξ2 < r2} will do.
Lemma 4.3. a) Qr is bounded in L
p for 1 ≤ p ≤ ∞ uniformly for r ≥ 1.
b) Qr is also bounded in X̃2(z) uniformly with respect to z ∈ C and r ≥ 1.
c) Qr satisfies the commutator estimate
(4.11) r−1‖|D|
2χru‖2L2 . (1 + |ℑz|)‖u‖2L2 + 〈(H − z)u,Qu〉
Proof. a) The Lp boundedness is straightforward and is left for the reader.
b) For the X̃2(z) boundedness we consider first the d terms, which without
any loss in generality we can write in the form
2‖(d2 +D2)
4χdu‖L2
Since Q is bounded in L2 it suffices to prove the commutator bound
‖[Qr, (d2 +D2)
4χd]u‖L2 . ‖u‖L2
But this is easily verified using the pdo calculus.
Next we consider the term
‖(H − z)u‖
yL2+∇L2+|z|
for which it suffices to prove the commutator bound
‖[Qr, H ]u‖
yL2+∇L2+|z|
. ‖u‖L2
or equivalently, by duality,
‖[Qr, H ]u‖L2 . ‖yu‖L2 + ‖∇u‖L2 + ‖|z|
2u‖L2
This follows again from the pseudodifferential calculus.
c) Since Qr is skewadjoint we have the identity
〈(H − z)u,Qru〉 = 〈[H,Qr]u, u〉+ ℑz〈iQru, u〉
therefore it suffices to insure that
(4.12) 〈[H,Qr]u, u〉 & ‖u‖2X̄2(r) +O(‖u‖
For this we compute the commutator [H,Q],
[H,Q] = Opw({ξ21 + y21, χ(y1r−1)χ(ξ1|y1|−1)}) +O(1)L2→L2
= Opw(2r−1χ′(y1r
−1)ξ1χ(ξ1|y1|−1)) +O(1)L2→L2
r−1(χ1r)
2(ξ21 + r
2 . r−1χ′(y1r
−1)ξ1χ(ξ1|y1|−1) + 1
and the conclusion follows by Garding’s inequality. �
We return to the proof of the proposition. By separation of variables the
bound (4.11) extends to higher dimensions and gives
r−1‖|Dj|
2χjru‖2L2 . (1 + |ℑz|)‖u‖2L2 + 2ℜ〈(H − z)u,Qjru〉
where Qjr the higher dimensional analogue of Qr with respect to the j variable.
TheX2(z) boundedness ofQr also extends easily to higher dimension. Hence
by Cauchy-Schwartz we obtain
r−1‖|Dj|
2χjru‖2L2 . (1 + |ℑz|)‖u‖2L2 + ‖(H − z)u‖X∗2 (z)‖u‖X2(z)
To conclude the proof of the estimate (4.10) it remains to show that
‖(H − z)u‖
2L2+yL2+∇L2
. ‖(H − z)u‖X∗2 (z)
which follows by duality from (4.4).
The final step in the L2 resolvent bounds is to replace the y′ derivatives by
angular derivatives. Let ∇⊥ = y|y| ∧∇ be the angular derivative and |D⊥|
the corresponding fractional derivative.
We split the coordinates into y = (y1, y
′) and use the notation ′ for coordi-
nates and derivatives in the obvious sense. For 1 ≤ d ≤ R ≤
λ we define the
sector
BR,d = {R < |y1| < 2R, |y′| ≤ d}
and χR,d a bump function inBR,d. Then we define the function spaceX2(λ,R, d)
‖u‖2X2(λ,R,d) = ‖u‖
L2 +R
−1/2λ−1/4‖∇(χR,du)‖2L2
+ R−1/2λ1/4‖χR,du‖2L2 + d−1/2‖|D⊥|
2χR,du‖2L2
and X∗2 (λ,R, d) as its dual.
Lemma 4.4. Suppose that n ≥ 2 and 1 ≤ d ≤ R. Then
‖u‖X2(λ,R,d) ≈ R−
4‖χR,du‖L2 +R−
4‖∇χR,du‖L2 + d−
2‖|D′|
2χR,du‖L2
Proof. Within BR,d the angular derivatives are close to the y
′ derivatives,
namely
|D⊥u| . |D′u|+
|∇u|, |D′u| . |D⊥u|+
|∇u|.
This implies the corresponding bounds for L2 norms, and the conclusion follows
by interpolation. �
¿From the above lemma we obtain
‖u‖X2(λ,R,d) . ‖u‖X̃2(λ,R,d)
Hence, we may replace X̃2 by X2 in (4.7) and (4.6):
Corollary 4.5. a) For λ in the spectrum of H we have
(4.13) ‖Πλf‖X2(λ,R,d) . ‖f‖L2, ‖Πλf‖L2 . ‖f‖X∗2 (λ,R,d)
b) For z away from the spectrum of H and 1 ≤ d ≤ R . ℜz we have
(4.14) ‖(H − z)−1−kf‖X2(λ,R,d) . (1 + |ℑz|)−k‖f‖X∗2 (λ,R,d), k ≥ 0
4.2. The Lp bounds of the resolvent. The Lp bounds for the spectral
projectors and the resolvent were proved in [16], [28] (see also [10]). For the
sake of completeness we also present them here in a simpler manner following
the approach in [19]. We refer the reader to the same paper for further results.
We consider pairs of exponents satisfying
(4.15)
where the range for p is
(4.16) p ≥ 4 for n = 1, p > 2 for n = 2, p ≥ 2 for n ≥ 3.
This leads to the following range3 for q:
(4.17)
q ∈ [2,∞] for n = 1, q ∈ [2,∞) for n = 2, q ∈ [2, 2n
] for n ≥ 3.
The dual exponents are denoted by p′ and q′ as usual.
Proposition 4.6. Let q be as in (4.17). Then
a) The spectral projectors Πλ satisfy
‖Πλ‖Lq′→L2 . 1, ‖Πλ‖L2→Lq . 1, n ≥ 2
‖Πλ‖Lq′→L2 . λ
p , ‖Πλ‖L2→Lq . λ−
p , n = 1
(4.18)
b) For z away from n+ N the resolvent (H − z)−1 satisfies
‖(H − z)−1‖Lq′→Lq . (1 + |ℑz|)
p′ , n ≥ 1(4.19)
Outline. To revisit the Lp bounds associated to the spectral projectors we recall
the approach in [19]. The first step there is to establish pointwise bounds for
the Schrödinger evolution4
(4.20) ‖eitH‖L1→L∞ . (sin t)−
This immediately (see also [18]) leads to Strichartz estimates for the solution
to the inhomogeneous equation
ivt −Hv = g, v(0) = v0
namely
(4.21) ‖v‖Lp([0,2π];Lq) . ‖v0‖L2 + ‖g‖Lp′([0,2π];Lq′)
where (p, q) are as described in (4.15), (4.16).
To obtain (4.18) we apply (4.21) to v = e−iλtΠλu, which yields L
2 → Lp
bounds, and hence by duality and selfadjointness all estimates of (4.18) for
3The exponent q = ∞ is actually allowed in the spectral projection bounds in dimension
n = 2. However, it is not allowed in any of the resolvent bounds.
4These bounds are very robust and are in effect established in [19] for a much larger class
of operators
n ≥ 2. The case n = 1 can be dealt with directly using the pointwise bounds
for the Hermite functions.
We note a consequence of the bounds (4.18), namely
(4.22) ‖(H − z)−1−k‖Lq′→Lq . (1 + |ℑz|)
, n ≥ 2, k ≥ 1
which is obtained by interpolating between q = 2 and q = 2n
Similarly we get
(4.23) ‖(H − z)−1‖Lq′→L2 . (1 + |ℑz|)
p′ , n ≥ 2.
Then we apply (4.21) to
v(x, t) = χ(t)e−iztu(x), g = χ′(t)e−iztu(x) + χ(t)e−izt(H − z)u
where χ is a unit bump function on an interval of size (1+ |ℑz|)−1. This yields
‖u‖Lq . (1 + |ℑz|)
p‖u‖L2 + (1 + |ℑz|)
p′ ‖(H − z)u‖Lq′ . ‖(H − z)u‖Lq′ .
concluding the proof of (4.19) for n ≥ 2. The case n = 1 is a variation on the
same theme. �
4.3. Combining the estimates. Here we combine the L2 and the Lp com-
ponents in the resolvent bounds:
Proposition 4.7. For z away from n+ 2N the resolvent (H − z)−1 satisfies
(4.24)
‖(H − z)−1‖Lq′→X2(ℜz,R,d) . (1 + |ℑz|)
2 , n ≥ 2, (n, q) 6= (2,∞)
with the obvious modification for n = 1.
Proof of Proposition 4.7. Taking into account the bounds (4.19) and (4.23), it
remains to prove the estimate
‖u‖X̃2(ℜz,R,d) .(1 + |ℑz|)
p′ ‖(H − z)u‖Lq′
+ (1 + |ℑz|)
p‖u‖Lq + (1 + |ℑz|)
2‖u‖L2
But this follows from (4.12) in the same way as for Proposition 4.2 since the
operator Q is bounded in Lq. �
5. Lp estimates in the flat case and parametrix bounds
In this section we begin with the mixed norm LpLq Carleman estimates
in the simplest case, i.e. with constant coefficients and a polynomial weight.
These were proved in [8] except for the endpoint which was obtained later in
[10] .
After a conformal change of coordinates and conjugation with respect to the
exponential weight the Carleman estimates reduce to proving LpLq estimates
for a parametrix K for ∂t −H + τ . In this article we need a stronger version
of these bounds, where we add in localized L2 norms.
In a simplified form, Escauriaza-Vega’s result in [10] has the form:
Theorem 4. [10] Let p and q be as above. Then
‖t−τe−
8t u‖L∞(L2)∩Lp(Lq) ≤ ‖t−τe−
8t (∂t +∆)u‖L1(L2)+Lp′(Lq′ ),
for all u with compact support in Rn × [0,∞) vanishing of infinite order at
(0, 0) uniformly with respect to 4τ with a positive distance from integers.
One can write the estimate in the (s, y) coordinates using the same trans-
formation as in Section 3:
(5.1) ‖eτsv‖L∞(L2)∩Lp(Lq) . ‖eτs(∂s +H)v‖L1(L2)+Lp′ (Lq′ )
Setting w = eτsv this becomes
(5.2) ‖w‖L∞(L2)∩Lp(Lq) . ‖(∂s +H − τ)w‖L1(L2)+Lp′ (Lq′ )
Denoting by Πλ the spectral projection onto the λ eigenspace ofH we obtain
a parametrix K for (∂t −H + τ),
K(∂t +H − τ) = I
where the s-translation invariant kernel of K is
K(s) =
s(τ−λ)1s(τ−λ)<0
Since w decays at ±∞ we have
w = K(∂s +H − τ)w
therefore (5.2) can be rewritten in the form
(5.3) ‖Kf‖L∞(L2)∩Lp(Lq) . ‖f‖L1(L2)+Lp′ (Lq′ )
The main result of this section is an improvement of (5.2), namely
Proposition 5.1. Assume that τ is away from n + N and that
1 ≤ d ≤ R . τ
(5.4) ‖Kf‖L∞(L2)∩Lp(Lq)∩L2X2(τ,R,d) . ‖f‖L1(L2)+Lp′ (Lq′ )+L2X∗2 (τ,R,d)
Proof. We work in dimension n ≥ 2; some obvious adjustments are needed in
dimension n = 1, which is slightly easier. We consider four endpoints:
A: The L1L2 → L∞L2 bound follows easily since the projectors Πλ are L2
bounded.
B: The L1L2 → LpLq bound. Here it suffices to prove
‖K(.)f‖LptLqx . ‖f‖L2
Splitting f into spectral projections and using (4.18) we obtain
‖K(t)f‖Lq .
e−|(λ−τ)t|‖Πλf‖L2
For |t| ≥ 1 we can use Cauchy-Schwartz to obtain
‖K(t)f‖Lq∩X2(R,d) . e−c|t|‖f‖L2
which suffices for all q. For |t| ≤ 1 we consider the most difficult case p = 2
and compute
‖K(t)f‖2L2([−1,1],Lq) .
e−|(λ−τ)t|‖Πλf‖L2
|λ− τ | + |µ− τ |
‖Πλf‖L2‖Πµf‖L2
0≤i,j
2−i−j
|λ−τ |≈2i
‖Πλf‖L2
|µ−τ |≈2i+j
‖Πµf‖L2
|λ−τ |≈2i
‖Πλf‖2L2
|µ−τ |≈2i+j
‖Πµf‖2L2
.‖f‖2L2
C: The L1L2 → X2(τ, R, d) bound for K follows in the same way from
‖Πu‖X2(τ,R,d) . ‖u‖L2.
D: The Lp
+ L2X∗2 (τ, R, d) → L∞L2 bound for K is equivalent to the
L1L2 → LpLq ∩X2(τ, R, d) bound for K∗. By reversing time this is seen to be
the same as the L1L2 → LpLq bound for K.
E: The Lp
+ L2(X∗2 (R, d)) → LpLq ∩ L2X2(R, d) bound. Using (4.18)
and (4.13) directly yields
‖K(s)‖
n+2+X∗2 (R,d)→L
n−2 ∩X2(R,d)
e|s(τ−λ)| . s−1e−cs
Similarly we obtain
‖K(s)‖
n−2 ∩X2(R,d)
2 e−cs, ‖K(s)‖
(R,d)
2 e−cs
Interpolation with the L2 estimate gives
‖K(s)‖Lq′→Lq . s
‖K(s)‖Lq′→X2(R,d) . s
2 , ‖K(s)‖X∗2 (R,d)→Lq . s
If p > 2 then the Hardy-Littlewood Sobolev inequality implies
‖K ∗ f‖LpLq . ‖f‖Lp′Lq′ .
‖K ∗ f‖L2X2(R,d) . ‖f‖Lp′Lq′ , ‖K ∗ f‖LpLq . ‖f‖L2X∗2 (R,d).
With obvious changes the analysis is similar if n = 1, 2.
It remains to prove the L2 → L2 type bounds, namely
‖Kf‖L2X2(R,d) . ‖f‖L2X∗2 (R,d) (n = 1, 2)
respectively
L2(X2(R,d)∩L
n−2 )
. ‖f‖
L2(X∗2 (R,d)+L
n+2 )
(n > 2)
For this, following an idea in [10], we consider a dyadic frequency decompo-
sition in time. By the Littlewood-Paley theory it suffices to prove the bound
for a single dyadic piece at frequency 2j, namely
(5.5) ‖Sj(Ds)Kf‖
L2(X2(R,d)∩L
n−2 )
. ‖f‖
L2(X∗2 (R,d)+L
n+2 )
(n > 2)
and its one and two dimensional counterpart.
Taking a time Fourier transform we can write (for f ∈ S(Rn))
ŜjK(σ)f = s(2
λ− τ − iσ
Πλf = s(2
−jσ)(H − τ − iσ)−1f
therefore by the inversion formula
(SjK)(t)f =
eitσs(2−jσ)(H − τ − iσ)−1fdσ
=− t−2
(s(2−jσ)(H − τ − iσ)−1)fdσ
Hence using the resolvent bounds (4.19) and (4.22) and the first line for |t| ≤
2−j and the second line for |t| ≥ 2−j we obtain
‖SjK(t)‖
(R,d)
n−2 ∩X2(R,d)
1 + 22jt2
and the similar estimate in one and two dimensions. The bound on the right
is integrable in t, therefore (5.5) follows. �
6. Modified weights and pseudoconvexity
The main result of this section, Theorem 5 is a considerable improvement
of Section 3. The weights t−τ in Section 3, while easy to use, satisfy merely
a degenerate pseudoconvexity condition, in the sense that the selfadjoint and
the skewadjoint parts of the operator in (3.4) commute. This is in contrast to
strong pseudoconvexity where one obtains better L2 bounds from the positivity
of the commutator. A perturbation argument easily implies an L2 Carleman
estimate for variable coefficients as soon as g = In+O(t). However, even arbi-
trarily small perturbations of g from In at t = 0 destroy the pseudoconvexity.
To obtain results for general variable coefficients we need a more robust
weight with additional convexity. A good way of doing this is by adding
convexity in t and by using a weight of the form
eh(− ln t)
with a convex function h. Then we obtain for the heat operator the strength-
ened L2 estimates of Proposition 3.2. The assumption of vanishing of infinite
order forces us to work with functions h with at most linear growth at infinity.
This in turn limits the convexity of h, and hence the gain from the convexity
in the L2 bounds.
These Carleman inequalities with the weight e−h(ln t) are more stable with
respect to perturbations. They can be obtained for coefficients satisfying
(6.1) |g − In|+ (t + |x|2)|∂tg|+ (t+ |x|2)1/2|∇g| . (t+ |x|2)ε.
with suitable functions h. It is not difficult to weaken (6.1) almost to our condi-
tions (1.15) and (1.16). This venue was pursued by Escauriaza and Fernández
In this paper we seek to obtain Lp Carleman inequalities and also to handle
Lp gradient potentials. Both require good spatial and temporal localization,
which depends on the strength of the L2 estimates. The weights eh(− ln t) seem
to be insufficient for this purpose.
Consequently we consider a larger class of weights of the form
eh(− ln t)+φ(xt
−1/2,− ln(t))
having some additional convexity in y = xt−1/2. Here we think of φ essen-
tially as a function of y with a milder dependence on s = − ln t. Obtaining
pseudoconvexity is not entirely straightforward because the Hamilton flow for
the Hermite operator H is periodic so no nonconstant function of y can be
convex along its orbits. We note that the projection of the orbits to the y
space are ellipses of size O(
τ ) where τ is the energy, centered at 0. Hence
we can choose φ to be convex in y for |y| ≪
τ . We compensate the lack of
convexity of φ when |y| ≈
τ by the s convexity of h. To elaborate this idea
we explain the precise setup.
Let δ1 be small positive constant. We begin with constants {αij}A (see (1.13)
and (1.14) for the notation) which control the regularity of the coefficients5
gkl − δkl, dkl and ekl of P given by (1.7) as in (1.15).
(6.2) δ1αij = ‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖Cmijt (Aij).
The condition (1.15) guarantee that for all τ ≥ 1
(6.3) ‖αij‖l1(A(τ)) ≤ 1. ‖αij‖l∞(A) ≤ 1.
We first adjust the αij ’s upward so that they vary slowly and do not con-
centrate in irrelevant regions. This readjustment depends on the choice of the
parameter τ .
Lemma 6.1. Let αij be a sequence satisfying (6.3). Then for each τ ≫ 1
there exists a double sequence (εij)A(τ) with the following properties:
5denoted generically by d here and later
(1) For each (i, j) ∈ A(τ)
αij ≤ εij
(2) We have εij ∈ l1(A(τ)),
‖εij‖l1(A(τ)) . 1.
(3) The sequence εij is slowly varying,
| ln εi1j1 − ln εi2j2| ≤
(|i1 − i2|+ |j1 − j2|), (i1, j1), (i2, j2) ∈ A(τ).
(4) The sequence (εi) defined by
j:(i,j)∈A(τ)
εij, i ≥ ln τ.
has the following properties
(6.4) | ln εi1 − ln εi2 | ≤
|i2 − i1|, εij . εi, εi[ln τ/2]+2 ≈ εi
(5) For each i ≥ ln τ there exists an unique 0 ≤ j(i) ≤ [ln τ/2]+2 with the
following properties:
(6.5) εij(i) ≈ εi.
εij ≤ e−jτ−1/2 if 0 ≤ j ≤ j(i), j(i) 6= 0
εij > e
−jτ−1/2 if j(i) < j ≤ [ln τ/2] + 2.
(6.6)
We shall see that j(i) is an important threshold. If j ≥ j(i) then we can
localize our estimates to the corresponding Aij and even to smaller sets. On
the other hand, we cannot localize to sets smaller than
(6.7) Bi0 =
j≤j(i)
Proof. To fulfill the conditions (1)-(4) we simply mollify the αij,
ε̃ij = max
(k,l)∈A(τ)
(|i−k|+|j−l|), ε̃i =
[ln τ/2]+2∑
ε̃ij.
For the last part of (4) we redefine
ε̃ij := ε̃ij + e
|j−([ln τ/2]+2)|ε̃i
This also increases ε̃i by a fixed factor.
For (5) we begin with a preliminary guess for j(i) which we call j0(i) ∈ R+.
We consider three cases.
j0(i) =
ln τ/2 if ε̃i < τ
− ln(ε̃iτ 1/2) if τ−1 ≤ ε̃i < τ−1/2,
0 if τ−1/2 ≤ ε̃i.
We define
(6.8) εij := max{ε̃ij, 2e−
|j−j0(i)|ε̃i},
which is still slowly varying because ε̃i is slowly varying.
We define j(i) according to (6.6). It is uniquely determined since the se-
quence εij is slowly varying compared to e
2 . Since ε̃ij is slowly varying we
must have ε̃ij ≤ ε̃i/2. This allows us to conclude that for j close to j0(i), the
second term in (6.8) is larger than the first one,
εij = 2e
|j−j0(i)|ε̃i for |j − j0(i)| ≤ 2.
If j0(i) = 0 then εi0 = 2ε̃i ≥ 2τ−1/2 and hence j(i) = 0. If 0 < j0(i) < ln τ/2
then for |j − j0(i)| ≤ 2 we have
εij = 2e
|j−j0(i)|e−j0(i)τ−1/2
therefore j0(i)− 2 < j(i) ≤ j0(i). If j0(i) = ln τ/2 then for |j − j0(i)| ≤ 2 we
εij ≤ 2e−
|j−j0(i)|e−j0(i)τ−1/2
and we arrive again at j0(i)−2 ≤ j(i). In all three cases we have |j0(i)−j(i)| ≤
2 therefore (6.5) holds. We observe that εi . ε̃i. �
The sequence (εij)A(τ) is used to describe the amount of spatial convexity
needed in the region Aij , which will be reflected in the construction of φ below.
The partial sums εi measure the amount of s-convexity needed in [i, i+1]. The
purpose of part (5) above is to correlate the two amounts in a region where
they have the same strength (where j is close to j(i)).
Our weights have the form
(6.9) ψ(s, y) = h(s) + φ(s, y)
Their choice is described in the next two lemmas:
Lemma 6.2. Let τ and (εi) be as in Lemma 6.1. Then there is a convex
function h with the following properties:
(1) h′ ∈ [τ, 2τ ].
(2) h′′(s) + dist(h′(s),N) > 1
(3) εiτ . h
′′(s) . εiτ + 1 for s ∈ [i, i+ 1].
(4) |h′′′| . h′′.
The proof of the lemma is fairly straightforward and uses only the fact that
(εi) is slowly varying and summable. The second part is needed in order to
avoid the eigenvalues of the Hermite operator.
Lemma 6.3. Let τ , (εij) and (εi) be as in Lemma 6.1. Then there exists a
smooth spherically symmetric function
φ : R× Rn → R
with the following properties:
(1) (Bounds) The function φ is supported in |y| ≤ 2τ 1/2 and satisfies
(6.10) 0 ≤ φ(s, y) . εiτ, |∂sφ(s, y))|+ |∂2sφ(s, y)| . εiτ
(6.11)
l,k=0
(1 + |y|)k|D1+ky ∂lsφ| . ǫiτ 1/2 for i ≤ s ≤ i+ 1,
(2) (Monotonicity)
(6.12) ∂rφ(s, y) ≈ εiτ
2 for (s, y) ∈ Aij , (i, j) ∈ A(τ), j ≥ j(i) + 1
(3) (Convexity)
(6.13) (1 + |y|)∂2rφ(s, y) ≈ εijτ
2 in Aij, (i, j) ∈ A(τ).
Proof. Let
φj(y) =
e2j + |y|2, j ≥ 0.
We fix a smooth partition of unity 1 =
η(s− i) and define
ln aj(s) =
η(s− i) ln εij .
These functions satisfy the bounds
(6.14) aj(s) ≈ εij, i ≤ s ≤ i+ 1, |a′j |, |a′′j |, |a′′′j | . aj.
Their sum satisfies
a(s) :=
[ln τ/2]+2∑
aj(s) ≈ εi, i ≤ s ≤ i+ 1.
We define
φ(s, y) = τ
2χ(|y|τ−1/2)
[ln τ/2]+2∑
aj(s)φj(|y|)
where χ is a smooth function supported in [−2, 2] and identically 1 in [−3
We verify the properties:
0 ≤ φ(s, y) . a(s)τ . εiτ, i ≤ s ≤ i+ 1.
The remaining part of (6.10) follows from (6.14). Estimate (6.11) is a conse-
quence of (6.14) and
|(1 + |y|)kDk+1φj(y)| . 1, 0 ≤ k ≤ 3.
The upper bound from (6.12) is covered by (6.11) and the lower one follows
∂rφ(s, y) & τ
j≤j(i)
aj ≈ τ 1/2
j(i)∑
εij ∼ εiτ 1/2
in Aij with j ≥ j(i) where we use εij(i) ≈ εi. The assertion (6.13) follows from
immediate bounds on second derivatives of the φj. �
Our aim in this section is to prove L2 Carleman estimates for the variable
coefficient operator P with the exponential weight
ψ(− ln t
ψ(s, y) = h(s) + δ2φ(s, y).
where δ2 is a small constant and h and φ are as in in Lemma 6.2 and 6.3.
The calculations are involved. For a first orientation we outline the key part
of the argument for the constant coefficient heat equation. Using the change
of coordinates of Section 3 we transform the problem to weighted estimates for
the operator P0 = ∂s +H and the exponential weight e
ψ(s,y). This translates
to obtaining bounds from below for the conjugated operator
P0,ψ = e
ψ(s,y)P0e
−ψ(s,y).
Lemma 6.4. Let τ be large enough. Let ψ be as in (6.9) with h, φ as in the
above two Lemmas 6.2,6.3 with δ2 ≪ 1 . Then the operator P0,ψ satisfies the
bound from below
‖(h′′)
2 v‖2 + δ2τ−1(‖a2int∇v‖2 + ‖a2⊥∇⊥v‖2) . ‖P0,ψv‖2L2
for all functions v supported in {|y|2 ≤ 9τ} where the weights aint, a⊥ are
defined by
a4int = εij(1 + |y|)−1τ
2 , a4⊥ = 1j≥j(i)εi(1 + |y|)−1τ
2 in Aij .
Proof. We decompose P0,ψ into its selfadjoint and its skewadjoint part
P0,ψ = L
0,ψ + L
where
(6.15) Lr0,ψ := −∆y + y2 − ψs − ψ2y , Li0,ψ := ∂s + ψy∂ + ∂ψy.
Expansion of the norm gives
(6.16) ‖(Lr0.ψ + Li0,ψ)v‖2L2 = ‖Lr0,ψv‖2L2 + ‖Li0,ψv‖2L2 + 〈[Lr0,ψ, Li0,ψ]v, v〉
The conclusion of the lemma follows from the commutator bound
(6.17) 〈[Lr0,ψ, Li0,ψ]v, v〉 & ‖(h′′)
2 v‖2 + δ2τ−1(‖a2int∇v‖2 + ‖a2⊥∇⊥v‖2)
The commutator is explicitly computed
[Lr0,ψ, L
0,ψ] = ψss + 4ψyψyyψy − 4∂ψyy∂ − 4yψy + 4ψyψsy −∆2ψ
Since δ2 ≪ 1 the first term has size h′′(s). The second one is nonnegative since
ψ is convex for |y|2 < 9τ .
The Hessian of the radial function ψ can be written in the form
(6.18) ψyy = ψrr
One can see that the radial and angular derivatives carry different weights.
Our construction of φ guarantees that
ψrr .
, ψyy & ψrrIn
hence the weight ψrr can be used for all derivatives. For the size of the two
weights we have
ψrr ≈ a4int,
& a4⊥
This gives the last two terms in (6.17).
It remains to see that the remaining terms in the commutator are negligible
compared to the first term on the right hand side of (6.17). For this we use
the bound (6.11) to conclude that in Aij we have
| − 4yψy + 4ψyψsy −∆2ψ| . δ2εiτ . δ2h′′
To switch to operators with variable coefficients it is convenient to extend
the weights to the full space and to regularize them. Precisely we shall assume
a4int(s, y) ≈ εiτ in Aij if |y|2 ≥ τ
a4int(s, y) ≈ εij(1 + |y|)−1τ
2 in Aij if |y|2 ≤ τ.
(6.19)
Observe that the two cases above match since εi ≈ εij in the region where
y2 ≈ τ . We also introduce a modification a of aint which is used to include the
effect of the spectral gap in regions where we have very little convexity:
a4(s, y) ≈ 1 + εiτ in Aij if |y|2 ≥ τ
a4(s, y) ≈ 1 + εij(1 + |y|)−1τ
2 in Aij if |y|2 ≤ τ.
(6.20)
Finally we choose a⊥ with the properties
supp a⊥ ⊂
{Aij : j(i)− 1 ≤ j ≤
ln τ}
a4⊥(s, y) .εi(1 + |y|)−1τ
2 in Aij
a4⊥(s, y) ≈εi(1 + |y|)−1τ
2 in Aij if j(i) ≤ j ≤
ln τ − 1
(6.21)
The bounds for the weights from above are assumed to remain true after
applying powers of the differential operators ∂s, ∂y and y∂y to them.
Consider now a the more general class of operators P with real variable
coefficients given by (1.7). We repeat the change of coordinates and write in
the (s, y) coordinates:
4e−4sP = − ∂
− 2y ∂
+ ∂ig
ij∂j + yid
ij∂j + ∂id
ijyj + yie
This further leads to
4e−(n+4)s−
2 Pens+
2 = −P̃
where P̃ is given by
− ∂igij∂j − yi(gij − 2δij + 2dij + eij)yj
− yi(gij − δij + dij)∂j − ∂i(gij − δij)yj
We rewrite it in the generic form
P̃ = P0 − ∂d∂ − ydy − yd∂ − ∂dy
with P0 = ∂s +H .
To simplify as much as possible the proof of the main L2 Carleman estimate
we introduce a stronger condition on the regularity of the coefficients:
|d|+ 〈y〉(|dy|+ τ−
2 |dyy|+ τ−1|dyyy|+ τ−
2 |ds|) . δ1εij in Aij
|d|+ 〈y〉(|dy|+ τ−
2 |dyy|+ τ−1|dyyy|+ τ−
2 |ds|) . δ1
(6.22)
This improved regularity will be gained later on by regularizing the coefficients.
We are now in the position to formulate the Carleman estimate.
Proposition 6.5. Let τ be large enough and δ1 ≪ δ2 ≪ 1. Let ψ be as in
(6.9) with h, φ as in Lemmas 6.2,6.3. Assume that the coefficients g − In, d
and e satisfy (6.22). Then the following L2 Carleman estimate holds for all
functions u supported in {y ≤ 9τ}:
j=0,1,2
2‖a2eψDju‖+ τ− 12‖a2⊥eψD⊥u‖
. ‖eψP̃ u‖.(6.23)
Proof. After conjugation
Pψ := e
ψ(s,y)P̃ e−ψ(s,y)
we decompose Pψ into its selfadjoint and its skewadjoint part
Pψ = L
ψ + L
which for y2 < 9τ can be expressed in the generic form (see also (6.15)):
Lrψ = L
0,ψ + ∂d∂ + τd
Liψ = L
0,ψ + τ
2 (d∂j + ∂jd)
with d satisfying (6.22). Then (6.23) follows from
(6.24)
j=0,1,2
τ−j(δ2‖a2intDjv‖2 + 〈h′′〉
2Djv‖2) + δ2τ−1‖a2⊥∇⊥v‖2 . ‖P̃ψv‖2.
The proof will consist of three steps.
Step 1: First we show that for v supported in {|y|2 ≤ 9τ} we have
(‖a2int∇v‖2+‖a2⊥∇⊥v‖2)+‖(h′′)
2v‖2+‖Lrψv‖2 . ‖P̃ψv‖2+ δ1‖a2intv‖2
(6.25)
We compute
‖Pψv‖2L2 = ‖Lrψv‖2L2 + ‖Liψv‖2L2 + 〈[Lrψ, Liψ]v, v〉
We expand the commutator
[Lrψ, L
ψ] = [L
0,ψ, L
0,ψ] + [∂d∂ + τd, L
0,ψ] + τ
2 [Lrψ, d∂j + ∂jd]
The main contribution in (6.25) comes from the first commutator, for which
we use (6.17) to obtain the terms on the left side of (6.25).
The second commutator is estimated by
|〈[M r, Li0,ψ]v, v〉| . δ1(‖a2intv‖2 + τ−1‖a2int∇v‖2)
and the second term on the right is negligible since δ1 ≪ δ2. Indeed, we write
[∂d∂ + τd, Li0,ψ] = −∂kqkl∂l + r
where the coefficients q, r have the generic form
q = ds + ψydy + ψyyd+ dψyy, r = ∂d∂∆ψ + τ(ds + ψydy)
Using the bounds (6.22) for d and (6.11) for φ we estimate
|q| . δ1τ−1a4int, |r| . δ1a4int.
Finally, the third commutator is estimated in a similar fashion. We write it
in the form
2 [Lrψ, d∂j + ∂jd] = −∂kqkl∂l + r
where the coefficients q, r have the generic form
q = τ
2 (dy + ddy), r = τ
2 (∆dy + ∂d∂dy + dψys + dψyyψy + τddy)
Using (6.22) and (6.11) we obtain the same bounds for q and r as in the
previous case. This concludes the proof of (6.25).
Step 2: We use an elliptic estimate to show that for functions v supported
in {|y|2 ≤ 9τ} we have
(6.26) δ2
τ−j‖a2intDjv‖2+τ−1‖a2⊥D⊥v‖2
+‖(h′′)
2v‖2+‖Lrψv‖2L2 . ‖P̃ψv‖2
The elliptic bound
‖D2v‖+ τ‖v‖ . τ
2‖Dv‖+ ‖(−∆− h′(s))v‖
can easily proven by a Fourier transform. It implies
‖D2v‖+ τ‖v‖ . τ
2‖Dv‖+ ‖(H − h′(s))v‖+ ‖y2v‖,
We can replace H − h′(s) by Lrψ due to the pointwise estimate
|(Lrψ − (H − h′(s)))v| . δ1(|D2v|+ τ
2 |Dv|+ τ |v|)
Then (6.26) follows from (6.25).
Step 3: Here we use the spectral gap condition to improve our bound when
h′′ ≪ 1 and show that (6.26) implies (6.23). It suffices to show that if h′′(s) < 1
‖v‖L2 + τ−1‖D2v‖ . ‖Lrψv‖L2
Indeed, let s ∈ [i, i + 1] so that h′′(s) < 1
. Then h′ has a positive distance
from the integers. Also εi . 1 which implies that at time s we must have
|g − In| . δ1τ−1, |Dg| . δ1τ−1, |ψr| . δ2τ−
Hence we may think of Lrψ as a small perturbation of H − h′(s) and compute
‖v‖+ τ−1‖D2v‖ . ‖(H − h′(s))v‖ . ‖Lrv‖+ (δ1 + δ22)‖v‖+ δ1τ−1‖D2v‖
where the last two terms on the right are negligible compared to the left hand
side. The proof of the proposition is concluded.
We want to reformulate the previous result in a more symmetric fashion.
To do this we weaken the estimates slightly by using a coarser partition of the
space. We distinguish three cases for i corresponding to the value of j(i) in
Lemma 6.1 (v).
Definition 6.6. We define the partition Bij as follows.
(1) If j(i) = 0 (which corresponds to εi & τ
2 ) we set
Bij = Aij, b ≈ a, b⊥ ≈ a⊥
(2) If 0 < j(i) < [ln τ/2+2] (which corresponds to τ−1 . εi . τ
2 ) we set
Bij = Aij, b ≈ a, b⊥ = a⊥ j ≥ j(i)
respectively
Bi0 =
j<j(i)
Aij , b ≈ a|Aij(i) b⊥ = 0 on Bi0
(3) If j(i) = [ln τ/2 + 2] (which corresponds to τ−1 . εi) we set
Bi0 =
[ln τ/2]+2⋃
Aij, b = 1, b⊥ = 0 on Bi0.
Heuristically the definition of the Bij partition is motivated by the fact that
in regions Aij with j < j(i) the weight φ is ineffective, i.e. it changes by at
most O(1). Thus the convexity there is useless, and instead we rely directly
on localized bounds for the Hermite operator.
Since the εij are slowly varying b . a and b⊥ . a⊥ and we may replace the
a’s by b’s in the above proposition.
To provide some bounds on the size of b and b⊥ we introduce a function
1 ≤ r(s) ≤ τ 12 which is smooth and slowly varying on the unit scale in s so
r(s) ≈ ej(i) s ∈ [i, i+ 1]
This describes the region where b is tapered off and b⊥ = 0. Precisely, consider
two cases corresponding to the three cases above.
(1) If r(s) ≈ 1 then we have the bounds
Mτ(1 + r)−
2 . b4(r, s) .Mτ(1 + r)−1
b4⊥(r, s) .Mτ(1 + r)
(6.27)
where the parameter M ≥ 1 is defined by M ≈ ε(s)τ 12 .
(23) If r(s) ≫ 1 then
τr(s)−
2 (r(s) + r)−
2 . b4(r, s) . τr(s)−1(r(s) + r)−1
b4⊥(r, s) . τr(s)
−1(r(s) + r)−1
(6.28)
with approximate equality when r . r(s) and approximate equality on the
right when r = τ
By slightly changing b and b⊥ we may and do assume that the functions b
and b⊥ are smooth with controlled derivatives. Thus b and b⊥ are smooth on
the unit scale in s and on the dyadic scale in y, and their derivatives satisfy
the bounds
(6.29) |bs|+ (rs + r)|br|+ (rs + r)2|brr| . b, r2 < 9τ
(6.30) |b⊥s|+ (rs + r)|b⊥r|+ (rs + r)2|b⊥rr| . b⊥ + b r2 < 9τ
In addition we have
(6.31) supp b⊥r ⊂ {r > rs}
Using the functions b and b⊥ we define the Banach space X
2 with norm
= ‖bv‖2L2 + τ−1/2‖b⊥D
Then the symmetrized version of Proposition 6.5 has the form
Proposition 6.7. Assume that the coefficients of P satisfy (6.22). Let ψ be
as in (6.9) with h, φ as in Lemmas 6.2,6.3. Then the following L2 Carleman
estimate holds for all functions u supported in {y ≤ 9τ}:
(6.32) ‖eψ(s,y)u‖X02 . ‖e
ψ(s,y)Pu‖(X02 )∗
Proof. Conjugating with respect to the exponential weight, the bound (6.32)
is rewritten in the form
(6.33) ‖v‖X02 . ‖Pψv‖(X02 )∗
Observing that
D2⊥ = −y−2∆Sn−1
we introduce the operator
Q = Q(|y|, (−∆Sn−1)
2 ), q(r, λ) = (b4(r) + r−2τ−1b4⊥(r)λ
Then the inequality (6.33) can be written as
‖Qv‖L2 . ‖Q−1Pψv‖L2
whereas inequality (6.23) implies
‖Q2w‖L2 . ‖Pψw‖L2.
Hence it is natural to apply (6.23) to the function w = Q−1v, which solves
Pψw = Q
−1Pψv +Q
−1[Q,Pψ]w
Thus (6.33) would follow provided that the commutator term is small,
‖Q−1[Q,Pψ]w‖L2 ≪ ‖b2w‖L2 + τ−1/2‖b2∇w‖L2 + τ−1‖b2D2yw‖L2
Unfortunately a direct computation shows that the smallness fails when j is
close to j(i) even in the flat case, i.e. with Pψ replaced by
P0,ψ = ∂s −∆+ y2 − ψs − ψ2y
To remedy this we introduce an additional small parameter δ and use it to
define a modification Qδ of Q. We modify r(s) to rδ(s) defined by
rδ(s)
−2 = δ8r(s)−2 + δ2τ−1
and use it to define the function
bδ(r, s)
4 = δ−12τ(r2 + rδ(s)
We can still compare it with b,
bδ(r, s)
4 . δ−4b(r, s)4
Its usefulness lies in the fact that it is larger than b exactly in the region where
the commutator term above is not small.
The modification Qδ of Q has symbol
qδ(r, s, λ) = q(r, s, λ) + bδ(r, s) = (b
4(r, s) + r−2τ−1b4⊥(r, s)λ
4 + bδ(r, s)
which satisfies
(6.34) q ≤ qδ . δ−1q
We claim that it satisfies the bound
(6.35) ‖Q−1δ [Qδ, Pψ]w‖L2 . (δ + c(δ)δ1)
j=0,1,2
2‖b2Djw‖L2
Suppose this is true. Then we fix δ sufficiently small, and for δ1 small enough
we apply (6.23) to w = Q−1δ v. By (6.35) he commutator term in the equation
for w can be neglected, and we obtain
‖Q2w‖L2 . ‖Q−1δ Pψv‖L2
which by (6.34) implies that
‖Qv‖L2 . δ−1‖Q−1Pψv‖L2
It remains to prove (6.35).
I. We first calculate the commutator in the flat case, i.e. with Pψ replaced
by P0,ψ. Due to the spherical symmetry the only contribution comes from the
radial part of the Laplacian and the s derivative. Hence using polar coordinates
we compute
Q−1δ [Qδ, P0,ψ] = Q
Qδrr +
Qδr + 2Qδr∂r −Qδs
Then is suffices to verify that on the symbol level we have
(6.36) |qδrr|+ r−1|qδr|+ |qδs|+ τ 1/2|qδr| . δqδb2(1 + τ−1r−2λ2)
I.(1). We begin with the q component of qδ. Using (6.29) and (6.30) one
obtains
|qrr|+ r−1|qr|+ |qs|+ τ 1/2|qr| . (r(s) + r)−1τ
Thus it remains to show that
(r(s) + r)−1τ
2 q . δqδb
2(1 + τ−1r−2λ2)
Optimizing with respect to λ it suffices to consider the cases λ = 0 respectively
λ = rτ
2 , where the above inequality becomes
(r(s) + r)−1τ
2 (b+ b⊥) . δ(bδ + b)b
or equivalently
(b+ b⊥)b
1 . δ(bδ + b)b
which is true since by (6.27) and (6.28) we have b⊥b1 . b
1 while b1 . δbδ.
I.(2). Next we consider the bδ component of qδ, for which it suffices to prove
(6.37) |bδrr|+ r−1|bδr|+ |bδs|+ τ 1/2|bδr| . δbδb2
I.(2).(a). For the s derivative we compute
(r2δ(s))s
r2 + r2δ(s)
therefore we want to show that
(r2δ(s))s . δ(r
2 + r2δ(s))b
We optimize the right hand side with respect to r. The minimum is attained
when r2 = min{r2δ(s), τ}. We need to consider two cases:
I.(2).(a).(i). If rδ(s) . τ
2 then r2δ(s) ≈ δ−8r2(s) and τ > δ−8r2(s). Hence
using the estimate from below in (6.27) and (6.28) we obtain b4(rδ) & δ
Then the above bound for r = rδ follows since |(rδ(s)−2)s| . rδ(s)−2.
I.(2).(a).(ii). If rδ(s) & τ
2 then by6 (6.28) we evaluate b2(τ
2 ) ≈ τ 14 r(s)− 12 .
Then the above bound becomes
δ8(r(s)−2)s . δr
4 r(s)−
Since |(r(s)−2)s| . r(s)−2 it suffices to show that
δ8r(s)−2 . δr−2δ τ
4 r(s)−
The worst case is r(s)2 = δ6τ , rδ(s) = δ
−2τ when it is verified directly.
I.(2).(b). For the r derivatives the last term is the worst. Since
r2 + r2δ(s)
we want to show that
2 r . δ(r2 + r2δ(s))b
Optimizing with respect to r the worst case is when r2 = min{r2δ(s), τ}.
I.(2).(b).(i). If r2δ(s) . τ then rδ(s) ≈ δ−4r(s) therefore for r = rδ(s) the
above relation becomes
2 . δ−3r(s)b2(rδ(s))
which follows from the bound from below in (6.27) and (6.28).
I.(2).(b).(ii). If r2δ(s) & τ then as before we evaluate b
2 ) ≈ τ 14 r(s)− 12 and
rewrite the above bound as
τ . δr2δ(s)τ
4 r(s)−
The right hand side is smallest either when rδ(s) = τ
2 and r(s) = δ4τ
when rδ(s) = δ
2 and r(s) = τ
2 . In both cases the inequality is easily
verified.
II. Now we deal with the general case, which we treat as a perturbation.
Since we do not care about the dependence of the constants on δ to keep the
notations simple we include bδ in b and work with Q instead of Qδ. Thus in
the computations below we allow the implicit constants to depend on δ.
Suppose that A is a pseudodifferential operator of order 1 and let η be any
Lipschitz function. Then
(6.38) ‖[A, η]f‖L2 . ‖f‖L2.
We write
q(λ) = b+ (b4 + r−2b4⊥λ
4 − b =: b+ q1(λ).
Even though q1 has order
, we treat it as an operator of order 1 and estimate
〈λ〉k−1|q1(k)(λ)| .
6the equality holds on the right when r2 = τ
Hence, for each r we obtain the bound on the sphere Sn−1
(6.39) ‖[Q, η]f‖L2 .
‖η‖Lip(Sn−1)‖f‖L2.
As a consequence, it also follows that
(6.40) ‖[Q, η∇jθ]f‖L2 .
‖η‖Lip(Sn−1)‖∇jθf‖L2.
where ∇θ stands for the vector fields xi∂j − xj∂i generating the tangent space
of Sn−1.
To use these bounds we write the difference Pψ − P 0ψ in polar coordinates,
Pψ − P 0ψ = P 0θ ∂2r + P 1θ ∂r + P 2θ
where P
θ are spherical differential operators of order j. Modulo zero homoge-
neous coefficients which are polynomials in xr−1 we can write
P 0θ = d, P
θ = dr
−1∇θ + τ
2d+ dy
P 2θ = dr
−2∇2θ + (τ
2d+ dy)r
−1∇θ + τd + τ
where d stands for coefficients satisfying (6.22). In the support of b⊥ we have
a ≈ b therefore our regularity assumptions on d show that for fixed r we have
‖d‖L∞ + ‖d‖Lip(Sn−1) . δb4τ−1
The coefficients involving dy satisfy better Lipschitz bounds and are neglected
in the sequel.
We expand the commutator
[Q,Pψ] =
j=0,1,2
r + P
θ (Qrr + 2Qr∂r) + P
Using the trivial b−1 bound for Q−1 and (6.17), (6.40) we estimate the first
term,
j=0,1,2
‖Q−1[Q,P jθ ]∂
r w‖L2 . b−1δb4τ−1
j=0,1,2
2‖Djw‖L2
This is bounded by the right hand side in (6.35) since
b2⊥ . rτ
The second term in the commutator is estimated by
‖Q−1P 0θ (Qrr + 2Qr∂r)w‖L2 . δ(‖Qrrw‖L2 + ‖Qr∂rw‖L2)
This is bounded by the right hand side in (6.35) provided that
|qrr|+ τ−
2 |qr| . b2(1 + τ−
2 r−1λ)
which follows from (6.36). The third term in the commutator is treated simi-
larly. This concludes the proof of the proposition. �
To conclude our study of the L2 Carleman estimates we need to also pay
some attention to elliptic estimates. The conjugated operator Pψ is elliptic in
the region {y2+ ξ2 ≥ 4τ}. Precisely, in this region we have the symbol bound
|Lrψ(s, y, ξ)| & y2 + ξ2
Consequently, we can improve our estimates in this region. We consider a
smooth symbol ae(y, ξ) with the following properties
supp ae ⊂ {y2 + ξ2 ≥ 8τ}
ae(y, ξ) = (y
2 + ξ2)
2 in {y2 + ξ2 ≥ 9τ}
We define the space X2 with norm
(6.41) ‖v‖2X2 = ‖v‖
+ ‖awe (y,D)v‖2
The dual space X∗2 has norm
(6.42) ‖f‖2X∗2 = inf{‖f1‖
(X02 )
∗ + ‖f2‖2; f = f1 + awe (y,D)f2}
We note that due to the elliptic bound for high frequencies, we also have the
dual bounds
(6.43) τ−
2‖bDv‖ . ‖v‖X2 , ‖∇f‖X∗2 . ‖b
Then our final L2 Carleman estimate is
Theorem 5. Assume that the coefficients of P satisfy (6.22). Let ψ be as
in (6.9) with h, φ as in Lemmas 6.2,6.3. Then the following L2 Carleman
estimate holds for all functions u for which the right hand side is finite:
(6.44) ‖eψ(s,y)u‖X2 . ‖eψ(s,y)Pu‖X∗2
Proof. We first prove the result using the stronger assumption (6.22) on the
coefficients. After conjugation we have to show that
(6.45) ‖v‖X02 . ‖Pψv‖X∗2
We consider two overlapping smooth cutoff symbols χi = χi(y
2 + ξ2) and
χe = χε(y
2+ξ2). The interior one χi is supported in {y2+ξ2 ≤ 7τ} and equals
1 in {y2 + ξ2 ≥ 6τ}. The exterior one χe is supported in {y2 + ξ2 ≥ 4τ} and
equals 1 in {y2 + ξ2 ≤ 5τ}. We need the following bounds for χi and χe:
Lemma 6.8. a) The operator χi(x,D) satisfies the bound
(6.46) ‖χi(x,D)f‖(X02 )∗ . ‖f‖X∗2
b) The operators χi(x,D) and χ
e (x,D) satisfy the following commutator
estimates:
(6.47) ‖b−1[χi(x,D), Pψ]v‖ . τ−
4‖bv‖+ ‖χev‖
(6.48) ‖[χwe , Pψ]v‖ . τ
8‖bv‖
Proof. a) By duality the bound (6.46) is equivalent to
‖χi(x,D)v‖X2 . ‖v‖X02
We have
‖ae(x,D)χi(x,D)v‖ . τ−N‖f‖
since the supports of the symbols (1 − χe(x, ξ)) and ae(x, ξ) are O(τ
2 ) sepa-
rated. Then it remains to show that
‖χi(x,D)v‖X02 . ‖v‖X02
which is fairly straightforward and is left for the reader.
b) We now consider the bound (6.47). Commute first χi with ∂s+H−h′(s).
We have
[χi(x,D), ∂s +H − h′(s)] = [χi(x,D), H ]
Since χe = 1 in the support of ∇x,ξχi and the Poisson bracket of χi and x2+ξ2
vanishes, by standard pdo calculus we obtain
‖[χi(x,D), ∂s +H − h′(s)]v‖ . ‖χev‖+ τ−N‖v‖
The difference Pψ − (∂s +H − h′(s)) can be expressed in the form
Pψ − (∂s +H − h′(s)) = ∂g∂ + τ
2 (g∂ + ∂g) + τg
where the function g satisfies the bounds
|g|+ 〈y〉|gy|+ 〈y〉|gyy| . εi
These lead to an estimate for fixed s ∈ [i, i+ 1],
‖b−1[χi(x,D), Pψ − (∂s +H − h′(s))]v‖ . εiτ
2‖〈y〉−1b−1v‖
Then (6.47) follows since
2 〈y〉−1 . τ−
Finally, the proof of the estimate (6.48) is similar but simpler.
We continue with the proof of the proposition. For the nonelliptic part we
apply (6.32) to the function χi(x,D)v which is supported in {y2 < 9τ}. This
gives
‖χi(x,D)v‖X02 . ‖χi(x,D)Pψv‖(X02 )∗ + ‖[χi(x,D), Pψ]v‖L2
For the first term on the right we use the bound (6.46) while for the second
we use (6.47). This yields
(6.49) ‖χi(x,D))v‖X02 . ‖Pψv‖X∗2 + τ
4‖bv‖+ ‖χev‖
On the other hand for the estimate in the elliptic region we compute
(6.50) 〈(χwe )2v, Pψv〉 = 〈χwe v, Lrψχwe v〉+ 〈χwe v, [χwe , P rψ]v〉
For the first term we split Lrψ into H−h′ plus a perturbation. Using pointwise
bounds for the coefficients of Pφ we obtain
|Lrψv − (H − h′)v| . δ1((τ + y2)|v|+ τ
2 |Dv|+ |D2v|)
which shows that
〈χwe v, Lrψχwe v〉 = 〈χwe v, (H − h′)χwe v〉+O(δ1〈χwe v, (H + τ)χwe v〉)
The symbol of H − h′ is elliptic in the support of χe, therefore a standard
elliptic argument yields
〈χwe v, (H + τ)χwe v〉 . 〈χwe v, (H − h′)χwe v〉+ Cτ−N‖v‖2
for a large constant C. This further gives
〈χwe v, (H + τ)χwe v〉 . 〈χwe v, Lrψχwe v〉+ Cτ−N‖v‖2
Returning to (6.50), we obtain
c〈χwe v, (H + τ)χwe v〉 ≤ −〈(χwe )2v, Pψv〉+ 〈χwe v, [χwe , P rψ]v〉+ Cτ−N‖v‖2
We use (6.48) and then the Cauchy-Schwartz inequality to obtain
〈χwe v, (H + τ)χwe v〉 . ‖(H + τ)−
2Pψv‖2 + τ−
4‖bv‖2
The first term on the right is properly controlled due to the straightforward
estimate
‖(H + τ)−
2f‖ . ‖f‖X∗2
Hence combining the above inequality with (6.49) we obtain
‖χ(ix,D))v‖X02 + ‖(H + τ)
2χwe (x,D)v‖ . ‖Pψv‖X∗2 + τ
4‖bv‖+ ‖χwe (x,D)v‖
The last two terms on the right are negligible compared to the left hand side,
therefore we obtain
(6.51) ‖v‖X02 + ‖(H + τ)
2χwe (x,D)v‖ . ‖Pψv‖X∗2
Then (6.45) follows since χe = 1 in the support of ae.
It remains to show that the assumption (6.22) on the coefficients for (6.7)
can be replaced by the weaker condition (1.15). This is a direct consequence
of (6.43) combined with the following regularization result:
Lemma 6.9. Let d be a function which satisfies (1.15). Then there is an
approximation g1 of it satisfying (6.22) so that
|g − g1| . b2τ−1
Proof. First we transfer (1.15) to the (s, y) coordinates. A short computation
yields the equivalent form
(6.52) ‖d‖L∞(Aij) + ej‖d‖Lipy(Aij) + ‖g‖Cmijt (Aij) . εij
where the new continuity modulus m̃ij is given by
m̃ij(ρ) = ρ+ e
Within Aij we regularize d in y on the δy = τ
2 scale and in s on the
δs = e
4 scale,
d1 = S
(Dy)S
(Ds)d
These localized regularizations are assembled together using a partition of unit
corresponding to Aij. In Aij we compute
|d− d1| . εij(e−jδy +mij(δs)) ≈ εij(e−jτ−
2 + e−
4 ) . ε
4 . b2τ−1
while
|∂sg1| . εij
mij(δs)
≈ εije−jτ
The bounds for higher order derivatives of g1 follow trivially due to the fre-
quency localization. �
7. Lp Carleman estimates for variable coefficient operators
The variable coefficient counterpart of Proposition 4 uses the more convex
weights constructed in Section 6. For convenience we write it in the (s, y)
coordinates. Let τ >> 1 and B(τ) be as in (1.25),(1.26) and (1.27).
We define the function space X through its norm
(7.1) ‖v‖X := ‖v‖X2 + ‖v‖l2(B(τ);L∞t L2x) + ‖v‖l2(B(τ);LptLqx)
where (p, q) is an arbitrary Strichartz pair, with X2 as defined in (6.41).
Its (pre)dual space has the norm
‖f‖X∗ = inf
f=f1+f2+f3
‖f1‖X∗2 + ‖f2‖L1L2 + ‖f3‖Lp′Lq′(7.2)
Then we have the following improvement of Theorem 5.
Theorem 6. There exists ψ as in (6.9) with h and φ as in Lemma 6.2 and
6.3. Then the following estimate holds for all compactly supported sufficiently
regular functions u.
(7.3) ‖eψu‖X . ‖eψP̃ u‖X∗
The relation between ψ and the partition B(τ) remains a bit mysterious at
this level. If we replace it by the empty partition then the statement remains
true for all ψ with h and φ as in Lemma 6.2. The same is true for a partition
into time slices of size 1. The convexity properties of φ allow a localization
to the finer partition Bij as in 6.6 (and, as we shall soon see, to an even finer
partition). q It is possible to choose φ and h so that the partition (Bij) is
finer than the one defined by B(τ). We assume in the sequel that ψ has been
chosen with these properties.
Proof. As usual this is equivalent to proving a bound from below for the con-
jugated operator,
(7.4) ‖v‖X . ‖Pψv‖X∗
The main step in the proof is to produce a parametrix for Pψ. The key
point is that the parametrix is allowed to have a fairly large L2 error. This
is because L2 errors can be handled by Theorem 5. The advantage in having
a large L2 error is that it permits to localize the parametrix construction to
relatively small sets, on which we can freeze the coefficients and eventually
reduce the problem to the case of the Hermite operator. The properties of the
parametrix are summarized in the following
Proposition 7.1. a) Under the assumptions of the theorem there exists a
parametrix T for Pψ with the following properties:
(7.5) ‖Tf‖X . ‖f‖X∗
(7.6) ‖PψTf − f‖X∗2 . ‖f‖X∗
b) The same result holds with Pψ replaced by P
We first use the proposition to conclude the proof of the Theorem. Let
Pψv = f + g, ‖f‖X∗2 + ‖g‖L1L2+Lp′Lq′ ≈ ‖Pψv‖X∗
With T as in part (a) of the proposition we set
w = v − Tg, Pψw = f + g − PψTg
By (7.5) we can bound Tg in X , therefore it suffices to bound w in X . On the
other hand by (7.6) we obtain
‖Pψw‖X∗2 . ‖f‖X∗2 + ‖g − PψTg‖X∗2 . ‖Pψv‖X∗
It remains to show that
‖w‖X . ‖Pψw‖X∗2
By Theorem 5 we can estimate the X2 norm of w and replace this with the
weaker bound
(7.7) ‖w‖L∞L2∩LpLq . ‖w‖X2 + ‖Pψw‖X∗2
This is proved using a duality argument and the parametrix T for P ∗ψ given
by part (b) of the proposition. For f ∈ X∗ we write
〈w, f〉 =〈w, P ∗ψTf〉+ 〈w, f − P ∗ψTf〉
=〈Pψw, Tf〉+ 〈w, f − P ∗ψTf〉
Using both (7.5) and (7.6) with Pψ replaced by P
ψ we obtain
|〈w, f〉| . (‖Pψw‖X∗2 + ‖w‖X2)‖f‖X∗
and (7.7) follows. This concludes the proof of Theorem 6. �
It remains to prove the Proposition 7.1.
Proof of Proposition 7.1. The strategy for the proof is simple: On sufficiently
small sets we can approximate the problem by one with constant coefficients
and the properties of the parametrix follow from Section 4. We use a partition
of unity to construct a global parametrix from local ones. We obtain L2 errors
(1) Commuting cutoff functions with the operator. Hence the partition
has to be sufficiently coarse.
(2) Approximating the variable coefficient operator by constant coefficient
operators. Hence the partition has to be sufficiently fine.
To elaborate on this we define the notion of a local parametrix:
Definition 7.2 (Local parametrix). Given a convex set B we call T a (B-)
local parametrix for Pψ if for all f supported in B
(7.8) ‖Tf‖X . ‖f‖X∗ ,
(7.9) ‖PψTf − f‖X∗2 . ‖f‖X∗
and Tf is supported in 2B.
If T is a parametrix and η is supported on 2B, η = 1 on B then ψT is a
local parametrix, but with constants depending on the commutator of Pψ and
η. Vice verse, if (Bj) is a covering, (ηj) a subordinate partition of 1 and Tj
are local parametrices then
is a global parametrix, because (7.8) is obtained by summation, and
Tj(ηjf)− f =
(PψTjηjf − ηjf
provided ∑
‖ηjf‖2X∗ . ‖f‖2X∗
and its adjoint
‖u‖2X .
‖ηju‖2X .
This is obvious for the L2 part and has to be checked for the other part. This
strategy of constructing local parametrices leads, if it is possible, to estimates
which are stronger than in Proposition 7.1 and Theorem 6, because we may
replace the function space X by l2X(Bj) respectively. l
2X∗(Bj).
In the first part of the proof we study the localization, and in the second
part we provide the local parametrices.
7.1. Localization scales. Here we introduce a localization scale which is finer
than the Bij partition of the space, and show that it suffices to construct the
parametrix in each of these smaller sets. Precisely, the sets Bkij introduced
below are the smallest sets to which one can localize the L2 estimates for the
operator Pψ. The choice of their size is not yet apparent at this point, but
will become clear in the very last step of the proof, where we estimate the
commutator of Pψ with cutoff functions on such sets. We consider three cases
depending on the size of εi.
(1) If εi ≤ τ−1 then we use Bi0 as it is.
(2) If τ−1 ≤ εi ≤ τ−
2 then we partition the set Bi0 into time slices B
i0 of
thickness
δs = b−2i0 .
(3) If τ−1 ≤ εi and j 6= 0 then we partition Bij into subsets Bkij which have
the time scale, radial scale and angular scale given by
δs = b−2ij , δy = τ
2 b−2ij , δy
⊥ = τ
2 b−2ij,⊥
This gives a decomposition of the space
R× Rn =
We also consider a subordinated partition of unity
Suppose that in each set Bkij we have a parametrix T
ij satisfying (7.5) and
(7.6). Then we define the global parametrix T by
T kijχ
We have by an iterated application of Minkowski’s inequality
‖χkijf‖l2(L1L2+Lp′Lq′) . ‖f‖L1L2+Lp′Lq′
and the dual bound
T kijχ
ijf‖L∞L2∩LpLq . ‖T kijχkijf‖l2(L∞L2∩LpLq).
Hence (7.5) for T would follow if we proved that
vkij‖X2 . ‖vkij‖l2X2
where vkij = T
ijf are supported in B
ij . This is trivial by orthogonality for
the L2 component of the X2 norm. It is also straightforward for the elliptic
part since the kernel of the operator in the following sense: Let χ0 ∈ C∞(R)
be supported in [−3/2, 3/2], identically 1 in [−1, 1]. We define χe(x, ξ) =
χ0(5− (x2 + ξ2)/τ 2). Then
χwe v
ij‖X2 . ‖χwe vkij‖l2X2
and the adjoint estimate holds since the kernel of χwe is rapidly decreasing
beyond the τ−
2 scale, which is much smaller than the smallest possible spatial
size for Bkij, namely δy & τ
It remains to consider the angular part of the X2 norm, which is best de-
scribed using the spherical multiplier Q appearing in Proposition 6.7. The
symbol of Q is smooth with respect to λ on the τ
2 rb2b−2⊥ scale therefore its
kernel is rapidly decaying on the angular scale δθ = τ−
2 r−1b−2b2⊥ which cor-
responds to δy⊥ = τ
2 b2⊥b
−2. But by (3) and (6.5) this can be no larger than
8 which is again much smaller than the smallest possible spatial size for Bkij .
Thus orthogonality arguments still apply.
Then we have
[P0, χ
ij]u = 2χijy uy + χ
yyu+ χ
We claim that the right hand side is negligible in the estimate. For this it
suffices to verify that
|χijy | ≪ τ−
2 b2ij , |χijyy| ≪ b2ij , |χijs | ≪ b2ij
The last relation is trivial. For the first two we consider three cases.
(1) If j = 0 and εj < Cτ
−1 then we need no spatial truncation. We are
allowed to truncate at |y| > Cτ 12 to separate the elliptic region, though.
(2) If j = 0 and εj > Cτ
−1 then
|χijy | . e−j(i), |χijyy| . e−2j(i)
while
b4i0 = a
ij(i) = εij(i)τ
2 e−j(i) ≈ Ce−2j(i)τ
(3) Otherwise,
|χijy | . e−j , |χijyy| . e−2j
while
b4ij = a
ij = εijτ
2 e−j & Ce−2j(i)τ
The results can be summarized by saying that it suffices to construct local
parametrices in the sets Bkij.
7.2. Freezing coefficients. Our first observation is that restricting the result
in the proposition to a single region Bkij allows us to freeze the weights b, b⊥
in the X2 norms.
Next we are interested in freezing the coefficients of P . We consider the
same three cases as above:
The case εi ≤ τ−1. In this case we are localized to
Bi0 = [i, i+ 1]× Rn
and we have
bi0 ≈ 1, εij . τ−1
By (6.2) the second relation leads to
|d| . τ−1
Then using also (6.43) we can estimate the terms involving d in the expression
(6.22) for Pψ,
(7.10) ‖∂d∂v‖X∗2 + τ‖dv‖X∗2 + τ
2‖(d∂ + ∂d)v‖X∗2 . ‖v‖X2
Hence without any restriction of generality we can assume that d = 0 in Pψ,
which corresponds to taking g = In.
We also observe that in this case we have
|φ| . 1, |φy| . τ−
Then we can also drop the φ component of ψ. Finally, since
|hss| . 1
we can replace h by its linearization at some point in the corresponding s
region.
Conclusion: It suffices to prove the result when d = 0, ψ(y, s) = τs.
We note that the separation of τ from integers is no longer needed due to
the localization to unit s intervals.
The case τ−1 ≤ εi. In this case we are localized to a region of the form
Bki0 = [s0, s0 + e
j(i)τ−
2 ]×B(0, ej(i))
and we have
b4i0 ≈ τe−2j(i), εij . e−j(i)τ−
The second relation leads to
|d| . e−j(i)τ−
Then (7.10) is still valid, so we can assume again that d = 0 in Pψ.
We also observe that in this case we have
|φ| . 1, |φy| . e−j(i)
Then we can also drop the φ component of ψ. Finally, since
|hss| . e−j(i)τ−
we can replace h by its linearization at some point in the corresponding s
region.
Conclusion: It suffices to prove the result when d = 0, ψ(y, s) = τs.
The case τ−1 ≤ εi. In this case we are localized to a region of the form
Bkij = [s0, s0 + τ
ij ]× B(y0, τ−
ij ), |y0| ≈ ej
and we have
b2ij ≈ ε
Using (6.2) it follows that
|g(s, y)− g(s0, y0)| . ε
Arguing as before, this allows us to freeze d within Bkij. However, we note that
we are no longer allowed to replace d by 0.
Next we turn our attention to the weight function ψ. First we have
|hss| . εjτ . εijτ
2 e−j
which allows us to replace h by its linearization in s at s0.
Secondly, we claim that we can replace φ by its linearization at y0. In the
radial direction we have weaker localization but a stronger bound
|φrr| . εijτ
2 e−j
In the transversal direction we have better localization but a weaker bound,
|φyy| . εiτ
2 e−j .
The first bound allows us to obtain the relation
|φ2y(y, s)− φ2y(y0, s0)| . ε
2 ≈ b2ij
Using also the second bound we can write
(φy(y, s)− φy(y0, s0))∂y = νr∂r + ν⊥∂⊥
where the coefficients νr and ν⊥ are smooth on the B
ij scale and satisfy the
bounds
|νr| . τ−
2 b2ij , |νr| . τ−
2 b2ij,⊥
Conclusion: It suffices to prove the result when d = g(s0, y0), ψ(y, s) =
τs + cy, |c| . εiτ
Additional simplification in the highly localized case. Given the above simpli-
fications we need to work with a constant coefficient operator Pψ which has
the form
Pψ = −∂t +H − τ + ∂d∂ + c∂, |d| . εij, |c| ≤ εi
We diagonalize the second order part with a linear change of variables to obtain
Pψ = −∂t +H − τ + c∂ +O(εij)y2
We can freeze the last term at y0 and add it into τ . To deal with c we make
the change of variable
y → y − (s− s0)c
Then our operator becomes
P̃ψ = −∂t +∆− (y − c(s− s0))2 + τ
and the s− s0 terms are negligible due to the s localization.
Conclusion: We can assume without any restriction in generality that g = In
and ψ = τs.
7.3. The localized parametrix. We begin with the global parametrix K
constructed in Section 5. Then we define the parametrix TB in B by
TB = χ2BK, B = B
and show that it satisfies (7.8) and (7.9).
The Lp part of (7.8) follows directly from (5.4). It remains to prove the X2
part,
‖TBf‖X2 . ‖f‖L1L2+Lp′Lq′
The elliptic part of the X2 bound, namely
‖awe (x,D)χ2BKf‖L2 . ‖f‖L1L2+Lp′Lq′ ,
is obtained by an argument which is similar to the one beginning with (6.50).
For the rest we consider two cases.
i) If j = 0 then B is a ball, and we can use (5.4) directly with R = d.
ii) If j > 0 then B is contained in a sector B ⊂ BR,d but may be shorter
than R. This is why we can use (5.4) for the angular part of the X2 norm, but
not for the L2 part. However, the L2 part can be always obtained by taking
advantage of the time localization,
‖bijTBf‖L2 . ‖TBf‖L∞L2 . ‖f‖L1L2+Lp′Lq′
It remains to consider the error estimate (7.9). We have
f − (∂s −H + τ)TBf = [χ2B, ∂s −H + τ ]Kf = [χ2B, ∂s −H + τ ]χ4BKf
But arguing as above χ4BK0f satisfies the same X2 bound as χ2BK0f . Hence
it suffices to show that
[χ2B , ∂s −H + τ ] : X2 → X∗2
This is where the dimensions of the set B are essential; they are chosen to be
minimal so that the above property holds. We have
[χ2B, ∂s −H + τ ] = −∂sχ2B + (∂yχ2B)∂y + ∂y(∂yχ2B)
= −∂sχ2B + (∂rχ2Br)∂r + ∂r(∂rχ2Br) + (∂⊥χ2By)∂⊥ + ∂⊥(∂⊥χ2B)
For the first factor we use the bound
|∂sχ2B| . b2ij
For the radial derivatives of χ2B we combine (6.43) with
|∂rχ2B| . τ−
2 b2ij
Finally, for the angular derivatives we use the angular H
2 norm in X2 and the
bound
|∂rχ2B| . τ−
2 b2ij⊥
8. The gradient term
In this section we consider the full problem, i.e. involving also the gradient
potential W . Ideally one might want to have a stronger version of Theorem 6
which includes additional bounds for the gradient, more precisely for
‖eψ(s,y)∇u‖L2
But such bounds cannot hold, for this would imply that one can improve the
Lp indices in a restriction type theorem. To overcome this difficulty we proceed
as in [17], using Wolff’s osculation Lemma. Wolff’s idea is that by varying the
weight one can ensure concentration in a sufficiently small set, in which the
gradient potential term is only as strong as the potential term. Thus we still
obtain a one parameter family of Carleman estimates, but with the weight
depending not only on the parameter but also on the function we apply the
estimate to.
Given a gradient potential W satisfying (1.19), we first readjust the param-
eters εij, εi constructed in Lemma 6.1 in order to insure that we have the
additional condition
‖W‖Ln+2(Aτi ) ≪ εi
Then we begin with the spherically symmetric weights ψ constructed in Sec-
tion 6 and modify them as follows:
(8.1) Ψ(s, y) = ψ(s, y) + δk(s, y)
where the perturbation k is supported in {|y| ≤ 9τ} and is subject to the
following conditions:
(8.2) |∂αs ∂βy ∂
⊥k(s, y)| . εiτ
2 s ∈ [i, i+ 1]
Here δ is a sufficiently small parameter.
In order to prove the strong unique continuation result in the presence of
the gradient potential W we need the following modification of Theorem (6):
Theorem 7. Assume that (1.15) holds. Then for each τ > 0 and W subject
‖W‖Ln+2(Aτi ) ≤ εi
and each function u vanishing of infinite order at (0, 0) and ∞ there exists a
perturbation k as in (8.2) so that
(8.3) ‖eΨu‖X+‖eΨW∇u‖X∗+‖eΨ∇(Wu)‖X∗+τ
2‖eΨWu‖X∗ . ‖eΨ(x)P̃ u‖X∗
Here and in the sequel we will omit indices for W . After returning to the
(x, t) coordinates and taking (1.19) into account this implies Theorem 3. The
reader should note that the choice of φ depends on both u and W . This is
essential since for fixed φ (8.3) cannot hold uniformly for all u and W .
Proof. Up to a point the proof follows the steps which were discussed in detail
before. We outline the main steps:
STEP 1: Show that the L2 Carleman estimate (6.23) holds with ψ replaced
by Ψ for all perturbations k as in (8.2). The new conjugated operator PΨ is
obtained from Pψ after conjugating with respect to the weight e
k(y,s). This
adds a few extra components to the selfadjoint and skewadjoint parts,
LrΨ = L
ψ + k
y + ks + 2ky(ψy + d)
LiΨ = L
ψ − ky(1 + d)∂ − ∂ky(1 + d)
Observing that we can write
k2y + ks + 2ky(ψy + d) = τd, ky(1 + d) = τ
with d as in (6.22) we conclude that the conjugated operator PΨ retains the
same form as Pψ, therefore the proof of (6.23) rests unchanged.
STEP 2: Show that the symmetric L2 Carleman estimate (6.23) holds with
ψ replaced by Ψ for all perturbations k as in (8.2). Since PΨ has the same
form as Pψ, this argument is identical.
STEP 3: Show that the symmetric mixed L2 ∩ Lp Carleman estimate in
Theorem 6 holds with ψ replaced by Ψ for all perturbations k as in (8.2).
Since PΨ has the same form as Pψ, this argument is also identical.
STEP 4: Decompose W into a low and a high Hermite-frequency part,
W =Wlow +Whigh, Wlow = χ
i (x,D)W
where the smooth symbol χ1i (x, ξ) is supported in {x2 + ξ2 ≤ 81τ} and equals
1 in the region {x2 + ξ2 ≤ 64τ}. Then we show that the high frequency part
of W satisfies the desired estimates for all perturbations k as in (8.2), namely
(8.4) ‖eΨWhigh∇v‖X∗ +‖∇Whighv‖X∗ + τ
2‖eΨWhighv‖X∗ . ‖eΨu‖X‖W‖Ln+2
After conjugation this becomes
‖Whigh∇v‖X∗ + ‖∇Whighv‖X∗ + τ
2‖Whighv‖X∗ . ‖v‖X‖W‖Ln+2
We only consider the first term on the left. The second one is equivalent
by duality, and the third one is similar but simpler. We divide v into two
components,
v = (1− χ1e)v + χ1ev
where the smooth symbol χ1e(x, ξ) is supported in {x2 + ξ2 ≥ 9τ} and equals
1 in the region {x2 + ξ2 ≥ 10τ}.
For the high frequency component of v we use the H1 part of the X2 norm
to estimate
‖Whigh∇χ1ev‖
2(n+2)
. ‖Whigh‖Ln+2‖∇χ1ev‖L2 . ‖W‖Ln+2‖v‖X
For the low frequency component of v it is still possible to estimate directly
the high frequency of the output,
‖χ1e(Whigh∇(1− χ1e)v)‖H−1 .τ−
2‖Whigh∇(1− χ1e)v‖L2
.‖Whigh‖Ln+2τ−
2‖∇(1− χ1e)v‖
2(n+2)
.‖W‖Ln+2‖v‖
2(n+2)
Finally, the last remaining part has a much better L2 estimate,
‖(1− χ1e)(Whigh∇(1− χ1e)v)‖ . τ−N‖W‖Ln+2‖v‖
which is due to the unbalanced frequency localizations of the two factors.
Due to the estimate (8.4), it suffices to prove (8.3) withW replaced byWlow.
This allows us to replace the term ∇(Wlowv) by
∇(Wlowv) =Wlow∇v + (∇Wlow)v
where we can estimate
‖∇Wlow‖Ln+2 . τ
2‖W‖Ln+2
Hence without any restriction in generality we can drop the third term in (8.3)
and show that we can choose the perturbation k so that
(8.5) ‖eΨW∇u‖X∗ + τ
2‖eΨWu‖X∗ . ‖eΨ(x)P̃ u‖X∗
STEP 5: Show that, given u and W , we can choose the perturbation k
so that (8.5) holds. At this stage we no longer need the full X∗ norm for
the W terms, it suffices instead to consider the L
2(n+2)
n+4 norm. Begin with the
unperturbed integral
Fψdxdt, Fψ = |eψW∇u|
2(n+2)
n+4 + |τ
2 eψWu|
2(n+2)
We can select a subset I of R consisting of time intervals of length 1 with unit
separation at least 8 so that
Fψdxdt .
Fψdxdt
By a small abuse of notation we label
Ii, Ii ⊂ [i− 1, i+ 1]
We define a family of perturbations k depending on parameters bi, σi by
k(y, s) =
100χ3τIi + χ2Iiχ|y|2≤τ (biy + σi(s− i))
|bi| ≤ τ
2 , |σi| ≤ τ
Due to the choice of the intervals Ii it is easy to see that after changing the
weight ψ to Ψ we retain the concentration to a dilate of I,
FΨdxdt .
FΨdxdt
The choice of the parameters bi, σi can be made independently for each i.
We consider two cases.
i) Suppose εi . τ
2 . Then the choice of the parameters is irrelevant since
in 3Ii we can estimate
‖eψW∇u‖
2(n+2)
2‖eψWu‖
2(n+2)
. ‖W‖Ln+2(‖eΨ∇u‖L2 + τ
2‖eΨu‖L2)
. (τ−1/2‖eΨ∇u‖L2 + ‖eΨu‖L2)
. ‖u‖X2
ii) Suppose εi ≫ τ−
2 . Then we need to choose the parameters bi, σi in a
favorable manner. This choice is made using Wolff’s Lemma:
Lemma 8.1 (Wolff’s Lemma [29]). Let µ be a measure in Rn and B a convex
set. Then one can find bk ∈ B and disjoint convex sets Ek ⊂ Rn so that the
measures exbkµ are concentrated in Ek,∫
exbkdµ &
exbkdµ
and ∑
|Ek|−1 & |B|
We apply the lemma for the measures
dµi = 13IiFψ
In our case we have
Bi = δεi
[−τ, τ ]×B(0, τ
, |Bi| ≈ εn+1i τ
Hence we can find parameters bki and σ
i and convex sets E
i ⊂ 3Ii×B(0, 3τ
so that the corresponding measures FΨk are concentrated in E
i with
|Eki |−1 & εn+1i τ
At the same time we have
|W |n+2dxdt . εn+2i
Hence we can choose k so that∫
|W |n+2dxdt . εiτ−
2 |Eki |−1
which by Holder’s inequality leads to
(8.6) ‖W‖
2 (Eki )
Denoting this index k by k(i) we can write
‖eΨW (∇, τ
2 )u‖
2(n+2)
. ‖eΨW (∇, τ
2 )u‖
2(n+2)
n+4 (Ii)
. ‖eΨW (∇, τ
2 )u‖
2(n+2)
n+4 (E
. ‖W (∇, τ
2 )(eΨu)‖
2(n+2)
n+4 (E
Decomposing the function v = eΨu into low and high frequencies we further
estimate
‖eΨW (∇, τ
2 )u‖
2(n+2)
. ‖W (∇, τ
2 )(1− χ1i (x,D))(eΨu)‖
2(n+2)
n+4 (Ii)
+ ‖W (∇, τ
2 )χ1i (x,D)(e
2(n+2)
n+4 (E
The first term on the right is estimated as in Step 4,
‖W (∇, τ
2 )(1− χ1i (x,D))(eΨu)‖
2(n+2)
n+4 (Ii)
. ‖W‖l∞Ln+2‖(1− χ1i (x,D))(eΨu)‖H1
. ‖W‖l∞Ln+2‖eΨu‖X
It is only for the second term on the right that we need to use (8.6):
‖W (∇, τ
2 )χ1i (x,D)(e
2(n+2)
n+4 (E
. ‖W‖
l∞i L
‖(∇, τ
2 )χ1i (x,D)(e
2(n+2)
n+4 (E
. ‖eΨu‖
2(n+2)
. ‖eΨu‖X
The proof of the Theorem is concluded. �
Appendix A. The change of coordinates
Suppose that the coefficients g satisfy (1.15). In this section we verify that
we can change coordinates so that (1.15) and (1.16) are both satisfied. Due
to the anisotropic character of the equation we must leave the time variable
unchanged and consider changes of coordinates which have the form
(t, x) → (s, y), s = t, y = χ(t, x).
The expression for the operator P in the new coordinates is
P = ∂t + ∂kg̃
kl(t, y)∂l +
d̃kl∂l
where the new coefficients g̃, d̃ are computed using the chain rule,
g̃kl =
d̃kl =
Dχ−1lm(∂
There is a price to pay for this, namely in the new coordinates we obtain
lower order terms which cannot be treated perturbatively. Instead we obtain
coefficients d̃k which have the same regularity and size as g1 − In.
The Lipschitz condition (1.15) ensures that g has a limit at (0, 0) so we
assume that g is continuous. After a linear change of coordinates we may and
do choose g with g(0, 0) = In. Again by (1.15) this implies
(A.1) |g(t, x)− In| ≪ 1.
Proposition A.1. Let g be a metric which satisfies (1.15) with g(0, 0) = In.
Then there is change of coordinates (t, y) = (t, χ(t, x)) which is close to the
identity
(A.2) ‖∂xχ− In‖L∞ ≪ 1
and has regularity
(A.3)
‖(t+ x2)−1/2(t∂t)α((t+ x2)1/2∂x)βχ‖l1(A(τ);L∞) ≪ 1, 2 ≤ 2α + |β| ≤ 4
so that in the new coordinates both functions g̃ and d̃ satisfy (1.15), while g̃−In
and d̃ satisfy (1.16).
Proof. Consider the covering of the [0, 2]× B(0, 2) = ∪Aij with an associated
smooth partition of unity ηij . We can assume that the functions ηij satisfy
(A.4) |∂αt ∂βxηij | . cαβt−α(t + x2)−
We choose the points
(ti, xij) = (e
−4i, e−2i+j) ∈ Aij.
and insure that ηij = 1 near (ti, xij). By (1.15) we have
(A.5) sup
(i,j)∈A(τ)
|g(ti, xij)−g(ti, xi,(j−1))|+ |g(ti, xij)−g(ti+1, x(i+1),j)| ≪ 1.
Within a fixed set Aij we consider the linear map defined by the matrix
χij = g
−1/2(ti, xij).
It transforms the coefficients at (ti, xij) to the identity and has the desired
properties within Aij . We assemble the maps defined by χij using the partition
of unity,
χ(t, x) =
ηij(t, x)χijx.
∇χ(t, x)− In =
(∇ηij)χijx+ ηij(χij − In),
Let (t, x) ∈ Ai0,j0. Since
∇ηij = 0 we have
∇xχ(t, x)− In =
ηij(t,x)>0
∇xηij(t, x)(χij − χi0,j0) + ηij(χij − In).
The first term on the right hand is small by (A.5) (for χij) and the second one
by (A.1) therefore the smallness of ∇χ− In follows.
For the second order spatial derivatives we write
D2xχ(t, x) =
D2ηij(t, x)χijx+ 2Dxηij(t, x)χij
D2xηij(t, x)(χij − χi0,j0)x+ 2Dxηij(χij − χi0,j0).
Hence by (A.4) and (A.5) (again for χij) we obtain
‖(|x|2 + t)1/2D2xχ‖l1(A(τ);L∞) ≪ 1.
∂tχ(t, x) =
∂tηij(χij − χi0j0)x
gives the desired bound for the time derivative. A similar computation yields
the bound for the higher order derivatives in (A.3).
Consider now the new metric g̃. Since both Dχ and (Dχ)−1 are Lipschitz on
the dyadic scale with l1(A(τ)) summability, from (1.15) for g we easily obtain
(1.15) for g̃. In addition, our construction insures that g̃(ti, xij) = In. This in
turn leads to the bound
‖g̃ − In‖L∞(Aij) . ‖g̃‖Lipx(Aij) + ‖g̃‖Cmijt (Aij)
which shows that (1.15) for g̃ implies (1.16) for g̃ − In.
It remains to consider the lower order terms. From ∂tχ we obtain coefficients
d̃ of the form
d̃ = t
∂tηijχij
Within Ai0,j0 this gives
d̃ = t
∂tηij(t, x)(χij − χi0j0)
The functions t∂tηij(t, x) are bounded and smooth on the dyadic scale, while
the l1(A(τ)) summability comes from the χij−χi0j0 factor due to (A.5). Hence
both (1.15) and (1.16) are satisfied.
The contribution of ∂2xχ to d̃ has the form
(∂xχ)
−1(∂2xχ)g
There is no singularity at x = 0 since χ is linear in x for x2 ≪ t. Then from
(A.2) and (A.3) we obtain
|d̃| .
with added l1(A(τ)) summability inherited from ∂2xχ. This is better than
(1.16), and in effect this term can be included inW and treated perturbatively.
The bound (1.16) is also easy to obtain from the similar bounds for g and
derivatives of χ.
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Mathematisches Institut der Universität Bonn, Beringstr.1, 53115 Bonn,
Germany
E-mail address : koch@math.uni-bonn.de
Department of Mathematics, University of California, Berkeley, CA 94720
E-mail address : tataru@math.berkeley.edu
	1. Introduction
	2. Proof of Theorem 1 and 2 
	3.  L2 bounds in the flat case and the Hermite operator
	4. Resolvent bounds for the Hermite operator 
	4.1. Weighted L2 bounds
	4.2. The Lp bounds of the resolvent
	4.3. Combining the estimates
	5. Lp estimates in the flat case and parametrix bounds
	6.  Modified weights and pseudoconvexity
	7. Lp Carleman estimates for variable coefficient operators
	7.1. Localization scales.
	7.2. Freezing coefficients
	7.3. The localized parametrix.
	8. The gradient term
	Appendix A. The change of coordinates
	References
