bluuebunny/crossref_metadata_2025_split huggingface dataset:
One-dimensional basic sets in the three-sphere
Authors: Gibbons Joel C. 
Published: July 2010

This paper is a continuation of Williams’ classification of one-dimensional attracting sets of a diffeomorphism on a compact manifold [Topology 6 (1967)]. After defining the knot presentation of a solenoid in

S
3

{S^3}

and some knottheoretic preliminaries, we prove Theorem: If

∑
1

,

h
1

{\sum _1},{h_1}

and

∑
2

,

h
2

{\sum _2},{h_2}

are shift classes of oriented solenoids admitting elementary presentations K,

K
,

g
1

K,{g_1}

and K,

K
,

g
2

K,{g_2}

, resp., where

g
1

∗

=

(

g
2

∗

)
t

:

H
1

(
K
)
→

H
1

(
K
)

{g_1}^ \ast = {({g_2}^ \ast )^t}:{H_1}(K) \to {H_1}(K)

, there is an Anosov-Smale diffeomorphism f of

S
3

{S^3}

such that

Ω
(
f
)

\Omega (f)

consists of a source

Λ
−

{\Lambda ^ - }

and a sink

Λ
+

{\Lambda ^ + }

for which

Λ
+

,
f

/

Λ
+

{\Lambda ^ + },f/{\Lambda ^ + }

and

Λ
−

,

f

−
1

/

Λ
−

{\Lambda ^ - },{f^{ - 1}}/{\Lambda ^ - }

are conjugate, resp., to

∑
1

,

h
1

{\sum _1},{h_1}

and

∑
2

,

h
2

{\sum _2},{h_2}

. (The author has proved [Proc. Amer. Math. Soc., to appear] that if f is an Anosov-Smale map of

S
3

,
Ω
(
f
)

{S^3},\Omega (f)

has dimension one, and contains no hyperbolic sets, then f has the above structure.) We also prove Theorem: there is a nonempty

C
1

{C^1}

-open set

F
2

{F_2}

in the class of such diffeomorphisms for which

K
=

S
1

K = {S^1}

and

g
1

=

g
2

{g_1} = {g_2}

is the double covering such that each f in

F
2

{F_2}

defines a loop t in

S
3

{S^3}

, stable up to

C
1

{C^1}

perturbations, for which at every x in t the generalized stable and unstable manifolds through x are tangent at x.
Stability theorems for some functional equations
Authors: MacCamy R. C. , Wong J. S. W. 
Published: July 2010

Functional-differential equations of the form
\[

u
˙

(
t
)
=
−

∫
0
t

A
(
t
−
τ
)
g
(
u
(
τ
)
)
d
τ
+
f
(
t
,
u
(
t
)
)

\dot u(t) = - \int _0^t {A(t - \tau )g(u(\tau ))d\tau + f(t,u(t))}

\]
are considered. Here

u
(
t
)

u(t)

is to be an element of a Hilbert space

H

,
A
(
t
)

\mathcal {H},A(t)

a family of bounded symmetric operators on

H

\mathcal {H}

and g an operator with domain in

H

\mathcal {H}

. g may be unbounded. A is called strongly positive if there exists a semigroup exp St, where S is symmetric and

(
S
ξ
,
ξ
)
≦
−
m

‖
ξ
‖

2

,
m
>
0

(S\xi ,\xi ) \leqq - m{\left \| \xi \right \|^2},m > 0

, such that

A
∗

=
A
−
exp

{A^ \ast } = A - \exp

St is positive, that is,
\[

∫
0
T

(

v
(
t
)
,

∫
0
t

A
∗

(
t
−
τ
)
v
(
τ
)

)

d
τ
≧
0
,

\int \nolimits _0^T \left ( {v(t),\int _0^t {{A^\ast }(t - \tau )v(\tau )} } \right )d\tau \geqq 0,

\]
for all smooth

v
(
t
)

v(t)

. It is shown that if A is strongly positive, and g and f are suitably restricted, then any solution which is weakly bounded and uniformly continuous must tend weakly to zero. Examples are given of both ordinary and partial differential-functional equations.
Iterated fine limits and iterated nontangential limits
Authors: Gowrisankaran Kohur 
Published: July 2010

Let

Ω
k

,
k
=
1

to

n

{\Omega _k},k = 1{\text { to }}n

, be harmonic spaces of Brelot and

u
k

>
0

{u_k} > 0

harmonic functions on

Ω
k

{\Omega _k}

. For each

w
w

in a class of multiply superharmonic functions it is shown that the iterated fine limits of

[
w

/

u
1

⋯

u
n

]

[w/{u_1} \cdots {u_n}]

exist up to a set of measure zero for the product of the canonical measures corresponding to

u
k

{u_k}

and are independent of the order of iteration. This class contains all positive multiply harmonic functions on the product of

Ω
k

{\Omega _k}

’s. For a holomorphic function

f
f

in the Nevanlinna class of the polydisc

U
n

{U^n}

, it is shown that the

n
n

th iterated fine limits exist and equal almost everywhere on

T
n

{T^n}

the

n
n

th iterated nontangential limits of

f
f

, for any fixed order of iteration. It is then deduced that, with the exception of a set of measure zero on

T
n

{T^n}

, the absolute values of the different iterated limits of

f
f

are equal. It is also shown that the

n
n

th iterated nontangential limits are equal almost everywhere on

T
n

{T^n}

for any

f
f

in

N
1

(

U
n

)

{N_1}({U^n})

.
Summability of Jacobi series
Authors: Askey Richard 
Published: July 2010

The positivity of some Cesàro mean is proven for Jacobi series

Σ

a
n

P
n

(
α
,
β
)

(
x
)
,
α
,
β
≧
−

1
2

\Sigma {a_n}P_n^{(\alpha ,\beta )}(x),\alpha ,\beta \geqq - \tfrac {1}{2}

. This has applications to the mean convergence of Lagrange interpolation at the zeros of Jacobi polynomials. The positivity of the

(
C
,
α
+
β
+
2
)

(C,\alpha + \beta + 2)

means is conjectured and proven for some

(
α
,
β
)

(\alpha ,\beta )

. One consequence of this conjecture would be the complete monotonicity of

x

−
c

(

x
2

+
1

)

−
c

,
c
≧
1

{x^{ - c}}{({x^2} + 1)^{ - c}},c \geqq 1

.
Oscillation, continuation, and uniqueness of solutions of retarded differential equations
Authors: Burton T. , Grimmer R. 
Published: July 2010

In this paper we present a number of results on continuation and uniqueńess of solutions of the n-dimensional system

x
′

(
t
)
=
f
(
t
,
x
(
t
)
)
+
g
(
t
,
x
(
t
−
τ
(
t
)
)
)

x’(t) = f(t,x(t)) + g(t,x(t - \tau (t)))

for

τ
(
t
)
≥
0

\tau (t) \geq 0

. We then give some necessary, some sufficient, and some necessary and sufficient conditions for oscillation of solutions of the second order equation

x
+
a
(
t
)
f
(
x
(
t
−
τ
(
t
)
)
)
=
0

x + a(t)f(x(t - \tau (t))) = 0

.
Completely seminormal operators with boundary eigenvalues
Authors: Clancey Kevin 
Published: July 2010

For

f
∈

L
2

(
E
)

f \in {L^2}(E)

we consider the singular integral operator

T
E

f
(
s
)
=
s
f
(
s
)
+

π

−
1

∫
E

f
(
t
)

(
t
−
s

)

−
1

d
t

{T_E}f(s) = sf(s) + {\pi ^{ - 1}}{\smallint _E}f(t){(t - s)^{ - 1}}dt

. These singular integral operators are a special case of operators acting on a Hilbert space with one dimensional self-commutator. We discover generalized eigenfunctions of the equation

T
E

f
=
0

{T_E}f = 0

and, for

p
>
2

p > 2

, we will give an

L
p

(
E
)

{L^p}(E)

solution of the equation

T
E

f
=

χ
E

{T_E}f = {\chi _E}

. The main result of the paper is an example of a nonzero

L
2

(
E
)

{L^2}(E)

solution of

T
E

f
=
0

{T_E}f = 0

, with

λ
=
0

\lambda = 0

a boundary point of the spectrum of

T
E

{T_E}

.
On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities
Authors: Landman Alan 
Published: July 2010

We consider a holomorphic family

{

V
t

}

t
∈
D

{\{ {V_t}\} _{t \in D}}

of projective algebraic varieties

V
t

{V_t}

parametrized by the unit disc

D
=
{
t
∈

C

:

|

t

|

>
1
}

D = \{ t \in {\mathbf {C}}:|t| > 1\}

and where

V
t

{V_t}

is smooth for

t
≠
0

t \ne 0

but

V
0

{V_0}

may have arbitrary singularities. Displacement of cycles around a path

t
=

t
0

e

i
θ

(
0
⩽
θ
⩽
2
π
)

t = {t_0}{e^{i\theta }}(0 \leqslant \theta \leqslant 2\pi )

leads to the Picard-Lefschetz transformation

T
:

H
∗

(

V

t
0

,

Z

)
→

H
∗

(

V

t
0

,

Z

)

T:{H_\ast }({V_{{t_0}}},{\mathbf {Z}}) \to {H_\ast }({V_{{t_0}}},{\mathbf {Z}})

on the homology of a smooth

V

t
0

{V_{t0}}

. We prove that the eigenvalues of

T
T

are roots of unity and obtain an estimate on the elementary divisors of

T
T

. Moreover, we give a global inductive procedure for calculating

T
T

in specific examples, several of which are worked out to illustrate the method.
Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group
Authors: Granirer Edmond E. 
Published: July 2010

We define for any locally compact group G, the space of bounded uniformly continuous functionals on Ĝ,

U
C
B
(

G
^

)

UCB(\hat G)

, in the context of P. Eymard [Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208] (for notations see next section). For

u
∈
A
(
G
)

u \in A(G)

let

u
⊥

=
{
ϕ
∈
V
N
(
G
)
;
ϕ
[
A
(
G
)
u
]
=
0
}

{u^ \bot } = \{ \phi \in VN(G);\phi [A(G)u] = 0\}

. Theorem. If for some norm separable subspace

X
⊂
V
N
(
G
)

X \subset VN(G)

and some positive definite

0
≠
u
∈
A
(
G
)
,
U
C
B
(

G
^

)
⊂

0 \ne u \in A(G),UCB(\hat G) \subset

norm closure

[
W
(

G
^

)
+
X
+

u
⊥

]

[W(\hat G) + X + {u^ \bot }]

then G is discrete. If G is discrete then

U
C
B
(

G
^

)
⊂
A
P
(

G
^

)
⊂
W
(

G
^

)

UCB(\hat G) \subset AP(\hat G) \subset W(\hat G)

.
Asymptotic solutions of linear Volterra integral equations with singular kernels
Authors: Wong J. S. W. , Wong R. 
Published: July 2010

Volterra integral equations of the form

u
′

(
t
)
=
−

∫
0
t

a
(
t
−
τ
)
u
(
τ
)
d
τ
,
u
(
0
)
=
1

u’(t) = - \smallint _0^ta(t - \tau )u(\tau )d\tau ,u(0) = 1

, are considered, where

a
(
t
)
∈
C
(
0
,
∞
)
∩

L
1

(
0
,
1
)

a(t) \in C(0,\infty ) \cap {L_1}(0,1)

. Explicit asymptotic forms are obtained for the solutions, when the kernels

a
(
t
)

a(t)

have a specific asymptotic representation.
Weighted norm inequalities for fractional integrals
Authors: Muckenhoupt Benjamin , Wheeden Richard 
Published: July 2010

The principal problem considered is the determination of all nonnegative functions,

V
(
x
)

V(x)

, such that

‖

T
γ

f
(
x
)
V
(
x
)
‖

q

≤
C

‖
f
(
x
)
V
(
x
)
‖

p

\left \|{T_\gamma }f(x)V(x)\right \|_q \leq C\left \|f(x)V(x)\right \|_p

where the functions are defined on

R
n

,
0
>
γ
>
n
,
1
>
p
>
n

/

γ
,
1

/

q
=
1

/

p
−
γ

/

n

{R^n},0 > \gamma > n,1 > p > n/\gamma ,1/q = 1/p - \gamma /n

, C is a constant independent of f and

T
γ

f
(
x
)
=
∫
f
(
x
−
y
)

|

y

|

γ
−
n

d
y

{T_\gamma }f(x) = \smallint f(x - y)|y{|^{\gamma - n}}dy

. The main result is that

V
(
x
)

V(x)

is such a function if and only if
\[

(

1

|

Q

|

∫
Q

[
V
(
x
)
]

q

d
x

)

1

/

q

(

1

|

Q

|

∫
Q

[
V
(
x
)
]

−

p
′

d
x

)

1

/

p
′

≤
K

{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^q}dx} } \right )^{1/q}}{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^{ - p’}}dx} } \right )^{1/p’}} \leq K

\]
where Q is any n dimensional cube,

|

Q

|

|Q|

denotes the measure of Q,

p
′

=
p

/

(
p
−
1
)

p’ = p/(p - 1)

and K is a constant independent of Q. Substitute results for the cases

p
=
1

p = 1

and

q
=
∞

q = \infty

and a weighted version of the Sobolev imbedding theorem are also proved.
Differentiability of solutions to hyperbolic initial-boundary value problems
Authors: Rauch Jeffrey B. , Massey Frank J. 
Published: July 2010

This paper establishes conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form

(
∂

/

∂
t
−
∑

A
j

∂

/

∂

x
j

−
B
)
u
=
F

(\partial /\partial t - \sum {A_j}\partial /\partial {x_j} - B)u = F

on

[
0
,
T
]
×
Ω
,
M
u
=
g

[0,T] \times \Omega ,Mu = g

on

[
0
,
T
]
×
∂
Ω
,
u
(
0
,
x
)
=
f
(
x
)
,
x
∈
Ω

[0,T] \times \partial \Omega ,u(0,x) = f(x),x \in \Omega

. Assuming that

L

2

{\mathcal {L}_2}

a priori inequalities are known for this equation, it is shown that if

F
∈

H
s

(
[
0
,
T
]
×
Ω
)
,
g
∈

H

s
+
1

/

2

(
[
0
,
T
]
×
∂
Ω
)
,
f
∈

H
s

(
Ω
)

F \in {H^s}([0,T] \times \Omega ),g \in {H^{s + 1/2}}([0,T] \times \partial \Omega ),f \in {H^s}(\Omega )

satisfy the natural compatibility conditions associated with this equation, then the solution is of class

C
p

{C^p}

from [0, T] to

H

s
−
p

(
Ω
)
,
0
≤
p
≤
s

{H^{s - p}}(\Omega ),0 \leq p \leq s

. These results are applied to mixed problems with distribution initial data and to quasi-linear mixed problems.
Regularly varying functions and convolutions with real kernels
Authors: Jordan G. S. 
Published: July 2010

Let

ϕ
\phi

be a positive, measurable function and k a real-valued function on

(
0
,
∞
)
,
k
∈

L
1

(
d
t

/

t
)

(0,\infty ),k \in {L^1}(dt/t)

. We give conditions on

ϕ
\phi

and k sufficient to deduce the regular variation of

ϕ
\phi

from the assumption that
\[

α
=

lim

x
→
∞

1

ϕ
(
x
)

∫
0
∞

ϕ
(
t
)
k

(

x
2

)

d
t

t

exits

(
α
≠
0
,
∞
)
.

\alpha = \lim \limits _{x \to \infty } \frac {1}{{\phi (x)}}\int _0^\infty {\phi (t)k\left ( {\frac {x}{2}} \right )} \;\frac {{dt}}{t}\;{\text {exits}}\;(\alpha \ne 0,\infty ).

\]
The general theorems extend in certain ways results of other authors and yield a new theorem on the relation between the radial growth and zero-distribution of those entire functions which are canonical products of nonintegral order with negative zeros.
A generalization of the 𝑐𝑜𝑠𝜋𝜌 theorem
Authors: Baernstein Albert 
Published: July 2010

Let f be an entire function, and let

β
\beta

and

λ
\lambda

be positive numbers with

β
≤
π

\beta \leq \pi

and

β
λ
>
π

\beta \lambda > \pi

. Let

E
(
r
)
=
{
θ
:
log
⁡

|

f
(
r

e

i
θ

)

|

>
cos
⁡
β
λ
log
⁡
M
(
r
)
}

E(r) = \{ \theta :\log |f(r{e^{i\theta }})| > \cos \beta \lambda \log M(r)\}

. It is proved that either there exist arbitrarily large values of r for which

E
(
r
)

E(r)

contains an interval of length at least

2
β

2\beta

, or else

lim

r
→
∞

r

−
λ

log
⁡
M
(
r
,
f
)

{\lim _{r \to \infty }}{r^{ - \lambda }}\log M(r,f)

exists and is positive or infinite. For

β
=
π

\beta = \pi

this is Kjellberg’s refinement of the cos

π
ρ

\pi \rho

theorem.
Analytic domination with quadratic form type estimates and nondegeneracy of ground states in quantum field theory
Authors: Sloan Alan D. 
Published: July 2010

We present a theorem concerning the analytic domination by a semi-bounded selfadjoint operator H of another linear operator A which requires only the quadratic form type estimates
\[

‖

H

−
1

/

2

(

(

ad

A
)

n

H
)

H

−
1

/

2

u

‖

≤

c
n

‖
u
‖

\left \| {{H^{ - 1/2}}({{({\text {ad}}\;A)}^n}H){H^{ - 1/2}}u} \right \| \leq {c_n}\left \| u \right \|

\]
instead of the norm estimates
\[

‖

(

ad

A
)

n

H
u

‖

≤

c
n

‖

H
u

‖

\left \| {{{({\text {ad}}\;A)}^n}Hu} \right \| \leq {c_n}\left \| {Hu} \right \|

\]
usually required for this type of theorem. We call the new estimates “quadratic form type", since they are sometimes equivalent to
\[

|

(

(

ad

A

)
n

H
u
,
u
)

|

≤

c
n

|

(
H
u
,
u
)

|

.

|({({\text {ad}}\;A)^n}Hu,u)| \leq {c_n}|(Hu,u)|.

\]
The theorem is then applied with H the Hamiltonian for the spatially cutoff boson field model with real, bounded below, even ordered polynomial self-interaction in one space dimension and

A
=
π
(
g
)

A = \pi (g)

, the conjugate momentum to the free field. When the underlying Hilbert space of this model is represented as

L
2

(
Q
,
d
q
)

{L^2}(Q,dq)

where dq is a probability measure on Q, the spectrum of the von Neumann algebra generated by bounded functions of certain field operators, then

e

−
t
H

{e^{ - tH}}

maximizes support in the sense that

e

−
t
H

f

{e^{ - tH}}f

is nonzero almost everywhere whenever f is not identically zero.
Riesz points of the spectrum of an element in a semisimple Banach algebra
Authors: Pearlman Lynn D. 
Published: July 2010

Let A be a semisimple Banach algebra with unit element and let

S
A

{S_A}

denote the socle of A. For an element y in A, let

L
y

[

R
y

]

{L_y}[{R_y}]

denote the operator of left [right] multiplication by y on A. The operational calculus and A. E. Taylor’s theory of the ascent

α
(
T
)

\alpha (T)

and descent

δ
(
T
)

\delta (T)

of an operator T on A are used to show that the following conditions on a number

λ
\lambda

in the spectrum of an element x in A are all equivalent. (1)

λ
\lambda

is a pole of the resolvent mapping

z
→

(
z
−
x

)

−
1

z \to {(z - x)^{ - 1}}

and the spectral idempotent f, for x at

λ
\lambda

is in

S
A

{S_A}

. (2)

λ
−
x
−
c

\lambda - x - c

is invertible in A for some c in the closure of

S
A

{S_A}

such that

c
x
=
x
c

cx = xc

. (3)

λ
−
x

\lambda - x

is invertible modulo the closure of

S
A

{S_A}

and

0
>
α
(

L

(
λ
−
x
)

)
=
δ
(

L

(
λ
−
x
)

)
>
∞

0 > \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) > \infty

. (4)

λ
−
x

\lambda - x

is invertible modulo the closure of

S
A

{S_A}

and

0
>
α
(

R

(
λ
−
x
)

)
=
δ
(

R

(
λ
−
x
)

)
=
α
(

L

(
λ
−
x
)

)
=
δ
(

L

(
λ
−
x
)

)
>
∞

0 > \alpha ({R_{(\lambda - x)}}) = \delta ({R_{(\lambda - x)}}) = \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) > \infty

. Such numbers

λ
\lambda

are called Riesz points. An element x is called a Riesz element of A if it is topologically nilpotent modulo the closure of

S
A

{S_A}

. It is shown that x is a Riesz element if and only if every nonzero number in the spectrum of x is a Riesz point.
Subgroups of groups of central type
Authors: Timmer Kathleen M. 
Published: July 2010

Let

λ
\lambda

be a linear character on the center Z of a finite group Z of a finite group H, such that (1)

λ
H

=

∑

i
=
1

p

ϕ
i

(
1
)

ϕ
i

{\lambda ^H} = \sum \nolimits _{i = 1}^p {{\phi _i}(1){\phi _i}}

where the

ϕ
i

{\phi _i}

’s are inequivalent irreducible characters on H of the same degree, and (2) if

∑

i
=
1

p

m
i

ϕ
i

(
x
)
=
0

\sum \nolimits _{i = 1}^p {{m_i}{\phi _i}(x) = 0}

for some

x
∈
H

x \in H

and nonnegative integers

m
i

{m_i}

, then either

ϕ
i

(
x
)
=
0

{\phi _i}(x) = 0

for all i or

m
i

=

m
j

{m_i} = {m_j}

for all i, j. The object of the paper is to describe finite groups which satisfy conditions (1) and (2) in terms of the multiplication of the group. If S is a p Sylow subgroup of the group H, and

R
=
S
⋅
Z

R = S \cdot Z

, then H satisfies conditions (1) and (2) if and only if (a)

{
x
∈
H
:

x

−
1

h

−
1

x
h
∈
Z
⇒
λ
(

x

−
1

h

−
1

x
h
)
=
1
,
h
∈
H
}

/

Z

\{ x \in H:{x^{ - 1}}{h^{ - 1}}xh \in Z \Rightarrow \lambda ({x^{ - 1}}{h^{ - 1}}xh) = 1,h \in H\} /Z

consists of elements of order a power of p in

H

/

Z

H/Z

, and these elements form p conjugacy classes of

H

/

Z

H/Z

, and (b) the elements of

{
x
∈
R
:

x

−
1

r

−
1

x
r
∈
Z
⇒
λ
(

x

−
1

r

−
1

x
r
)
=
1
,
r
∈
R
}

/

Z

\{ x \in R:{x^{ - 1}}{r^{ - 1}}xr \in Z \Rightarrow \lambda ({x^{ - 1}}{r^{ - 1}}xr) = 1,r \in R\} /Z

form p conjugacy classes of

R

/

Z

R/Z

.
Multiplier transformations on compact Lie groups and algebras
Authors: Strichartz Robert S. 
Published: July 2010

Let G be a semisimple compact Lie group and

T
f
=
∑
ϕ
(
m
)

d

m
χ
m

∗
f

Tf = \sum \phi (m){d_{m \chi m}} \ast f

a bi-invariant operator on

L
2

(
G
)

{L^2}(G)

, where

χ
m

{\chi _m}

and

d
m

{d_m}

are the characters and dimensions of the irreducible representations of G, which are indexed by a lattice of points m in the Lie algebra

G

\mathfrak {G}

in a natural way. If

Φ
\Phi

is a bounded ad-invariant function on

G

\mathfrak {G}

and

(

∗

)

ϕ

(

m

) =

Φ

(

m

+

β

)

or

\begin{equation}\tag {$\ast $} \phi {\text {(}}m{\text {) = }}\Phi {\text {(}}m{\text { + }}\beta {\text {)}}\quad {\text {or}}\end{equation}

(

∗
∗

)

ϕ

(

m

) =

∫
G

Φ
(
m
+
β
−

ad

g
β
)
d
g

\begin{equation}\tag {$\ast \ast $} \phi {\text {(}}m{\text {) = }}\int _G {\Phi (m + \beta - {\text {ad}}\;g\beta )dg} \end{equation}

β
\beta

being half the sum of the positive roots, then various properties of T are related to properties of the Fourier multiplier transformation on

G

\mathfrak {G}

with multiplier

Φ
\Phi

. These properties include boundedness on

L
1

{L^1}

, uniform boundedness on

L
p

{L^p}

of a family of operators, and, in the special case

G
=

SO

(
3
)

G = {\text {SO}}(3)

, boundedness in

L
p

{L^p}

for ad-invariant functions with

1
≤
p
>
3

/

2

1 \leq p > 3/2

.
Analytic capacity, Hölder conditions and 𝜏-spikes
Authors: O’Farrell Anthony G. 
Published: July 2010

We consider the uniform algebra

R
(
X
)

R(X)

, for compact

X
⊂

C

X \subset {\text {C}}

, in relation to the condition

I

p
+
α

=

Σ
1
∞

2

(
p
+
α
+
1
)
n

γ
(

A
n

(
x
)
∖
X
)
>
+
∞

{I_{p + \alpha }} = \Sigma _1^\infty {2^{(p + \alpha + 1)n}}\gamma ({A_n}(x)\backslash X) > + \infty

, where

0
≤
p
∈

Z

,
0
>
α
>
1
,
γ

0 \leq p \in {\mathbf {Z}},0 > \alpha > 1,\gamma

is analytic capacity, and

A
n

(
x
)

{A_n}(x)

is the annulus

{
z
∈

C

:

2

−
n
−
1

>

|

z
−
x

|

>

2

−
n

}

\{ z \in {\text {C}}:{2^{ - n - 1}} > |z - x| > {2^{ - n}}\}

. We introduce the notion of

τ
\tau

-spike for

τ
>
0

\tau > 0

, and show that

I

p
+
α

=
+
∞

{I_{p + \alpha }} = + \infty

implies

x
x

is a

p
+
α

p + \alpha

-spike. If

X
˚

\mathring {X}

satisfies a cone condition at

x
x

, and

I

p
+
α

>
+
∞

{I_{p + \alpha }} > + \infty

, we show that the

p
p

th derivatives of the functions in

R
(
X
)

R(X)

satisfy a uniform Hölder condition at

x
x

for nontangential approach. The structure of the set of non-

τ
\tau

-spikes is examined and the results are applied to rational approximation. A geometric question is settled.
The existence, characterization and essential uniqueness of solutions of 𝐿^{∞} extremal problems
Authors: Fisher S. D. , Jerome J. W. 
Published: July 2010

Let

I
=
(
a
,
b
)

I = (a,b)

be an interval in R and let

H

n
,
∞

{H^{n,\infty }}

consist of those real-valued functions f such that

f

(
n
−
1
)

{f^{(n - 1)}}

is absolutely continuous on I and

f

(
n
)

∈

L
∞

(
I
)

{f^{(n)}} \in {L^\infty }(I)

. Let L be a linear differential operator of order n with leading coefficient

1
,
a
=

x
1

>
⋯
>

x
m

=
b

1,a = {x_1} > \cdots > {x_m} = b

be a partition of I and let the linear functionals

L

i
j

{L_{ij}}

on

H

n
,
∞

{H^{n,\infty }}

be given by
\[

L

i
j

f
=

∑

v
=
0

n
−
1

a

i
j

(
v
)

f

(
v
)

(

x
i

)
,

j
=
1
,
⋯
,

k
i

,
i
=
1
,
⋯
,
m
,

{L_{ij}}f = \sum \limits _{v = 0}^{n - 1} {a_{ij}^{(v)}{f^{(v)}}({x_i}),\quad j = 1, \cdots ,{k_i},i = 1, \cdots ,m,}

\]
where

1
≤

k
i

≤
n

1 \leq {k_i} \leq n

and the

k
i

{k_i}

n-tuples

(

a

i
j

(
0
)

,
⋯
,

a

i
j

(
n
−
1
)

)

(a_{ij}^{(0)}, \cdots ,a_{ij}^{(n - 1)})

are linearly inde pendent. Let

r

i
j

{r_{ij}}

be prescribed real numbers and let

U
=
{
f
∈

H

n
,
∞

:

L

i
j

f
=

r

i
j

,
j
=
1
,
⋯
,

k
i

,
i
=
1
,
⋯
,
m
}

U = \{ f \in {H^{n,\infty }}:{L_{ij}}f = {r_{ij}},j = 1, \cdots ,{k_i},i = 1, \cdots ,m\}

. In this paper we consider the extremal problem

(

∗

)

‖

L
s

‖

L
∞

=
α
=
inf
{

‖

L
f

‖

L
∞

:
f
∈
U
}
.

\begin{equation}\tag {$\ast $}{\left \| {Ls} \right \|_{{L^\infty }}} = \alpha = \inf \{ {\left \| {Lf} \right \|_{{L^\infty }}}:f \in U\} .\end{equation}

We show that there are, in general, many solutions to

(
∗
)

( \ast )

but that there is, under certain consistency assumptions on L and the

L

i
j

{L_{ij}}

, a fundamental (or core) interval of the form

(

x
i

,

x

i
+

n
0

)

({x_i},{x_{i + {n_0}}})

on which all solutions to

(
∗
)

( \ast )

agree;

n
0

{n_0}

is determined by the

k
i

{k_i}

and satisfies

n
0

≥
1

{n_0} \geq 1

. Further, if s is any solution to

(
∗
)

( \ast )

then on

(

x
i

,

x

i
+

n
0

)
,

|

L
s

|

=
α

({x_i},{x_{i + {n_0}}}),|Ls| = \alpha

a.e. Further, we show that there is a uniquely determined solution

s
∗

{s_ \ast }

to

(
∗
)

( \ast )

, found by minimizing

‖

L
f

‖

L
∞

{\left \| {Lf} \right \|_{{L^\infty }}}

over all subintervals

(

x
j

,

x

j
+
1

)
,
j
=
1
,
⋯
,
m
−
1

({x_j},{x_{j + 1}}),j = 1, \cdots ,m - 1

, with the property that

|

L

s
∗

|

|L{s_ \ast }|

is constant on each subinterval

(

x
j

,

x

j
+
1

)

({x_j},{x_{j + 1}})

and

L

s
∗

L{s_ \ast }

is a step function with at most

n
−
1

n - 1

discontinuities on

(

x
j

,

x

j
+
1

)

({x_j},{x_{j + 1}})

. When

L
=

D
n

,

s
∗

L = {D^n},{s_ \ast }

is a piecewise perfect spline. Examples show that the results are essentially best possible.