Littlewood-Paley and Lusin functions of α-parabolic typeYôsuke H ISHIKAWA and Masahiro YAMADAABSTRACT. For 0 < α ≤ 1, we consider the L(α) -harmonic extensions of L2 -functions
on the Euclidean space Rn . In this paper, we study Littlewood-Paley and Lusin functions
for L(α) -harmonic extensions, and we give some identities concerning L2 -norms of thierfunctions.

1. Introduction
Let n ≥ 1 and H be the upper half-space of the (n + 1)-dimensional Euclidean space, thatis, H = {X = (x, t) ∈ Rn+1 : x = (x1 , . . . , xn ) ∈ Rn , t > 0}. For 0 < α ≤ 1, the parabolic
operator L(α) is defined byL(α) := ∂t + (−Δx )α ,(1.1)

where ∂t = ∂/∂t, ∂j = ∂/∂xj , and Δx = ∂12 + · · · + ∂n2 . Let C(H) be the set of all real-valued
continuous functions on H. A function u ∈ C(H) is said to be L(α) -harmonic if L(α) u = 0 in
the sense of distributions (for details, see Section 2). In this paper, we study Littlewood-Paleyand Lusin functions for L(α) -harmonic extensions, and we give some identities concerning L2 norms of thier functions.
To state our main results, we give some definitions. For 1 ≤ p ≤ ∞, the Lebesgue spacep
L := Lp (Rn , dVn ) is defined to be the Banach space of Lebesgue measurable (real-valued)
functions on Rn with norm  · Lp , where dVn is the Lebesgue measure on Rn . We denote byW (α) the fundamental solution of L(α) (see Section 2 for the definition). We define an L(α) (α)
harmonic extension Hf of f ∈ Lp by(α)
W (α) (x − y, t)f (y)dVn (y), (x, t) ∈ H.
(1.2)
Hf (x, t) =Rn(α)

It is shown that the function Hf is L(α) -harmonic on H (see [4, Theorem 5.2]). It is wellknown that when α = 1/2, the fundamental solution W (1/2) coincides with the Poisson kernel(1/2)
for H (see [5, Section 2]). Therefore, the function Hfis the usual harmonic extensions of f .
κκ
For a real number κ, let Dt = (−∂t ) be a fractional differential operator, and FC κ the classof functions ϕ on R+ = (0, ∞) such that Dtκ ϕ is well-defined (for the explicit definitions of Dtκ
2010 Mathematics Subject Classification: Primary 42B25; Secondary 42B10, 35K05.
Keywords and phrases: harmonic extension, Littlewood-Paley function, Lusin function, parabolic operator of
fractional orderThis work was supported in part by Grant-in-Aid for Scientific Research (C) (No. 16K05198), Japan Society
for the Promotion of Science.

1and FC κ , see Section 2). For a function u on H, let ∇x u = (∂1 u, · · · , ∂n u) and |∇x u(x, t)|2 =n2
j=1 |∂j u(x, t)| . Furthermore, let Γ be the gamma function.
In this paper, we show the following theorem, which are identities of Littlewood-Paley typefor L(α) -harmonic extensions. When α = 1/2, the following identities are well known (see [7,pp. 82–83]).
T HEOREM 1. Let 0 < α ≤ 1 and f ∈ L2 . Then the following identities hold: ∞
0andRn ∞
01

11Rn1(α)

t α −1 |Dt2α Hf (x, t)|2 dVn (x) dt = 2− α Γ(α−1 )f 2L21(α)

t α −1 |∇x Hf (x, t)|2 dVn (x) dt = 2− α Γ(α−1 )f 2L2 .

We also show the following theorem, which are identities of Lusin type for L(α) -harmonic
extensions. When α = 1/2, the following identities are well known (see [6]).
For ξ ∈ Rn and ρ > 0, letCρ(α) (ξ) := {(x, t) ∈ H : |x − ξ|2α ≤ ρ−1 t}.
We define Lusin functions for L(α) -harmonic extensions. Let (α)
Sf,t (ξ) =and(α)
Sf,x (ξ) =(α)

Cρ (ξ)t 12α1/21n−1− 2αα|Dt1n−1− 2αα(α)|∇x Hf (x, t)|2 dVn (x) dt(α)
Hf (x, t)|2 dVn (x) dt

1/2(α)Cρ (ξ)t.

T HEOREM 2. Let 0 < α ≤ 1 and f ∈ L2 . Furthermore, let dn be the volume of the unit ballof Rn . Then the following identities hold:
Rnand
Rn(α)

n1(α)

n1|Sf,t (ξ)|2 dVn (ξ) = dn ρ− 2α 2− α Γ(α−1 )f 2L2|Sf,x (ξ)|2 dVn (ξ) = dn ρ− 2α 2− α Γ(α−1 )f 2L2 .

We describe the construction of this paper. In Section 2, we recall definitions of the L(α) harmonic functions and fractional differential operators. In Section 3, we show the identities ofLittlewood-Paley type in Theorem 1 (see Theorem 3.3 in Section 3). In Section 4, we also show
the identities of Lusin type in Theorem 2 (see Theorem 4.1 in Section 4).

2

2. Preliminaries
In this section, we recall some basic properties. We begin with describing the operator(−Δx )α and the L(α) -harmonic functions. Since the case α = 1 is trivial, we only describe
the case 0 < α < 1. Let C ∞ (H) denote the set of all infinitely differentiable functions on
H. Furthermore, let Cc∞ (H) be the set of all functions in C ∞ (H) with compact support. For0 < α < 1, (−Δx )α is the convolution operator defined byψ(x + y, t) − ψ(x, t)
αdVn (y)(2.1)(−Δx ) ψ(x, t) := −Cn,α lim
ε→+0 |y|>ε|y|n+2α
for all ψ ∈ Cc∞ (H) and (x, t) ∈ H, where Cn,α = −4α π −n/2 Γ (n + 2α)/2 /Γ(−α) > 0. Let
(α) := −∂t + (−Δx )α be the adjoint operator of L(α) . Then, a function u ∈ C(H) is said toLbe L(α) -harmonic if u satisfies L(α) u = 0 in the sense of distributions, that is,(α)
 ψ|dVn+1 < ∞ and(α) ψdVn+1 = 0 for all ψ ∈ C ∞ (H).
|u LuLcHH(α)

We describe the fundamental solution of L . For (x, t) ∈ H, let1(α)
exp(−t|ξ|2α + i x · ξ) dVn (ξ)
W (x, t) =(2π)n Rn
2α(2.2)e−t|2πξ| e2πix·ξ dVn (ξ).
=Rnwhere x · ξ denotes the inner product on Rn and |ξ| = (ξ · ξ)1/2 . The function W (α) is thefundamental solution of L(α) and it is L(α) -harmonic on H. Furthermore, W (α) ∈ C ∞ (H).
We also recall definitions of the fractional integral and differential operators for functions
on R+ = (0, ∞) (for details, see [2]). For a real number κ > 0, let(2.3)

FC −κ := ϕ ∈ C(R+ ) : ϕ(t) = O(t−κ ) (t → ∞) for some κ > κ}.

For a function ϕ ∈ FC −κ , we can define the fractional integral Dt−κ ϕ of ϕ by
 ∞
1−κ
(2.4)
Dt ϕ(t) :=τ κ−1 ϕ(τ + t)dτ, t ∈ R+ .
Γ(κ) 0We put FC 0 := C(R+ ) and Dt0 ϕ := ϕ. Moreover, let(2.5)

κFC κ := {ϕ ; ∂t ϕ ∈ FC −(κ−κ) },where κ is the smallest integer greater than or equal to κ. Then, we can also define thefractional derivative Dtκ ϕ of ϕ ∈ FC κ by−(κ−κ) (2.6)
Dtκ ϕ(t) := Dt(−∂t )κ ϕ (t), t ∈ R+ .
Clearly, when κ ∈ N0 := N ∪ {0}, the operator Dtκ coincides with the ordinary differential
operator (−∂t )κ . For a real number κ, we may call both (2.4) and (2.6) the fractional derivatives3

of ϕ with order κ. And, we call Dtκ the fractional differential operator with order κ. Here, wegive some examples of fractional derivatives of elementary functions.
E XAMPLE 2.1. Let κ > 0 and ν be real numbers. Then, we have the following.
(1) Dtν e−κt = κν e−κt .
(2) If −κ < ν, then Dtν t−κ =Γ(κ + ν) −κ−ν
.
t
Γ(κ)

3. Littlewood-Paley functions of α-parabolic typeFor a function f ∈ L2 , we denote by fˆ or F(f ) the Fourier transform of f , that is,fˆ(ξ) = F(f )(ξ) =f (y)e−2πiξ·y dVn (y), ξ ∈ Rn .
RnLet n ≥ 1 and 0 < α ≤ 1 be fixed. For γ ∈ Nn0 and 1 ≤ p ≤ ∞, define the interval I(γ, p)byI(γ, p) :={ν ∈ R : ν > −(n/2α)(1/p) − |γ|/2α} (p = ∞){ν ∈ R : ν > −|γ|/2α} ∪ {0}(p = ∞).

L EMMA 3.1. ([3, Theorem 3.4]) Let 0 < α ≤ 1, 1 ≤ p ≤ ∞, and γ ∈ Nn0 . If f ∈ Lp and(α)ν ∈ I(γ, p), then the derivative Dtν ∂xγ Hf (x, t) is well defined, andν γ (α)Dtν ∂xγ W (α) (x − y, t)f (y)dVn (y).
Dt ∂x Hf (x, t) =Rn

Furthermore, there exists a constant C = C(n, α, p, γ, ν) > 0 such that(α)|Dtν ∂xγ Hf (x, t)| ≤ Ct−(n/2α)(1/p)−|γ|/2α−ν f Lp
for all (x, t) ∈ H.
We give properties of fractional derivatives of L(α) -harmonic extensions.
L EMMA 3.2. Let 0 < α ≤ 1 and f ∈ L2 . Then the following statements hold:n,(1) For a real number ν > − 2αDtν W (α) (x, t) =

2α

Rn|2πξ|2αν e−t|2πξ| e2πix·ξ dVn (ξ).

Furthermore, for integers 1 ≤ j ≤ n and ∈ N0 ,2α(α)∂j W (x, t) =(2πiξj ) e−t|2πξ| e2πix·ξ dVn (ξ).
Rn4n(2) For a real number ν > − 4α,ν (α)
Dt Hf (x, t) =Rn|2πξ|2αν fˆ(ξ)e−t|2πξ| e2πix·ξ dVn (ξ).
2αFurthermore, for integers 1 ≤ j ≤ n and ∈ N0 ,
2α(α)(2πiξj ) fˆ(ξ)e−t|2πξ| e2πix·ξ dVn (ξ).
∂j Hf (x, t) =RnnP ROOF. (1) Since ν > − 2α, we have ∞2ατ ν−ν−1 |2πξ|2αν e−(τ +t)|2πξ| dVn (ξ) dτ < ∞.
0Rn

Differentiating through the integral (2.2) with respect to t, the Fubini theorem and Example 2.1(1) imply that ∞12ανν(α)
ν−ν−1τ
Dt e−(τ +t)|2πξ| e2πix·ξ dVn (ξ) dτ
Dt W (x, t) =Γ( ν − ν) 0Rn ∞12αντ ν−ν−1 Dt e−(τ +t)|2πξ| dτ e2πix·ξ dVn (ξ)=Γ( ν − ν) 0nR2αDtν e−t|2πξ|e2πix·ξ dτ dVn (ξ)=nR2α=|2πξ|2αν e−t|2πξ| e2πix·ξ dVn (ξ).
Rn

Furthermore, differentiating through the integral (2.2) with respect to x, we have
2α(α)(2πiξj ) e−t|2πξ| e2πix·ξ dVn (ξ).
∂j W (x, t) =Rn(2) By Lemma 3.1 and Lemma 3.2 (1), we haveν (α)
Dt Hf (x, t) =Dtν W (α) (x − y, t)f (y)dVn (y)
n
R
2α=
f (y)
|2πξ|2αν e−t|2πξ| e2πi(x−y)·ξ dVn (ξ) dVn (y)nRnR2α2αν=|2πξ|
f (y)e−2πiy·ξ dVn (y) e−t|2πξ| e2πix·ξ dVn (ξ)
nRn
R2α=|2πξ|2αν fˆ(ξ)e−t|2πξ| e2πix·ξ dVn (ξ).
RnFurthermore, we have(α)∂j W (α) (x − y, t)f (y)dVn (y)∂j Hf (x, t) =Rn5

2α(2πiξj ) e−t|2πξ| e2πi(x−y)·ξ dVn (ξ) dVn (y)nnR
R
2α(2πiξj )f (y)e−2πiy·ξ dVn (y) e−t|2πξ| e2πix·ξ dVn (ξ)=nRnR2α=(2πiξj ) fˆ(ξ)e−t|2πξ| e2πix·ξ dVn (ξ).

=f (y)Rn

This completes the proof.
We give identities of Littlewood-Paley type for L(α) -harmonic extensions.
T HEOREM 3.3. Let 0 < α ≤ 1 and f ∈ L2 . Then the following identities hold:
 ∞
0Rn

 ∞
(3.2)
011(3.1)

Rn1(α)t α −1 |Dt2α Hf (x, t)|2 dVn (x) dt = 2− α Γ(α−1 )f 2L211(α)

t α −1 |∇x Hf (x, t)|2 dVn (x) dt = 2− α Γ(α−1 )f 2L2 .

P ROOF. We show the identity (3.1). By Lemma 3.2 (2), we have12α(α)
2αDt Hf (x, t) =|2πξ|fˆ(ξ)e−t|2πξ| e2πix·ξ dVn (ξ) = F −1 (ϕt )(x),Rn2αwhere ϕt (ξ) = |2πξ|fˆ(ξ)e−t|2πξ| . Therefore, we obtain ∞ ∞111(α)−12−12αt α |Dt Hf (x, t)| dVn (x) dt =tα|F −1 (ϕt )(x)|2 dVn (x) dtnn0R ∞ 0 R∞
112αt α −1|ϕt (ξ)|2 dVn (ξ) dt =t α −1|2πξ|2 |fˆ(ξ)|2 e−2t|2πξ| dVn (ξ) dt=Rn0Rn ∞0112 ˆ
2−1 −2t|2πξ|2α−α−1α|2πξ| |f (ξ)|
tedt dVn (ξ) = 2 Γ(α )|fˆ(ξ)|2 dVn (ξ).
=Rn

Rn0

We show the identity (3.2). By Lemma 3.2 (2), for 1 ≤ j ≤ n, we have
2α(α)∂j Hf (x, t) =(2πiξj )fˆ(ξ)e−t|2πξ| e2πix·ξ dVn (ξ) = F −1 (ψt,j )(x),Rn2αwhere ψt,j (ξ) = (2πiξj )fˆ(ξ)e−t|2πξ| . Therefore, we obtain ∞ ∞n 
11(α)−12−1t α |∇x Hf (x, t)| dVn (x) dt =tα|F −1 (ψt,j )(x)|2 dVn (x) dt0Rn

 ∞n 

1−1tα=0

j=1

Rn0

j=1Rn

 ∞n 

12−1|ψt,j (ξ)| dVn (ξ) dt =tα0

j=16Rn|2πiξj |2 |fˆ(ξ)|2 e−2t|2πξ| dVn (ξ) dt2α ∞1−1=tαRn0|2πξ| |fˆ(ξ)| e2 −2t|2πξ|2α21−αdVn (ξ) dt = 2−1Γ(α )

Rn|fˆ(ξ)|2 dVn (ξ).
This completes the proof.
4. Lusin functions of α-parabolic type

We recall the definitions of Lusin functions for L(α) -harmonic extensions. For ξ ∈ Rn andρ > 0, letCρ(α) (ξ) := {(x, t) ∈ H : |x − ξ|2α ≤ ρ−1 t}.
Lusin functions for L(α) -harmonic extensions are defined by (α)

Sf,t (ξ) =(4.1)

1(α)
Cρ (ξ)

1/2

12αn(α)t α −1− 2α |Dt Hf (x, t)|2 dVn (x) dtand(α)
Sf,x (ξ) =(4.2)

 

1/2(α)Cρ (ξ)t1n−1− 2αα(α)|∇x Hf (x, t)|2 dVn (x) dt.

We give identities of Lusin type for L(α) -harmonic extensions.
T HEOREM 4.1. Let 0 < α ≤ 1 and f ∈ L2 . Furthermore, let dn be the volume of the unit
ball of Rn . Then the following identities hold:(4.3)
Rn
(4.4)
Rn(α)

n1(α)

n1|Sf,t (ξ)|2 dVn (ξ) = dn ρ− 2α 2− α Γ(α−1 )f 2L2|Sf,x (ξ)|2 dVn (ξ) = dn ρ− 2α 2− α Γ(α−1 )f 2L2 .

P ROOF. We show the identity (4.3). Let Φξ (x, t) be the characteristic function of the set(α)
Cρ (ξ). The Fubini theorem implies that(α)|Sf,t (ξ)|2 dVn (ξ)
Rn
 ∞

11n(α)
Φξ (x, t) t α −1− 2α |Dt2α Hf (x, t)|2 dVn (x) dt dVn (ξ)=nnR∞ 0 R11n(α)−1− 2ααtΦx (ξ, t) dVn (ξ) |Dt2α Hf (x, t)|2 dVn (x) dt.
=0Rn

Rn7

SinceRnnn

Φx (ξ, t) dVn (ξ) = Vn (Cρ(α) (x)) = Vn (Cρ(α) (0)) = dn ρ− 2α t 2α ,Theorem 3.3 implies that ∞
n(α)
2− 2α|Sf,t (ξ)| dVn (ξ) = dn ρRn= dn ρ11(α)

t α −1 |Dt2α Hf (x, t)|2 dVn (x) dt0Rnn
1− 2α−α−12Γ(α )f 2L2 .

The proof of the identity (4.4) is similar. This completes the proof.

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Yôsuke Hishikawa
Department of Mathematics, Faculty of Education, Gifu University
Yanagido 1–1, Gifu 501–1193, Japanyhishik@gifu-u.ac.jp
andMasahiro Yamada
Department of Mathematics, Faculty of Education, Gifu University
Yanagido 1–1, Gifu 501–1193, Japan
yamada33@gifu-u.ac.jp8