Grand Triebel-Lizorkin-Morrey spaces

Kwok-Pun Ho

doi:10.1515/dema-2024-0085

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Grand Triebel-Lizorkin-Morrey spaces

Kwok-Pun Ho

Published/Copyright: February 11, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This article studies the Triebel-Lizorkin-type spaces built on grand Morrey spaces on Euclidean spaces. We establish a number of characterizations on the grand Triebel-Lizorkin-Morrey spaces such as the Peetre maximal function characterizations, the Lusin area function characterizations, and the g λ ∗ function characterizations. The boundedness of the Fourier multipliers on the grand Triebel-Lizorkin-Morrey spaces is also obtained.

Keywords: grand Lebesgue spaces; Morrey spaces; Triebel-Lizorkin spaces; Fourier multiplier

MSC 2010: 42B20; 42B30; 42B35; 46E30

1 Introduction

In this article, we introduce and study the grand Triebel-Lizorkin-Morrey spaces. They are the Triebel-Lizorkin-type spaces built on the grand Morrey spaces on Euclidean spaces.

The grand Morrey spaces on Euclidean spaces are extensions of the grand Lebesgue spaces and Morrey spaces. The Morrey spaces were introduced by Morrey [1] to study the solutions of some quasi-linear elliptic partial differential equations. For the studies of the Morrey spaces, the reader may consult previous studies [2,3]. The grand Lebesgue spaces were introduced by Iwaniec and Sbordone [4] to investigate the integrability of Jacobian. Note that the grand Lebesgue spaces introduced in [4] are defined on domain with finite measure. Since the introductions of these function spaces, a number of results from the harmonic analysis had been extended to the grand Lebesgue spaces [5–13].

The grand Morrey spaces on domains with finite measure were introduced in [14–16]. It had been extended to the grand Morrey spaces and the grand Hardy-Morrey spaces on Euclidean spaces in [17,18]. In addition, the boundedness and the mapping properties of the Calderon-Zygmund operators, the spherical maximal functions, the geometric maximal function, the fractional integral operators, and the geometric fractional functions on the grand Morrey spaces were established in [17–19].

The Triebel-Lizorkin spaces were introduced to unify the studies of the Lebesgue spaces, the Sobolev spaces, and the Hardy spaces [20,21]. They were generalized to the Triebel-Lizorkin-Morrey spaces in [22–25], the Triebel-Lizorkin spaces with variable exponents in [26–28], and the weighted Triebel-Lizorkin-Morrey spaces with variable exponent in [29]. Some general approaches on the Triebel-Lizorkin-type spaces are given in [30–32].

The Triebel-Lizorkin spaces also provide important applications on partial differential equation, such as the heat equations and the Navier-Stokes equations [33,34]. The above results on the grand Morrey spaces and the Triebel-Lizorkin spaces motivate us to investigate the Triebel-Lizorkin spaces built on the grand Morrey spaces.

In this article, we generalize the study of the Triebel-Lizorkin-type spaces to the grand Triebel-Lizorkin-Morrey spaces. We show that they possess several real variable characterizations, namely, the Peetre maximal function characterizations, the Lusin area function characterizations, and the g λ ∗ function characterizations. We also show the boundedness of the Fourier multipliers on the grand Triebel-Lizorkin-Morrey spaces. We obtain the above results by the extrapolation theory for the grand Morrey spaces in [17].

This work is organized as follows. The definition and some preliminary results of the grand Morrey space are presented in Section 2. The extrapolation theory for the grand Morrey spaces is also recalled in Section 2. The grand Triebel-Lizorkin-Morrey spaces are defined and studied in Section 3. The Peetre maximal function characterizations, the Lusin area function characterizations, the g λ ∗ function characterizations of the grand Triebel-Lizorkin-Morrey spaces, and the boundedness of the Fourier multipliers on the grand Triebel-Lizorkin-Morrey spaces are also established in Section 3.

2 Preliminaries and definitions

Let ℳ ( R n ) and L loc 1 be the class of Lebesgue measurable functions and the class of locally integrable functions on R n , respectively. For any Lebesgue measurable set F , the Lebesgue measure of F is denoted by ∣ F ∣ . For any x ∈ R n and r > 0 , define B ( x , r ) = { y ∈ R n : ∣ y − x ∣ < r } . Define B = { B ( x , r ) : x ∈ R n , r > 0 } . For any B = B ( x , r ) ∈ B and s ∈ ( 0 , ∞ ) , write s B = B ( x , s r ) . We denote the center of B by c B .

Let S and S ′ be the class of Schwartz functions and tempered distributions, respectively. Let P denote the class of polynomials on R n .

We recall the definition and review some properties of the grand Lebesgue spaces in this section. For any f ∈ L loc 1 and B ∈ B , write

f B = ⨍ B f ( x ) d x = 1 ∣ B ∣ ∫ B f ( x ) d x .

For any p ∈ [ 0 , ∞ ) and B ∈ B , L p ( B ) consists of all Lebesgue measurable function f satisfying

‖ f ‖ L p ( B ) = ⨍ B ∣ f ( x ) ∣ p d x 1 p < ∞ .

For any p ∈ [ 1 , ∞ ] , let p ′ be the conjugate of p . That is, 1 p + 1 p ′ = 1 .

Let B ∈ B , f ∈ ℳ ( R n ) , and s > 0 . Define d f , B ( s ) = 1 ∣ B ∣ ∣ { x ∈ B : ∣ f ( x ) ∣ > s } and f B ∗ ( t ) = inf { s > 0 : d f , B ( s ) ≤ t } , t > 0 .

Definition 2.1

Let p ∈ ( 0 , ∞ ) and B ∈ B . The grand Lebesgue space L p ) ( B ) consists of all f ∈ ℳ ( R n ) satisfying

‖ f ‖ L p ) ( B ) = sup 0 < t < 1 ( 1 − ln t ) − 1 p ∫ t 1 ( f B ∗ ( s ) ) p d s 1 p < ∞ .

The small Lebesgue space L ( p ( B ) consists of all f ∈ ℳ ( R n ) satisfying

‖ f ‖ L ( p ( B ) = ∫ 0 1 ( 1 − ln t ) − 1 p ∫ 0 t ( f B ∗ ( s ) ) p d s 1 p d t t < ∞ .

According to [9, Theorem 2.3], whenever p ∈ ( 1 , ∞ ) , L p ) ( B ) and L ( p ( B ) are rearrangement-invariant Banach function spaces. When p ∈ ( 0 , 1 ) , the grand Lebesgue spaces and the small Lebesgue spaces are rearrangement-invariant quasi-Banach function spaces. The reader is referred to [35, Definition 2.1] for the definition of rearrangement-invariant quasi-Banach function spaces.

When p ∈ ( 1 , ∞ ) , the grand Lebesgue spaces and the small Lebesgue spaces are initially defined in terms of the following norms:

‖ f ‖ L p ) ( B ) ∗ = sup 0 < ε < p − 1 ε ⨍ B ∣ f ( x ) ∣ p − ε d x 1 p − ε ‖ g ‖ L ( p ( B ) ∗ = inf g = ∑ g k ∑ k = 1 ∞ inf 0 < ε < p − 1 ε − 1 p − ε ⨍ B ∣ g ( x ) ∣ ( p − ε ) ′ d x 1 ( p − ε ) ′ .

According to [36, Corollary 3.3 (23)] and [36, Theorem 4.2 (30)], whenever p ∈ ( 1 , ∞ ) , ‖ ⋅ ‖ L p ) ( B ) ∗ and ‖ ⋅ ‖ L ( p ( B ) ∗ are equivalent norms of ‖ ⋅ ‖ L p ) ( B ) and ‖ ⋅ ‖ L ( p ( B ) , respectively.

Since ‖ ⋅ ‖ L p ) ( B ) and ‖ ⋅ ‖ L p ) ( B ) ∗ are mutually equivalent, there is a constant C > 0 such that for any Lebesgue measurable set F

(1) ‖ χ F ‖ L p ) ( B ) ≤ C ‖ χ F ‖ L p ) ( B ) ∗ ≤ C ‖ χ F ‖ L p ( B ) = C ∣ F ∩ B ∣ ∣ B ∣ 1 p .

Additionally, the embedding L p − ε ( B ) ↪ L 1 ( B ) , ε ∈ ( 0 , p − 1 ) guarantees that there are constants C 0 , C 1 > 0 independent of B such that for any f ∈ L p ) ( B ) ,

(2) ‖ f ‖ L 1 ( B ) ≤ C 0 ‖ f ‖ L p ) ( B ) ∗ ≤ C 1 ‖ f ‖ L p ) ( B ) .

We use the quasi-norms ‖ ⋅ ‖ L p ) ( B ) and ‖ ⋅ ‖ L ( p ( B ) instead of ‖ ⋅ ‖ L p ) ( B ) ∗ and ‖ ⋅ ‖ L ( p ( B ) ∗ to define the grand Lebesgue spaces and the small Lebesgue spaces because ‖ ⋅ ‖ L p ) ( B ) and ‖ ⋅ ‖ L ( p ( B ) are well defined for all p ∈ ( 0 , ∞ ) .

Let p , q ∈ ( 0 , ∞ ) . We have

‖ ∣ f ∣ q ‖ L p ) ( B ) 1 q = sup 0 < t < 1 ( 1 − ln t ) − 1 p q ∫ t 1 ( f B ∗ ( s ) ) p q d s 1 p q = ‖ f ‖ L p q ) ( B ) .

Consequently, the q -convexification of L p ) ( B ) is L p q ) ( B ) .

The results in [8] and [9, Section 3] assert that the associate space of L p ) ( B ) is L ( p ′ ( B ) and vice versa. Particularly, we have the Hölder inequality [9, Theorem 2.5]

(3) ⨍ B ∣ f ( x ) g ( x ) ∣ d x ≤ C ‖ f ‖ L p ) ( B ) ‖ g ‖ L ( p ′ ( B )

and the norm conjugate formula [9, Corollary 2.10]

(4) C 0 ‖ f ‖ L p ) ( B ) ≤ sup ⨍ B ∣ f ( x ) g ( x ) ∣ d x : ‖ g ‖ L ( p ′ ( B ) ≤ 1 ≤ C 1 ‖ f ‖ L p ) ( B )

for some C 0 , C 1 > 0 .

For any f ∈ L loc 1 , the Hardy-Littlewood maximal operator M is defined as

M f ( x ) = sup B ∋ x ⨍ B ∣ f ( y ) ∣ d y , x ∈ R n ,

where the supremum is taken over all ball B containing x . It is well known that for any p ∈ ( 1 , ∞ ) , M is bounded on L p ( R n ) , see [37, Chapter 1, Theorem 1].

We now state the definition of the grand Morrey space [17, Definition 3.1].

Definition 2.2

Let p ∈ ( 0 , ∞ ) and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. The grand Morrey space M u p ) ( R n ) consists of all Lebesgue measurable functions f satisfying

‖ f ‖ M u p ) ( R n ) = sup B ( x , r ) ∈ B 1 u ( x , r ) ‖ f ‖ L p ) ( B ( x , r ) ) < ∞ .

For any B = B ( x , r ) ∈ B , we also write u ( B ) = u ( x , r ) .

The grand Morrey space is an extension of the grand Lebesgue spaces and the classical Morrey spaces. The classical Morrey spaces were introduced by Morrey [1] for the study of quasi-linear elliptic partial differential equation. Since then, a huge number of extensions of Morrey spaces had been given [3,14–16,38,39].

The following proposition assures that the characteristic functions of balls belong to the grand Morrey space.

Proposition 2.1

Let p ∈ ( 0 , ∞ ) and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. If there exists a constant C > 0 such that for any x ∈ R n ,

(5) C r − n ⁄ p < u ( x , r ) , r > 1 ,

(6) C ≤ u ( x , r ) , r ≤ 1 ,

then χ B ∈ M u p ) ( R n ) .

For the proof of the above result, the reader is referred to [17, Proposition 3.1].

We recall the definition of small block space from [17, Definition 3.2].

Definition 2.3

Let p ∈ ( 1 , ∞ ) and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. A Lebesgue measurable function b is a small ( p , u ) -block if there exists a B ∈ B such that supp b ⊂ B and

(7) ‖ b ‖ L ( p ( B ) ≤ 1 u ( B ) ∣ B ∣ .

We write b ∈ b u ( p if b is a small ( p , u ) -block.

The small block space B u ( p ( R n ) consists of all f ∈ ℳ ( R n ) satisfying

‖ f ‖ B u ( p ( R n ) = inf ∑ i = 1 ∞ ∣ λ i ∣ : f = ∑ i = 1 ∞ λ i b i , { b i } i = 1 ∞ ⊂ b u ( p < ∞ .

We recall an operator from [17, Definition 4.2] used in the extrapolation theorem of the grand Morrey spaces.

Definition 2.4

Let p ∈ ( 1 , ∞ ) and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. Suppose that u satisfies (6),

C r − n ⁄ p ′ < u ( x , r ) , r > 1 , x ∈ R n , ∑ k = 0 ∞ u ( 2 k B ) ≤ C u ( B ) , ∀ B ∈ B .

For any f ∈ B u ( p ( R n ) , define

ℛ p , u f = ∑ k = 0 ∞ M k f 2 k ‖ M k ‖ B u ( p ( R n ) → B u ( p ( R n )

where M 0 f = ∣ f ∣ , M k is the k -iterations of M and ‖ M k ‖ B u ( p ( R n ) → B u ( p ( R n ) is the operator norm of M on B u ( p ( R n ) .

The above operator is also named as the Rubio de Francia operator.

We are now ready to present the extrapolation theorem of the grand Morrey space from [17, Theorem 4.1].

Theorem 2.2

Let p ∈ ( 0 , ∞ ) and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. Suppose that there is a s ∈ ( 0 , p ) such that u satisfies (5), (6), and

(8) ∑ k = 0 ∞ u ( 2 k B ) s ≤ C u ( B ) s , ∀ B ∈ B

for some C > 0 . If F ∈ M u p ) ( R n ) and G is a Lebesgue measurable function such that for every

(9) ω ∈ { ℛ ( p ⁄ s ) ′ , u s h : h ∈ b u s ( ( p ⁄ s ) ′ } ,

we have

(10) ∫ R n ∣ G ( x ) ∣ s ω ( x ) d x ≤ C 0 ∫ R n ∣ F ( x ) ∣ s ω ( x ) d x < ∞ ,

for some C 0 > 0 , then G ∈ M u p ) ( R n ) and there is a constant C 1 such that

(11) ‖ G ‖ M u p ) ( R n ) ≤ C 1 ‖ F ‖ M u p ) ( R n ) .

Note that we modify the presentation of the above result from [17, Theorem 4.1] in order to match with the notations used in this work.

3 Grand Triebel-Lizorkin-Morrey spaces

In this section, we introduce and study the grand Triebel-Lizorkin-Morrey spaces. We establish the Peetre maximal function characterizations, the Lusin area function characterizations, and the g λ ∗ function characterizations of the grand Triebel-Lizorkin-Morrey spaces and the boundedness of the Fourier multipliers on the grand Triebel-Lizorkin-Morrey spaces.

We begin with the definition of the grand Triebel-Lizorkin-Morrey spaces.

Definition 3.1

Let α ∈ R , p , q ∈ ( 0 , ∞ ) , and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. The grand Triebel-Lizorkin-Morrey space F ˙ q , u α , p ) consists of those f ∈ S ′ ⁄ P satisfying

(12) ‖ f ‖ F ˙ q , u α , p ) = ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q M u p ) ( R n ) < ∞ ,

where φ ∈ S satisfies

(13) supp φ ˆ ⊆ { x ∈ R n : 1 ⁄ 2 ≤ ∣ x ∣ ≤ 2 }

(14) ∣ φ ˆ ( ξ ) ∣ ≥ C , 3 ⁄ 5 ≤ ∣ x ∣ ≤ 5 ⁄ 3

for some C > 0 and φ j ( x ) = 2 j n φ ( 2 j x ) .

Let p , q ∈ ( 0 , ∞ ) , α ∈ R , and ω : R n → ( 0 , ∞ ) be a Lebesgue measurable function, the weighted Triebel-Lizorkin space F ˙ p α , q ( ω ) consists of all f ∈ S ′ ⁄ P satisfying

‖ f ‖ F ˙ p α , q ( ω ) = ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q L p ( ω ) < ∞ ,

where φ satisfies (13) and (14), refer [40,41] and [42, p. 124].

We are going to show that the grand Triebel-Lizorkin-Morrey spaces are well defined. That is, the grand Triebel-Lizorkin-Morrey space is independent of the functions φ ∈ S satisfying (13) and (14).

Theorem 3.1

Let α ∈ R , p , q ∈ ( 0 , ∞ ) , and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. Let φ , ψ ∈ S satisfy (13)–(14). If there exists an s ∈ ( 0 , p ) such that u satisfies (5), (6), and (8), then there exist constants C 0 , C 1 > 0 such that for any f ∈ S ′ ⁄ P , we have

(15) C 0 ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q M u p ) ( R n ) ≤ ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ ψ j ∣ ) q 1 q M u p ) ( R n ) ≤ C 1 ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q M u p ) ( R n ) .

Proof

It suffices to show that whenever f ∈ S ′ / P satisfies

(16) ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q M u p ) ( R n ) < ∞ ,

then

(17) ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ ψ j ∣ ) q 1 q ∈ M u p ) ( R n )

and

(18) ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ ψ j ∣ ) q 1 q M u p ) ( R n ) ≤ C 1 ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q M u p ) ( R n ) .

The above result gives the right-hand side of (15). The left-hand side of (15) follows from interchanging the functions φ and ψ .

According to [42, Remark 2.6 and Proposition 10.14], for any ω ∈ A 1 , we have

(19) K 0 ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q L p ( ω ) ≤ ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ ψ j ∣ ) q 1 q L p ( ω ) ≤ K 1 ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q L p ( ω )

for some K 0 , K 1 > 0 .

Let f ∈ S ′ / P satisfy (16). In view of [17, (5.1)], we have the embedding

(20) M u p ) ( R n ) ↪ ⋂ h ∈ b u s ( ( p ⁄ s ) ′ L s ( ℛ ( p ⁄ s ) ′ , u s h ) .

Therefore, we find that

∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q ∈ ⋂ h ∈ b u s ( ( p ⁄ s ) ′ L s ( ℛ ( p ⁄ s ) ′ , u s h ) .

According to [17, (4.3)], we have

{ ℛ ( p ⁄ s ) ′ , u s h : h ∈ b u s ( ( p ⁄ s ) ′ } ⊂ A 1 ⊆ A max { 1 , p } .

Therefore, (10) is fulfilled with

F = ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q and G = ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ ψ j ∣ ) q 1 q .

Thus, we are allowed to apply Theorem 2.2. Consequently, Theorem 2.2 yields (17) and (18).□

We now turn to the real variable characterization of the grand Triebel-Lizorkin-Morrey spaces. We begin with the Peetre maximal function characterization of the grand Triebel-Lizorkin-Morrey spaces.

Let f ∈ S ′ ⁄ P and

φ ∈ S 0 = h ∈ S : ∫ R n x γ h ( x ) d x = 0 , γ ∈ ( N ∪ { 0 } ) n .

Let a ∈ ( 0 , ∞ ) . For any j ∈ Z , the Peetre maximal function of f is defined as

( φ j ∗ f ) a ( x ) = sup y ∈ R n φ j ∗ f ( y ) ( 1 + 2 j ∣ x − y ∣ ) a , x ∈ R n .

Theorem 3.2

Let α ∈ R , p , q ∈ ( 0 , ∞ ) , and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. Let a ∈ ( n min ( 1 , p , q ) + 1 , ∞ ) . If φ ∈ S satisfies (13)–(14) and there exists a s ∈ ( 0 , p ) such that u satisfies (5), (6), and (8), then there exist constants C 0 , C 1 > 0 such that for any f ∈ S ′ ⁄ P , we have

C 0 ‖ f ‖ F ˙ q , u α , p ) ≤ ∑ j = − ∞ ∞ ( 2 j α ( φ j ∗ f ) a ) q 1 q M u p ) ( R n ) ≤ C 1 ‖ f ‖ F ˙ q , u α , p ) .

According to [43, Theorem 3.1], we have the Peetre maximal function characterization of the weighted Triebel-Lizorkin spaces, therefore, the above result follows from Theorem 3.1 in [43] Theorem 2.2. For brevity, we omit the details. For the maximal function characterization of the weighted Triebel-Lizorkin spaces, the reader may also refer [41].

In addition, we also have the Lusin area function characterization and the g λ ∗ function characterization of F ˙ q , u α , p ) .

Theorem 3.3

Let α ∈ R , p , q ∈ ( 0 , ∞ ) , and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. If φ ∈ S satisfies (13)–(14) and there exists a s ∈ ( 0 , p ) such that u satisfies (5), (6), and (8), then there exist constants C 0 , C 1 > 0 such that for any f ∈ S ′ ⁄ P , we have

C 0 ‖ f ‖ F ˙ q , u α , p ) ≤ ∑ j = − ∞ ∞ 2 j α q 1 ∣ B ( ⋅ , 2 − j ) ∣ ∫ B ( ⋅ , 2 − j ) ∣ φ j ∗ f ( y ) ∣ q d y 1 q M u p ) ( R n ) ≤ C 1 ‖ f ‖ F ˙ q , u α , p ) .

Theorem 3.4

Let α ∈ R , p , q ∈ ( 0 , ∞ ) , and u : R n × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. Let λ ∈ ( n min ( 1 , p , q ) + 1 n , ∞ ) . If φ ∈ S satisfies (13)–(14) and there exists a s ∈ ( 0 , p ) such that u satisfies (5), (6), and (8), then there exist constants C 0 , C 1 > 0 such that for any f ∈ S ′ ⁄ P , we have

C 0 ‖ f ‖ F ˙ q , u α , p ) ≤ ∑ j = − ∞ ∞ 2 j α q 2 j n ∫ R n ∣ φ j ∗ f ( y ) ∣ q ( 1 + 2 j ∣ ⋅ − y ∣ ) λ n q d y 1 q M u p ) ( R n ) ≤ C 1 ‖ f ‖ F ˙ q , u α , p ) .

The above characterizations follow from [43, Theorems 3.11 and 3.14], respectively. For brevity, we omit the details.

We now turn to the study of Fourier multiplier on the grand Triebel-Lizorkin-Morrey spaces.

Let l ∈ N and m ∈ C l ( R n \ { 0 } ) . The Fourier multiplier T m is defined as

T m f ^ = m f ˆ , f ∈ S .

We recall the boundedness of the Fourier multipliers on the weighted Triebel-Lizorkin spaces from [43, Theorem 4.8]. Note that the following theorem is a special case of [43, Theorem 4.8].

Theorem 3.5

Let α ∈ R , p , q ∈ ( 0 , ∞ ) , ω ∈ A max { 1 , p } , and l ∈ ( 1 min { 1 , p , q } + 1 + n 2 , ∞ ) . If m ∈ C l ( R n \ { 0 } ) , then the Fourier multiplier T m can be extended to be a bounded operator on F ˙ p α , q ( ω ) .

Theorem 3.5 follows from [43, Lemmas 2.9, 2.12, and Theorem 4.8]. For simplicity, we refer the reader to [43] for the details.

Theorem 3.6

Let α ∈ R , p , q ∈ ( 0 , ∞ ) , and l ∈ ( 1 min { 1 , p , q } + 1 + n 2 , ∞ ) . Suppose that u satisfies (5) and (6). If m ∈ C l ( R n \ { 0 } ) and there exists a s ∈ ( 0 , p ) satisfying l ∈ ( 1 min { 1 , s , q } + 1 + n 2 , ∞ ) and (8), then the Fourier multiplier T m can be extended to be a bounded operator on the grand Triebel-Lizorkin-Morrey space F ˙ q , u α , p ) .

Proof

Let f ∈ F ˙ q , u α , p ) . In view of (20), we obtain

f ∈ ⋂ h ∈ b u s ( ( p ⁄ s ) ′ F ˙ s α , q ( ℛ ( p ⁄ s ) ′ , u s h ) .

As { ℛ ( p ⁄ s ) ′ , u s h : h ∈ b u s ( ( p ⁄ s ) ′ } ⊂ A 1 ⊆ A max { 1 , p } , Theorem 3.5 guarantees that T m f is well defined. Furthermore, for any ω ∈ { ℛ ( p ⁄ s ) ′ , u s h : h ∈ b u s ( ( p ⁄ s ) ′ } , we have a constant C > 0 such that for any f ∈ F ˙ q , u α , p )

∫ R n ∑ j = 0 ∞ ( 2 j α ∣ T m f ∗ φ j ∣ ) q s q ω ( x ) d x ≤ C ∫ R n ∑ j = 0 ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q s q ω ( x ) d x .

Therefore, (10) is fulfilled with

F = ∑ j = − ∞ ∞ ( 2 j α ∣ f ∗ φ j ∣ ) q 1 q and G = ∑ j = 0 ∞ ( 2 j α ∣ T m f ∗ φ j ∣ ) q 1 q .

We are allowed to apply Theorem 2.2, which yields a constant C > 0 such that for any f ∈ F ˙ q , u α , p )

□ ‖ T m f ‖ F ˙ q , u α , p ) ≤ C ‖ f ‖ F ˙ q , u α , p ) .

When p ∈ ( 1 , ∞ ) , we see that whenever u satisfies (5), (6), and

(21) ∑ k = − ∞ ∞ u ( 2 k B ) ≤ C u ( B ) , ∀ B ∈ B ,

the conditions in the above theorem are fulfilled because we can select a s ∈ ( 1 , p ) such that l > 1 min { 1 , s , q } + 1 + n 2 and (21) with s > 1 yields

∑ k = − ∞ ∞ u ( 2 k B ) s ≤ ∑ k = − ∞ ∞ u ( 2 k B ) s ≤ C u ( B ) s , ∀ B ∈ B .

Acknowledgments

The author thanks the reviewers for their valuable suggestions which improved the presentation of this paper.

Funding information: Not applicable.
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Conflict of interest: Not applicable.
Data availability statement: Not applicable.

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Received: 2023-08-15

Revised: 2024-05-26

Accepted: 2024-10-30

Published Online: 2025-02-11

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2024-0085

Keywords for this article

grand Lebesgue spaces; Morrey spaces; Triebel-Lizorkin spaces; Fourier multiplier

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