Atomic Decomposition for Mixed Morrey Spaces
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Atomic Decomposition for Mixed Morrey Spaces Toru Nogayama1 · Takahiro Ono2 · Daniel Salim3 · Yoshihiro Sawano2 Received: 10 March 2020 / Accepted: 3 September 2020 © Mathematica Josephina, Inc. 2020
Abstract In this paper, we consider some norm estimates for mixed Morrey spaces considered by the first author. Mixed Lebesgue spaces are realized as a special case of mixed Morrey spaces. What is new in this paper is a new norm estimate for mixed Morrey spaces that is applicable to mixed Lebesgue spaces as well. An example shows that the condition on parameters is optimal. As an application, the Olsen inequality adapted to mixed Morrey spaces can be obtained. Keywords Mixed Morrey spaces · Fractional integral operators · Atoms Mathematics Subject Classification 41A17 · 42B35
1 Introduction We obtain some decomposition results for mixed Lebesgue spaces and mixed Morrey spaces. Let us first recall the definition of mixed Lebesgue spaces. Let 0 < q1 , q2 , . . . , qn ≤ ∞ be constants. Write q = (q1 , q2 , . . . , qn ). Then define the mixed Lebesgue norm · L q by
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Toru Nogayama [email protected] Daniel Salim [email protected] Yoshihiro Sawano [email protected]
1
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji-shi, Tokyo 192-0397, Japan
2
Department of Mathematics, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan
3
Analysis and Geometry Group, Bandung Institute of Technology, Bandung, Indonesia
123
T. Nogayama et al.
⎛ f L q ≡ ⎝
R
···
R
qq3
q2 R
| f (x1 , x2 , . . . , xn )|q1 dx1
q1
2
dx2
⎞ q1n · · · dxn ⎠
.
A natural modification for xi is made when qi = ∞. We define the mixed Lebesgue space L q (Rn ) to be the set of all measurable function f on Rn with f L q < ∞. Here and below we use the notation q = (q1 , q2 , . . . , qn ), q∗ = (q1∗ , q2∗ , . . . , qn∗ ), t = (t1 , t2 , . . . , tn ) to denote the vectors in Rn . The aim of this paper is to develop a theory of decompositions based on the following boundedness of the maximal operator. Here and below, for 0 ≤ a ≤ b ≤ ∞, a ≤ q ≤ b means that a ≤ qi ≤ b for all i = 1, 2, . . . , n. Theorem 1 Assume that 1 ≤ tk < min{q1 , . . . , qk } ≤ ∞ (k = 1, . . . , n). Define χ Q (x) f χQ L t χ Q L t Q∈Q
M (t) f (x) = sup
for a measurable function f . Then for all measurable functions f M (t) f L q f L q . For 0 < p < ∞, and 0 < q < ∞ satisfying 1 n ≤ p qj n
j=1
recall that mixed Morrey spaces are defined by the norm given by f
p Mq
≡
sup
Q∈D (Rn )
|Q|
1 1 p−n
n
1 j=1 q j
f χQ L q
for measurable functions f : Rn → C, where D(Rn ) denotes the set of all dyadic cubes. We denote by Q(Rn ) the set of all cubes whose edges are parallel to the coordinate axes. If there is no confusion, we substitute D and Q for D(Rn ) and Q(Rn ), respectively. Using Theorem 1, we seek to prove the following decomposition result about the functions in mixed Morrey spaces. This result extends [26, Chapter 8, Lemma
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