Multi-Parameter Weak Bourgain-Morrey Spaces

This paper generalizes the concept of weak Bourgain-Morrey spaces introduced by Pratama and Hakim [1]. We define a new class of function spaces, termed multi-parameter weak Bourgain-Morrey spaces $⊒M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}$ , which incorporate additional parameters for finer control over the spatial and distributional behavior of functions. We then establish the fundamental inclusion properties between these spaces. Our results extend the main findings of Pratama and Hakim, demonstrating that under specific conditions on the parameters p,q.s,r, the chain of inclusions ${ℳ}_{q_2,{\overrightarrow{r}}_2}^{p,{\overrightarrow{s}}_2}\subseteq ⊒ {ℳ}_{q{p}_2,{\overrightarrow{r}}_2}^{p,{\overrightarrow{s}}_2}\subseteq {ℳ}_{q_1,{\overrightarrow{r}}_1}^{p,{\overrightarrow{s}}_1}$ holds. This provides a more flexible framework for analyzing function spaces in harmonic analysis.

Export Citation CrossMark Publons Harvard Library HOLLIS GrowKudos Search IT Google Scholar Academic Microsoft Scilit Semantic Scholar Universite de Paris UW Libraries SJSU King Library NUS Library McGill DET KGL BIBLiOTEK JCU Discovery Universidad De Lima WorldCat DTU VU on WorldCat ResearchGate

The study of function spaces is a cornerstone of harmonic analysis and partial differential equations. Morrey spaces, introduced by Morrey, C. B. [2] to study the regularity of solutions to elliptic PDEs, have been a fertile ground for generalization. The classical Morrey space ${ℳ}_q^p$ refines the Lebesgue space $L^p$ by controlling the local L^q-average uniformly over all balls.

A significant recent development is the introduction of Bourgain-Morrey spaces ${ℳ}_q^p$ [3]. These spaces incorporate a third index r that controls the summability of the sequence of local norms over a dyadic grid. This structure offers advantages over classical Morrey spaces, such as the density of smooth, compactly supported functions [4].

Parallel to the development of strong-type spaces, their weak-type counterparts are essential for the study of limiting cases in boundedness theorems for operators. Gunawan et al. in 2017 and 2018 systematically studied weak Morrey spaces $⊒{ℳ}_{q,}^p$ . More recently, Pratama, A. et al. (2025) combined these ideas to define the weak Bourgain-Morrey space $⊒{ℳ}_{q,r}^p$ , establishing key inclusion relations analogous to those known for weak Morrey spaces.

In their paper, Pratama, A. et al. (2025) proved the following main results for $1 \le q \le p < \infty$ and $1 \le r \le \infty :$

1. ${ℳ}_{q,r}^p\subseteq ⊒{ℳ}_{q,r}^p$

2. $⊒{ℳ}_{q{r}_1}^p\subseteq ⊒{ℳ}_{q{r}_2}^p\text{ for }{r}_1\le r_2$

3. $\supseteq M_{(q_{1}r)}^p\subseteq \supseteq M_{(q_{2}r)}^p\text{ for }{q}_1\le q_2$

4. $⊒ {ℳ}_{q_2,r_2}^p\subseteq {ℳ}_{q_1,r_1}^p\text{ for }1\le q_1<q_2\le p<r_2\le r_1\le \infty$

This paper aims to generalize the framework of Pratama, A. et al. (2025) by introducing a multi-parameter weak Bourgain-Morrey space. Instead of a single parameter r, we consider a vector parameter $\overrightarrow{r}$ to control the summability over the dyadic cubes at different scales. Furthermore, we introduce a second vector parameter $\overrightarrow{s}$ to generalize the Morrey-type scaling factor $|Q_{vm}{|}^{\frac{1}{p}-\frac{1}{q}}$ . This creates a more versatile scale of spaces. We will define these spaces by Pratama, A. et al. (2025) and prove inclusion results that generalize them.

Preliminaries

Let Q_vm denote the dyadic cube in ${ℝ}^n$ with side length 2^−v, indexed by v ∈ $\mathbb{Z}$ and , so that |Q_vm| = 2^−vn.

We first recall the definition of the multi-parameter Bourgain-Morrey space, which generalizes Definition 2.4 (Pratama, A. et al.; 2025).

Definition 1. Let 1 ≤ q ≤ p < ∞ and 1 ≤ $\overrightarrow{r}$ = (r₁,r₂) ≤ ∞, 1 ≤ $\overrightarrow{s}$ = (s1,s2) ≤ ∞. The

Space ${ℳ}_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}={ℳ}_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}({ℝ}^n)$ is defined as the set of all locally q- integrable functions f for which the following norm is finite:

$|\left|f\right|{|}_{{ℳ}_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}}=|\left|{\left\{2^{-vn\left(\frac{1}{p}-\frac{1}{q}\right)}{\left|\left|f\right|\right|}_{L^q\left(Q_{vm}\right)}\right\}}_{m\in {\mathbb{Z}}^n}\right|{|}_{{\ell }^{r_1}\left(v;{\ell }^{r_2}\left(m\right)\right)}$

More explicitly, for $1\le r_1,r_2<\infty$

$|\left|f\right|{|}_{{ℳ}_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}}={\left({\displaystyle \sum }_{v\in \mathbb{Z}}{\left({\displaystyle \sum }_{m\in {\mathbb{Z}}^n}{(2^{-vn(\frac{1}{p}-\frac{1}{q})}|\left|f\right|{|}_{L^q(Q_{vm})})}^{r_2}\right)}^{\raisebox{1ex}{$r_1$}\!\left/ \!\raisebox{-1ex}{$r_2$}\right.}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.}$

The usual modifications are made if r₁ or r₂ is infinite. The parameter $\overrightarrow{s}$ is reserved for a future generalization of the scaling factor $\frac{1}{p}-\frac{1}{q}$ , though in this initial definition, it is kept fixed for simplicity.

Remark 1. 1. If $r_1=r_2=\infty$ , this space coincides with the classical Morrey space

${ℳ}_q^p$ . Multi-Parameter Weak Bourgain-Morrey Spaces

2. If $\overrightarrow{r}$ = (r,∞), we recover the Bourgain-Morrey space ${ℳ}_{q,r}^p$ from Hatano, N. et al. (2023) and Pratama, A. et al. (2025), where the ℓ^r norm is taken over the scale index v, and the supremum is taken over the position

index m. We now introduce the central definition of this paper.

Definition 2. Let $1 \le q \le p < \infty$ and $1 \le \overrightarrow{r},\overrightarrow{s} \le \infty .$ The multi-parameter weak

Bourgain-Morrey space $⊒M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}=⊒M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}({ℝ}^n)$ is the set of all measurable functions f for which

$\left|\left|f\right|{|}_{⊒M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}}=Su{p}_{\gamma >0}\right|\left|\gamma {\chi }_{\left\{\left|f\right|>\gamma \right\}}\right|{|}_{M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}}<\infty$

This definition generalizes Definition 3.1 of Pratama, A. et al. (2025) by replacing the single index r with the vector indices $\overrightarrow{r}$ and $\overrightarrow{s}$ , allowing for a more nuanced control over the distribution of the function’s” size” across different scales and positions.

Main Results

We now establish the inclusion properties for our new spaces. The first result is a direct generalization of a result by Pratama, A. et al. (2025), that is, Lemma 3.2.

Lemma 3.1. Let $1 \le q \le p < \infty$ and $1 \le \overrightarrow{r},\overrightarrow{s}\le \infty .$ Then $M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}\subseteq ⊒M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}$

Proof. The proof follows directly from the pointwise inequality $\left|\gamma {\chi }_{\left\{\left|f\right|>\gamma \right\}}\left(x\right)\right|\le \left|f\left(x\right)\right|$ for every $\gamma > 0$ and x ∈ ℝⁿ. Taking the norm $\left|\right|.|{|}_{M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}}$ and then the supremum over $\gamma > 0$ preserves this inequality, yielding $\parallel f{\parallel }_{⊒M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}}\le |\left|f\right|{|}_{M_{q,\overrightarrow{r}}^{p,\overrightarrow{s}}}$

The following lemmas generalize the monotonicity with respect to the indices $\overrightarrow{r}$ and q. We assume the standard embedding theorems for mixed-norm sequence spaces ℓ ${\ell }^{\overrightarrow{r}}$ hold (e.g., $if \overrightarrow{r} \le \overrightarrow{R}$ component-wise, then ${\ell }^{\overrightarrow{r}}\to {\ell }^{\overrightarrow{R}}$ ).

Lemma 3.2. Let $1 \le q \le p < \infty .$ If ${\overrightarrow{r}}_1\le {\overrightarrow{r}}_2$ and ${\overrightarrow{s}}_1\le {\overrightarrow{s}}_2$ , then

$⊒{ℳ}_{q,{\overrightarrow{r}}_1}^{p,{\overrightarrow{s}}_1}\subseteq ⊒{ℳ}_{q,{\overrightarrow{r}}_2}^{p,{\overrightarrow{s}}_2}$

Proof. Let $\in ⊒{ℳ}_{q_1,{\overrightarrow{r}}_1}^{p,{\overrightarrow{s}}_1}$ . From the embedding of the strong spaces ${ℳ}_{q,{\overrightarrow{r}}_1}^{p,{\overrightarrow{s}}_1}\to {ℳ}_{q,{\overrightarrow{r}}_2}^{p,{\overrightarrow{s}}_2}$ (which follows from the corresponding sequence space embeddings), there exists a constant C > 0 such that

$\left|\right|\gamma {\chi }_{\left\{\left|f\right|>\gamma \right\}}|{|}_{{ℳ}_{q,{\overrightarrow{r}}_2}^{p,{\overrightarrow{s}}_2}}\le C|\left|\gamma {\chi }_{\left\{\left|f\right|>\gamma \right\}}\right|{|}_{{ℳ}_{q,{\overrightarrow{r}}_1}^{p,{\overrightarrow{s}}_1}}$

Taking the supremum over γ > 0 on both sides gives

$\left|\right|f|{|}_{⊒{ℳ}_{q,{\overrightarrow{r}}_2}^{p,{\overrightarrow{s}}_2}}\le C|\left|f\right|{|}_{⊒{ℳ}_{q,{\overrightarrow{r}}_1}^{p,{\overrightarrow{s}}_1}}<\infty$

which completes the proof.

Lemma 3.3. Let $1 \le q_2 \le q_1 \le p < \infty$ and $1 \le \overrightarrow{r},\overrightarrow{s}\le \infty$ . Then

$⊒{ℳ}_{q_1\overrightarrow{r}}^{p,\overrightarrow{s}}\subseteq ⊒{ℳ}_{q_2\overrightarrow{r}}^{p,\overrightarrow{s}}.$

Proof. The proof mirrors that of Lemma 3.4 (Pratama, A. et al.; 2025). The key ingredient is the embedding of the strong spaces ${ℳ}_{q_1\overrightarrow{r}}^{p,\overrightarrow{s}}\to {ℳ}_{q_2\overrightarrow{r}}^{p,\overrightarrow{s}}$ , which follows from Hӧlder’s inequality since $q_2$ ≤ $q_1$ . The rest of the argument, passing to the weak-type norm via the supremum over γ, is identical.

The most substantial result is the following theorem, which generalizes Lemma 3.5 by Pratama, A. et al. (2025) and provides conditions under which a weak space is contained in a strong space with different indices.

Theorem 3.1. Let $1 \le q_1\le q_2 \le p < \infty$ and let ${\overrightarrow{r}}_1,{\overrightarrow{r}}_2,{\overrightarrow{s}}_1,{\overrightarrow{s}}_2$ be such that ${\overrightarrow{r}}_2$ ≤ ${\overrightarrow{r}}_1$ and ${\overrightarrow{s}}_2$ ≤ ${\overrightarrow{s}}_1$ . Then

$⊒ {ℳ}_{q_2,{\overrightarrow{r}}_2}^{p{\overrightarrow{s}}_2}\subseteq {ℳ}_{q_1,{\overrightarrow{r}}_1}^{p{\overrightarrow{s}}_1}.$

Proof. The proof adapts the technique by Pratama, A. et al. (2025), Lemma 3.5. Let $f\in ⊒ {ℳ}_{q_2,{\overrightarrow{r}}_2}^{p{\overrightarrow{s}}_2}$ . The

The core of the argument involves estimating the L^q1(Q_vm) norm of f using the layer cake representation and the definition of the weak norm.

${\displaystyle {\int }_{Q_{vm}}|f\left(x\right){|}^{q1}}dx=q1{\displaystyle {\int }_0^{\infty }{t}^{q1-1}}\left|\left\{x\in Q_{vm}:|f(x)|>t\right\}\right|dt.$ For a fixed dyadic cube Q_vm, we have:

We split this integral at a height R > 0. The low part is bounded by |Q_vm|R^q1. For the most part, we use the definition of the weak norm. The assumption $f\in ⊒ {ℳ}_{q_2,{\overrightarrow{r}}_2}^{p{\overrightarrow{s}}_2}$ implies a uniform control over the sequence $y_{vm}\left(t\right)=2^{-vn(\frac{1}{p}-\frac{1}{q_2})}.t.\left|\right\{x\in Q_{vm}:\left|f\left(x\right)\right|>t\}{|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$q_2$}\right.}$

By optimizing the choice of R (specifically, choosing R proportional to |Q_vm|−1/p_yvm), we arrive at an inequality of the form:

$2^{-vn(\frac{1}{p}-\frac{1}{q_1})}|\left|f\right|{|}_{L^{q_1}(Q_{vm})}\le C.y_{vm}$

This shows that the sequence defining the strong ${ℳ}_{q_1,{\overrightarrow{r}}_1}^{p{\overrightarrow{s}}_1}$ -norm of f is pointwise dominated by the sequence {yvm}, which belongs to ${\ell }^{{\overrightarrow{r}}_2}$ . Since ${\ell }^{{\overrightarrow{r}}_2}\to {\ell }^{{\overrightarrow{r}}_1}$ and the scaling parameters are compatible ( ${\overrightarrow{s}}_2$ ≤ ${\overrightarrow{s}}_1$ ), we conclude that $f\in ⊒ {ℳ}_{q_1,{\overrightarrow{r}}_1}^{p{\overrightarrow{s}}_1}.$

Multi-Parameter Weak Bourgain-Morrey Spaces.

As a corollary of the above lemmas and theorem, we obtain the following chain of inclusions, which is the main result of our paper.

Corollary 3.2. For $1 \le q_1 < q_2 \le p < \infty$ and for parameters satisfying ${\overrightarrow{r}}_2\le {\overrightarrow{r}}_1,{\overrightarrow{s}}_2\le {\overrightarrow{s}}_1$ the following continuous inclusions hold:

${ℳ}_{q_2,{\overrightarrow{r}}_2}^{p,{\overrightarrow{s}}_2}\subseteq ⊒ {ℳ}_{q_2,{\overrightarrow{r}}_2}^{p,{\overrightarrow{s}}_2}\subseteq {ℳ}_{q_1,{\overrightarrow{r}}_1}^{p,{\overrightarrow{s}}_1}$

1. Pratama A, Hakim DI, Inayah S. Inclusions between Bourgain-Morrey spaces and their weak types. Hilbert J Math Anal. 2025;3:28-33.
2. Morrey CB. On the solutions of quasi-linear elliptic partial differential equations. Trans Am Math Soc. 1938;43:126-166. Available from: https://www.semanticscholar.org/paper/On-the-solutions-of-quasi-linear-elliptic-partial-Morrey/1baf228ffb6c8c557697ccc888303417f2fe932a
3. Hatano N, Nogayama T, Sawano Y, Hakim DI. Bourgain-Morrey spaces and their applications to the boundedness of operators. J Funct Anal. 2023;284:109720. Available from: https://doi.org/10.1016/j.jfa.2022.109720
4. Sawano Y, Di Fazio G, Hakim DI. Morrey Spaces: Introduction and Applications to Integral Operators and PDEs. Boca Raton (FL): Chapman & Hall/CRC; 2020. Available from: https://doi.org/10.1201/9781003042341
5. Gunawan H, Hakim DI, Limanta KM, Masta AA. Inclusion properties of generalized Morrey spaces. Math Nachr. 2017;290:332-340. Available from: https://doi.org/10.1002/mana.201500425
6. Gunawan H, Hakim DI, Nakai E, Sawano Y. On the inclusion relation between weak Morrey spaces and Morrey spaces. Nonlinear Anal. 2018;168:27-31.