arXiv subjects

Zongze Zeng

Publications and source records attributed to Zongze Zeng.

4 recordsLinked to original sources

arXiv2026

Atomic Characterization and Its Applications of Matrix-Weighted Variable Hardy Spaces

In this article, by means of the matrix-weighted grand maximal function we first introduce the variable Hardy space $H^{p(\cdot)}_W$ on $\mathbb{R}^n$ with the $\mathscr{A}_{p(\cdot),\infty}$ matrix weight $W$ and with the variable exponent $p(\cdot)$ having globally log-H\"older continuity, and then via using several different convex body valued maximal functions we establish its various maximal function equivalent characterizations. Combining a refined Whitney decomposition with both the convex body valued maximal function and its corresponding convex-body reducing operator, we obtain the atomic characterization of $H^{p(\cdot)}_W$. As applications, we give its dual space and establish the boundedness of Calder\'on--Zygmund operators from $H^{p(\cdot)}_W$ to the matrix-weighted variable Lebesgue space $L^{p(\cdot)}_W$ and to itself. This approach to establishing atomic characterization differs from all previous ones.

math.FA↗

arXiv2026

Variable Muckenhoupt $A_\infty$ Weights

In this article, with introducing concepts of variable scalar $\mathcal{A}_{p(\cdot),\infty}$ weights and variable matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we seek a comprehensive theory of $A_\infty$ weights within the framework of variable exponent spaces. We first show that a weight belongs to $\mathcal{A}_{p(\cdot),\infty}$ if and only if its $p(\cdot)$-th power is an $A_\infty$ weight. Using this, we characterize the $\mathcal{A}_{p(\cdot),\infty}$ condition by the minimal operator. Then we establish the reverse Hölder's inequality for $\mathcal{A}_{p(\cdot),\infty}$ weights in variable Lebesgue spaces with explicit constants and, combining this with the previously established relationship between $\mathcal{A}_{p(\cdot),\infty}$ weights and $A_\infty$ weights, we prove that, for any weight $w$, the reverse Hölder's inequality holds in variable Lebesgue spaces if and only if $w$ is an $\mathcal{A}_{p(\cdot),\infty}$ weight. For the matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we first show the existence of the reducing operators for matrix $\mathscr{A}_{p(\cdot),\infty}$ weights and then, combining the matrix $\mathscr{A}_{p(\cdot),\infty}$ weights with the scalar $\mathcal{A}_{p(\cdot),\infty}$ weights, we establish the reverse Hölder's inequality for $\mathscr{A}_{p(\cdot),\infty}$ weights in variable Lebesgue spaces. Finally, for further applications to variable matrix-weighted function spaces, we introduce the upper and the lower dimensions for $\mathscr{A}_{p(\cdot),\infty}$ weights and use these concepts to establish the sharp estimate involving reducing operators.

math.FA↗

arXiv2026

Variable Matrix-Weighted Besov Spaces

In this article, using variable matrix ${\mathscr{A}}_{p(\cdot),\infty}$ weights, we introduce the matrix-weighted variable Besov space $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$ and the corresponding averaging variable Besov space $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{A})$ and prove that they are equivalent. Applying this, we establish the $φ$-transform characterization of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$. By this and via first establishing the boundedness of $α$-convexification $η$-type operators on variable Lebesgue spaces, we obtain the boundedness of almost diagonal operators on the sequence space $b^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$ related to $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, which is further used to establish various decomposition characterizations of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, respectively, in terms of molecules, wavelets, and atoms. Applying the wavelet decomposition of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, we obtain the trace theorem and the extension properties of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, and, applying the molecular characterization, we obtain the boundedness of Calderón--Zygmund operators on $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$.

math.FA↗

arXiv2021

Nontriviality of Riesz--Morrey Spaces

In this article, the authors completely answer an open question, presented in [Banach J. Math. Anal. 15 (2021), no. 1, 20], via showing that the Riesz--Morrey space is truly a new space larger than a particular Lebesgue space with critical index. Indeed, this Lebesgue space is just the real interpolation space of the Riesz--Morrey space for suitable indices. Moreover, the authors further show the aforementioned inclusion is also proper, namely, this embedding is sharp in some sense, via constructing two nontrivial spare functions, respectively, on $\mathbb{R}^n$ and any given cube $Q_0$ of $\mathbb{R}^n$ with finite side length. The latter constructed function is inspired by the striking function constructed by Dafni et al. [J. Funct. Anal. 275 (2018), 577--603]. All the proofs of these results strongly depend on some exquisite geometrical analysis on cubes of $\mathbb{R}^n$. As an application, the relationship between Riesz--Morrey spaces and Lebesgue spaces is completely clarified on all indices.

math.FA↗