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Dachun Yang

Publications and source records attributed to Dachun Yang.

At least 73 records · Page 4Linked to original sources
arXiv2024

Parabolic Muckenhoupt Weights Characterized by Parabolic Fractional Maximal and Integral Operators with Time Lag

In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) via these weights under an extra mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned two-weighted boundedness of the uncentered parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón--Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

math.AP↗
arXiv2024

Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains

Let $n \ge 2$ and $Ω\subset \mathbb{R}^n$ be a bounded Lipschitz domain. In this article, we establish first-order global regularity estimates in the scale of BMO spaces on $Ω$ for weak solutions to the second-order elliptic equation $\mathrm{div}(A \nabla u) = \mathrm{div} , \boldsymbol{f}$ in $Ω$. This is achieved under minimal regularity assumptions on $Ω$ and the coefficient matrix $A$, utilizing the pointwise multiplier characterization of the BMO space on $Ω$. As an application, we also obtain global estimates of $\nabla u$ in the Lebesgue space $L^1(Ω)$ when $\boldsymbol{f}$ belongs to the Hardy space on $Ω$.

math.AP↗
arXiv2024

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces II: Sharp Boundedness of Almost Diagonal Operators

This article is the second one of three successive articles of the authors on the matrix-weighted Besov-type and Triebel--Lizorkin-type spaces. In this article, we obtain the sharp boundedness of almost diagonal operators on matrix-weighted Besov-type and Triebel--Lizorkin-type sequence spaces. These results not only possess broad generality but also improve several existing related results in various special cases covered by this family of spaces. This improvement depends, on the one hand, on the notion of $A_p$-dimensions of matrix weights and their properties introduced in the first article of this series and, on the other hand, on a careful direct analysis of sequences of averages avoiding maximal operators. While a recent matrix-weighted extension of the Fefferman--Stein vector-valued maximal inequality would provide an alternative route to some of our results in the restricted range of function space parameters $p,q\in(1,\infty)$, our approach covers the full scale of exponents $p\in(0,\infty)$ and $q\in(0,\infty]$ that is relevant in the theory of function spaces.

math.FA↗
arXiv2024

A compact extension of Journé's $T1$ theorem on product spaces

We prove a compact version of the $T1$ theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator $T$ admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal $\mathrm{CMO}$ condition, and the product $\mathrm{CMO}$ condition, then $T$ can be extended to a compact operator on $L^p(w)$ for all $1<p<\infty$ and $w \in A_p(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$. Even in the unweighted setting, it is the first time to give a compact extension of Journé's $T1$ theorem on product spaces.

math.CA↗
arXiv2024

New approach to affine Moser-Trudinger inequalities via Besov polar projection bodies

We extend the affine inequalities on $\mathbb{R}^n$ for Sobolev functions in $W^{s,p}$ with $1 \leq p < n/s$ obtained recently by Haddad-Ludwig [16, 17] to the remaining range $p \geq n/s$. For each value of $s$, our results are stronger than affine Moser-Trudinger and Morrey inequalities. As a byproduct, we establish the analog of the classical $L^p$ Bourgain-Brezis-Mironescu inequalities related to the Moser-Trudinger case $p=n$. Our main tool is the affine invariant provided by Besov polar projection bodies.

math.MG↗
arXiv2023

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces III: Characterizations of Molecules and Wavelets, Trace Theorems, and Boundedness of Pseudo-Differential Operators and Calderón--Zygmund Operators

This is the last one of three successive articles by the authors on matrix-weighted Besov-type and Triebel--Lizorkin-type spaces $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$. In this article, the authors establish the molecular and the wavelet characterizations of these spaces. Furthermore, as applications, the authors obtain the optimal boundedness of trace operators, pseudo-differential operators, and Calderón--Zygmund operators on these spaces. Due to the sharp boundedness of almost diagonal operators on their related sequence spaces obtained in the second article of this series, all results presented in this article improve their counterparts on matrix-weighted Besov and Triebel--Lizorkin spaces $\dot B^{s}_{p,q}(W)$ and $\dot F^{s}_{p,q}(W)$. In particular, even when reverting to the boundedness of Calderón--Zygmund operator on unweighted Triebel--Lizorkin spaces $\dot F^{s}_{p,q}$, these results are still better.

math.FA↗
arXiv2023

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces I: $A_p$-Dimensions of Matrix Weights and $φ$-Transform Characterizations

Let $s\in{\mathbb R}$, $q\in (0,\infty]$, and $τ\in[0,\infty)$. It is well known that Besov-type spaces $\dot B^{s,τ}_{p,q}$ with $p\in (0,\infty]$ and Triebel--Lizorkin-type spaces $\dot F^{s,τ}_{p,q}$ with $p\in (0,\infty)$ when $τ\in [0,\infty)$ or with $p\in (0,\infty]$ when $τ=0$ on $\mathbb{R}^n$ consist of a general family of function spaces that cover not only the well-known Besov and Triebel--Lizorkin spaces $\dot B^{s}_{p,q}$ and $\dot F^{s}_{p,q}$ (when $τ=0$) but also several other function spaces of interest, such as Morrey spaces and $Q$ spaces. In three successive articles, the authors develop a complete real-variable theory of matrix-weighted Besov-type spaces $\dot B^{s,τ}_{p,q}(W)$ and matrix-weighted Triebel--Lizorkin-type spaces $\dot F^{s,τ}_{p,q}(W)$ on $\mathbb{R}^n$, where $W$ is a matrix-valued Muckenhoupt $A_p$ weight. This article is the first one, whose main novelty exists in that the authors introduce the new concept, $A_p$-dimensions of matrix weights, and intensively study their properties, especially those elaborate properties expressed via reducing operators. The authors then introduce the spaces $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$ and, using $A_p$-dimensions and their nice properties, the authors establish the $φ$-transform characterization of $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$. The $A_p$-dimensions of matrix weights and their properties also enable the authors to obtain the sharp boundedness of almost diagonal operators on related sequence spaces in the subsequent second article and the optimal characterizations of molecules and wavelets, trace theorems, and the optimal boundedness of pseudo-differential operators and Calderón--Zygmund operators in the subsequent third article.

math.FA↗
arXiv2023

New Characterizations and Properties of Matrix $A_\infty$ Weights

We provide several new characterizations of $A_{p,\infty}$-matrix weights, originally introduced by A. Volberg as matrix-valued substitutes of the classical $A_\infty$ weights. In analogy with the notion of $A_p$-dimension of matrix weights introduced in our previous work, we introduce the concepts of the lower and the upper dimensions of $A_{p,\infty}$-matrix weights, which enable us to obtain sharp estimates related to their reducing operators. In a follow-up work, these results will play a key role in the study of function spaces with $A_{p,\infty}$-matrix weights, which extends earlier results in the more restricted class of $A_p$-matrix weights.

math.FA↗
arXiv2023

Parabolic John-Nirenberg spaces with time lag

We introduce a parabolic version of the so-called John-Nirenberg space that is a generalization of functions of parabolic bounded mean oscillation. Parabolic John-Nirenberg inequalities, which give weak type estimates for the oscillation of a function, are shown in the setting of the parabolic geometry with a time lag. Our arguments are based on a parabolic Calderón-Zygmund decomposition and a good lambda estimate. Chaining arguments are applied to change the time lag in the parabolic John-Nirenberg inequality.

math.CA↗
arXiv2023

A unified approach to self-improving property via K-functionals

In this paper we obtain new quantitative estimates that improve the classical inequalities: Poincaré-Ponce, Gaussian Sobolev, and John-Nirenberg. Our method is based on the K-functionals and allows one to derive self-improving type inequalities. We show the optimality of the method by obtaining new Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limiting formulas. In particular, we derive these formulas for fractional powers of infinitesimal generators of operator semigroups on Banach spaces.

math.FA↗
arXiv2023

Brezis--Seeger--Van Schaftingen--Yung-Type Characterization of Homogeneous Ball Banach Sobolev Spaces and Its Applications

Let $γ\in\mathbb{R}\setminus\{0\}$ and $X(\mathbb{R}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $Ω=\mathbb{R}^n$ or $Ω\subset\mathbb{R}^n$ is an $(\varepsilon,\infty)$-domain for some $\varepsilon\in(0,1]$. In this article, the authors prove that a function $f$ belongs to the homogeneous ball Banach Sobolev space $\dot{W}^{1,X}(Ω)$ if and only if $f\in L_{\mathrm{loc}}^1(Ω)$ and $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{y\inΩ:\ |f(\cdot)-f(y)|>λ|\cdot-y|^{1+\fracγ{p}}\}} \left|\cdot-y\right|^{γ-n}\,dy \right]^\frac{1}{p}\right\|_{X(Ω)}<\infty, $$ where $p\in[1,\infty)$ is related to $X(\mathbb{R}^n)$. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain--Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when $X(Ω):=L^q(\mathbb{R}^n)$ with $1<p=q<\infty$, while it is still new even when $X(Ω):=L^q(\mathbb{R}^n)$ with $1\leq p<q<\infty$.

math.FA↗
arXiv2023

Extension Theorem and Bourgain--Brezis--Mironescu-Type Characterization of Ball Banach Sobolev Spaces on Domains

Let $Ω\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space satisfying some extra mild assumptions, and $\{ρ_ν\}_{ν\in(0,ν_0)}$ with $ν_0\in(0,\infty)$ a $ν_0$-radial decreasing approximation of the identity on $\mathbb{R}^n$. In this article, the authors establish two extension theorems, respectively, on the inhomogeneous ball Banach Sobolev space $W^{m,X}(Ω)$ and the homogeneous ball Banach Sobolev space $\dot{W}^{m,X}(Ω)$ for any $m\in\mathbb{N}$. On the other hand, the authors prove that, for any $f\in\dot{W}^{1,X}(Ω)$, $$ \lim_{ν\to0^+} \left\|\left[\int_Ω\frac{|f(\cdot)-f(y)|^p}{ |\cdot-y|^p}ρ_ν(|\cdot-y|)\,dy \right]^\frac{1}{p}\right\|_{X(Ω)}^p =\frac{2π^{\frac{n-1}{2}}Γ(\frac{p+1}{2})}{Γ(\frac{p+n}{2})} \left\|\,\left|\nabla f\right|\,\right\|_{X(Ω)}^p, $$ where $Γ$ is the Gamma function and $p\in[1,\infty)$ is related to $X(\mathbb{R}^n)$. Using this asymptotics, the authors further establish a characterization of $W^{1,X}(Ω)$ in terms of the above limit. To achieve these, the authors develop a machinery via using a method of the extrapolation, two extension theorems on weighted Sobolev spaces, and some recently found profound properties of $W^{1,X}(\mathbb{R}^n)$ to overcome those difficulties caused by that the norm of $X(\mathbb{R}^n)$ has no explicit expression and that $X(\mathbb{R}^n)$ might be neither the reflection invariance nor the translation invariance. This characterization has a wide range of generality and can be applied to various Sobolev-type spaces, such as Morrey [Bourgain--Morrey-type, weighted (or mixed-norm or variable), local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, all of which are new.

math.FA↗
arXiv2023

Fourier Transform of Anisotropic Hardy Spaces Associated with Ball Quasi-Banach Function Spaces and Its Applications to Hardy--Littlewood Inequalities

Let $A$ be a general expansive matrix and $X$ be a ball quasi-Banach function space on $\mathbb R^n$, whose certain power (namely its convexification) supports a Fefferman--Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy--Littlewood maximal operator. Let $H_X^A(\mathbb{R}^n)$ be the anisotropic Hardy space associated with $A$ and $X$. The authors first prove that the Fourier transform of $f\in H^A_{X}(\mathbb{R}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. Moreover, the authors obtain a pointwise inequality that the function $F$ is less than the product of the anisotropic Hardy space norm of $f$ and a step function with respect to the transpose matrix of the expansive matrix $A$. Applying this, the authors further induce a higher order convergence for the function $F$ at the origin and give a variant of the Hardy--Littlewood inequality in $H^A_{X}(\mathbb{R}^n)$. All these results have a wide range of applications. Particularly, the authors apply these results, respectively, to classical (variable and mixed-norm) Lebesgue spaces, Morrey spaces, Lorentz spaces, Orlicz spaces, Orlicz-slice spaces, and local generalized Herz spaces and, even on the last five function spaces, the obtained results are completely new.

math.FA↗
arXiv2023

Hardy Spaces Associated with Non-Negative Self-Adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Let $(\mathcal{X},d,μ)$ be a doubling metric measure space in the sense of R. R. Coifman and G. Weiss, $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ satisfying the Davies--Gaffney estimate, and $X(\mathcal{X})$ a ball quasi-Banach function space on $\mathcal{X}$ satisfying some mild assumptions. In this article, the authors introduce the Hardy type space $H_{X,\,L}(\mathcal{X})$ by the Lusin area function associated with $L$ and establish the atomic and the molecular characterizations of $H_{X,\,L}(\mathcal{X}).$ As an application of these characterizations of $H_{X,\,L}(\mathcal{X})$, the authors obtain the boundedness of spectral multiplies on $H_{X,\,L}(\mathcal{X})$. Moreover, when $L$ satisfies the Gaussian upper bound estimate, the authors further characterize $H_{X,\,L}(\mathcal{X})$ in terms of the Littlewood--Paley functions $g_L$ and $g_{λ,\,L}^\ast$ and establish the boundedness estimate of Schrödinger groups on $H_{X,\,L}(\mathcal{X})$. Specific spaces $X(\mathcal{X})$ to which these results can be applied include Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces. This shows that the results obtained in the article have extensive generality.

math.FA↗
arXiv2023

Anisotropic Ball Campanato-Type Function Spaces and Their Applications

Let $A$ be a general expansive matrix and let $X$ be a ball quasi-Banach function space on $\mathbb R^n$, which supports both a Fefferman--Stein vector-valued maximal inequality and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce some anisotropic ball Campanato-type function spaces associated with both $A$ and $X$, prove that these spaces are dual spaces of anisotropic Hardy spaces $H_X^A(\mathbb R^n)$ associated with both $A$ and $X$, and obtain various anisotropic Littlewood--Paley function characterizations of $H_X^A(\mathbb R^n)$. Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both $A$ and $X$, further induces their Carleson measure characterizations. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of quasi-norm $\|\cdot\|_X$, the authors embed $X$ under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space.

math.FA↗
arXiv2023

Generalized Brezis--Van Schaftingen--Yung Formulae and Their Applications in Ball Banach Sobolev Spaces

Let $X$ be a ball Banach function space on $\mathbb{R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on both $X$ and the associate space of its convexification, the authors successfully recover the homogeneous ball Banach Sobolev semi-norm $\|\,|\nabla f|\,\|_X$ via the functional $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{y\in\mathbb{R}^n:\ |f(\cdot)-f(y)|>λ|\cdot-y|^{1+\fracγ{q}}\}} \left|\cdot-y\right|^{γ-n}\,dy\right]^\frac{1}{q}\right\|_X $$ for any distributions $f$ with $|\nabla f|\in X$, as well as the corresponding limiting identities with the limit for $λ\to\infty$ when $γ\in(0,\infty)$ or the limit for $λ\to0^+$ when $γ\in(-\infty,0)$, where $γ\in\mathbb{R}\setminus\{0\}$ and where $q\in(0,\infty)$ is related to $X$. In particular, some of these results are still new even when $X:=L^p(X)$ with $p\in[1,\infty)$. As applications, the authors obtain some fractional Sobolev-type and some fractional Gagliardo--Nirenberg-type inequalities in the setting of $X$. All these results are of quite wide generality and are applied to various specific function spaces, including Morrey, mixed-norm (or variable or weighted) Lebesgue, Lorentz, and Orlicz (or Orlicz-slice) spaces, some of which are new even in all these special cases. The novelty of this article is to use both the method of the extrapolation and the boundedness of the Hardy--Littlewood maximal operator on both $X$ and the associate space of its convexification to overcome the essential difficulties caused by the deficiency of both the translation and the rotation invariance and an explicit expression of the norm of $X$.

math.FA↗
arXiv2023

Fractional Fourier Transforms Meet Riesz Potentials and Image Processing

Via chirp functions from fractional Fourier transforms, the authors introduce fractional Riesz potentials related to chirp functions, establish their relations with fractional Fourier transforms, fractional Laplace operators related to chirp functions, and fractional Riesz transforms related to chirp functions, and obtain their boundedness on rotation invariant spaces related to chirp functions. Finally, the authors give the numerical image simulation of fractional Riesz potentials related to chirp functions and their applications in image processing. The main novelty of this article is to propose a new image encryption method for the double phase coding based on the fractional Riesz potential related to chirp functions. The symbol of fractional Riesz potentials related to chirp functions essentially provides greater degrees of freedom and greatly makes the information more secure.

math.FA↗
arXiv2023

Products and Commutators of Martingales in $H_1$ and ${\rm BMO}$

Let $f:=(f_n)_{n\in \mathbb{Z}_+}$ and $g:=(g_n)_{n\in \mathbb{Z}_+}$ be two martingales related to the probability space $(Ω,\mathcal F,\mathbb P)$ equipped with the filtration $(\mathcal F_n)_{n\in \mathbb{Z}_+}.$ Assume that $f$ is in the martingale Hardy space $H_1$ and $g$ is in its dual space, namely the martingale $\rm BMO.$ Then the semi-martingale $f\cdot g:=(f_ng_n)_{n\in \mathbb{Z}_+}$ may be written as the sum $$f\cdot g=G(f, g)+L( f,g).$$ Here $L( f,g):=(L( f,g)_n)_{n\in\mathbb{Z}_+}$ with $L( f,g)_n:=\sum_{k=0}^n(f_k-f_{k-1})(g_k-g_{k-1)})$ for any $n\in\mathbb{Z}_+$, where $f_{-1}:=0=:g_{-1}$. The authors prove that $L( f,g)$ is a process with bounded variation and limit in $L^1,$ while $G(f,g)$ belongs to the martingale Hardy-Orlicz space $H_{\log}$ associated with the Orlicz function $$Φ(t):=\frac{t}{\log(e+t)},\quad \forall\, t\in[0,\infty).$$ The above bilinear decomposition $L^1+H_{\log}$ is sharp in the sense that, for particular martingales, the space $L^1+H_{\log}$ cannot be replaced by a smaller space having a larger dual. As an application, the authors characterize the largest subspace of $H_1$, denoted by $H^b_1$ with $b\in {\rm BMO}$, such that the commutators $[T, b]$ with classical sublinear operators $T$ are bounded from $H^b_1$ to $L^1$. This endpoint boundedness of commutators allow the authors to give more applications. On the one hand, in the martingale setting, the authors obtain the endpoint estimates of commutators for both martingale transforms and martingale fractional integrals. On the other hand, in harmonic analysis, the authors establish the endpoint estimates of commutators both for the dyadic Hilbert transform beyond doubling measures and for the maximal operator of Cesàro means of Walsh--Fourier series.

math.PR↗