arXiv subjects

Dachun Yang

Publications and source records attributed to Dachun Yang.

At least 37 records · Page 2Linked to original sources

arXiv2026

Sobolev--Morrey Spaces and Divergence-Form Degenerate Second-Order Elliptic Equations on Domains with Higher Co-Dimensional Boundaries

In this article, we study the weighted homogeneous Sobolev--Morrey spaces on domains in $\mathbb{R}^n$ with higher co-dimensional boundaries. Precisely, we systematically establish a real-variable theory of these spaces, including completeness, embedding theorems, Riesz potential characterizations, continuity, trace and extension theorems, and complex interpolation. Applying the boundedness of the trace and the extension operators, we obtain sharp weighted a priori estimates for solutions to the Dirichlet problem of divergence-form degenerate second-order elliptic equations on such domains in weighted Lebesgue spaces. The absence of a boundary manifold structure of these domains poses some essential difficulties, which are overcome by using some tools, such as the intrinsic properties of distance weights and the geometric structure of domains, different from those available in Lipschitz domains.

math.AP↗

arXiv2026

Stable Image Reconstruction via Two-Parameter Power-Scale Variation Minimization

In this article, we introduce a power-scale variation (PSV$_{a,p}$) with two tunable parameters: the sparsity-inducing exponent $p\in(0,1]$ and the scaling factor $a\in(0,\infty)$. By minimizing the PSV$_{a,p}$, we establish stable reconstructions in both the gradient and the image domains under the restricted isometry property (RIP) framework. Furthermore, we design an iteratively re-weighted least squares algorithm IRLSPSV to solve the unconstrained PSV$_{a,p}$ minimization. Numerical experiments demonstrate its superior performance and broad applicability. The main novelties are: (i) the PSV$_{a,p}$ minimization enjoys great flexibility and wide applicability due to its two tunable parameters $a$ and $p$, (ii) as $a\to\infty$, the PSV$_{a,p}$ minimization reduces to the $p$-th power total variation (TV$_p$) minimization and, even in this limiting case, the established RIP condition for image reconstruction is also new, (iii) the derived RIP upper bound $\overlineδ$ is proved to be asymptotically optimal in $a$ for gradient recovery, (iv) sensitivity analysis confirms the distinct roles of $a$ and $p$, thereby motivating a practical parameter tuning scheme for the proposed model.

cs.IT↗

arXiv2026

Compactness and Its Applications of Sobolev Spaces Associated with Ball Banach Function Spaces

Let $N\in\mathbb{N}\cap[2,\infty),$ $Ω$ be a bounded Lipschitz domain in $\mathbb{R}^N$, and $X(Ω)$ be a ball Banach function space on $Ω.$ In this article, under some mild assumptions, we establish a compactness theorem for Sobolev spaces associated with $X(Ω)$. Different from the fractional Sobolev space, our proof is based on an elaborate decomposition of bounded Lipschitz domains and its corresponding weighted fractional Poincaré inequality on each piece. As applications, we obtain the fractional Poincaré inequality in $X(Ω)$ that, for any $s\in (s_0,1)$ and $f\in X(Ω)$, \begin{align*} \|f-f_Ω\|_{X(Ω)} \lesssim(1-s)^{\frac 1q} \left\|\left[\int_Ω\frac{|f(\cdot)-f(y)|^q}{|\cdot-y|^{N+sq}}\,dy \right]^{\frac{1}{q}} \right\|_{X(Ω)}, \end{align*} where $s_0$ is a given positive constant and the implicit positive constant is independent of $s$ and $f$. Using this, we further establish the well-posedness of a weighted Triebel--Lizorkin type nonlocal variational problem. These results are of wide generality and, even when they are applied to Morrey spaces, weighted Lebesgue spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, and Orlicz-slice spaces, the obtained results are also new.

math.FA↗

arXiv2026

A Novel Two-Parameter Penalty: Relaxation Degree Analysis and Sparse Signal Recovery

In this article, we introduce a nonconvex two-parameter penalty function $P_{a,p}$, parameterized by $a\in(0,\infty)$ and $p\in(0,1]$, and the relaxation degree RD$_P$ for a separable nonconvex penalty function $P$. Based on $P_{a,p}$, we further propose the $P_{a,p}$ minimization framework for sparse signal recovery. This framework generalizes the TL1 minimization model established by S. Zhang and J. Xin (corresponding to the special case $p=1$) and provides a unified and flexible family of nonconvex penalty functions for sparse signal recovery. Using the sparse convex-combination technique, we establish both exact and stable sparse signal recovery under the restricted isometry property (RIP). To efficiently solve the resulting nonconvex optimization problem, we apply a modified iteratively re-weighted least squares method and the difference of convex functions algorithm (DCA) to develop the IRLSTLp algorithm for unconstrained $P_{a,p}$ minimization and prove some convergence results. Finally, some numerical experiments are conducted to show the flexibility of the $P_{a,p}$ minimization framework, the robustness of the IRLSTLp, and also the utility of the relaxation degree.

math.FA↗

arXiv2026

A Counterexample to the Necessity of the Vanishing Carleson Condition for VMO Poisson Kernels

In [Problem 3.2.23, CBMS Regional Conference Series in Mathematics 83, 1994], Kenig asked whether the vanishing Carleson condition is the necessary and sufficient for the logarithm of the Poisson kernel of a perturbation of the Laplacian on the unit ball in $\mathbb{R}^n$ belonging to the space VMO. The sufficiency was proved by Escauriaza [Israel J. Math. 1996] and extended by Milakis, Pipher, and Toro [Contemp. Math. 2014] to more general setting. In this article, using the technique of bi-Lipschitz mappings, we construct a counterexample to show that the vanishing Carleson condition is not necessary and hence give a negative answer to the aforementioned problem.

math.AP↗

arXiv2026

A Counterexample to Kenig's Interpolation Problem for Sobolev Spaces with Zero Boundary Conditions

Let $n\in \mathbb N\cap[2,\infty)$. In this article, we show that there exists a bounded $C^1$ domain $Ω\subset \mathbb R^n$ such that, for any given $s\in(1,2)\setminus\{\frac32\}$, \begin{align*} \left[H_0^1(Ω),H^2(Ω)\cap H_0^1(Ω)\right]_{s-1} =H^s(Ω)\cap H_0^1(Ω)=H_0^s(Ω) \end{align*} with equivalent norms, but \begin{align*} \left[H_0^1(Ω),H^2(Ω)\cap H_0^1(Ω)\right]_{\frac12} \subsetneqq H^{\frac32}(Ω)\cap H_0^1(Ω), \end{align*} which provides a counterexample to Problem 3.3.19 of Kenig in [CBMS Regional Conf. Ser. in Math. 83, 1994]. As applications, we prove that for such a domain $Ω$ \begin{align*} H^2(Ω)\cap H_0^1(Ω)\subsetneqq D(-Δ_D) \end{align*} (the domain of the Dirichlet Laplacian operator $-Δ_D$ on $Ω$) and construct a solution of the homogeneous heat equation with zero Dirichlet boundary condition, which does not belong to $L^2((0,T);H^2(Ω)\cap H_0^1(Ω))$ for any given $T\in(0,\infty)$.

math.AP↗

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-Hölder 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 Calderón--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

Limit Properties at Critical Indices of Linear Canonical Riesz Potentials and Their Applications to Security of Multi-Image Encryption

In this article we introduce the linear canonical Riesz potential (for short, LCRP) and give its symbol in terms of linear canonical transforms. Driven by image processing, we establish the convergence/divergence of these LCRPs for different kinds of functions. Concretely, for grating functions, we prove that their classical Riesz potentials diverge, whereas their LCRP converge due to the key role of chirp functions. For the characteristic function ${\mathbf 1}_P$ of a convex polygon $P$, we show that the limit of its Riesz potential at any non-boundary point $\boldsymbol{x}$ equals ${\mathbf 1}_P(\boldsymbol{x})$, but its limit at the boundaries differ from ${\mathbf 1}_P$, while it is known that, for any Schwartz function $f$, the limit of its Riesz potential at any point $\boldsymbol{x}$ always equals $f(\boldsymbol{x})$. Based on these and the inverse operator of the LCRP (namely the linear canonical Laplacian operator), we propose an asymmetric cascaded LCRP method for the multi-image encryption and create an efficient and secure cryptosystem. Systematic security evaluations, including sensitivity, statistical, noise attack, and occlusion attack analyses, demonstrate its robustness and its security. Even for a single image, the proposed method is more efficient than the known encryption approach based on the fractional Riesz potential. The novelty of these results lies in that the convergence and the divergence of LCRTs at the critical indices, respectively, for ``good" Schwartz functions and for ``bad" discrete image functions essentially affect the security of image encryption and decryption.

cs.CR↗

arXiv2026

Real-variable theory of function spaces with operator-valued $A_p$ weights in Banach spaces

While the theory of matrix-weighted function spaces is well established, the majority of previous results in the infinite-dimensional operator-valued setting deal with "no go" theorems, showing the impossibility of some prospective generalizations. However, we show that a complete real-variable theory of Besov and Triebel-Lizorkin spaces with operator-valued Muckenhoupt $A_p$ weights can still be developed, once correctly formulated. This covers operator-weighted extensions of results like the $φ$-transform characterization in terms of discrete sequence spaces, the boundedness of almost diagonal operators, and applications to the $T(1)$ theorem and trace/extension theorems. A key tool is a version of the reverse Hölder inequality, which is weak enough to follow from the operator-valued $A_p$ condition (unlike a variant that had to be imposed as an additional assumption in some previous works), yet strong enough to be used much like its classical counterpart. In contrast to the established scalar and matrix-weighted theories, our approach cannot build on operator-weighted $L^p$ results, as these fail in a very definite way. We also strengthen the existing "no go" statements in Hilbert spaces, showing (among other counterexamples) that every infinite-dimensional Banach space has an operator-valued $A_p$ weight $V$ for which the Hilbert transform is unbounded on $L^p(V)$. This is nontrivial, since the lack of Hilbert space structure also complicates the construction of $A_p$ weights. Building on results from Banach space theory, we achieve this unboundedness by combining two distinct methods in two different classes of spaces (so-called $K$-convex ones and those that are {\em not} UMD), whose union covers all Banach spaces.

math.FA↗

arXiv2026

Real-Variable Theory of Hardy--Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property

Let $(X,\mathbf{q},μ)$ be an ultra-RD-space with upper dimension $n\in(0,\infty)$; i.e., it is a quasi-ultrametric space of homogeneous type whose measure $μ$ satisfies an additional reverse doubling property. Let $\mathrm{ind\,}(X,\mathbf{q})\in(0,\infty]$ denote its lower smoothness index, as introduced by Mitrea et al. In this monograph, the authors first construct a new approximation of the identity on quasi-ultrametric spaces of homogeneous type, achieving a maximal degree of smoothness $0<\varepsilon\preceq\mathrm{ind\,}(X,\mathbf{q})$. This fundamental tool is then used to derive sharp homogeneous (as well as inhomogeneous) continuous/discrete Calderón reproducing formulae on ultra-RD-spaces. As applications, the authors establish Littlewood--Paley function characterizations for both Hardy spaces and Triebel--Lizorkin spaces on ultra-RD-spaces. The authors further introduce Hardy--Lorentz spaces $H^{p,q}_\ast(X)$ via the grand maximal function, with the sharp range $p\in(\frac{n}{n+\mathrm{ind\,}(X,\mathbf{q})},\infty)$ and $q\in(0,\infty]$, and provide their real-variable characterizations using radial/non-tangential maximal functions, (finite) atoms, molecules, and various Littlewood--Paley functions. Based on these characterizations, the authors prove a duality theorem between Hardy--Lorentz spaces and Campanato--Lorentz spaces, establish a real interpolation theorem for Hardy--Lorentz spaces, and derive boundedness results for Calderón--Zygmund operators on them. It should be emphasized that many of the main results in this monograph are indeed established in the more general setting of quasi-ultrametric spaces of homogeneous type.

math.FA↗

arXiv2026

Real-variable theory of matrix-weighted multi-parameter Besov--Triebel--Lizorkin-type spaces

We develop a comprehensive theory for a general class of multi-parameter function spaces of Besov-Triebel-Lizorkin type, with a matrix weight. We prove the equivalence of different quasi-norms, the identification of function and sequence spaces via the $φ$-transform, the boundedness of almost diagonal operators and multi-parameter singular integrals under minimal assumptions, molecular and wavelet characterisations, and Sobolev-type embedding theorems. We identify matrix-weighted $L^p$ spaces, Sobolev spaces, and multi-parameter BMO spaces as examples of our general scale of spaces. Thus, our result on the boundedness of multi-parameter singular integrals on these spaces is seen as an extension, with a different method, of a recent theorem of Domelevo et al. [J. Math. Anal. Appl. 2024] on matrix-weighted $L^p$ spaces. For this theory, we develop several tools of independent interest. Many previous results were restricted to integrability exponents $p\in(1,\infty)$, while Besov-Triebel-Lizorkin spaces naturally involve the full range $p\in(0,\infty)$. We extend the definition of multi-parameter $A_p$ matrix weights to $p\in(0,1]$ and establish their basic properties, culminating in the $L^p$-boundedness of a matrix-weighted strong maximal operator (suitably rescaled when $p\in(0,1]$) for all $p\in(0,\infty)$. For $p\in(1,\infty)$, this is due to Vuorinen [Adv. Math. 2024] by convex-set-valued techniques of Bownik and Cruz-Uribe [arXiv 2022; Math. Ann. (to appear)]; the lack of convexity requires us to develop a new approach that works for all $p\in(0,\infty)$. We also need and prove a multi-parameter extension of Carleson-type embeddings from Frazier and Roudenko [Math. Ann. 2021] but attributed by them to F. Nazarov. We prove the necessity of the conditions of the new embedding using a nontrivial elaboration of Carleson's classical counterexample [Mittag-Leffler Rep. 1974].

math.FA↗

arXiv2026

Product Hardy Spaces on Spaces of Homogeneous Type: Discrete Product Calderón-Type Reproducing Formula, Atomic Characterization, and Product Calderón--Zygmund Operators

Let $i\in\{1,2\}$ and $X_i$ be a space of homogeneous type in the sense of Coifman and Weiss with the upper dimension $ω_i$. Also let $η_i$ be the smoothness index of the Auscher--Hytönen wavelet function $ψ^{k_i}_{α_i}$ on $X_i$. In this article, for any $p\in(\max\{\frac{ω_1}{ω_1+η_1},\frac{ω_2}{ω_2+η_2}\}, 1]$, by regarding the product Carleson measure space $\mathrm{CMO}^p_{L^2}(X_1\times X_2)$ as the test function space and its dual space $(\mathrm{CMO}^p_{L^2}(X_1\times X_2))'$ as the corresponding distribution space, we introduce the product Hardy space $H^p(X_1\times X_2)$ in terms of wavelet coefficients. Moreover, we establish an atomic characterization of this product Hardy space and, as an application, obtain a criterion for the boundedness of linear operators from product Hardy spaces to corresponding Lebesgue spaces. To escape the wavelet reproducing formula, which is not useful for this atomic characterization because the wavelets have no bounded support, we establish a new discrete product Calderón-type reproducing formula, which holds in the product Hardy space and has bounded support. This reproducing formula also leads to the boundedness of product Calderón--Zygmund operators on the product Hardy space.

math.FA↗

arXiv2026

Matrix-Weighted Poincaré-Type Inequalities with Applications to Logarithmic Hajłasz--Besov Spaces on Spaces of Homogeneous Type

Let ${\mathcal {X}}$ be a space of homogeneous type. In this article, based on the reducing operators of matrix $A_p$-weights, the authors introduce the vector-valued Hajłasz gradient sequences and establish some related matrix-weighted Poincaré-type inequalities on ${\mathcal {X}}$. As an application, the authors introduce the matrix-weighted logarithmic Besov spaces on ${\mathcal {X}}$ and establish their pointwise characterization via Hajłasz gradient sequences. The novelty of this article lies in that, by means of both the $A_p$ dimension and its properties of matrix $A_p$-weights and the wavelet reproducing formula with exponential decay of P. Auscher and T. Hytönen, all the main results get rid of the dependence on the reverse doubling conditions of both weights and ${\mathcal {X}}$ under consideration and these results are also completely new even for unweighted logarithmic Besov spaces on ${\mathcal {X}}$.

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↗

arXiv2026

One-Sided and Parabolic BLO Spaces with Time Lag and Their Applications to Muckenhoupt $A_1$ Weights and Doubly Nonlinear Parabolic Equations

In this article, we first introduce the one-sided BLO space $\mathrm{BLO}^+(\mathbb{R})$ and characterize it, respectively, in terms of the one-sided Muckenhoupt class $A_1^+(\mathbb{R})$ and the one-sided John--Nirenberg inequality. Using these, we establish the Coifman--Rochberg type decomposition of $\mathrm{BLO}^+(\mathbb{R})$ functions and show that $\mathrm{BLO}^+(\mathbb{R})$ is independent of the distance between the two intervals, which further induces the characterization of this space in terms of the one-sided BMO space $\mathrm{BMO}^+(\mathbb{R})$ (the Bennett type lemma). As applications, we prove that any $\mathrm{BMO}^+(\mathbb{R})$ function can split into the sum of two $\mathrm{BLO}^+(\mathbb{R})$ functions and we provide an explicit description of the distance from $\mathrm{BLO}^+(\mathbb{R})$ functions to $L^\infty(\mathbb{R})$. Finally, as a higher-dimensional analogue we introduce the parabolic BLO space $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ with time lag, and we extend all the above one-dimensional results to $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$; furthermore, as applications, we not only establish the relationships between $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ and the solutions of doubly nonlinear parabolic equations, but also provide a necessary condition for the negative logarithm of the parabolic distance function to belong to $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ in terms of the weak porosity of the set.

math.AP↗

arXiv2026

A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications

In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\in (1,\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp. The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist.

math.FA↗

arXiv2025

Maz'ya--Shaposhnikova Representation of Quasi-Norms of Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type with Weak Reverse Doubling Property

Let $Y(\mathcal{X})$ be a ball quasi-Banach function space on the space of homogeneous type $(\mathcal{X},ρ,μ)$ satisfying some mild additional assumptions, $q\in(0,\infty)$, and $\dot{W}^{s,q}_Y(\mathcal{X})$ with $s\in(0,1)$ be the homogeneous fractional Sobolev space associated with $Y(\mathcal{X})$. In this article, we show that, for any $f\in Y(\mathcal{X})\cap\bigcup_{s\in(0,1)} \dot{W}^{s,q}_Y(\mathcal{X})$, \begin{align*} \|f\|_{Y(\mathcal{X})} &\lesssim\varliminf_{s \to 0^+} s^{\frac{1}{q}}\left\| \left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[ρ(\cdot,y)]^{sq}} \, dμ(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}}\\ &\leq \varlimsup_{s \to 0^+} s^{\frac{1}{q}}\left\|\left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[ρ(\cdot,y)]^{sq}} \, dμ(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}} \lesssim\|f\|_{Y(\mathcal{X})}, \end{align*} where $U(x,y):=\min\{μ(B(x,ρ(x,y))),\,μ(B(y,ρ(x,y)))\}$ for any $x,y\in\mathcal{X}$ and the implicit positive constants are independent of $f$, which is applied to ten specific ball quasi-Banach function spaces and hence is of wide generality. In particular, when $Y(\mathcal{X})=L^q(\mathbb{R}^n)$ with $q\in[1,\infty)$, the above formula is closely related to the celebrated result of Maz'ya and Shaposhnikova in 2002. We also establish the above representation formula on domains of $\mathcal{X}$. The main novelty lies in proposing two new concepts, namely the weak reverse doubling condition (for $\mathcal{X}$) and the weak measure density condition (for domains of $\mathcal{X}$), which are proved to be necessary in some sense. In addition, we find an interesting fact that, when the underlying space under consideration is bounded, the above Maz'ya--Shaposhnikova-type limit always tends to zero.

math.FA↗