In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterizations in terms of several local maximal functions. Next, the new atomic and the finite atomic decomposition for the local variable Hardy spaces are established. As an application, we also introduce the local variable Campanato space which is showed to be the dual space of the local variable Hardy spaces. Analogous to the homogeneous case, some equivalent definitions of the dual of local variable Hardy spaces are also considered. Finally, we show that the boundedness of inhomogeneous Calderón-Zygmund singular integrals and local fractional integrals on local variable Hardy spaces and their duals.
In this paper, by using the atomic decomposition theory of Hardy space and weak Hardy space, we discuss the boundedness of parameterized Littlewood-Paley operator with variable kernel on these spaces.
In this paper, by using the atomic decomposition theory of Hardy space and weak Hardy space, the author establishes the boundedness of parameterized Marcinkiewicz integral with variable kernel on these spaces, under the Dini condition or Hörmander condition imposed on kernel.
In this article, we show that multilinear fractional type operators are bounded from product Hardy spaces with variable exponents into Lebesgue spaces with variable exponents via the atomic decomposition theory. We also study continuity properties of commutators of multilinear fractional type operators on product of certain Hardy spaces with variable exponents.
In this paper, we first introduce $L^{σ_1}$-$(\log L)^{σ_2}$ conditions satisfied by the variable kernels $Ω(x,z)$ for $0\leqσ_1\leq1$ and $σ_2\geq0$. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators $T_Ω$, fractional integrals $T_{Ω,α}$ and parametric Marcinkiewicz integrals $μ^ρ_Ω$ with variable kernels on the Hardy spaces $H^p(\mathbb R^n)$ and weak Hardy spaces $WH^p(\mathbb R^n)$. Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy…
We develop the theory of variable exponent Hardy spaces. Analogous to the classical theory, we give equivalent definitions in terms of maximal operators. We also show that distributions in these spaces have an atomic decomposition including a "finite" decomposition; this decomposition is more like the decomposition for weighted Hardy spaces due to Stromberg and Torchinsky than the classical atomic decomposition. As an application of the atomic decomposition we show that singular integral operators are bounded on variable Hardy spaces with minimal regularity assumptions on the exponent function…
We study the boundedness of certain fractional integral operators from Hp(.) into Lq(.). We also obtain the Hp(.)- Hq(.) boundedness of the Riesz potential.
In this paper, by using the decomposition theorem for weak Hardy spaces, we will obtain the boundedness properties of some integral operators with variable kernels on these spaces, under some Dini type conditions imposed on the variable kernel $Ω(x,z)$.
Let $p(\cdot):\ \mathbb{R}^n\to(0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable anisotropic Hardy space associated with $A$ defined via the radial maximal function. In this article, via the known atomic characterization of $H_{A}^{p(\cdot)}(\mathbb{R}^n)$ and establishing two useful estimates on anisotropic variable atoms, the author shows that the Fourier transform $\widehat{f}$ of $f\in H_A^{p(\cdot)}(\mathbb{R}^n)$ coincides with a continuous…
In this paper a two weight criterion for multidimensional geometric mean operator in variable exponent Lebesgue space is proved. Also, we found a criterion on weight functions expressing one-dimensional Hardy inequality via a certain nonlinear differential equation. In particular, considered nonlinear differential equation is nonlinear integro-differential equation.
We give new proofs of Hardy space estimates for fractional and singular integral operators on weighted and variable exponent Hardy spaces. Our proofs consist of several interlocking ideas: finite atomic decompositions in terms of $L^\infty$ atoms, vector-valued inequalities for maximal and other operators, and Rubio de Francia extrapolation. Many of these estimates are not new, but we give new and substantially simpler proofs, which in turn significantly simplifies the proofs of the Hardy spaces inequalities.
We study the boundedness from Hp(.) into Lq(.) of certain generalized Riesz potentials and the Hp(.)-Hq(.) boundedness of the Riesz potential. Both results are achieved via the finite atomic decomposition developed in [4].
We characterize the sufficient conditions which three weight functions $u$ and $v_{1}, v_{2}$ satisfy ensure the boundedness of the Hardy operator with variable limits on product space. The corresponding bound is explicitly worked out. Moreover, as application, we can obtain an explicit scale of bound for the Pólya-Knopp type operator with certain weights.
In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak type and strong type inequalities on Doob's maximal operator and get a $(1,p(\cdot),\infty)$-atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John-Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with probabilistic characterization of $p(\cdot)$ to replace the so-called lo…
Let $T_Ω$ be the singular integral operator with variable kernel $Ω(x,z)$. In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of $T_Ω$ on these spaces, under some Dini type conditions imposed on the variable kernel $Ω(x,z)$.
In this article we define the Calderón-Hardy spaces with variable exponents on $\mathbb{R}^{n}$, $\mathcal{H}^{p(.)}_{q, γ}(\mathbb{R}^{n})$, and we show that for $m \in \mathbb{N}$ the operator $Δ^{m}$ is a bijective mapping from $\mathcal{H}^{p(.)}_{q, 2m}(\mathbb{R}^{n})$ onto $H^{p(.)}(\mathbb{R}^{n})$.
In this paper a weighted variable exponent Lebesgue spaces $L_{p(x), ω}$ for $0< p(x)< 1$ is investigated. We show that this spaces is a quasi-Banach spaces. Note that embedding theorem between weight variable Lebesgue spaces is proved. In particular, we show that $L_{p(x), ω}(Ω)$ for $0< p(x)< 1$ isn't locally convex. Also, in this paper a some two-weight estimates for Hardy operator are proved.
Let $A$ be an expansive dilation on $\mathbb{R}^n$, and $p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. Let $\mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors obtain that the Fourier transform of $f\in \mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. As applications, the authors further conclude a highe…
We show that the Hardy-Littlewood maximal operator is bounded on a reflexive variable Lebesgue space $L^{p(\cdot)}$ over a space of homogeneous type $(X,d,μ)$ if and only if it is bounded on its dual space $L^{p'(\cdot)}$, where $1/p(x)+1/p'(x)=1$ for $x\in X$. This result extends the corresponding result of Lars Diening from the Euclidean setting of $\mathbb{R}^n$ to the setting of spaces of homogeneous type $(X,d,μ)$.
In this paper we introduce Hardy-Lorentz spaces with variable exponents associated to dilation in ${\Bbb R}^n$. We establish maximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.
Let $\T (0\leq α<n)$ be the singular and fractional integrals with variable kernel $Ω(x,z)$, and $[b,\T]$ be the commutator generated by $\T$ and a Lipschitz function $b$. In this paper, the authors study the boundedness of $[b,\T]$ on the Hardy spaces, under some assumptions such as the $L^r$-Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution operators $\tT (0\leq α<n)$. The smoothness conditions imposed on $\tOmega$ are weaker than the corresponding known results.
We prove norm estimates for multilinear fractional integrals acting on weighted and variable Hardy spaces. In the weighted case we develop ideas we used for multilinear singular integrals [7]. For the variable exponent case, a key element of our proof is a new multilinear, off-diagonal version of the Rubio de Francia extrapolation theorem.
Let $A$ be an expansive dilation on $\mathbb{R}^n$, and $p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. Let $H^{p(\cdot)}_A({\mathbb {R}}^n)$ be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors establish its molecular decomposition, which is still new even in the classical isotropic setting (in the case $A:=2\mathrm I_{n\times n}$). As applications, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from $H^{p(\c…
In this paper we introduce molecules associated to Hardy spaces with pointwise variable anisotropy, and prove that each molecule can be represented as a sum of atoms.
In this paper we introduce variable exponent local Hardy spaces associated with a non-negative self-adjoint operator L. We define them by using an area square integral involving the heat semigroup associated to L. A molecular characterization is established and as an aplication of the molecular characterization we prove that our local Hardy space coincides with the (global) variable exponent Hardy space associated to L, provided that 0 does not belong to the spectrum of L. Also, we show that it coincides with the global variable exponent Hardy space associated to L+I.
In this article, the authors consider the Schrödinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q(\mathbb{R}^n)$ with $q\in(n/2,\,\infty)$. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,\infty)$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition. When $p(\cdot):\ \mathbb{R}^n\to(1,\,\infty)$, the authors prove that the operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded on va…
In this paper, we prove the Hardy-Leray inequality and related inequalities in variable Lebesgue spaces. Our proof is based on a version of the Stein-Weiss inequality in variable Lebesgue spaces derived from two weight inequalities due to Melchiori and Pradolini. We also discuss an application of our results to establish an existence result for the degenerate $\pp$-Laplace operator.
In this paper, the author obtain the continuity of a class of linear operators on variable anisotropic Hardy-Lorentz spaces. In addition, the author also obtain that the dual space of variable anisotropic Hardy-Lorentz spaces is the anisotropic BMO-type spaces with variable exponents. This result is still new even when the exponent function $p(\cdot)$ is $p$.
We give the optimal embedding relations between local Hardy space and $α$-modulation spaces, which extend the results for the embedding relations between local Hardy and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79-96].
In this article, the authors consider the Schrödinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ is symmetric and satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q(\mathbb{R}^n)$ with $q\in(n/2,\,\infty)$. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition. The authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$ associated to $L$ and establish its atomic characterization. The a…
We establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multil…
The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor: \[ \sum_{z\i…
Let $L$ be a linear operator on $L^2(\mathbb R^n)$ generating an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ with kernels having pointwise upper bounds and $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(\cdot)}(\mathbb R^n)$, and the BMO-type space ${\mathrm{BMO}}_{p(\cdot),L}(\mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of…
We present a unified strategy to derive Hardy-Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. The idea also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.
Let $λ>0$, $p\in((2\lz+1)/(2\lz+2), 1]$, and $\triangle_λ\equiv-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces $H^p((0, \infty), dm_λ)$ associated with $\triangle_λ$ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood-Paley $g$-function and the Lusin-area function, where $dm_λ(x)\equiv x^{2λ}\,dx$. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.
We investigate the class $\mathcal{B}^{loc}(\mathbb{R}^{n})$ of exponents $p(\cdot)$ for with local Hardy-Littlewood maximal operator is bounded in $L^{p(\cdot)}(\mathbb{R}^{n})$ space. Littlewood-Paley square-function characterization of $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces with the above class of exponent are also obtained.
Let $X$ be a separable Banach function space on the unit circle $\mathbb{T}$ and $H[X]$ be the abstract Hardy space built upon $X$. We show that the set of analytic polynomials is dense in $H[X]$ if the Hardy-Littlewood maximal operator is bounded on the associate space $X'$. This result is specified to the case of variable Lebesgue spaces.
Let $p(\cdot):\mathbb R^n\rightarrow(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of para-product operators $π_b$ on variable Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$, where $b\in BMO(\mathbb R^n)$. As an application, we show that non-convolution type Calderón-Zygmund operators $T$ are bounded on $H^{p(\cdot)}(\mathbb R^n)$ if and only if $T^\ast1=0$, where $\frac{n}{n+ε}<\mbox{essinf}_{x\in\mathbb R^n} p\le \mbox{esssup}_{x\in\mathbb R^n} p\le 1$, $ε$ is the regular exponent of kernel of $T$. Our a…
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying that there exists a constant $p_0\in(0,p_-)$, where $p_-:=\mathop{\mathrm {ess\,inf}}_{x\in \mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(\cdot)/p_0}(\mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A…
This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective which means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. To be precise, in this book, the authors first study some basic properties of generalized Herz spaces and obtain boundedness and compactness characterizations of commutators on them. Then the authors introduce the associated Herz-Hardy spaces, local Herz-…
In this paper, we characterize the weighted local Hardy spaces $h^p_ρ(ω)$ related to the critical radius function $ρ$ and weights $ω\in A_{\infty}^{ρ,\,\infty}(\mathbb{R}^{n})$ which locally behave as Muckenhoupt's weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with $h^{p}_ρ(ω)$. Furthermore, we establish boundedness in $h^p_ρ(ω)$ of quasi- Banach-valued sublinear operators. As their application…
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a measurable function satisfying some decay condition and some locally log-Hölder continuity. In this article, via first establishing characterizations of the variable exponent Hardy space $H^{p(\cdot)}(\mathbb R^n)$ in terms of the Littlewood-Paley $g$-function, the Lusin area function and the $g_λ^\ast$-function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_λ^\ast$-function. The $p(\cdot)$-Carleson measure charac…
In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors introduce a new kind of Hardy type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz-Hardy space of A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson in…
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the anisotropic variable Hardy-Lorentz space $H_A^{p(\cdot),q}(\mathbb R^n)$ associated with $A$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain characterizations of $H_A^{p(\cdot),q}(\mathbb R^n)$, respectively, in terms of the atom and th…
Let $φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight. The Musielak-Orlicz Hardy space $H^φ(\mathbb R^n)$ is defined to be the space of all $f\in{\mathcal S}'({\mathbb R^n})$ such that the grand maximal function $f^*$ belongs to the Musielak-Orlicz space $L^φ(\mathbb R^n)$. Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of $H^φ(\mathbb R^n)$ in terms of the vertical or the non-tangential maximal f…
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on $\mathbb R^n$, $W\!H^{p(\cdot)}(\mathbb R^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W\!H^{p(\cdot)}(\mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_λ^\ast…
Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition and $L$ a non-negative self-adjoint operator on $L^2(\mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estimates. Let $H_L^{p(\cdot)}(\mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels $\{e^{-t^2L}\}_{t\in (0,\infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(\cdot)}(\mathbb R^n)$; using this, the authors then obtain its non-tangential ma…
Let $L$ be a one-to-one operator of type $ω$ in $L^2(\mathbb{R}^n)$, with $ω\in[0,\,π/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H^{p(\cdot)}_L(\mathbb{R}^n)$ associated with $L$. By means of variable tent spaces, the authors establish the molecular characterization of $H^{p(\cdot)}_L(\mathbb{R}^n)$. Then the authors show that the dual space of $…
Let $Ω$ be a strongly Lipschitz domain of $\mathbb{R}^n$, whose complement in $\mathbb{R}^n$ is unbounded. Let $L$ be a second order divergence form elliptic operator on $L^2 (Ω)$ with the Dirichlet boundary condition, and the heat semigroup generated by $L$ have the Gaussian property $(G_{\mathrm{diam}(Ω)})$ with the regularity of their kernels measured by $μ\in(0,1]$, where $\mathrm{diam}(Ω)$ denotes the diameter of $Ω$. Let $Φ$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_Φ\in(n/(n+μ),1]$. In t…
We prove fractional Sobolev-Poincaré inequalities, capacitary versions of fractional Poincaré inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.
Recently, both the bilinear decompositions $h^1(\mathbb{R}^n)\times \mathrm{\,bmo}(\mathbb{R}^n) \subset L^1 (\mathbb{R}^n)+h_\ast^Φ(\mathbb{R}^n)$ and $h^1(\mathbb{R}^n) \times \mathrm{bmo}(\mathbb{R}^n) \subset L^1 (\mathbb{R}^n) + h^{\log}(\mathbb{R}^n)$ were established. In this article, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains the variant $h_\ast^Φ(\mathbb{R}^n)$ of the local Orlicz Hardy space introduced by A. Bonami and J. Feuto as a special case, and obta…
