In 2011, Dekel et al. developed highly geometric Hardy spaces $H^p(Θ)$, for the full range $0<p\leq 1$, which are constructed by continuous multi-level ellipsoid covers $Θ$ of $\mathbb{R}^n$ with high anisotropy in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. In this article, if the cover $Θ$ is pointwise continuous, then the authors further obtain some real-variable characterizations of $H^p(Θ)$ in terms of the radial, the non-tangential and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.
We establish simple pointwise characterizations of functions in the Hardy-Sobolev spaces within the range n/(n+1)<p <=1. In addition, classical Hardy inequalities are extended to the case p <= 1.
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.
We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formulation of the problem. We further compute directional derivatives for this functional as a computational means to approach the issue. We then consider a finite-dimensional polynomial version of the boun…
We study multipliers of Hardy-Orlicz spaces $\mH_Φ$ which are strictly contained between $\bigcup_{p>0}H^p$ and so-called ``big'' Hardy-Orlicz spaces. Big Hardy-Orlicz spaces, carrying an algebraic structure, are equal to their multiplier algebra, whereas in classical Hardy spaces $H^p$, the multipliers reduce to $H^{\infty}$. For Hardy-Orlicz spaces $\mH_Φ$ between these two extremal situations and subject to some conditions, we exhibit multipliers that are in Hardy-Orlicz spaces the defining functions of which are related to $Φ$. Even if the results do not entirely characterize the multiplie…
We obtain a pointwise description of functions belonging to function spaces with the lattice property. In particular, it is valid for Banach function spaces provided that the Hardy-Littlewood maximal operator is bounded. We also study weighted grand Sobolev spaces, defined on arbitrary open sets (of finite or infinite measure), and provide pointwise description of them.
For a pointwise multiplier $φ$ of the Hardy-Sobolev space $H^2_β$ on the open unit ball $\bn$ in $\cn$, we study spectral properties of the multiplication operator $M_φ: H^2_β\to H^2_β$. In particular, we compute the spectrum and essential spectrum of $M_φ$ and develop the Fredholm theory for these operators.
For a real-valued function $f$ on a metric measure space $(X,d,μ)$ the Hardy-Littlewood maximal-function of $f$ is given by the following `supremum-norm': $$Mf(x):=\sup_{r>0}\frac{1}{μ(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|dμ.$$ In this note, we replace the supremum-norm on parameters $r$ by $\mathcal{L}_p$-norm with weight $w$ on parameters $r$ and define Hardy-Littlewood integral-function $I_{p,w}f$. It is shown that $I_{p,w}f$ converges pointwise to $Mf$ as $p\to\infty$. Boundedness of the sublinear operator $I_{p,w}$ and continuity of the function $I_{p,w}f$ in case that $X$ is a L…
Let $Γ$ be a rectifiable Jordan curve, let $X$ and $Y$ be two reflexive Banach function spaces over $Γ$ such that the Cauchy singular integral operator $S$ is bounded on each of them, and let $M(X,Y)$ denote the space of pointwise multipliers from $X$ to $Y$. Consider the Riesz projection $P=(I+S)/2$, the corresponding Hardy type subspaces $PX$ and $PY$, and the Toeplitz operator $T(a):PX\to PY$ defined by $T(a)f=P(af)$ for a symbol $a\in M(X,Y)$. We show that if $X\hookrightarrow Y$ and $a\in M(X,Y)\setminus\{0\}$, then $T(a)\in\mathcal{L}(PX,PY)$ has a trivial kernel in $PX$ or a dense image…
We consider the optimizers $u$ in the Hardy-Sobolev inequality for the space $\dot{W}^{s,p}({\mathbb R}^N)$ with order of differentiability $s\in ]0,1[$. After proving existence through concentration-compactness, we derive the pointwise asymptotic $u(x)\simeq |x|^{-\frac{N-ps}{p-1}}$ for large $|x|$ and the summability estimate $u\in \dot{W}^{s,γ}({\mathbb R}^N)$ for all $γ>\frac{N(p-1)}{N-s}$. These estimates are optimal in the limit $s\to 1^-$, in which case optimizers are explicitly known.
Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory
College of Mathematics and Informatics, South China Agricultural University（华南农业大学数学与信息学院）；Faculty of Innovation Engineering, Macau University of Science and Technology（澳门科技大学创新工程学院）
Let $\vec{p}\in(0,1]^n$ be a $n$-dimensional vector and $A$ a dilation. Let $H_A^{\vec{p}}(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H_{A}^{\vec{p}}(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f\in H_A^{\vec{p}}(\mathbb{R}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. Moreover, the function $F$ can be controlled pointwisely by the product of the Hardy spac…
We introduce the class of variable anisotropic singular integral operators associated to a continuous multi-level ellipsoid cover $Θ$ of $\mathbb{R}^n$ introduced by Dahmen, Dekel, and Petrushev \cite{ddp}. This is an extension of the classical isotropic singular integral operators on $\mathbb{R}^n$ of arbitrary smoothness and their anisotropic analogues for general expansive matrices introduced by the first author \cite{b}. We establish the boundedness of variable anisotropic singular integral operators $T$ on the Hardy spaces with pointwise variable anisotropy $H^p(Θ)$, which were developed …
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…
In this paper it is shown that the Hardy-Littlewood maximal operator $M$ is not bounded on Zygmund-Morrey space $\mathcal{M}_{L(\log L),λ}$, but $M$ is still bounded on $\mathcal{M}_{L(\log L),λ}$ for radially decreasing functions. The boundedness of the iterated maximal operator $M^2$ from $\mathcal{M}_{L(\log L),λ}$ to weak Zygmund-Morrey space ${\mathcal {W \! M}}_{L(\log L),λ}$ is proved. The class of functions for which the maximal commutator $C_b$ is bounded from $\mathcal{M}_{L(\log L),λ}$ to ${\mathcal {W \! M}}_{L(\log L),λ}$ are characterized. It is proved that the commutator of the …
A result of Helson on general Dirichlet series $\sum a_{n} e^{-λ_{n}s}$ states that, whenever $(a_{n})$ is $2$-summable and $λ=(λ_{n})$ satisfies a certain condition introduced by Bohr, then for almost all homomorphism $ω\colon (\mathbb{R},+) \to \mathbb{T}$ the Dirichlet series $\sum a_{n} ω(λ_{n})e^{-λ_{n}s}$ converges on the open right half plane $[Re>0]$. For ordinary Dirichlet series $\sum a_n n^{-s}$ Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces $\mathcal…
Let $\mathbb B_n$ be the open unit ball in $\mathbb C^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb B_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A^p_{α,β}$, Bloch-type spaces $\mathcal B_α$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H^2_β$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.
We introduce a new class of Hardy spaces $H^{φ(\cdot,\cdot)}(\mathbb R^n)$, called Hardy spaces of Musielak-Orlicz type, which generalize the Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of García-Cuerva, Strömberg, and Torchinsky. Here, $φ: \mathbb R^n\times [0,\infty)\to [0,\infty)$ is a function such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight. A function $f$ belongs to $H^{φ(\cdot,\cdot)}(\mathbb R^n)$ if and only if its maximal function $f^*$ is so that $x\mapsto φ(x,|f^*(x)|)$ is integrable. Such a space arises naturally for …
We study the boundary behavior of functions in the Hardy spaces on the infinite dimensional polydisk. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz-Zygmund type theorems for radial convergence. As a consequence one obtains easy new proofs of the brothers F. and M. Riesz theorems in infinite dimension. Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.
We define by interpolation a scale analogous to the Hardy $H^p$ scale for complete Pick spaces, and establish some of the basic properties of the resulting spaces, which we call $\mathcal{H}^p$. In particular, we obtain an $\mathcal{H}^p-\mathcal{H}^q$ duality and establish sharp pointwise estimates for functions in $\mathcal{H}^p$.
Let $ϕ: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz)$ be a function such that $ϕ(x,\cdot)$ is an Orlicz function and $ϕ(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights introduced by V. S. Rychkov). In this paper, the authors introduce a local Hardy space $h_ϕ(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand maximal function, and a local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_ϕ(\mathbb{R}^n)$ which is further proved to be the dual space of $h_ϕ(\mathbb{R}^n)$. As an application, the authors prove that the class of pointwise multipli…
We show that the boundedness of the Hardy-Littlewood maximal operator on a Köthe function space ${\mathbb{X}}$ and on its Köthe dual ${\mathbb{X}}'$ is equivalent to the well-posedness of the $\mathbb{X}$-Dirichlet and $\mathbb{X}'$-Dirichlet problems in $\mathbb{R}^{n}_{+}$ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versio…
Given a frequency $λ= (λ_n)$ and $\ell \ge 0$, we introduce the scale of Banach spaces $H_{\infty,\ell}^λ[Re > 0]$ of holomorphic functions $f$ on the open right half-plane $[Re > 0]$, which satisfy $(A)$ the growth condition $|f(s)| = O((1 + |s|)^\ell)$, and $(B)$ have a Riesz germ, i.e. on some open subset and for some $m \ge 0$ the function $f$ coincides with the pointwise limit (as $x \to \infty$) of the so-called $(λ,m)$-Riesz means $\sum_{λ_n < x} a_n e^{-λ_n s}\big( 1-\frac{λ_n}{x}\big)^m ,\,x >0$ of some $λ$-Dirichlet series $\sum a_n e^{-λ_n s}$. Reformulated in our terminology, an im…
