We characterize the (essentially) decreasing sequences of positive numbers $β$ = ($β$ n) for which all composition operators on H 2 ($β$) are bounded, where H 2 ($β$) is the space of analytic functions f in the unit disk such that $\infty$ n=0 |c n | 2 $β$ n < $\infty$ if f (z) = $\infty$ n=0 c n z n. We also give conditions for the boundedness when $β$ is not assumed essentially decreasing.
We obtain boundedness from a product of Lebesgue or Hardy spaces into Hardy spaces under suitable cancellation conditions for a large class of multilinear operators that includes the Coifman-Meyer class, sums of products of linear Calderon-Zygmund operators and combinations of these two types.
We give conditions for boundedness of Hausdorff operators on real Hardy spaces $H^1$ over homogeneous spaces of locally compact groups with local doubling property. The special case of the hyperbolic plane is considered.
In this paper, we estimate an operator norm of dilation operators on block spaces ($\mathfrak{B}_{r,α}(\mathbb{Q}_p)$) over $p$-adic field. With this estimate, we establish the boundedness of $p$-adic Hardy-Hilbert type integral operator on $\mathfrak{B}_{r,α}(\mathbb{Q}_p)$. Moreover as application to our result, we obtain the $p$-adic Hilbert inequality, $p$-adic Hardy inequality and $p$-adic Hardy-Littlewood-Pólya inequality on $\mathfrak{B}_{r,α}(\mathbb{Q}_p)$.
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.
In this paper, we prove strong type, weak type inequalities of Hardy-Littlewood maximal operator and fractional Hardy-Littlewood maximal operator on variable sequence spaces lp(Z). This is achieved using Calderon-Zygmund decomposition for sequences, properties of modular functional and Log Holder continuity.
The aim of this note is to give the boundedness conditions for Hausdorff operators on Hardy spaces $H^{1}$ with the norm defined via $(1,q)$ atoms over homogeneous spaces of Lie groups with doubling property and to apply results we obtain to generalized Delsarte operators and to Hausdorff operators over multidimensional spheres.
Results of Liflyand and collaborators on the boundedness of Hausdorff operators on the Hardy space $H^1$ over finite-dimensional real space generalized to the case of locally compact groups that are spaces of homogeneous type. Special cases and examples of compact Lie groups, homogeneous groups (in particular the Heisenberg group) and finite-dimensional spaces over division rings are considered.
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
We study the weighted boundedness of the Cauchy singular integral operator $S_\Gm$ in Morrey spaces $L^{p,λ}(\Gm)$ on curves satisfying the arc-chord condition, for a class of "radial type" almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces $L^{p,λ}(0,\ell), \ell>0$. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood max…
We give an alternative proof of recent results by the authors on uniform boundedness of dyadic averaging operators in (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type. This result served as the main tool to establish Schauder basis properties of suitable enumerations of the univariate Haar system in the mentioned spaces. The rather elementary proof here is based on characterizations of the respective spaces in terms of orthogonal compactly supported Daubechies wavelets.
We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.
In the present work, we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calderón-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $σ(ξ_1,\dots , ξ_m)$ and their derivatives vanish on the hyperplane $ξ_1+\cdots+ξ_m=0$.
Let $\mathcal{M}$ be the bilinear Hardy-Littlewood maximal function and $\vec{b}=(b,b)$ be a collection of locally integrable functions. In this paper, the authors establish characterizations of the weighted {\rm BMO} space in terms of several different commutators of bilinear Hardy-Littlewood maximal function, respectively; these commutators include the maximal iterated commutator $\mathcal{M}_{Π\vec{b}}$, the maximal linear commutator $\mathcal{M}_{Σ\vec{b}}$, the iterated commutator $[Π\vec{b},\mathcal{M}]$ and the linear commutator $[Σ\vec{b},\mathcal{M}]$.
In this paper the boundedness of the weighted iterated Hardy-type operators $T_{u,b}$ and $T_{u,b}^*$ involving suprema from weighted Lebesgue space $L_p(v)$ into weighted Cesàro function spaces ${\operatorname{Ces}}_{q}(w,a)$ are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator $R_u$ from $L^p(v)$ into ${\operatorname{Ces}}_{q}(w,a)$ on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator $P_{u,b }$ from $L^p(v)$ into ${\oper…
In this paper, we investigate weighted composition, Volterra and Integral operators on second derivative Hardy spaces. Some equivalent conditions for boundedness of the operators will be given using the boundedness on the Hardy spaces. Also we give a criteria for compactness of weighted composition operators.
In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.
One defines a non-homogeneous space $(X, μ)$ as a metric space equipped with a non-doubling measure $μ$ so that the volume of the ball with center $x$, radius $r$ has an upper bound of the form $r^n$ for some $n> 0$. The aim of this paper is to study the boundedness of a Calderón-Zygmund operator $T$ as well as the boundedness of certain related singular integrals associated with $T$ on various function spaces on $(X, μ)$ such as the Hardy spaces, the $L^p$ spaces and the regularized BMO spaces. This article thus extends the work of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n…
Let $p(\cdot):\ \mathbb{R}^n\to(0,\infty]$ 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 non-tangential grand maximal function. In this article, via the known atomic characterization of $H_A^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its molecular characterization with the known best possible decay of molecules. As an application, the author obtains a criterion on the boundedness of …
We discuss the Hardy-Littlewood maximal operator on discrete Morrey spaces of arbitrary dimension. In particular, we obtain its boundedness on the discrete Morrey spaces using a discrete version of the Fefferman-Stein inequality. As a corollary, we also obtain the boundedness of some Riesz potentials on discrete Morrey spaces.
We study new weighted estimates for the 2-fold product of Hardy-Littlewood maximal operators defined by $M^{\otimes}(f,g):= MfMg$. This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón-Zygmund operators, the bilinear Hardy-Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures $$ \Vert fg \Vert_{L^{p,\infty}\left(w_1^{p/p_1} w_2^{p/p_2}\right)} \le C \Vert f \Vert_{L^{p_1,\infty}(w_1 )} \Vert g \Vert_{L^{p_2,\…
In this paper we prove that the Hardy-Littlewood maximal operator is bounded on Morrey spaces $\mathcal{M}_{1,λ}(\rn)$, $0 \le \la < n$ for radial, decreasing functions on $\rn$
In this paper, the fractional Hardy-type operator of variable order $β(x)$ is shown to be bounded from the variable exponent Herz-Morrey spaces $M\dot{K}_{p_{_{1}},q_{_{1}}(\cdot)}^{α(\cdot),λ}(\R^{n})$ into the weighted space $M\dot{K}_{p_{_{2}},q_{_{2}}(\cdot)}^{α(\cdot),λ}(\R^{n},ω)$, where $α(x)\in L^{\infty}(\mathbb{R}^{n})$ be log-Hölder continuous both at the origin and at infinity, $ω=(1+|x|)^{-γ(x)}$ with some $γ(x)>0$ and $ 1/q_{_{1}}(x)-1/q_{_{2}}(x)=β(x)/n$ when $q_{_{1}}(x)$ is not necessarily constant at infinity.
In this paper, the fractional Hardy-type operator of variable order $β(x)$ is shown to be bounded from the Herz-Morrey spaces $M\dot{K}_{p_{_{1}},q_{_{1}}(\cdot)}^{α,λ}(\mathbb{R}^{n})$ with variable exponent $q_{1}(x)$ into the weighted space $M\dot{K}_{p_{_{2}},q_{_{2}}(\cdot)}^{α,λ}(\mathbb{R}^{n},ω)$, where $ω=(1+|x|)^{-γ(x)}$ with some $γ(x)>0$ and $ 1/q_{_{1}}(x)-1/q_{_{2}}(x)=β(x)/n$ when $q_{_{1}}(x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_{_{1}}(x)$ satisfies the logarithmic continuity condition both locally and at infinity that $1< q_{1}(\infty)\l…
The main goal of this paper is to prove a two-weight criteria for multidimensio-nal Hardy type operator from weighted Lebesgue spaces into $p$-convex weighted Banach function spaces. Analogously problem for the dual operator is considered. As application we prove a two-weight criteria for boundedness of multidimensional geometric mean operator and sufficient condition on the weights for boundedness of certain sublinear operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.
We study sharp growth conditions for the boundedness of the Hardy-Littlewood maximal function in the generalized Orlicz spaces. We assume that the generalized Orlicz function $ϕ(x, t)$ satisfies the standard continuity properties (A0), (A1) and (A2). We show that if the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space to itself then $ϕ(x,t)/ t^p$ is almost increasing for large $t$ for some $p>1$. Moreover we show that the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space $L^ϕ(\mathbb{R}^n)$ to itself if and only if $ϕ$ is weakly equiva…
We analyse Morrey spaces, generalised Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators and generalised fractional integral operators in generalised Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalised fractional integral operator and Olsen type inequalities in Campanato spaces and generalised Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidea…
We consider the products of composition and differentiation operators on the Hardy space. We provide a complete characterization of the boundedness and compactness of these operators. Furthermore, we obtain the explicit condition for these operators to be Hilbert-Schmidt operators.
In this paper, we define for the first time the local Morrey-type space associated with ball quasi-Banach function spaces and show the related series of properties. In addition, Hardy-Littlewood maximal operator's boundedness is proved. We investigate nonsmooth decomposition of the local Morrey-type space associated with ball quasi-Banach function spaces via the Hardy local Morrey-type spaces associated with ball quasi-Banach function spaces. And we consider Hardy operator's boundedness.
This paper is motivated by Phong and Stein's paper on non-standard singular integrals with mixed homogeneities. Our purpose is to study these new non-standard convolution singular integrals and establish the boundedness of these singular integrals on the Hardy spaces.
A simple normal form for Hardy operators is introduced that unifies and simplifies the theory of weighted Hardy inequalities. A straightforward transition to normal form is given that applies to the various Hardy operators and their duals, whether defined on Lebesgue spaces of sequences, of functions on the half-line, or of functions on $\mathbb R^n$ or more general metric spaces. This is done by introducing an abstract formulation of Hardy operators, more general than any of these, and showing that the normal form transition applies to all operators formulated in this way. The transition to n…
In this paper the complete solution of the restricted inequalities for supremal operators are given. The boundedness of the composition of supremal operators with the Hardy and Copson operators in weighted Lebesgue spaces are characterized.
We study the composition operators of the Hardy space on D $\infty$ $\cap$ {\ell} 1 , the {\ell} 1 part of the infinite polydisk, and the behavior of their approximation numbers.
In this paper, we carry on with the study of the Hardy-Amalgam spaces $\mathcal{H}_{\mathrm{loc}}^{(q,p)}$ spaces introduced in \cite{AbFt}. We investigate their dual spaces and establish some results of boundedness of pseudo-differential operators in these spaces.
In this paper, we investigate the conditions under which the Toeplitz Composition operator on the Hardy space $\mathcal{H}^2$ becomes complex symmetric with respect to a certain conjugation. We also study various normality conditions for the Toeplitz Composition operator on $\mathcal{H}^2$.
In this paper we study the self-adjoint and co-isometry composition and weighted composition operator on $H_E(ζ).$ We also discuss the conditions under which adjoint operator of a composition and weighted composition operator on $H_E(ζ)$ to be some composition and weighted composition operator, respectively.
Let $(\mathcal{X},d,μ)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and $b\in\rm{\widetilde{RBMO}(μ)}$ (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by $T$ and $b$ from the atomic Hardy space $\widetilde{H}^1(μ)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(μ)$. The b…
In the first part, we obtain sharp results for L^2 boundedness of strongly singular operators on the Heisenberg group. We also define the oscillating convolution operators on the Heisenberg group and study their boundedness properties. In the second part, we obtain the boundedness on Hardy spaces.
This paper defines local weighted Hardy spaces with variable exponent. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As an application, we obtain the boundedness of singular integral operators, the Littlewood--Paley characterization and wavelet decomposition.
In this paper, new equivalence theorems for the boundedness of the composition of a quasilinear operator $T$ with the Hardy and Copson operators in weighted Lebesgue spaces are proved. The usefulness of the obtained results is illustrated in the case of weighted Hardy-type and weighted iterated Hardy-type inequalities.
We prove that for operators satistying weighted inequalities with $A_p$ weights the boundedness on a certain class of Morrey spaces holds with weights of the form $|x|^αw(x)$ for $w\in A_p$. In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N.~Samko for the Hilbert transform, by H.~Tanaka for the Hardy-Littlewood maximal operator, and by S.~Nakamura and Y.~Sawano for Calderón-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Mo…
Let $L_1$ be a nonnegative self-adjoint operator in $L^2({\mathbb R}^n)$ satisfying the Davies-Gaffney estimates and $L_2$ a second order divergence form elliptic operator with complex bounded measurable coefficients. A typical example of $L_1$ is the Schrödinger operator $-Δ+V$, where $Δ$ is the Laplace operator on ${\mathbb R}^n$ and $0\le V\in L^1_{\mathop\mathrm{loc}} ({\mathbb R}^n)$. Let $H^p_{L_i}(\mathbb{R}^n)$ be the Hardy space associated to $L_i$ for $i\in\{1,\,2\}$. In this paper, the authors prove that the Riesz transform $D (L_i^{-1/2})$ is bounded from $H^p_{L_i}(\mathbb{R}^n)$ …
Let $p\in(0, 1]$. In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ to some quasi-Banach space ${\mathcal B}$ if and only if $T$ maps all $(p, 2, s_1, s_2)$-atoms into uniformly bounded elements of ${\mathcal B}$. Here $s_1\ge\lfloor n(1/p-1)\rfloor$ and $s_2\ge\lfloor m(1/p-1)\rfloor$. As usual, $\lfloor n(1/p-1)\rfloor$ denotes the maximal integer no more than $n(1/p-1)$. Applying this result, t…
In this paper, we study the basic properties such as boundedness and compactness of composition operators on discrete analogue of generalized Hardy space defined on a homogeneous rooted tree. Also, we compute the operator norm of composition operator when inducing symbol is automorphism of a homogenous tree.
