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In this paper, we characterize weighted composition operators that preserve frames on the weighted Hardy spaces in the unit disk. In particular, we obtain the symbol properties of the bounded invertible weighted composition operators. Moreover, we establish the equivalence between bounded invertible operators and frame-preserving operators. Furthermore, we show that weighted composition operator preserves frames if and only if it preserves the Riesz bases property. Additionally, we investigate the weighted composition operators that preserve tight or normalized tight frames on the Dirichlet space. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterization 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 the boundedness of inhomogeneous Calderón-Zygmund singular integrals and local fractional integrals on local variable Hardy spaces and their duals. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the weak Lipschitz spaces WLipq,α, 1 < q < ∞, 0 < α < 1 which are the analog of weak Lebesgue spaces Lq,∞ in the setting of Lipschitz space. We obtain the equivalence between the norm and . As an application, we show that the commutator Mβb is bounded from Lp to Lq,∞ for some p ∈ (1, ∞) and if and only if b is in Lipα. We also introduce the weak central bounded mean oscillation space WCBMOq,α and give a characterization of WCBMOq,α via the boundedness of the commutators of Hardy type operators. [ABSTRACT FROM AUTHOR]

Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ. [ABSTRACT FROM AUTHOR]

It is known that the Invariant Subspace Problem for Hilbert spaces is equivalent to the statement that all minimal non-trivial invariant subspaces for a universal operator are one dimensional. In this paper, we characterize all linear fractional composition operators and their adjoints that have universal translates on the space S 2 (D) . Moreover, we characterize all adjoints of linear fractional composition operators that have universal translates on the Hardy space H 2 (D) . In addition, we consider the minimal invariant subspaces of the composition operator C φ a on S 2 (D) , where ϕa(z) = az + 1 − a, a ∈ (0, 1). Finally, some relationships between complex symmetry and universality for bounded linear operators and commuting pairs of operators on a complex separable, infinite dimensional Hilbert space are explored. [ABSTRACT FROM AUTHOR]

In this paper, we completely characterize the positive Borel measures μ on the unit ball B n such that the differential type operator ℛ m of order m ∈ ℕ is bounded from Hardy type tent space ℋ T q , α p (B n) into Ls(μ) for full range of p, q, s, α. Subsequently, the corresponding compact description of differential type operator ℛ m is also characterized. As an application, we obtain the boundedness and compactness of integration operator Jg from ℋ T q , α p (B n) to ℋ T s , β t (B n) , and the methods used here are adaptable to the Hardy spaces. [ABSTRACT FROM AUTHOR]

Let H 2 (D 2) be the Hardy space over the bidisk D 2 , and let Mφ = [(z − φ(w))2] be the submodule generated by (z − φ(w))2, where φ(w) is a function in H∞(w). The related quotient module is denoted by N φ = H 2 (D 2) Θ M φ . In the present paper, we study the Fredholmness of compression operators Sz, Sw on Nφ. When φ(w) is a nonconstant inner function, we prove that the Beurling type theorem holds for the fringe operator Fw on [(z − w)2] ⊖ z[(z − w)2] and the Beurling type theorem holds for the fringe operator Fz on Mφ ⊖ wMφ if φ(0) = 0. Lastly, we study some properties of Fw on [(z − w2)2] ⊖ z[(z − w2)2]. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for −1/2 ≤ β ≤ 0, suppose M is an invariant subspaces of the Hardy-Sobolev spaces H (D) for T , then M ⊖ zM is a generating wandering subspace of M, that is, $$M = {\left[ {M \ominus zM} \right]_{T_z^\beta }}$$ . Moreover, any non-trivial invariant subspace M of H (D) is also generated by the quasi-wandering subspace P T M , that is, $$M = {\left[ {{P_M}T_z^\beta {M^ \bot }} \right]_{T_z^\beta }}$$ . [ABSTRACT FROM AUTHOR]

In this paper, we generalize the concept of asymptotic Hankel operators on H 2 (D) to the Hardy space H 2 (D n) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce ith-partial Hankel operators on H 2 (D n) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on H 2 (D n) . It is also shown that a Toeplitz operator with symbol ϕ is asymptotic Hankel if and only if ϕ is holomorphic function in L ∞ (T n) . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish, using the Littlewood-Paley-Stein theory (in particular, the Littlewood-Paley-Stein square functions), a Calderón-Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces Hp (ℝn; A) associated with expensive dilation A: T m f (x) = ∫ ℝ n m (ξ) f ^ (ξ) e 2 π i x ⋅ ξ d ξ . Our main Theorem is the following: Assume that m(ξ) is a function on ℝn satisfying sup j ∈ ℤ ‖ m j ‖ W s (A ∗) < ∞ with s > ζ − − 1 (1 p − 1 2) Then Tm is bounded from Hp(ℝn; A) to Hp(ℝn; A) for all 0 < p ≤ 1 and ‖ T m ‖ H A p → H A p ≲ sup j ∈ ℤ ‖ m j ‖ W s (A ∗) , where A* denotes the transpose of A. Here we haveusedthe notations mj (ξ)= m(A*jξ)φ(ξ)and is a suitable cut-off function on ℝn, and Ws(A*) is an anisotropic Sobolev space associated with expansive dilation A* on ℝn. [ABSTRACT FROM AUTHOR]

Usually, the condition that T is bounded on L2(ℝn) is assumed to prove the boundedness of an operator T on a Hardy space. With this assumption, one only needs to prove the uniformly boundness of T on atoms, since T(f)= ∑i λiT(ai), provided that f = ∑i λiai in L2 (ℝn), where ai is an L2 atom of this Hardy space. So far, the L2 atomic decomposition of local Hardy spaces hp(ℝn), 0 > p ≤ 1, hasn't been established. In this paper, we will solve this problem, and also show that hp(ℝn) can also be characterized by discrete Littlewood-Paley functions. [ABSTRACT FROM AUTHOR]

A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact. [ABSTRACT FROM AUTHOR]

In this paper, we establish the boundedness of parameterized Littlewood-Paley operator μλ*,ρ and parameterized area integral μΩ,Sρ with kernel satisfying the logarithmic type Lipschitz condition on the weak Hardy space. [ABSTRACT FROM AUTHOR]

In this paper, we obtain that a strongly singular integral operator is bounded on Lpw space for 1 < p < ∞. We also obtain that a strongly singular integral operator is a bounded operator from Hpw to Lpw for some weight w and 0 < p ≤ 1. And by an atomic decomposition, we obtain that a strongly singular integral operator is a bounded operator on Hpw for some w and 0 < p ≤ 1. [ABSTRACT FROM AUTHOR]

In this paper, we will study the nonelementary groups of Möbius transformations in $$ \overline{R} ^{n} $$ and some properties are obtained. Also in this paper we will prove several theorems about discreteness criteria and group convergence of nonelementary groups of M( $$ \overline{R} ^{n} $$). [ABSTRACT FROM AUTHOR]

Letbe a unit sphere of thed-dimensional Euclidean space Rd and let(0 < p= 1) denote the real Hardy space onFor 0 < p= 1 andletEj(f,Hp) (j= 0, 1, ...) be the best approximation offby spherical polynomials of degree less than or equal toj, in the spaceGiven a distributionfonits Cesàro mean of order d>-1 is denoted byFor 0p. In this paper, the following result is proved:TheoremLet0N(f)˜BN(f)means that there’s a positive constant C, independent of N and f, such thatIn the cased= 2,this result was proved by Belinskii in 1996. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the boundedness of Littlewood-Paley functions and pseudo-differential operators on weighted Herz-type Hardy spaces over locally compact Vilenkin groups. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the Hörmander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k ≥ 3 using the multi-parameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k ≥ 3: where $$x = \left( {{x_1},{x_2},{x_3}} \right) \in {\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}$$ and $$\xi = \left( {{\xi _1},{\xi _2},{\xi _3}} \right) \in {\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}$$ . One of our main results is the following: Assume that m( ξ) is a function on $${\mathbb{R}^{{n_1} + {n_2} + {n_3}}}$$ satisfying $$\mathop {\sup }\limits_{j,k,l \in \mathbb{Z}} {\left\| {{m_{j,k,l}}} \right\|_{{W^{\left( {{s_1},{s_2},{s_3}} \right)}}}} < \infty $$ with $$s_{i} > n_{i}(\frac{1}{p}-\frac{1}{2})$$ for 1 ≤ i ≤ 3. Then T is bounded from $${H^p}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}} \right)$$ to $${H^p}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}} \right)$$ for all 0 < p ≤ 1 and $$\left\| {{T_m}} \right\|{}_{{H^P} \to {H^P}} \lesssim \mathop {\sup }\limits_{j,k,l \in \mathbb{Z}} {\left\| {{m_{j,k,l}}} \right\|_{{W^{\left( {{s_1},{s_{2,}}{s_3}} \right)}}}}$$ . Moreover, the smoothness assumption on s for 1 ≤ i ≤ 3 is optimal. Here we have used the notations m ( ξ) = m(2 ξ , 2 ξ , 2 ξ )Ψ( ξ )Ψ( ξ )Ψ( ξ ) and Ψ( ξ ) is a suitable cut-off function on $${\mathbb{R}^{{n_i}}}$$ for 1 ≤ i ≤ 3, and $${W^{\left( {{s_1},{s_2},{s_3}} \right)}}$$ is a three-parameter Sobolev space on $${\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}$$ . Because the Fefferman criterion breaks down in three parameters or more, we consider the L boundedness of the Littlewood-Paley square function of T f to establish its boundedness on the multi-parameter Hardy spaces. [ABSTRACT FROM AUTHOR]

In this paper, we give counterexamples to show that variation and oscillation operators related to rough truncations of the Hilbert transform are not bounded from H to L , and prove variation, oscillation and λ-jump operators related to smooth truncations are bounded from H to L . [ABSTRACT FROM AUTHOR]

Let ( X, d, μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Under the weak reverse doubling condition, the authors prove that the generalized homogeneous Littlewood-Paley g-function ġ ( r ∈ [2,∞)) is bounded from Hardy space H ( μ) into L ( μ). Moreover, the authors show that, if f ∈ RBMO( μ), then [ ġ ( f)] is either infinite everywhere or finite almost everywhere, and in the latter case, [ ġ ( f)] belongs to RBLO( μ) with the norm no more than | f| multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of ġ from RBMO( μ) into RBLO( μ). The vector valued Calderón-Zygmund theory over ( X, d, μ) is also established with details in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce $${L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}$$ conditions satisfied by the variable kernels Ω( x, z) for 0 ≤ σ ≤ 1 and σ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T, fractional integrals T and parametric Marcinkiewicz integrals μ with variable kernels on the Hardy spaces H(R) and weak Hardy spaces WH(R). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy-Lorentz spaces H(R) for all p < q < ∞. [ABSTRACT FROM AUTHOR]

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H. The sharp bounds for the strong type ( p, p) (1 ≤ p ≤ ∞) estimates of n-dimensional Hausdorff operators on H are obtained. The sharp bounds for strong ( p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on H. The weak type ( p, p) (1 ≤ p≤∞) estimates are also obtained. Keywords Hausdorff operator, Heisenberg group, multilinear, sharp estimate [ABSTRACT FROM AUTHOR]

In this paper, we show that two Toeplitz operators T and T on the Hardy space of the polydisk can commute if and only if the Berezin transform of the commutator [ T, T] is n-harmonic. [ABSTRACT FROM AUTHOR]

Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite differently and, therefore, seem to be a worth studying new class of Toeplitz-type operators. The purpose of this paper is to introduce and start a systematic investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space of the unit sphere in ℂ. In particular, we establish a corresponding spectral inclusion theorem and a Brown-Halmos type theorem. On the other hand, we characterize commuting dual Toeplitz operators as well as normal and quasinormal ones. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space H( X × X) for 0 < p < p ≤ 1 for some p and all 1 < q < ∞, where X × X is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L( X × X) (for 1 < q < ∞) and Hardy space H( X × X) (for 0 < p ≤ 1). As an application, we prove that an operator T, which is bounded on L( X × X) for some 1 < q < ∞, is bounded from H( X × X) to L( X × X) if and only if T is bounded uniformly on all ( p, q)-product atoms in L( X × X). The similar boundedness criterion from H( X × X) to H( X × X) is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, the authors establish the boundedness of the multilinear Calderón-Zygmund operator from products of Hardy spaces into Hardy spaces. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider a class of bilinear pseudo-differential operators with symbols of order 0 and type (1, 0) in the sense of Hörmander and use the atomic decompositions of local Hardy spaces to establish the boundedness of the bilinear pseudo-differential operators and the bilinear singular integral operators on the product of local Hardy spaces. [ABSTRACT FROM AUTHOR]

Let $$\left( {\mathcal{Y},d,d\lambda } \right)$$ be (ℝ, |·|, µ), where |·| is the Euclidean distance, µ is a nonnegative Radon measure on ℝ satisfying the polynomial growth condition, or the Gauss measure metric space (ℝ, |·|, d), or the space ( S, d, ρ), where S ≡ ℝ ⋉ ℝ is the ( ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces $$\left\{ {X_s \left( \mathcal{Y} \right)} \right\}_{0 < s \leqslant \infty }$$ and the BMO-type spaces $$\left\{ {BMO\left( {\mathcal{Y}, s} \right)} \right\}_{0 < s \leqslant \infty }$$. Let H $$\left( \mathcal{Y} \right)$$ be the known atomic Hardy space and L $$\left( \mathcal{Y} \right)$$ the subspace of f ∈ L $$\left( \mathcal{Y} \right)$$ with integral 0. The authors prove that the dual space of X $$\left( \mathcal{Y} \right)$$ is $$BMO\left( {\mathcal{Y},s} \right)$$ when s ∈ (0,∞), X $$\left( \mathcal{Y} \right)$$ = H $$\left( \mathcal{Y} \right)$$ when s ∈ (0, 1], and X $$\left( \mathcal{Y} \right)$$ = L $$\left( \mathcal{Y} \right)$$ (or L $$\left( \mathcal{Y} \right)$$). As applications, the authors show that if T is a linear operator bounded from H $$\left( \mathcal{Y} \right)$$ to L $$\left( \mathcal{Y} \right)$$ and from L $$\left( \mathcal{Y} \right)$$ to L $$\left( \mathcal{Y} \right)$$, then for all r ∈ (1,∞) and s ∈ ( r,∞], T is bounded from X $$\left( \mathcal{Y} \right)$$ to the Lorentz space L $$\left( \mathcal{Y} \right)$$, which applies to the Calderón-Zygmund operator on (ℝ, |·|, µ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝ, |·|, d) and the spectral operator associated with the spectral multiplier on ( S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to prove that for any 0 < p ≤ 2/3 there exists a martingale f ∈ H such that Marcinkiewicz-Fejér means of the two-dimensional conjugate Walsh-Fourier series of the martingale f is not uniformly bounded in the space L. [ABSTRACT FROM AUTHOR]

In this paper an atomic decomposition theorem for Banach-space-valued weak Hardy regular martingale space w H( X) is given. As an application, p-smoothable Banach spaces are characterized in terms of bounded sublinear operators defined on Banach-space-valued weak Hardy regular martingale space w H( X). [ABSTRACT FROM AUTHOR]

In the present paper, we obtain three independent results on the Besov-Morrey spaces and the Triebel-Lizorkin-Morrey spaces. That is, we are going to obtain the characterization of local means, the boundedness of pseudo-differential operators and the characterization of the Hardy-Morrey spaces. By using the maximal estimate and the molecular decomposition, we shall integrate and extend the known results on these spaces. [ABSTRACT FROM AUTHOR]

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ n are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces. [ABSTRACT FROM AUTHOR]

Let X = ( X, d, µ) be a space of homogeneous type in the sense of Coifman and Weiss. The purpose of this paper is to generalize the definition of Hardy space H p ( X) and prove that the generalized Hardy spaces have the same property as H p ( X). Our definition includes a kind of Hardy-Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the ℝ n case. Let ( X, δ, µ) be the normalized space of ( X, d, µ) in the sense of Macías and Segovia. We also study the relations of our function spaces for ( X, d, µ) and ( X, δ, µ). [ABSTRACT FROM AUTHOR]

In this paper, the authors prove that if Ω satisfies a class of the integral Dini condition, then the parametrized area integral $$ \mu ^{\rho }_{{\Omega ,S}} $$ is a bounded operator from the Hardy space H 1(ℝ n ) to L 1(ℝ n ) and from the weak Hardy space H 1,∞(ℝ n ) to L 1,∞(ℝ n ), respectively. As corollaries of the above results, it is shown that $$ \mu ^{\rho }_{{\Omega ,S}} $$ is also an operator of weak type (1, 1) and of type ( p, p) for 1 < p < 2, respectively. These conclusions are substantial improvement and extension of some known results. [ABSTRACT FROM AUTHOR]

In this paper, we treat a class of non–standard commutators with higher order remainders in the Lipschitz spaces and give ( L p , L q ), ( H p , L q ) boundedness and the boundedness in the Triebel– Lizorkin spaces. Our results give simplified proofs of the recent works by Chen, and extend his result. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the boundedness properties of $$ \mu ^{m}_{{\Omega ,b}} $$ generated by the function b ∈ Lip β (ℝn)(0 < β ≤ 1/ m) and the Marcinkiewicz integrals operator μ Ω . The boundednesses are established on the Hardy type spaces $$ H^{p}_{{b^{m} ,s}} {\left( {\mathbb{R}^{n} } \right)} $$ and the Herz–Hardy type spaces $$ H_{{b^{m} }} \dot{K}^{{\alpha ,p}}_{q} {\left( {\mathbb{R}^{n} } \right)} $$ . [ABSTRACT FROM AUTHOR]

Let G be a locally compact Vilenkin group. In this paper the authors study the boundedness of bilinear operators B ( f, g) given by finite sums of products of Calderón–Zygmund operators in Herz space and Herz–type Hardy space on G. And an example, the boundedness from the products of Herz space to Herz–type Hardy space is given in the last section. [ABSTRACT FROM AUTHOR]

In this paper, two classes of closely related multilinear singular and fractional integrals, which include the commutators as special cases, are studied and their boundedness on Herz type spaces is discussed. In fact, it is proved that these operators are actually not bounded in certain extreme cases. [ABSTRACT FROM AUTHOR]

In this paper, the autors study the continuity properties of higher order commutators generated by the homogeneous fractional integral and BMO functions on certain Hardy spaces, weak Hardy spaces and Herz-type Hardy spaces. [ABSTRACT FROM AUTHOR]

We first study the Volterra operator V acting on spaces of Dirichlet series. We prove that V is bounded on the Hardy space ℋ 0 p for any 0 < p ≤ ∞, and is compact on ℋ 0 p for 1

g. We prove that if Tg is bounded on the Hardy space ℋ p , then it is bounded on the Bergman space ℋ w p . [ABSTRACT FROM AUTHOR]

Let (χ, ρ, μ) be a space of homogeneous type in the sense of Coifman and Weiss, and Y(χ) a ball quasi-Banach function space on χ, 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 the Hardy space HY(χ) associated with Y(χ), via the Lusin-area function, and then establish its various equivalent characterizations, respectively, in terms of atoms, molecules, and Littlewood—Paley g-functions and gλ*-functions. As an application, the authors obtain the boundedness of Calderón—Zygmund operators from HY (χ) to Y(χ), or to HY(χ) via first establishing a boundedness criterion of linear operators on HY(χ). All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this article exist in that, to escape the reverse doubling condition of μ and the triangle inequality of ρ, the authors subtly use the wavelet reproducing formula, originally establish an admissible molecular characterization of HY(χ), and fully apply the geometrical properties of χ expressed by dyadic reference points or dyadic cubes. [ABSTRACT FROM AUTHOR]

Let T be a strongly singular Calderón-Zygmund operator and b ∈ Lloc(ℝn). This article finds out a class of non-trivial subspaces BMOω,p,n(ℝn)of BMO(ℝn) for certain ω ∈ A1, 0 < p ≤ 1 and 1 < u ≤ ∞, such that the commutator [b, T] is bounded from weighted Hardy space H ω p (ℝ n) to weighted Lebesgue space L ω p (ℝ n) if b ∈ BMOω,p,∞(ℝn), and is bounded from weighted Hardy space H ω p (ℝ n) to itself if T*1 = 0 and b ∈ BMOω,p,u(ℝn)for 1 < u < 2. [ABSTRACT FROM AUTHOR]

Abstract Basic properties of the Herz-type Hardy spaces HK[sup alpha, p]sub q], such as the boundedness of singular integral operators and the fractional integration operators, atomic decomposition, dense subspaces, etc., are established in the full range 0 < q < infinity. Keywords Herz-type Hardy space. Singular integral, Fractional integration, Multilinear operator [ABSTRACT FROM AUTHOR]

Let B be the unit disc in ℝ2, ℋ be the completion of C 0 ∞ (B) under the norm ∥ u ∥ H = (∫ B | ▽ u | 2 d x − ∫ B u 2 (1 − | x | 2) 2 d x) 1 2 , ∀ u ∈ C 0 ∞ (B). By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric d v H = d x (1 − | x | 2 ) 2 , we prove that, for any fixed α , 0 ≤ α < λ p (B) = inf u ∈ H , u ≠ 0 ∥ u ∥ H 2 ∥ u ∥ p 2 , the supremum sup u ∈ H , ∥ u ∥ H = 1 ∫ B e 4 π (1 + α ∥ u ∥ p 2) u 2 d x < + ∞ , ∀ p > 1. This is an analog of early results of Lu—Yang (Discrete Contin. Dyn. Syst., 2009) and Yang (Trans. Amer. Math. Soc., 2007), and extends those of Wang—Ye (Adv. Math., 2012) and Yang—Zhu (Ann. Global Anal. Geom., 2016). [ABSTRACT FROM AUTHOR]

On the Dirichlet space of the unit ball, we study some algebraic properties of Toeplitz operators. We give a relation between Toeplitz operators on the Dirichlet space and their analogues defined on the Hardy space. Based on this, we characterize when finite sums of products of Toeplitz operators are of finite rank. Also, we give a necessary and sufficient condition for the commutator and semi-commutator of two Toeplitz operators being zero. [ABSTRACT FROM AUTHOR]

Let A be an expansive dilation on ℝ and φ : ℝ × [0, ∞) → [0, ∞) an anisotropic Musielak-Orlicz function. Let H (ℝ) be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of H (ℝ). The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case (namely, A := 2 I ) coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space H(ℝ) with p ∈ (0, 1] (in this case, A := 2 I , φ( x, t) := t for all x ∈ ℝ and t ∈ [0, ∞)), this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from H (ℝ) to L (ℝ) or from H (ℝ) to itself. [ABSTRACT FROM AUTHOR]

The dual of B-valued martingale Hardy space H( B) with small index 0 < r ≤ 1, which is associated with the conditional p-variation of B-valued martingale, is characterized. In order to obtain the results, a new type of Campanato spaces for B-valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of the underlying Banach space. [ABSTRACT FROM AUTHOR]

We study the weighted composition operators W on Hardy space $$H^2 (\mathbb{B})$$ whenever h ∈ BMOA (resp. h ∈ VMOA). Analogous results are given for $$H^p (\mathbb{B})$$ spaces and the scale of weighted Bergman spaces. In the latter case, BMOA is replaced by the Bloch space (resp. VMOA by the little Bloch space). [ABSTRACT FROM AUTHOR]

By bounded vector-valued functions and block matrix representations of Hankel operators, we completely characterize the hyponormality of Toeplitz operators on the Hardy space of the polydisk. [ABSTRACT FROM AUTHOR]

We apply discrete Littlewood-Paley-Stein theory, developed by Han and Lu, to establish Calderón-Zygmund decompositions and interpolation theorems on weighted Hardy spaces H for ω ∈ A in both the one-parameter and two-parameter cases. [ABSTRACT FROM AUTHOR]