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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]

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 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]

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, 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]

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 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]

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W( h p, ℓ∞) to W( L p, ℓ∞). This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W( L1, ℓ∞) ⊃ L1. [ABSTRACT FROM AUTHOR]

Let A be an expansive dilation. We define weakly strongly singular integral kernels and study the action of the operators induced by these kernels on anisotropic Hardy spaces associated with A. [ABSTRACT FROM AUTHOR]

By the weak atomic decompositions of weak martingale Hardy spaces, we investigate the interpolation spaces between weak martingale Hardy spaces and martingale Hardy spaces. [ABSTRACT FROM AUTHOR]

Some estimates for the convolution operators with kernels of type ( l, r) on Lebesgue spaces with power weights and Herz–type spaces in the setting of homogeneous groups are established. [ABSTRACT FROM AUTHOR]