Herz Spaces Research Articles

Commutators of generalized Hardy operators

AbstractLetHa,b be the commutator generated by the generalized Hardy operator and the CMO function. The (Lp, Lp) boundedness of Ha,b is discussed in this paper. Furthermore, the authors consider the boundedness of Ha,b on the weighted homogeneous Herz spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Pseudo-differential operators of class $S_{0,0}^m$ on the Herz-type spaces

In this paper, we show the boundedness of pseudo-differential operators of class $S_{0,0}^m$ on the Herz spaces $\dot{K}_q^{\alpha,p}$ and the Herz-type Hardy spaces $H\dot{K}_q^{\alpha,p}$.

Homogeneous Herz spaces and regularity results

In the note the author treats second order divergence form operators L with coefficients in V M O and proves that a solution of L u = d i v f , where f is in homogeneous Herz spaces K ̇ q α , p , has derivatives in K ̇ q α , p .

Boundedness of several operators on weighted Herz spaces

We consider the boundedness of singular integral operators and fractional integral operators on weighted Herz spaces. For this purpose we introduce generalized Herz space. Our results are the best possible.

Boundedness for multilinear fractional integral operators on Herz type spaces

In this paper the boundedness for the multilinear fractional integral operator Iα(m) on the product of Herz spaces and Herz-Morrey spaces are founded, which improves the Hardy-Littlewood-Sobolev inequality for classical fractional integral Iα. The method given in the note is useful for more general multilinear integral operators.

Pointwise Summability of Gabor Expansions

A general summability method, the so-called θ-summability method is considered for Gabor series. It is proved that if the Fourier transform of θ is in a Herz space then this summation method for the Gabor expansion of f converges to f almost everywhere when f∈L1 or, more generally, when f∈W(L1,l∞) (Wiener amalgam space). Some weak type inequalities for the maximal operator corresponding to the θ-means of the Gabor expansion are obtained. Hardy-Littlewood type maximal functions are introduced and some inequalities are proved for these.

Multipliers and Herz type spaces

Hormander condition for boundedness of multiplier operators will be replaced by a weaker condition described by certain weighted or non-weighted Herz spaces. Some results on boundedness of multiplier operators are then established. As direct corollaries of main theorems in this paper, several celebrated results on boundedness of multiplier operators will be improved or deduced.

Wiener amalgams, Hardy spaces and summability of Fourier series

AbstractA general summability method, the so called θ-summability is considered for multi-dimensional Fourier series, where θ is in the Wiener algebraW(C,ℓ1)($\mathbb{R}$d). It is based on the use of Wiener amalgam spaces, weighted Feichtinger's algebra, Herz and Hardy spaces. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from theHpHardy space toLp(orHp). This implies some norm and almost everywhere convergence results for the θ-means. Large number of special cases of the θ-summation are considered.

The norm inequalities for the weighted Cesaro mean operators

This paper gives some necessary and sufficient conditions for the weighted Cesaro mean operators to be bounded on Herz spaces. The corresponding new operator norm inequalities are obtained.

New hardy spaces associated with some anisotropic Herz spaces and their applications

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝn 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-Calderon-Zygmund operators on the anisotropic Herz-type Hardy spaces.

Fourier multipliers on power‐weighted Hardy spaces

AbstractUsing Herz spaces, we obtain a sufficient condition for a bounded measurable function on ℝn to be a Fourier multiplier on Hpα (ℝn ) for 0 < p < 1 and –n < α ≤ 0. Our result is sharp in a certain sense and generalizes a recent result obtained by Baernstein and Sawyer. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Herz spaces and restricted summability of Fourier transforms and Fourier series

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy–Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type ( 1 , 1 ) , provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that σ T θ f → f over a cone-like set a.e. for all f ∈ L 1 ( R d ) . Moreover, σ T θ f ( x ) converges to f ( x ) over a cone-like set at each Lebesgue point of f ∈ L 1 ( R d ) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.

Herz spaces and summability of Fourier transforms

AbstractA general summability method is considered for functions from Herz spaces Kαp,r (ℝd ). The boundedness of the Hardy–Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ ‐means σθ T f is also bounded on the corresponding Herz spaces and σθ T f → f a.e. for all f ∈ K–d /p p,∞ (ℝd ). Moreover, σθ T f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K–d /p p,∞ (ℝd ) if and only if the Fourier transform of θ is in the Herz space Kd /p p ′,1 (ℝd ). Norm convergence of the θ ‐means is also investigated in Herz spaces. As special cases some results are obtained for weighted Lp spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Convolution operators on Lorentz spaces over locally compact Vilenkin groups

Let G be a locally compact Vilenkin group. Using Herz spaces, we give sufficient conditions for a distribution on G to be a convolution operator on certain Lorentz spaces. Our results generalize Hormander's multiplier theorem on G. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Toeplitz Operators and Herz Spaces on the Half-Space

On the setting of the half-space we introduce the Schatten-Herz classes of Toeplitz operators and obtain characterizations for positive Toeplitz operators to belong to those classes. We also prove results concerning the boundedness and compactness of Toeplitz operators with Herz symbols. Such a study has been recently done on the ball. At a critical step of the proofs we employ a much simplified argument to extend the range of parameters for Herz spaces on which the Berezin transform is bounded. Our results show not only that most of results on the ball continue to hold, but also that there is some pathology caused by the unboundedness of the domain.

Boundedness of vector-valued Calderón-Zygmund operators on Herz spaces with non-doubling measures

In the paper we obtain vector-valued inequalities for Calderón-Zygmund operator, simply CZO on Herz space and weak Herz space. In particular, we obtain vector-valued inequalities for CZO on Lq(℞d, |x|αdμ) space, with 1 < q < ∞, −n < α < n(q − 1), and on L1,∞(℞d, |x|αdμ) space, with −n < α < 0.

Boundedness for commutators of fractional Hardy operators on Herz spaces

Some boundedness results are established for commutators generated by classical fractional Hardy operators and one-sided CMO functions on the homogeneous Herz spaces, which extends some known results.

Some multiplier theorems for anisotropic hardy spaces

Let A be a symmetric expansive matrix and Hp (ℝ n) be the anisotropic Hardy space associated with A. For a function m in L∞ (ℝ n), an appropriately chosen function η in C ∞c (ℝ n) and j ∈ ℤ define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 < p < 1 and \(\hat m_j \) belongs to the anisotropic non-homogeneous Herz space K 1/p−1,p 1 (ℝ n), then m is a Fourier multiplier from Hp (ℝ n) to Lp (ℝ n). For p = 1, a similar result is obtained if the space K 0,1 1 (ℝ n) is replaced by a slightly smaller space K(ω). Moreover, the authors show that if 0 < p ≤ 1 and if the sequence {(mj)∨} belongs to certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp (ℝ n) to Lp (ℝ n).

Strang‐Fix theory for approximation order in weighted Lp‐spaces and Herz spaces

In this paper, we study the Strang‐Fix theory for approximation order in the weighted Lp ‐spaces and Herz spaces.

Bilinear Operators on Herz?type Hardy Spaces on Locally Compact Vilenkin Groups

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 Calderon–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.