New Lorentz spaces for the restricted weak-type Hardy's inequalities

Associated to the class of restricted weak-type weights for the Hardy operator R_p, we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal function.     

Weighted convolution inequalities for radial functions

Annali di Matematica Pura ed Applicata (1923 -) - We obtain convolution inequalities in Lebesgue and Lorentz spaces with power weights when the functions involved are assumed to be radially...     

Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation

We exhibit the optimal constant for Sobolev inequalities in Lorentz spaces for a mean oscillation, and its relation with a boundedness of the Hardy–Littlewood maximal operator in Sobolev spaces. Some applications to a scale invariant form of Hardy?s inequality in a limiting case are also considered.     

Hankel convolution operators on entire functions and distributions

In this paper we study the Hankel convolution operators on the space of even and entire functions and on Schwartz distribution spaces. We characterize the Hankel convolution operators as those ones that commute with Hankel translations and with a Bessel operator. Also we prove that the Hankel convolution operators are hypercyclic and chaotic on the spaces under consideration.     

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

Bounded composition operator on Lorentz spaces

We study composition operators on Lorentz spaces. In particular, we obtain necessary and sufficient conditions under which a measurable mapping induces a bounded composition operator.     

Some remarks on short‐time Fourier integral operators and classes of rapidly decaying functions

In this article, we extend the space of rapidly decaying functions to a space of rapidly decaying Boehmians. We provide convolution products, convolution theorems and generate their associated spaces of Boehmian. Then, we define the short‐time Fourier integral operator on the Boehmian spaces. Moreover, we show that the short‐time Fourier integral operator of the Boehmian is a sequentially continuous mapping that preserves certain desired properties. An inversion formula and some injections have also been obtained.     

Interpolative construction and the generalized cotype of abstract Lorentz spaces

The Banach operator ideals generated by an interpolative construction depending on concave functions are studied. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. The abstract notion of Rademacher type and cotype for operators between Banach spaces is introduced. It is shown that abstract interpolation Lorentz spaces that appeared in the factorization theorem are of the generalized Rademacher cotype determined by Orlicz sequence spaces.     

Compactness and essential norms of composition operators on Orlicz–Lorentz spaces

In this work, we present necessary and sufficient conditions for compactness of the composition operator on Orlicz–Lorentz spaces and determine upper and lower estimates for the essential norm of the composition operator on these spaces.     

Compactness of Some Operators of Convolution Type in Generalized Morrey Spaces

Sufficient conditions for the compactness in generalized Morrey spaces of the composition of a convolution operator and the operator of multiplication by an essentially bounded function are obtained. Very weak conditions on the function are also obtained under which the commutator of the operator of multiplication by such a function and a convolution operator is compact. The compactness of convolution operators in domains of cone type is investigated.     

Multiplicative convolutions of functions from Lorentz spaces and convergence of series of Fourier–Vilenkin coefficients

Let f and g be functions from different Lorentz spaces L ^{p, q} [0, 1), h be theirmultiplicative convolution and xxxx be Fourier coefficients of h with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of xxxx with multiplicators of type k ^b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.     

Elementary reverse Hölder type inequalities with application to operator interpolation theory

We give a very elementary proof of the reverse Hölder type inequality for the classes of weights which characterize the boundedness on of the Hardy operator for nonincreasing functions. The same technique is applied to Calderón operator involved in the theory of interpolation for general Lorentz spaces. This allows us to obtain further consequences for intermediate interpolation spaces.

     

A quadratic-phase integral operator for sets of generalized integrable functions

In this paper, we aim to discuss the classical theory of the quadratic-phase integral operator on sets of integrable Boehmians. We provide delta sequences and derive convolution theorems by using certain convolution products of weight functions of exponential type. Meanwhile, we make a free use of the delta sequences and the convolution theorem to derive the prerequisite axioms, which essentially establish the Boehmian spaces of the generalized quadratic-phase integral operator. Further, we nominate two continuous embeddings between the integrable set of functions and the integrable set of Boehmians. Furthermore, we introduce the definition and the properties of the generalized quadratic-phase integral operator and obtain an inversion formula in the class of Boehmians.     

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented. The author is supported by the National Science Foundation under grant DMS 0099881. The author is supported by KBN Grant 1 P03A 013 26.     

Interpolation theorems on weighted Lorentz martingale spaces

In this paper several interpolation theorems on martingale Lorentz spaces are given.The proofs are based on the atomic decompositions of martingale Hardy spaces over weighted measure spaces.Applying the interpolation theorems,we obtain some inequalities on martingale transform operator.     

Fourier Series in Weighted Lorentz Spaces

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LogL and more general Lorentz–Zygmund spaces.     

Chaotic dynamics of the heat semigroup on the Damek-Ricci spaces

We consider the rank one Riemannian symmetric spaces of noncompact type and their non-symmetric generalization, namely the Damek-Ricci spaces. We show that the heat semigroup generated by a certain perturbation of the Laplace-Beltrami operator of these spaces is chaotic on their L _p -spaces when p > 2. The range of p and the corresponding perturbation are sharp. A precursor to this result is due to Ji and Weber [19] where it was shown that under identical conditions the heat operator is subspace-chaotic on the Riemannian symmetric spaces, which is weaker than it being chaotic. We also extend the results to the Lorentz spaces L _p,q , which are generalizations of the Lebesgue spaces. This enables us to point out that the chaoticity degenerates to subspace-chaoticity only when q = ∞.