Boundedness and compactness in weighted Lebesgue spaces of integral operators with variable integration limits

We give criteria for the boundedness of integral operators with nonnegative monotone kernel in the weighted Lebesgue spaces on a half-axis.

Schur-type Banach modules of integral kernels acting on mixed-norm Lebesgue spaces

In the framework of rational Dunkl analysis on the real line, we introduce Dunkl-type Besov-Fofana spaces which include Besov-Dunkl spaces and Dunkl-type Besov-Morrey spaces.As applications, we establish the boundedness of Dunkl-type fractional integral and fractional maximal operators in these spaces.

<p>In the setting of a Heisenberg group, we first studied the sharp weak estimate for the $ n $-dimensional fractional Hardy operator from $ L^p $ to $ L^{q, \infty} $. Next, we studied the sharp bounds for the $ m $-linear $ n $-dimensional integral operator with a kernel on weighted Lebesgue spaces. As an application, the sharp bounds for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on weighted Lebesgue spaces were obtained. Finally, according to the previous steps, we also found the estimate for the Hausdorff operator on weighted $ L^p $ spaces.</p>

In this paper a two-weight criterion for multidimensional Hardy type operator and its dual operator acting from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces is proved. As application we prove the boundedness of multidimensional geometric mean operator in the weighted Musielak-Orlicz spaces. In particular, from obtained results implies the boundedness of multidimensional Hardy operator and its dual operator acting from usual weighted Lebesgue spaces into weighted variable Lebesgue spaces. In this paper we establish integral-type neces- sary and sufficient condition on weights, which provides the boundedness of the multidimensional Hardy type operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.

The main aim of this paper is to show that certain Banach spaces, defined via integral kernel operators, are Banach modules (with respect to some known Banach algebras and convolution products onℝ+). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spacesLωpℝ+forp≥1. We also show new inequalities in these weighted Lebesgue spaces. These results are applied to several concrete function spaces, for example, weighted Sobolev spaces and fractional Sobolev spaces defined by Weyl fractional derivation.

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions.
 The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences.
 The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

Problems of solving different linear difference equation is given to study the properties of the matrix operators in various functional spaces. One of the important problems of functional analysis is to establish criteria of boundedness of the linear operators in functional spaces. Question of the boundedness of matrix operators in sequence spaces is a classic problem of functional analysis and there are many unsolved problems in it. For example, in the general case it is impossible to establish the boundedness of the matrix operator in the spaces of sequences by the given matrix. Therefore, various classes of matrix operators are considered for which the criteria of their boundedness are known. Due to the variety of encountered problems in practice, it is necessary to have various alternative criteria for the boundedness of matrix operators. In this paper, we establish a new alternative criterion for the boundedness of one class of matrix operators.

Let [Formula: see text], [Formula: see text] and [Formula: see text] be a singular or fractional integral operator with homogeneous kernel [Formula: see text]. In this paper, a CMO type space [Formula: see text] is introduced and studied. In particular, the relationship between [Formula: see text] and the Lipchitz space [Formula: see text] is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator [Formula: see text] on weighted Lebesgue spaces via functions in [Formula: see text], and an equivalent characterization of the compactness for [Formula: see text] via functions in [Formula: see text] are obtained. Some results are new even in the unweighted setting for the first-order commutators.

In this paper the authors give a sufficient condition such that the commutator generated by the weighted Lipschitz function and the fractional integral operator with homogeneous kernel satisfying certain Dini condition is bounded on weighted Lebesgue spaces.

We prove necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces $L^{p(\cdot)}_w$ (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever $w$ is a Khvedelidze weight, and the variable exponent $p(t)$ satisfies the estimate $|p(\tau)-p(t)|\le A/(-\log|\tau-t|)$.

In this paper, we consider the boundedness of integrals of fractional Hadamard integration and Hadamard-type integration (mixed and directional) in Lebesgue spaces with mixed norm. We prove Sobolev-type theorems of boundedness of one-dimensional and multidimensional Hadamard-type fractional integration in weighted Lebesgue spaces with mixed norm.

For weighted spaces of functions of positive smoothness on irregular domains, embedding theorems into weighted Lebesgue spaces are proved.

In this paper, we consider convolution operators, integral operators with weak singularity, Riesz potentials, in particular, those with kernelsacting in special classes of Banach function spaces X () and their subspaces X s ()), and we prove some representation theorems for the functions from Banach-Sobolev spaces.We also prove the boundedness of Riesz potential in additive-invariant spaces.