On Convolution Type Operators with Piecewise Slowly Oscillating Data

ABSTRACTLet μ be a non-negative Ahlfors n-dimensional measure on . In this context we shall consider convolution type operators , , where the kernels are supposed to satisfy certain size and regularity conditions. We prove Welland's type inequalities for the operator and its commutator , with , that include the case . As far as we know both estimates are new even in the case of the Lebesgue measure. We shall also give sufficient conditions on a pair of weights that guarantee the boundedness of between two different weighted Lebesgue spaces when the underlying measure is Ahlfors n-dimensional.

The study of some signal processing problems within Bayesian frameworks and semigroups theory, in the case where the Banach space under consideration may be nonseparable, was discussed in the paper published in 2022 by two of the authors. For the Cauchy problem for the diffusion equation, which is a typical example of evolution equation, there was studied the question of what can be saved in the study of the problem in the case with the initial data in the nonseparable classical Morrey space. In this paper we develop this approach to an essentially more general case of generalized Morrey spaces with general weights. The study is based on emersion of such spaces into weighted Lebesgue spaces with Muckenhoupt weights.

In this paper, we obtain the compactness of commutators of pseudo-differential operators with smooth symbols on weighted $$L^{p}$$ spaces where the weight functions belong to a class of new weights which includes the classical Muckenhoupt weights and the results are new even on weighted $$L^{p}$$ spaces where the weight functions belong to Muckenhoupt weights. In addition, a strong sufficient condition for compactness in new weighted $$L^{p}(1<p<\infty )$$ spaces is given, which has interest in its own right.

We give quantitative estimates of the rate of convergence of equiuniform approximation processes of integral operators in Banach spaces in terms of the modulus of continuity of functions to be approximated and higher order absolute moments of approximate kernels. Furthermore, applications are presented for various equi-uniform summation processes, interpolation type operators, convolution type operators, and several concrete examples of approximating operators are also provided.

We establish Fredholm criteria and index formulas for one-dimensional zero-order pseudodifferential operators with piecewise continuous generating functions onLp spaces with Muckenhoupt weights. The Fredholm symbol of such operators is shown to be a matrix function defined on a set which, roughly speaking, is a cylinder with a certain collection of horn shaped handles. The presence of these horns implies that, unlike the case ofLp spaces without weight or with so-called power weights, the spectrum may contain heavy parts, i. e. the set of the interior points of the spectrum need not be empty. Our proof makes essential use of recent results by Finck, Roch, Silbermann, Gohberg, and Krupnik on the inverse closedness of certain Banach algebras.

In this paper we give a further development of the results of the paper [1] and apply it to convolution operators in the spaces Lp(x) . We consider the question of extendability of the Young theorem: well known for constant p and q, to the case when they may be variable. We also potential type operators with the Kernel . In section 1 we develop some estimates for Lp(x) -norms of power functions of distance truncated to exterior of a ball of radius . Section 2 deals with convolution operators in the spaces Lp(x) and Section 3 is devoted to potential type operators.

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.

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.

Let $H$ be a compact subgroup of a locally compact group $G$‎. ‎In this paper we provide a (convolution type) multiplication on $L^1(G/H)$‎, ‎the space of all integrable functions on the homogenous space $G/H$ equipped with a relatively invariant measure‎. ‎Among other things‎, ‎we show that $L^1(G/H)$ endowed with this multiplication is a Banach algebra possessing a bounded left approximate identity‎.

Vertical square functions and other operators associated with an elliptic operator

&lt;p&gt;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.&lt;/p&gt;

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.&#x0D; 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.&#x0D; 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.

Lebesgue spaces are considered with Muckenhoupt weights. Fractional order mixed difference operator is investigated to obtain mixed fractional modulus of smoothness in these spaces. Using this modulus of smoothness we give the proof of direct and inverse estimates of angular trigonometric approximation. Also we obtain an equivalence between fractional mixed modulus of smoothness and fractional mixed K-functional.********************************************************************************************************************

Let X be a Banach function space over the unit circle T and let H[X] be the abstract Hardy space built upon X. If the Riesz projection P is bounded on X and a∈L∞, then the Toeplitz operator Taf=P(af) is bounded on H[X]. We extend well-known results by Brown and Halmos for X=L2 and show that, under certain assumptions on the space X, the Toeplitz operator Ta is bounded (resp., compact) if and only if a∈L∞ (resp., a=0). Moreover, aL∞≤TaB(H[X])≤PB(X)aL∞. These results are specified to the cases of abstract Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and Nakano spaces with radial oscillating weights.

The automorphisms of convolution algebras of C∞ functions