Some sufficient conditions for embedding a weighted Sobolev class on a John domain

Order estimates are obtained for the Kolmogorov and linear widths of weighted Sobolev classes on a John domain. The conditions imposed on the weights are such that the orders of the widths have the same form as in the case of constant weights and a domain with Lipschitz boundary.

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 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.

<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>

A sufficient condition for higher‐order Sobolev‐type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement‐invariant function spaces. The condition takes form of a one‐dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.

According to a known characterization, a function $$f$$ belongs to the Sobolev space $$W^{p,1}(\mathbb{R}^n)$$ of functions contained in $$L^p(\mathbb{R}^n)$$ along with their generalized first-order derivatives precisely when there is a function $$g\in L^p(\mathbb{R}^n)$$ such that $$|f(x)-f(y)|\le |x-y|(g(x)+g(y))$$ for almost all pairs $$(x,y)$$ . An analogue of this estimate is also known for functions from the Gaussian Sobolev space $$W^{p,1}(\gamma)$$ in infinite dimension. In this paper the converse is proved; moreover, it is shown that the above inequality implies membership in appropriate Sobolev spaces for a large class of measures on finite-dimensional and infinite-dimensional spaces.

A Constructive Approach to Strong Tractability Using Quasi-Monte Carlo Algorithms

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.

We establish lower norm bounds for multivariate functions within weighted Lebesgue spaces, characterised by a summation of functions whose components solve a system of nonlinear integral equations. This problem originates in portfolio selection theory, where these equations allow one to identify mean-variance optimal portfolios, composed of standard European options on several underlying assets. We elaborate on the Smirnov property—an integrability condition for the weights that guarantees the uniqueness of solutions to the system. Sufficient conditions on weights to satisfy this property are provided, and counterexamples are constructed, where either the Smirnov property does not hold or the uniqueness of solutions fails.

The method of using rearrangements to give sufficient conditions for Fourier inequalities between weighted Lebesgue spaces is revisited, a comparison between two known sufficient conditions is completed, and the method is extended to provide sufficient conditions for a new range of indices. When $$1<q<p<\infty $$ , simple conditions on weights ensure that the Fourier transform will map a weighted $$L^p$$ space into a weighted $$L^q$$ space. These are established in Theorems 1 and 4 of Benedetto and Heinig (J Fourier Anal Appl 9(1):1–37, 2003). The proofs apply when $$2<q<p$$ and $$1<q<p<2$$ but not in the remaining case, $$1<q<2<p$$ . Here, counterexamples are given to show that these simple conditions are no longer sufficient when $$1<q<2<p$$ . Also, various additional conditions are presented, any of which will restore sufficiency in that case.

Necessary and sufficient conditions are given for generalized Calderon and Hilbert operators to be bounded from weighted Lebesgue spaces into suitable weighted BMO and Lipschitz spaces. Moreover, we have obtained new results on the boundedness of these operators from $$L^{\infty }$$ into BMO, even in the unweighted case for the Hilbert operator. The class of weights involved are close to the doubling and reverse Holder conditions related to the Muckenhoupt’s classes.

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.

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.

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.