WeightedBMOEstimates for Toeplitz Operators on Weighted Lebesgue Spaces

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.

<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>

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.

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.

<abstract> <p>In the paper, some weighted maximal inequalities for the Toeplitz operator related to the singular integral transform with variable CalderȮn-Zygmund kernel are proved. As an application, the boundedness of the operator on weighted Lebesgue space are obtained.</p> </abstract>

Consider a bilinear interaction between two linear dispersive waves with a generic resonant structure (roughly speaking, space and time resonant sets intersect transversally). We derive an asymptotic equivalent of the solution for data in the Schwartz class, and bilinear dispersive estimates for data in weighted Lebesgue spaces. An application to water waves with infinite depth, gravity and surface tension is also presented.

This paper is devoted to the study of Fourier series multipliers . An analog of the Marcinkiewicz theorem on multipliers of Fourier series in weighted Lebesgue spaces is obtained.

Our aim in this paper is to deal with the boundedness of the central Hardy-Littlewood maximal operator on Herz spaces with radial variable exponent. As a special case, we show the boundedness of the maximal operator in weighted Lebesgue spaces with radial variable exponent. We also discuss the integrability of the maximal functions in such spaces.

<abstract><p>We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of $ p $-Laplacian type with $ p > \frac{2n}{n+2} $ in weighted Lebesgue spaces $ L^q_w $. We introduce a new condition on the weight $ w $ which depends on the intrinsic geometry concerned with the parabolic $ p $-Laplace problems. Our condition is weaker than the one in <sup>[<xref ref-type="bibr" rid="b13">13</xref>]</sup>, where similar estimates were obtained. In particular, in the case $ p = 2 $, it is the same as the condition of the usual parabolic $ A_q $ weight.</p></abstract>

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.

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

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.

We obtain some sufficient conditions for embedding weighted Sobolev classes on a John domain into weighted Lebesgue spaces. The weights are functions of the distance to some h-set on the boundary of the domain. The problem reduces to estimating the norms of two-weighted summation operators on a certain tree. We then apply our previous results on these estimates.

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.