Abstract:
A singular operator with Cauchy kernel on the subspaces of weight Lebesgue space is considered. A sufficient condition for a bounded action of this operator from a subspace to another subspace of weight Lebesgue space of functions is found. These conditions are not identical with Muckenhoupt conditions. Moreover, the completeness, minimality, and basicity of sines and cosines systems are considered. 1. Introduction Consider the following singular operator with Cauchy kernel: where , , is an appropriate density, is a weight function of the form ( for ), are real numbers. Under we understand a Lebesgue weight space with the norm Bounded action of the operator in the spaces plays an important role in many problems of mathematics including the theory of bases. This direction has been well developed and treated in the known monographs. We will need the following. Statement 1. The operator is bounded in if and only if the inequalities are fulfilled. Concerning this fact a one can see the monograph [1] and papers [2–4]. Inequalities (1.4) are Muckenhoupt condition with respect to the weight function with degrees . It is known that the classic system of exponents ( are integers) forms a basis in if and only if inequalities (1.4) hold (see, e.g., [3, 4]). It turns that if you consider the singular operator acting on the subspace of the weighted Lebesgue space, then inequality (1.4) is not necessary for the bounded action. At different points of degeneration the change interval of the corresponding exponent is expanded. This paper is devoted to studying these issues. 2. Some Necessary Facts Let , —a weight function of the form where , . Denote the space of even (odd) functions in by ( ), that is, We'll need the following identity: Indeed, we have For compactness of the notation, we assume . Thus, From this identity, we can easily get the following relations: The authors of the papers [4–7] used these relations earlier while establishing the basicity criterion of the system of sines and cosines with linear phases in . Thus, the following is valid. Lemma 2.1. The following identities are true: In the similar way, we obtain Further, we must take into account the following relation: As a result, we have Assume Thus, In sequel the following lemma is valid. Lemma 2.2. The following identities are true: 3. Boundedness of Singular Operators on Subspace of Even Functions Let . We have where the kernels , , are determined by the expressions Continue the weight to the interval by parity and denote by : It is obvious that are the degeneration points of . Thus, , , that is,

Abstract:
In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion.) The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic H\"older singularities demonstrate sharpness of our machinery comparatively with known results.

Abstract:
We prove that the pointwise multipliers acting in a pair of fractional Sobolev spaces form the space of boundary traces of multipliers in a pair of weighted Sobolev space of functions in a domain.

Abstract:
In this article we consider an elliptic system in R3 and prove a theorem on unique solvability in a special scale of weighted Sobolev spaces.

Abstract:
In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

Abstract:
We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$. Denoting by $\rho$ the weight function, we compare the space $W^{1,p}(X,d,\rho m)$ (which always concides with the closure $H^{1,p}(X,d,\rho m)$ of Lipschitz functions) with the weighted Sobolev spaces $W^{1,p}_\rho(X,d,m)$ and $H^{1,p}_\rho(X,d,m)$ defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that $W^{1,p}(X,d,\rho m)=H^{1,p}_\rho(X,d, m)$. We also adapt results by Muckenhoupt and recent work by Zhikov to the metric measure setting, considering appropriate conditions on $\rho$ that ensure the equality $W^{1,p}_\rho(X,d,m)=H^{1,p}_\rho(X,d,m)$.

Abstract:
We define weighted variable Sobolev capacity and discuss properties of capacity in the space 1,(？)(？,). We investigate the role of capacity in the pointwise definition of functions in this space if the Hardy-Littlewood maximal operator is bounded on the space 1,(？)(？,). Also it is shown the relation between the Sobolev capacity and Bessel capacity.

Abstract:
We study in this paper a class of second-order linear elliptic equations in weighted Sobolev spaces on unbounded domains of ？, ≥3. We obtain an a priori bound, and a regularity result from which we deduce a uniqueness theorem.

Abstract:
In this paper, we survey a number of recent results obtained inthe study of weighted Sobolev spaces (with power-type weights, A_p -weights, p-admissible weights, regular weights and the conjecture of De Giorgi) and the existence of entropy solutions for degenerate quasilinear elliptic equations.

Abstract:
In the present paper a criterion for basicity of exponential system with linear phase is obtained in Sobolev weight space . In solving mathematical physics problems by the Fourier method, there often arise the systems of exponents of the form where and are continuous or piecewise-continuous functions. Substantiation of the method requires studying the basis properties of these systems in Lebesgue and Sobolev spaces of functions. In the case when and are linear functions, the basis properties of these systems in , , were completely studied in the papers [1–9]. The weighted case of the space was considered in the papers [10, 11]. The basis properties of these systems in Sobolev spaces were studied in [12–14]. It should be noted that the close problems were also considered in [15]. In the present paper we study basis properties of the systems (1) and (2) in Sobolev weight spaces when , , where is a real parameter. Therewith the issue of basicity of system (2) in Sobolev spaces is reduced to the issue of basicity of system (1) in respective Lebesgue spaces. Let and be weight spaces with the norms respectively, where , . Denote by the direct sum , where is the complex plane. The norm in this space is defined by the expression , where . The following easily provable lemmas play an important role in obtaining the main results. It holds the following. Lemma 1. Let , ; . Then the operator realizes an isomorphism between the spaces and ; that is, the spaces and are isomorphic. Proof. At first prove the boundedness of the operator . We have Having applied the Holder inequality, hence we get where Consequently where Let us show that . Put ; that is, where , . By differentiating this equality, we get , a.e. on . Hence it follows that . From (11) it directly follows that a.e. on , and so . Show that ( is the range of values of the operator ). Let be an arbitrary function. Let . It is clear that and . Then from the Banach theorem we get that the operator is boundedly invertible. The lemma is proved. The following lemma is also valid. Lemma 2. Let and , . Then for all , where Proof. Let , . We have Since and , then , . It is easy to see that and moreover . The lemma is proved. In obtaining the basic results we need the following main lemma. Lemma 3. Let , and , , be a real parameter, . Let have the expansion in the space . Then it is valid Proof. As it follows from Lemma 2, . At first consider the case when , . In this case the system is minimal in (see [4]). Then from the results of the paper [16], the Hausdorff-Young inequality is valid for this system; that is,