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Abstract. This paper deal with a new Sobolev type function space called the space with multiweighted derivatives. This space is a generalization of the usual one dimensional Sobolev space. As basis for this space serves some differential operators containing weight functions.We establish necessary and sufficient conditions for the boundedness and compactness of the embedding between the spaces with multiweighted derivatives with different weights and different metrics.Mathematics subject classification (2010): 46E35, 46E30.

Abstract. We derive necessary and sufficient conditions (of Muckenhoupt-Bradley type) for the validity of q -analogs of (r, p) -weighted Hardy-type inequalities for all possible positive values of the parameters r and p . We also point out some possibilities to further develop the theory of Hardy-type inequalities in this new direction.Mathematics subject classification (2010): 26D10, 26D15, 33D05, 39A13.

The description of the closure of finite or smooth finite functions in functional spaces are classical tasks of functional space theory. This task is important in smooth functional spaces such as those of Sobolev, Nikolski, Besov and in their various generalizations. Usually, in a weightless space of smooth functions, the set of compactly finite functions, generally speaking, is not dense. But in the weighted space of smooth functions, for example, in the Sobolev weighted space, with strong degeneracy of the weight, many compactly finite functions can be dense. Therefore, an important issue is the problem of characterizing the closure of compactly finite functions in the weight space under consideration. Here we consider a weighted space of Sobolev type of the second order with three weights and it describes the closure of the set of functions with compact supports.

Abstract. Inequalities of the formare considered, where K is an integral operator of Volterra type and H is the Hardy operator. Under some assumptions on the kernel K we give necessary and sufficient conditions for such an inequality to hold.Mathematics subject classification (2010): 26D10, 39B62.