Abstract. In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A pcondition. 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 nonsmooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Hölder singularities demonstrate the sharpness of our machinery comparatively with known results.

ABSTRACT. We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M, g) and the L q,p -cohomology of that manifold.The L q,p -cohomology of (M, g) is defined to be the quotient of the space of closed differential forms in L p (M) modulo the exact forms which are exterior differentials of forms in L q (M).

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Abstract. In 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The "free vibrating membrane" corresponds to the Neumann-Laplace operator in bounded plane domains. In this paper we obtain estimates for the first non-trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains. This case includes convex domains, John domains etc... On the base of our estimates we conjecture that the eigenvalues of the Neumann-Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates which is based on weighted Poincaré-Sobolev inequalities, obtained by the authors recently.

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