We consider the monomial weight |x1|1 · · · |xn|n in R, where Ai ≥ 0 is a real number for each i = 1, ..., n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by |x1|1 · · · |xn|ndx, and they contain the best or critical exponent (which depends on A1, ..., An). More i...

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and the electrorheological fluids. also get singular limit formula extending Nguyen results to anisotropic case.

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 withMuckenhoupt conditions. Moreover, the completeness, minimality, and basicity of sines and cosines systems are conside...

We consider the sigma-finite measures in the space of vector-valued distributions on the manifold X with characteristic functional