Conductor inequalities and criteria for Sobolev type two-weight imbeddings

A typical inequality handled in this article connects the LpLp-norm of the gradient of a function to a one-dimensional integral of the pp-capacitance of the conductor between two level surfaces of the same function. Such conductor inequalities   lead to necessary and sufficient conditions for multi-dimensional and one-dimensional Sobolev type inequalities involving two arbitrary measures. Compactness criteria and two-sided estimates for the essential norm of the related imbedding operator are obtained. Some counterexamples are presented to illustrate the peculiarities arising in the case of higher derivatives. Criteria for two-weight inequalities with fractional Sobolev norms of order l<2l<2 are found.