No abstract

The purpose of this paper is to establish the weighted norm inequalities of one-sided oscillatory integral operators by the aid of interpolation of operators with change of measures.

For 1 ≤ p ≤ ∞ and n/p < α < 1 + n/p, the fractional capacity C α,p (K, ) of a compact subset K with respect to a bounded open Lipschitz set ⊃ K of R n is systematically investigated and effectively applied to the fractional Sobolev space W α,p ( ) which plays an important role in calculus of variations, harmonic analysis, potential theory, partial differential equations, mathematical physics and so forth. In particular, we explore:• the basic properties of C α,p (K, ); the existence of a capacitary potential, and the relationship between C α,p (K, ) and the fractional Laplace equation. • the C α,p (K, )-based nature of a nonnegative Radon measure μ given on that induces an embedding of W α,p ( ) into the Lebesgue space L q ( , μ) as 1 ≤ p ≤ q < ∞ or the exponentially integrable Lebesgue space exp(c|f | 2n 2n−α ) ∈ L 1 ( , μ) as p = 2n/α or the Lebesgue space L ∞ ( , μ) as 2n/α < p ≤ ∞; the geometric characterization of a fractional Sobolev embedding domain via C α,p (K, ).• some of the essential behaviours of K with C α,p (K, ) = 0.