Norm estimates and integral kernel estimates for a bounded operator in Sobolev spaces

We show that a bounded linear operator from the Sobolev space W-r(-m)(Omega) to W-r(m)(Omega) is a bounded operator from L-p(Omega) to L-q(Omega), and estimate the operator norm, if p, q, r is an element of [1, infinity] and a positive integer m satisfy certain conditions, where Omega is a domain in R-n. We also deal with a bounded linear operator from W-p'(-m)(Omega) to W-p(m)(Omega) with p' = p/(p - 1), which has a bounded and continuous integral kernel. The results for these operators are applied to strongly elliptic operators.