Bilinear forms on non-homogeneous Sobolev spaces

Abstract

In this paper we show that if $b\in L^2(\R^n)$, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces $H_s^2(\R^n)\times H_s^2(\R^n)$, $0<s<1$ by $$ (f,g)\in H_s^2(\R^n)\times H_s^2(\R^n) \to \int_{\R^n} (Id-\Delta)^{s/2}(fg)({\bf x}) b({\bf x})d{\bf x}, $$ is continuous if and only if the positive measure $|b({\bf x})|^2d{\bf x} $ is a trace measure for $H_s^2(\R^n)$.