Traces for fractional Sobolev spaces with variable exponents

Abstract

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $p \colon \overline{\Omega}\times \overline{\Omega} \rightarrow (1,\infty)$ and $q \colon \partial \Omega \rightarrow (1,\infty )$ are continuous functions such that $$\frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) \qquad \mathrm{in} \hspace{1em} \partial \Omega \cap \lbrace x \in \overline{\Omega}\colon n-sp(x,x) >0 \rbrace,$$ then the inequality $$||f||_{L^{q(\cdot)}(\partial\Omega)} \leq C \left\lbrace ||f||_{L^{\bar{p}(\cdot)}(\Omega)} + [f]_{s,p ( \cdot , \cdot )} \right\rbrace $$ holds. Here $\bar{p}(x)=p(x,x)$ and $\lbrack f \rbrack_{s,p(\cdot,\cdot)} $ denotes the fractional seminorm with variable exponent, that is given by $$[f]_{s,p(\cdot , \cdot)} := \mathrm{inf} \left\lbrace \lambda > 0: \int_{\Omega}\int_{\Omega }\frac{|f(x)-f(y)|^{p(x,y)}}{\lambda ^{p(x,y)} |x-y|^{n+sp(x,y)}}dxdy \lt 1 \right\rbrace$$ and $||f||_{L^{q(\cdot)}(\partial\Omega)}$ and $||f||_{L^{\bar{p}(\cdot)}(\Omega)}$ are the usual Lebesgue norms with variable exponent.