Traces for fractional Sobolev spaces with variable exponents (1704.02599v1)

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