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 $p:\overline{\Omega }\times \overline{\Omega }\to (1,\infty )$ and $q:\partial \Omega \to (1,\infty )$ are continuous functions such that \[ \frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) \qquad \mbox{in} \partial Ω\cap \{x\in\overlineΩ\colon n-sp(x,x) >0\}, \] then the inequality $$\Vert f{\Vert }_{L^{q(\cdot )}(\partial \Omega )}\le C\left\{\Vert f{\Vert }_{L^{\bar{p}(\cdot )}(\Omega )}+[f{]}_{s,p(\cdot ,\cdot )}\right\}$$ holds. Here $\bar{p}(x)=p(x,x)$ and $[f{]}_{s,p(\cdot ,\cdot )}$ denotes the fractional seminorm with variable exponent, that is given by \[ \lbrack f\rbrack_{s,p(\cdot,\cdot)} := \inf \left\{λ>0\colon \int_Ω\int_Ω\frac{|f(x)-f(y)|^{p(x,y)}}{λ^{p(x,y)} |x-y|^{n+sp(x,y)}}dxdy<1\right\} \] and $\Vert f{\Vert }_{L^{q(\cdot )}(\partial \Omega )}$ and $\Vert f{\Vert }_{L^{\bar{p}(\cdot )}(\Omega )}$ are the usual Lebesgue norms with variable exponent.

10 pages