Traces for fractional Sobolev spaces with variable exponents

TL;DR

This paper establishes a trace theorem for fractional Sobolev spaces with variable exponents, providing conditions under which boundary norms are controlled by interior fractional seminorms and Lebesgue norms.

Contribution

It introduces a new trace inequality for fractional Sobolev spaces with variable exponents, extending classical results to more general variable exponent settings.

Findings

Proved a trace inequality for fractional Sobolev spaces with variable exponents.

Derived conditions relating boundary and interior variable exponents.

Established bounds connecting boundary Lebesgue norms to interior fractional seminorms.

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.