Del Pezzo, L., Rossi, J. (2017). Traces for fractional Sobolev spaces with variable exponents. Advances in Operator Theory, 2(4), 435-446. doi: 10.22034/aot.1704-1152
Leandro Del Pezzo; Julio D. Rossi. "Traces for fractional Sobolev spaces with variable exponents". Advances in Operator Theory, 2, 4, 2017, 435-446. doi: 10.22034/aot.1704-1152
Del Pezzo, L., Rossi, J. (2017). 'Traces for fractional Sobolev spaces with variable exponents', Advances in Operator Theory, 2(4), pp. 435-446. doi: 10.22034/aot.1704-1152
Del Pezzo, L., Rossi, J. Traces for fractional Sobolev spaces with variable exponents. Advances in Operator Theory, 2017; 2(4): 435-446. doi: 10.22034/aot.1704-1152
Traces for fractional Sobolev spaces with variable exponents
2Universidad de Buenos Aires Facultad de Ciencias Exactas y Naturales Depto Matematica Ciudad Universitaria, pab 1, Buenos Aires, Argentina
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 \[ \frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) \qquad \mbox{ in } \partial \Omega \cap \{x\in\overline{\Omega}\colon n-sp(x,x) >0\}, \] 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 \begin{equation*} \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\} \end{equation*} 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.