### 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.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 435-446.

Dates
Accepted: 21 June 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431720

Digital Object Identifier
doi:10.22034/aot.1704-1152

Mathematical Reviews number (MathSciNet)
MR3730039

Zentralblatt MATH identifier
06804220

#### Citation

Del Pezzo, Leandro; Rossi, Julio D. Traces for fractional Sobolev spaces with variable exponents. Adv. Oper. Theory 2 (2017), no. 4, 435--446. doi:10.22034/aot.1704-1152. https://projecteuclid.org/euclid.aot/1512431720

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