A fractional modular Poincar\'e--Wirtinger-type inequality in Sobolev spaces with variable exponents

Abstract

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain and let $p:\Omega\to[1,\infty)$ be a measurable, essentially bounded exponent; no regularity of $p$ -- in particular, no log-H\"older continuity -- is assumed. For a general non-negative measurable kernel $\mathcal{K}(x,y)$ admitting a radial bounding function $\kappa(x,r)$ in the second variable, we establish the fractional modular Poincar\'e--Wirtinger-type inequality $$ \iint_{\Omega\times\Omega}\mathcal{K}(x,y)\,|u(x)-u(y)|^{p(x)} d y\, d x \leqslant C\int_\Omega c(x)\left(\int_\Omega \frac{|\nabla u(z)|^{p(x)}}{|z-x|^{n-1+\theta}}\, d z\right) d x, $$ valid for every $u\in W^{1,1}(\Omega)$, in particular every $u\in W^{1,p(\cdot)}(\Omega)$, and every $\theta\in[0,1)$. Here $$ c(x)=\int_0^{D}r^{n+p(x)-2+\theta}\kappa(x,r)\,d r, $$ $D$ is the diameter of any fixed ball containing $\overline\Omega$, and the constant $C$ depends only on $n,p^+,\theta$ and the fixed domain $\Omega$. For the fractional kernel $\mathcal{K}(x,y)=|x-y|^{-n-\lambda p(x)}$ we obtain a variable-exponent Gagliardo-modular estimate. The proof replaces the starshaped chain decompositions used there by a combination of a Morrey-type estimate on starshaped domains, a positive-kernel proof of weighted Poincar\'e inequalities, and one reflection extension for the weights $|z-x|^{-(n-1+\theta)}$. The constants are uniform down to exponent $1$, and the inequalities are understood in the extended nonnegative sense when either side is infinite.