A fractional modular Poincaré--Wirtinger-type inequality in Sobolev spaces with variable exponents

Abstract

Let $Ω\subset\mathbb{R}^n$ be a bounded Lipschitz domain and let $p:Ω\to[1,\infty)$ be a measurable, essentially bounded exponent; no regularity of $p$ -- in particular, no log-Hölder continuity -- is assumed. For a general non-negative measurable kernel $\mathcal{K}(x,y)$ admitting a radial bounding function $κ(x,r)$ in the second variable, we establish the fractional modular Poincaré--Wirtinger-type inequality $$ \iint_{Ω\timesΩ}\mathcal{K}(x,y)\,|u(x)-u(y)|^{p(x)} d y\, d x \leqslant C\int_Ωc(x)\left(\int_Ω \frac{|\nabla u(z)|^{p(x)}}{|z-x|^{n-1+θ}}\, d z\right) d x, $$ valid for every $u\in W^{1,1}(Ω)$, in particular every $u\in W^{1,p(\cdot)}(Ω)$, and every $θ\in[0,1)$. Here $$ c(x)=\int_0^{D}r^{n+p(x)-2+θ}κ(x,r)\,d r, $$ $D$ is the diameter of any fixed ball containing $\overlineΩ$, and the constant $C$ depends only on $n,p^+,θ$ and the fixed domain $Ω$. For the fractional kernel $\mathcal{K}(x,y)=|x-y|^{-n-λp(x)}$ 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 Poincaré inequalities, and one reflection extension for the weights $|z-x|^{-(n-1+θ)}$. The constants are uniform down to exponent $1$, and the inequalities are understood in the extended nonnegative sense when either side is infinite.