preprint

Inserted: 29 oct 2026
Last Updated: 29 oct 2026

Year: 2026

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.

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