A modular Poincaré–Wirtinger inequality for Sobolev spaces with variable exponents

Abstract

In the context of Sobolev spaces with variable exponents, Poincaré–Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form

$$\begin{aligned} \int _\Omega \left| f(x)-\langle f\rangle _{\Omega }\right| ^{p(x)} \ {\textrm{d} x} \leqslant C \int _\Omega |\nabla f(x)|^{p(x)}{\textrm{d} x}, \end{aligned}$$

are known to be false. As a result, all available modular versions of the Poincaré–Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. This obstruction is commonly believed to be an unavoidable anomaly of the variable exponent setting. The main aim of this paper is to show that this is not the case, i.e., that a consistent generalization of the Poincaré–Wirtinger inequality to the variable exponent setting is indeed possible. Our contribution is threefold. First, we establish that a modular Poincaré–Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if \(\Omega \subset \mathbb {R}^n \) is a bounded Lipschitz domain, and if \(p\in L^\infty (\Omega )\), \(p \geqslant 1\), then for every \(f\in C^\infty (\bar{\Omega })\) the following generalized Poincaré–Wirtinger inequality holds

$$\begin{aligned} \int _\Omega \left| f(x)-\langle f\rangle _{\Omega }\right| ^{p(x)} \ {\textrm{d} x} \leqslant C \int _\Omega \int _\Omega \frac{|\nabla f(z)|^{p(x)}}{|z-x|^{n-1}}\ {\textrm{d} z}{\textrm{d} x}, \end{aligned}$$

where \(\langle f\rangle _{\Omega }\) denotes the mean of \(f\) over \(\Omega \), and \(C>0\) is a positive constant depending only on \(\Omega \) and \(\Vert p\Vert _{L^\infty (\Omega )}\). Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincaré–Wirtinger constant on Lipschitz domains.