Variable Exponent Regularity via Muckenhoupt Condition

Daviti Adamadze; Anna Kh. Balci; Lars Diening

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arXiv · 2608.12095

Variable Exponent Regularity via Muckenhoupt Condition

Daviti Adamadze·Anna Kh. Balci·Lars Diening

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Abstract

For the first time, we establish higher integrability of the gradient and local $L^\infty$ estimates of weak solutions to the $p(x)$-Laplacian without assuming $\log$-H\"older continuity of $p$. Instead, we establish these results for exponents satisfying a generalized Muckenhoupt condition. This admits discontinuous exponents acting as pointwise multipliers of the BMO class. Our framework bridges the gap between classical $p(x)$-regularity and weighted Muckenhoupt theory, providing a new foundation for the analysis of differential equations with highly irregular non-standard growth.

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Daviti Adamadze, Anna Kh. Balci, Lars Diening. 2026-08-12. Variable Exponent Regularity via Muckenhoupt Condition. https://arxiv.org/abs/2608.12095

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