arXiv subjects

Daviti Adamadze

Publications and source records attributed to Daviti Adamadze.

5 recordsLinked to original sources

arXiv2026

Variable Exponent Regularity via Muckenhoupt Condition

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 $\log$-Hölder 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.

math.AP↗

arXiv2026

Double phase meets Muckenhoupt

In this paper we generalize the famous result of [FKS] to the double phase model. In particular, we work with minimal assumptions on the modulating coefficient by introducing a Muckenhoupt-type condition on generalized Orlicz spaces. We develop a complete theory equivalent to that of classical Muckenhoupt weights, including the boundedness of the maximal operator and Sobolev-Poincare estimates. We combine this with the De~Giorgi technique to show Hölder continuity of the solutions.

math.AP↗

arXiv2025

Vilenkin-Fourier series in variable Lebesgue spaces

Let $S_{n}f$ denote the $n$th partial sum of the Vilenkin-Fourier series of a function $f \in L^{1}(G)$. For $1 < p_{-} \leq p_{+} < \infty$, we characterize all exponents $p(\cdot)$ for which the convergence of $S_{n}f$ to $f$ in $L^{p(\cdot)}(G)$ holds whenever $f \in L^{p(\cdot)}(G)$.

math.FA↗

arXiv2025

Maximal operator on variable exponent spaces

We explore the boundedness of the Hardy-Littlewood maximal operator $M$ on variable exponent spaces. Our findings demonstrate that the Muckenhoupt condition, in conjunction with Nekvinda's decay condition, implies the boundedness of $M$ even for unbounded exponents. This extends the results of Lerner, Cruz-Uribe and Fiorenza for bounded exponents. We also introduce a novel argument that allows approximate unbounded exponents by bounded ones while preserving the Muckenhoupt and Nekvinda conditions.

math.FA↗

arXiv2020

On singular extensions of continuous functionals from C([0,1]) to variable Lebesgue spaces

Valadier and Hensgen proved independently that the restriction of functional $ϕ(x)=\int_{0}^{1}x(t)dt,\,\,x\in L^{\infty}([0,1])$ on the space of continuous functions $C([0,1])$ admits a singular extension back to the whole space $L^{\infty}([0,1]).$ Some general results in this direction for the Banach lattices were obtained by Abramovich and Wickstead. In present note we investigate analogous problem for variable exponent Lebesgue spaces, namely we prove that if the space of continuous functions $C([0,1])$ is closed subspace in $L^{p(\cdot)}([0,1]),$ then every bounded linear functional on $C([0,1])$ is the restriction of a singular linear functional on $L^{p(\cdot)}([0,1])$.

math.FA↗