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arXiv · 2511.13286

Logarithmic double phase embeddings with variable exponents: Necessary and Sufficient Conditions

Ankur Pandey·Nijjwal Karak
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Abstract

In this paper, we study the necessary and sufficient conditions in the domain for Sobolev-type embedding of the space $W^{1,\Phi(\cdot,\cdot)}(\Omega)$ where $\Phi(x,t):=t^{p(x)}+ a(x) t^{q(x)}\log^{r(x)}(e+t)$ with $1\leq p(x)\leq q(x).$ We have established subcritical embedding in bounded John domains under some regularity assumptions on exponents $p,$ $q,$ $r$, and $a$. Conversely, we have proved that if the embedding holds in any domain $\Omega$ in $\mathbb{R}^n,$ then $\Omega$ must satisfy the log-measure density condition.

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Ankur Pandey, Nijjwal Karak. 2025-11-17. Logarithmic double phase embeddings with variable exponents: Necessary and Sufficient Conditions. https://arxiv.org/abs/2511.13286

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