Logarithmic double phase embeddings with variable exponents: Necessary and Sufficient Conditions
Ankur Pandey, Nijjwal Karak
TL;DR
This work characterizes Sobolev-type embeddings for the variable-exponent logarithmic double-phase Musielak–Orlicz–Sobolev spaces $W^{1,Φ(\cdot,\cdot)}(Ω)$ with $Φ(x,t)= t^{p(x)}+a(x)t^{q(x)}(\log(e+t))^{r(x)}$. It proves a subcritical embedding on bounded John domains under detailed regularity assumptions on the exponents and data, and shows that the domain must satisfy a log-measure density condition if the embedding holds in any domain. The sufficient part leverages the hypothesis $(A0),(A1),(A2)$ and related Inc/Dec properties for $Φ$, together with a Hardy–Littlewood–type framework from HH19 to obtain $W^{1,Φ(\cdot,\cdot)}(Ω)\hookrightarrow L^{Ψ(\cdot,\cdot)}(Ω)$. The necessary part translates embedding properties into geometric density constraints on $Ω$, showing a dichotomy: log-measure density when $r(⋅)\ge0$ somewhere, or standard measure density when $r(⋅)\le0$ everywhere, with Nekvinda’s decay handling unbounded domains.
Abstract
In this paper, we study the necessary and sufficient conditions in the domain for Sobolev-type embedding of the space $W^{1,Φ(\cdot,\cdot)}(Ω)$ where $Φ(x,t):=t^{p(x)}+ a(x) t^{q(x)}\log^{r(x)}(e+t)$ 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 $Ω$ in $\mathbb{R}^n,$ then $Ω$ must satisfy the log-measure density condition.