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

2025/11/17 by Ankur Pandey, Pandey, Ankur, Nijjwal Karak +1
Mathematics · #Advanced Harmonic Analysis Research #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2511.13286

openalex publication_date 2025/11/17 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the necessary and sufficient conditions in the domain for Sobolev-type embedding of the space W^{1,Φ(⋅,⋅)}(Ω) where Φ(x,t):=t^p(x)+ a(x) t^q(x)log^r(x)(e+t) with 1≤ p(x)≤ 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 ℝⁿ, then Ω must satisfy the log-measure density condition.

Citations

• On logarithmic double phase problems
• New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems
• A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type
• Optimal Embeddings for Triebel-Lizorkin and Besov Spaces on Quasi-Metric Measure Spaces
• Pointwise Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type
• A new class of double phase variable exponent problems: Existence and uniqueness
• Orlicz Sobolev Inequalities and the Doubling Condition
• Lower bound of measure and embeddings of Sobolev, Besov and\n Triebel-Lizorkin spaces

Related

• New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems
• On logarithmic double phase problems
• Logarithmic double phase problems with generalized critical growth
• Musielak-Orlicz-Sobolev embeddings: Necessary and Sufficient Conditions
• Double phase problems with variable exponents depending on the solution and the gradient in the whole space ℝ^N