Littlewood-Paley Characterizations of Fractional Sobolev Spaces via Averages on Balls

2015/11/24 by Feng Dai, Jun Liu, Dai, Feng +5 · 1 citation
Mathematics · Medicine · #42B20 #42B35 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Hidradenitis Suppurativa and Treatments #Nonlinear Partial Differential Equations #Primary 46E35 #Secondary 42B25

paper · pdf · doi:10.48550/arxiv.1511.07598

openalex publication_date 2015/11/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/02

Abstract

In this paper, the authors characterize Sobolev spaces W^α,p(\mathbb Rⁿ) with the smoothness order α∈(0,2] and p∈(max\1, (2n)/(2α+n)\,∞), via the Lusin area function and the Littlewood-Paley g_λ^∗-function in terms of centered ball averages. The authors also show that the condition p∈(max\1, (2n)/(2α+n)\,∞) is nearly sharp in the sense that these characterizations are no longer true when p∈ (1,max\1, (2n)/(2α+n)\). These characterizations provide a new possible way to introduce fractional Sobolev spaces with smoothness order in (1,2] on metric measure spaces.

Citations

• Some Maximal Inequalities

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• Equivalent Characterizations and Applications of Fractional Sobolev Spaces with Partially Vanishing Traces on (ε,δ,D)-Domains Supporting D-Adapted Fractional Hardy Inequalities

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