L^p-Taylor approximations characterize the Sobolev space W^1,p

2014/06/03 by Daniel Spector, Spector, Daniel E.
Computer Science · Decision Sciences · Engineering · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.1406.0672

openalex publication_date 2014/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we introduce a variant of Calderón and Zygmund's notion of L^p-differentiability - an L^p-Taylor approximation. Our first result is that functions in the Sobolev space W^1,p(ℝ^N) possess a first order L^p-Taylor approximation. This is in analogy with Calderón and Zygmund's result concerning the L^p-differentiability of Sobolev functions. In fact, the main result we announce here is that the first order L^p-Taylor approximation characterizes the Sobolev space W^1,p(ℝ^N), and therefore implies L^p-differentiability. Our approach establishes connections between some characterizations of Sobolev spaces due to Swanson using Calderón-Zygmund classes with others due to Bourgain, Brezis, and Mironescu using nonlocal functionals with still others of the author and Mengesha using nonlocal gradients. That any two characterizations of Sobolev spaces are related is not surprising, however, one consequence of our analysis is a simple condition for determining whether a function of bounded variation is in a Sobolev space.

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