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- W1987319048 abstract "In this note, we introduce a variant of Calderon and Zygmund's notion of $L^p$-differentiability - an emph{$L^p$-Taylor approximation}. Our first result is that functions in the Sobolev space $W^{1,p}(mathbb{R}^N)$ possess a first order $L^p$-Taylor approximation. This is in analogy with Calderon 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 $W^{1,p}(mathbb{R}^N)$, and therefore implies $L^p$-differentiability. Our approach establishes connections between some characterizations of Sobolev spaces due to Swanson using Calderon-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." @default.
- W1987319048 created "2016-06-24" @default.
- W1987319048 creator A5010246813 @default.
- W1987319048 date "2014-06-03" @default.
- W1987319048 modified "2023-09-27" @default.
- W1987319048 title "$L^p$-Taylor approximations characterize the Sobolev space $W^{1,p}$" @default.
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