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Book Sections Year : 2018

Sum-intersection property of Sobolev spaces

Abstract

We may always decompose any function $f\in W^{1,p}({\mathbb R}^N)$ as $f=g+h$ with $g\in W^{\lambda, p/\lambda}$ and $h\in W^{p,1}$. A stronger property, natural in the context of functional calculus in Sobolev spaces, is that we may choose $g\in W^{\lambda, p/\lambda}\cap W^{1,p}$ and $h\in W^{p,1}\cap W^{1,p}$. We address here the question of the validity of a similar result for three Sobolev spaces $W^{s_1, p_1}$, $W^{s, p}$, $W^{s_2, p_2}$ satisfying the proportionality relations\begin{equation*}(1)\ s=\theta s_1+(1-\theta)s_2,\ \frac 1p=\frac\theta{p_1}+\frac{1-\theta}{p_2}\ \text{for some }\theta\in (0,1).\end{equation*}For most of $s_1,\ldots, p_2$ satisfying (1), we prove that\begin{equation*}(2)\ W^{s,p}({\mathbb R}^N)=(W^{s_1, p_1} ({\mathbb R}^N)\cap W^{s,p} ({\mathbb R}^N))+(W^{s_2, p_2}({\mathbb R}^N)\cap W^{s,p} ({\mathbb R}^N)).\end{equation*}In some exceptional situations, this equality does not hold, and we derive an alternative decomposition. We also establish the validity of (2) when the first equality in (1) is replaced by the suboptimal condition $s>\theta s_1+(1-\theta)s_2$.
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Dates and versions

hal-01574338 , version 1 (13-08-2017)
hal-01574338 , version 3 (31-10-2017)

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  • HAL Id : hal-01574338 , version 3

Cite

Petru Mironescu. Sum-intersection property of Sobolev spaces. Current Research in Nonlinear Analysis, 135, Springer, pp.203-228, 2018, Springer Optimization and its Applications. ⟨hal-01574338v3⟩
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