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Journal Articles Journal of Functional Analysis Year : 2019

Where Sobolev interacts with Gagliardo-Nirenberg

Abstract

We investigate the validity of the fractional Gagliardo-Nirenberg-Sobolev inequality (1) $\displaystyle \|f\|_{W^{r,q}(\Omega)}\lesssim\| f\|_{W^{s_1,p_1}(\Omega)}^\theta\|f\|_{W^{s_2,p_2}(\Omega)}^{1-\theta},\ \forall\, f\in W^{s_1, p_1}(\Omega)\cap W^{s_2, p_2}(\Omega)$. Here, $s_1, s_2, r$ are non-negative numbers (not necessarily integers), $1\le p_1, p_2,q\le \infty$, and we assume, for some $\theta\in (0,1)$, the standard relations (2) $\displaystyle r$<$s:=\theta s_1+(1-\theta)s_2$ and $ 1/q=(\theta/p_1+(1-\theta)/p_2)-(s-r)/N$. Formally, estimate (1) is obtained by combining the ``pure'' fractional Gagliardo-Nirenberg style interpolation inequality (3) $\displaystyle \|f\|_{W^{s, p}(\Omega)}\lesssim\| f\|_{W^{s_1,p_1}(\Omega)}^\theta\|f\|_{W^{s_2,p_2}(\Omega)}^{1-\theta}$ (with $1/p:=\theta/p_1+(1-\theta)/p_2$) with the fractional Sobolev style embedding (4) $\displaystyle W^{s, p}(\Omega)\hookrightarrow W^{r,q}(\Omega), \ 0\le r$<$s,\, 1\le p$<$q\le\infty,\, 1/q=1/p-(s-r)/N, \, p(s-r)\le N$. Estimates (3) and (4) are true ``most of the time'', but not always; the exact range of validity of (3) and (4) has been known. Combining these results, we infer that (1) is valid ``most of the time''. However, the validity of (1) when (3) and/or (4) fail was unclear. The goal of this paper is to characterize the values of $s_1, s_2, r, p_1, p_2, q, \theta, N$ such that (1) holds.
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Dates and versions

hal-01982813 , version 1 (16-01-2019)
hal-01982813 , version 2 (08-03-2019)

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Haim Brezis, Petru Mironescu. Where Sobolev interacts with Gagliardo-Nirenberg. Journal of Functional Analysis, In press, ⟨10.1016/j.jfa.2019.02.019⟩. ⟨hal-01982813v2⟩
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