Where Sobolev interacts with Gagliardo–Nirenberg

By Haim Brezis, Petru Mironescu. Journal of Functional Analysis, 2019

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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.

Subfields: Applied Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/8da804b96b7c6c93b1f792c8558ea153ea8ea57c

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In Applied Mathematics

  1. Lenka Slav'ikov'a
  2. L. Pick
  3. Andrea Cianchi
  4. A. Alberico
  5. Iwona Chlebicka
  6. Cristina Tarsi
  7. Joaquim Martín
  8. Robert J. Berman

In Mathematical Physics

  1. Lenka Slav'ikov'a
  2. L. Pick
  3. Andrea Cianchi
  4. A. Alberico
  5. Iwona Chlebicka
  6. P. Hasto
  7. Joaquim Martín
  8. Giulio Ciraolo

In Mechanics of Materials

  1. L. Pick
  2. Lenka Slav'ikov'a
  3. A. Alberico
  4. Andrea Cianchi
  5. Cristina Tarsi

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