Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that

$||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ||f||_{L^1({\mathbb R})}^{1-\frac{\alpha}{\beta}} ||D^\beta f||_{L^2({\mathbb R})}^{\frac{\alpha}{\beta}}$

for $0 < \alpha \leq \beta$ arbitrary (i.e., fractional)? This is a special case of a Gagliardo-Nirenberg-Sobolev inequality, but Nirenberg's 1959 proof in Ann. Scuola Norm. Sup. Pisa only holds for integer $\alpha, \beta$. There are many authors who prove slightly different GNS inequalities, but I need this exact one and it won't budge. One can play with Nirenberg's idea to remove the constraint that $\beta$ is an integer. Knowing that, you can work even harder and get $1 \leq \alpha \leq \beta$. But for $0 < \alpha < 1$, the proof breaks seemingly irreparably. It's also unfortunately true that the $L^1$ breaks Littlewood-Paley proofs (at least without some replacement for Nirenberg's magic).

Has anyone seen this particular family of inequalities anywhere? If it's a folk theorem, what's the trick?

  • $\begingroup$ It is a very interesting question. Maybe a good start point is Lemma 5.3 in arxiv.org/pdf/1104.0306.pdf and Lemma 6.6 in arxiv.org/pdf/0812.4979v1.pdf I would also suggest to think on homogeneous Sobolev spaces, the embeding operator and the Riesz-Thorin Theorem. $\endgroup$
    – guacho
    Sep 1 '14 at 6:15

The inequality is valid. The precise answer to your questions (in much more general form) is given in the resent article

Brezis, H., & Mironescu, P. (2017). Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Annales de l'Institut Henri Poincare (C) Non Linear Analysis. DOI:10.1016/j.anihpc.2017.11.007


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