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?
