Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO

Abstract

Using elementary arguments based on the Fourier transform we prove that for $1\le q<p<\infty$ and $s\ge 0$ with $s>n(1/2-1/p)$, if $f\in L^{q,\infty }({\mathbb{R}}^n)\cap {\dot{H}}^s({\mathbb{R}}^n)$ then $f\in L^p({\mathbb{R}}^n)$ and there exists a constant $c_{p,q,s}$ such that \[ \|f\|_{L^p} \leq c_{p,q,s} \|f\|_{L^{q,\infty}}^θ\|f\|_{\dot H^s}^{1-θ}, \] where $1/p=\theta /q+(1-\theta )(1/2-s/n)$. In particular, in ${\mathbb{R}}^2$ we obtain the generalised Ladyzhenskaya inequality $\Vert f{\Vert }_{L^4}\le c\Vert f{\Vert }_{L^{2,\infty }}^{1/2}\Vert f{\Vert }_{{\dot{H}}^1}^{1/2}$. We also show that for $s=n/2$ the norm in $\Vert f{\Vert }_{{\dot{H}}^{n/2}}$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon-Zygmund decompositions.