Generalised Gagliardo–Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO

Abstract

Using elementary arguments based on the Fourier transform we prove that for ${1 \leq q < p < \infty}$ and ${s \geq 0}$ 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

$$\| f \|_{L^{p}} \leq c_{p,q,s} \| f \|^\theta _{L^{q,\infty}} \| f \|^{1-\theta}_{\dot{H}^s},$$

where 1/p = θ/q + (1−θ)(1/2−s/n). In particular, in ${\mathbb{R}^2}$ we obtain the generalised Ladyzhenskaya inequality ${\| f \| _{L^4} \leq c \| f \|^{1/2}_{L^{2,\infty}} \| f \|^{1/2}_{\dot{H}^1}}$.We also show that for s = n/2 and q > 1 the norm in ${\| f \|_{\dot{H}^{n/2}}}$ 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.