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.