Generalised Gagliardo-Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO

Abstract

Using elementary arguments based on the Fourier transform we prove that for (Formula presented.) and (Formula presented.) with s > n(1/2 - 1/p), if (Formula presented.), then (Formula presented.) and there exists a constant c p,q,s such that (Formula presented.) where 1/p = θ/q + (1-θ)(1/2-s/n). In particular, in (Formula presented.). We also show that for s = n/2 and q > 1 the norm in (Formula presented.) 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. © 2013 Springer Basel.