Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO

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Using elementary arguments based on the Fourier transform we prove that for 1 ≤ q < p < ∞ and s ≥ 0 with s > n(1/2-1/p), if f ∈ Lq,∞(ⁿ) ∩ Hˢ(ⁿ) then f ∈ Lᵖ(ⁿ) and there exists a constant cₚ,q,ₛ such that fₗₚ ≤ cₚ,q,ₛ fLq,∞θf Hˢ¹⁻θ, where 1/p = θ/q + (1-θ)(1/2-s/n). In particular, in ² we obtain the generalised Ladyzhenskaya inequality fₗ₄≤ cfₗ²,∞¹/²f H¹¹/². We also show that for s=n/2 the norm in f Hⁿ/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.