Generalised Gagliardo–Nirenberg inequalities using weak Lebesgue spaces and BMO
DS McCormick, JC Robinson, JL Rodrigo - Milan Journal of Mathematics, 2013 - Springer
DS McCormick, JC Robinson, JL Rodrigo
Milan Journal of Mathematics, 2013•SpringerUsing elementary arguments based on the Fourier transform we prove that for 1 ≦ q< p< ∞
1≤ q< p<∞ and s ≧ 0 s≥ 0 with s> n (1/2− 1/p), if f ∈ L^ q, ∞ (R^ n) ∩ ̇ H^ s (R^ n) f∈ L
q,∞(R n)∩ H˙ s (R n), then f ∈ L^ p (R^ n) f∈ L p (R n) and there exists a constant cp, q, s
such that ‖ f ‖ _ L^ p ≦ c_ p, q, s ‖ f ‖^ θ _ L^ q, ∞ ‖ f ‖^ 1-θ _ ̇ H^ s,‖ f‖ L p≤ cp,
q, s‖ f‖ L q,∞ θ‖ f‖ H˙ s 1-θ, where 1/p= θ/q+(1− θ)(1/2− s/n). In particular, in R^ 2 R 2
we obtain the generalised Ladyzhenskaya inequality ‖ f ‖ _ L^ 4 ≦ c ‖ f ‖^ 1/2 _ L^ 2, ∞ …
1≤ q< p<∞ and s ≧ 0 s≥ 0 with s> n (1/2− 1/p), if f ∈ L^ q, ∞ (R^ n) ∩ ̇ H^ s (R^ n) f∈ L
q,∞(R n)∩ H˙ s (R n), then f ∈ L^ p (R^ n) f∈ L p (R n) and there exists a constant cp, q, s
such that ‖ f ‖ _ L^ p ≦ c_ p, q, s ‖ f ‖^ θ _ L^ q, ∞ ‖ f ‖^ 1-θ _ ̇ H^ s,‖ f‖ L p≤ cp,
q, s‖ f‖ L q,∞ θ‖ f‖ H˙ s 1-θ, where 1/p= θ/q+(1− θ)(1/2− s/n). In particular, in R^ 2 R 2
we obtain the generalised Ladyzhenskaya inequality ‖ f ‖ _ L^ 4 ≦ c ‖ f ‖^ 1/2 _ L^ 2, ∞ …
Abstract
Using elementary arguments based on the Fourier transform we prove that for and with s > n(1/2 − 1/p), if , then and there exists a constant c p,q,s such that where 1/p = θ/q + (1−θ)(1/2−s/n). In particular, in we obtain the generalised Ladyzhenskaya inequality .We also show that for s = n/2 and q > 1 the norm in 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.
Springer