Optimal orders of the best constants in the Littlewood-Paley inequalities

Abstract

Let {Pt}t>0 be the classical Poisson semigroup on Rd and GP the associated Littlewood-Paley g-function operator:GP(f)=(∫0∞t|∂∂tPt(f)|2dt)12. The classical Littlewood-Paley g-function inequality asserts that for any 1<p<∞ there exist two positive constants Lt,pP and Lc,pP such that(Lt,pP)−1‖f‖p≤‖GP(f)‖p≤Lc,pP‖f‖p,f∈Lp(Rd). We determine the optimal orders of magnitude on p of these constants as p→1 and p→∞. We also consider similar problems for more general test functions in place of the Poisson kernel.The corresponding problem on the Littlewood-Paley dyadic square function inequality is investigated too. Let Δ be the partition of Rd into dyadic rectangles and SR the partial sum operator associated to R. The dyadic Littlewood-Paley square function of f isSΔ(f)=(∑R∈Δ|SR(f)|2)12. For 1<p<∞ there exist two positive constants Lc,p,dΔ and Lt,p,dΔ such that(Lt,p,dΔ)−1‖f‖p≤‖SΔ(f)‖p≤Lc,p,dΔ‖f‖p,f∈Lp(Rd). We show thatLt,p,dΔ≈d(Lt,p,1Δ)d and Lc,p,dΔ≈d(Lc,p,1Δ)d.All the previous results can be equally formulated for the d-torus Td. We prove a de Leeuw type transference principle in the vector-valued setting.