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

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