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

Let {P-t}(t>0) be the classical Poisson semigroup on R-d and G(P) the associated Littlewood-Paley g-function operator: G(P)(f) = {integral(infinity)(0) t vertical bar partial derivative/partial derivative t P-t (f)vertical bar(2) dt)1/2. The classical Littlewood-Paley g-function inequality asserts that for any 1 < p < infinity there exist two positive constants L-t,p(P) and L-c,p(P) such that (L-t,p(P))(-1) parallel to f parallel to(p) <= parallel to G(P)(f)parallel to(p) <= L-c,p(P) parallel to f parallel to(p), f is an element of L-p (R-d). We determine the optimal orders of magnitude on pof these constants as p -> 1 and p -> infinity. 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 Delta be the partition of R-d into dyadic rectangles and SRthe partial sum operator associated to R. The dyadic Littlewood-Paley square function of f is S-Delta(f) = (Sigma(R is an element of Delta)vertical bar S-R(f)vertical bar(2))(1/2). For 1 < p < infinity there exist two positive constants L-c, p,d(Delta) and L-t, p,d(Delta) such that (L-t,p,d(Delta))(-1) parallel to f parallel to(p) <= parallel to S-Delta(f)parallel to p <= L-c,p,d(Delta) parallel to f parallel to(p), f is an element of L-p (R-d). We show that L-t,p,d(Delta) approximate to(d) (L-t,p,1(Delta))(d) and L-c,p,d(Delta) approximate to(d) (L-c,p,1(Delta))(d). All the previous results can be equally formulated for the dtorus T-d. We prove a de Leeuw type transference principle in the vector-valued setting. (C) 2022 Elsevier Inc. All rights reserved.