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: [Formula presented] The classical Littlewood-Paley g-function inequality asserts that for any 1<p<∞ there exist two positive constants L_t,p^P and L_c,p^P such that (L_t,p^P)⁻¹‖f‖_p≤‖G^P(f)‖_p≤L_c,p^P‖f‖_p,f∈L_p(R^d). 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 R^d into dyadic rectangles and S_R the partial sum operator associated to R. The dyadic Littlewood-Paley square function of f is [Formula presented] For 1<p<∞ there exist two positive constants L_c,p,d^Δ and L_t,p,d^Δ such that (L_t,p,d^Δ)⁻¹‖f‖_p≤‖S^Δ(f)‖_p≤L_c,p,d^Δ‖f‖_p,f∈L_p(R^d). We show that L_t,p,d^Δ≈_d(L_t,p,1^Δ)^d and L_c,p,d^Δ≈_d(L_c,p,1^Δ)^d. All the previous results can be equally formulated for the d-torus T^d. We prove a de Leeuw type transference principle in the vector-valued setting.