Square function in L^p(w), p>2: duality + Carleson loses Lerner's sharp exponent

Paper check before formalizing. Let A2 f = sum_{Q in S} <|f|>_Q^2 1_Q be the squared sparse form and r = p/2. Test against g in L^{r'}(w) and write f = h*sigma. The A_p condition gives exactly sigma_Q^2 w_Q |Q| <= [w]^{2/p} sigma(Q)^{2/p} w(Q)^{1/r'}, because 2/p + 1/r' = 1. Then Hoelder and weighted Carleson embeddings (Fujii-Wilson constants C_sigma, C_w) give ||S||_{L^p(w)} <~ [w]^{1/p} C_sigma^{1/p} C_w^{1/2-1/p}. At p=2 this is the sharp mixed L^2 bound, but for p>2 it only yields the power 1/2 + 1/(p(p-1)) of [w], not Lerner's sharp max(1/2, 1/(p-1)). So do not expect this route to formalize the sharp p>2 result; it is a valid but non-sharp mixed bound.