Littlewood-Paley theory for subharmonic functions on the unit ball in RN.

Let B denote the unit ball in RN with boundary S. For a non-negative C2 subharmonic function f on B and ∈S, we define the Lusin square area integral Sα(,f) by Sα(f)=[Γα(1-|x|)2-NΔf2(x)dx]12,where for α>1, Γα={x∈B:|x-|<α(1-|x|)} is the non-tangential approach region at ∈S, and Δ is the Laplacian in RN. In the paper we will prove the following: Let f be a non-negative subharmonic function such thatfpois subharmonic for somepo>0. Iffpp=sup0<r<1Sfp(r)dσ()<∞for somep>po, then for everyα>1,Sα(•,f)p≤Aα,pfpfor some constantAα,pindependent of f. The above result includes the known results for harmonic or holomorphic functions in the Hardy Hp spaces, as well as for a system F=(u1,...,uN) of conjugate harmonic functions for which it is known that |F|p=(Σuj2)p/2 is subharmonic for p≥(N-2)/(N-1),N≥3. We also consider analogues of the functions g and g\* of Littlewood-Paley, and introduce the function gλ\*, λ>1, defined bygλ\*(,f)=[B(1-|y|)Δf2(y)Kλ(y,)dy]12,whereKλ(y,)=(1-|y|)(λ-1)(N-1)|y-|λ(N-1). In the paper we prove that the inequality gλ\*(•,f)p≤Cpfp holds for all λ≥N/(N-1) when p≥2, and for λ>3-p whenever 1<p<2. Taking λ=N/(N-1) proves that g\*(•,f)p≤Cpfp for all p>(2N-3)/(N-1). [ABSTRACT FROM AUTHOR]