Littlewood–Paley theory for subharmonic functions on the unit ball in [formula omitted

Abstract

Let B denote the unit ball in RN with boundary S. For a non-negative C2 subharmonic function f on B and ζ∈S, we define the Lusin square area integral Sα(ζ,f) bySα(ζ,f)=[∫Γα(ζ)(1−|x|)2−NΔf2(x)dx]12,where for α>1, Γα(ζ)={x∈B:|x−ζ|<α(1−|x|)} is the non-tangential approach region at ζ∈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. If‖f‖pp=sup0<r<1∫Sfp(rζ)dσ(ζ)<∞for somep>po, then for everyα>1,‖Sα(⋅,f)‖p≤Aα,p‖f‖pfor 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=(∑uj2)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⁎(ζ,f)=[∫B(1−|y|)Δf2(y)Kλ(y,ζ)dy]12,whereKλ(y,ζ)=(1−|y|)(λ−1)(N−1)|y−ζ|λ(N−1). In the paper we prove that the inequality ‖gλ⁎(⋅,f)‖p≤Cp‖f‖p 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≤Cp‖f‖p for all p>(2N−3)/(N−1).