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

Let B denote the unit ball in R N with boundary S. For a non-negative C 2 subharmonic function f on B and ζ ∈ S , we define the Lusin square area integral S α ( ζ , f ) by S α ( ζ , f ) = [ ∫ Γ α ( ζ ) ( 1 − | x | ) 2 − N Δ f 2 ( x ) d x ] 1 2 , where for α > 1 , Γ α ( ζ ) = { x ∈ B : | x − ζ | < α ( 1 − | x | ) } is the non-tangential approach region at ζ ∈ S , and Δ is the Laplacian in R N . In the paper we will prove the following: Let f be a non-negative subharmonic function such that f p o is subharmonic for some p o > 0 . If ‖ f ‖ p p = sup 0 < r < 1 ∫ S f p ( r ζ ) d σ ( ζ ) < ∞ for some p > p o , then for every α > 1 , ‖ S α ( ⋅ , f ) ‖ p ≤ A α , p ‖ f ‖ p for some constant A α , p independent of f. The above result includes the known results for harmonic or holomorphic functions in the Hardy H p spaces, as well as for a system F = ( u 1 , … , u N ) of conjugate harmonic functions for which it is known that | F | p = ( ∑ u j 2 ) 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 g λ ⁎ , λ > 1 , defined by g λ ⁎ ( ζ , f ) = [ ∫ B ( 1 − | y | ) Δ f 2 ( y ) K λ ( y , ζ ) d y ] 1 2 , where K λ ( y , ζ ) = ( 1 − | y | ) ( λ − 1 ) ( N − 1 ) | y − ζ | λ ( N − 1 ) . In the paper we prove that the inequality ‖ g λ ⁎ ( ⋅ , f ) ‖ p ≤ C p ‖ 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 ≤ C p ‖ f ‖ p for all p > ( 2 N − 3 ) / ( N − 1 ) .