Invariant Potential Theory in the Unit Ball of Cⁿ

Hyperbolic mean growth of bounded holomorphic functions in the ball

“…where the last equality comes from [R,Theorem 5.4.8 or Theorem 5.4.9], or [Sto1,Proposition 5.6]. But we already proved that (4.15) holds for p = 2 and p = 4 respectively in Case 1 and Case 2.…”

“…Other undefined notation and terminology in this paper will follow the books by W. Rudin [R] and M. Stoll [Sto1].…”

On the asymptotic formula for the solution of degenerate elliptic partial differential equations

“…When D is the unit ball B n in C n and g is the Bergman metric, the existence and regularity theory were studied by R. Graham [1], M. Stoll [6], Krantz [3] and the references therein. In [1], Graham proved that if g is the Bergman metric in the unit ball B n in C n , f = 0 and φ ∈ C ∞ (∂B n ), then (1.7) has a C ∞ (D) solution if and only if φ has a pluriharmonic extension in B n .…”

Asymptotic behaviour of means of nonpositive M-subharmonic functions

“…For z, w ∈ B, define the involutive automorphism φ w of the unit ball B given by φ w (z) = w − P w z − (1 − |w| 2 ) 1/2 Q w z 1 − ⟨z, w⟩ where P 0 z = 0, P w z = ⟨z,w⟩ |w| 2 w, w ̸ = 0, is the orthogonal projection of C n onto the subspace generated by w and Q w = I − P w ( [8,9]). …”

“…It is known that∆ is invariant with respect to any holomorphic automorphism of B, i.e.,∆(f • ψ) = (∆f ) • ψ for all ψ ∈ M, the group of holomorphic automorphisms of B ( [8,Chap.4], [9]). …”