Further regularity results in $$P-$$extremal setting

In our paper, Pluripotential theory and convex bodies: a Siciak-Zaharjuta theorem (Computational Methods and Function Theory (2020), 20(3-4), 571-590) by Bayraktar et al., we proved a regularity result of $$P-$$ P - extremal function. In this paper, we prove some other stronger regularity results analogue to those in the standard setting. Specifically we focus on proving the following proposition: Proposition: Let K=D̄⊂ Cᵈ K = D ¯ ⊂ C d be the closure of a bounded domain D with ∂ D ∂ D of class C1,1 C 1 , 1 . Then, for any Q∈ C_α (K) Q ∈ C α ( K ) , where C_α (K) C α ( K ) is the Hölder class α α on K , we have VP,K,Q∈ C_α (Cᵈ) for a convex body P⊂ (R⁺)ᵈ. V P , K , Q ∈ C α ( C d ) for a convex body P ⊂ ( R + ) d .