Continuous solutions of Dirichlet problem to Hessian type equations for ( ω , m ) − β -subharmonic functions on a ball in CnCⁿ

In this paper, we investigate the continuity of solutions to the Dirichlet problem for complex Hessian-type equations associated with ( ω , m ) − β -subharmonic functions on a ball in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> Cⁿ , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>β</m:mi> <m:mo>=</m:mo> <m:mi>d</m:mi> <m:msup> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>c</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">‖</m:mo> <m:mi>z</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">‖</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:msubsup> <m:mrow> <m:mo form="prefix" movablelimits="false">∑</m:mo> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:mi mathvariant="normal">d</m:mi> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>∧</m:mo> <m:mi mathvariant="normal">d</m:mi> <m:msub> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> β =ddᶜz²=i/2∑ ⱼ₌₁ⁿdzⱼ dz̄ⱼ is denoted the flat metric on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> Cⁿ .