On boundary H\"older gradient estimates for solutions to the linearized Monge-Amp\`ere equations

In this paper, we establish boundary H"older gradient estimates for solutions to the linearized Monge-Amp`ere equations with $L^p$ ($n<p\le \mathrm{\infty }$) right hand side and $C^{1,\gamma }$ boundary values under natural assumptions on the domain, boundary data and the Monge-Amp`ere measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Amp`ere equations with bounded right hand side and $C^{1,1}$ boundary data.