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 $L^{p}$ ($n<p\leq\infty$) 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.