Universal regularity estimates for solutions to fully nonlinear elliptic equations with oblique boundary data

In this work, we establish universal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations with oblique boundary conditions, whose general model is given by $$\{\begin{array}{rcl}F(D^{2}u,x)& =& f(x)in\mathrm{\Omega }\\ \beta (x)\cdot Du(x)+\gamma (x)u(x)& =& g(x)on\mathrm{\partial }\mathrm{\Omega }\mathrm{.}\end{array}$$ Such regularity estimates are achieved by exploring the integrability properties of $f$ based on different scenarios, making a $\text{VMO}$ assumption on the coefficients of $F$, and by considering suitable smoothness properties on the boundary data $\beta ,\gamma$ and $g$. Particularly, we derive sharp estimates for borderline cases where $f\in L^n(\mathrm{\Omega })$ and $f\in p-\text{BMO}(\mathrm{\Omega })$. Additionally, for source terms in $L^p(\mathrm{\Omega })$, for $p\in (n,\mathrm{\infty })$, we obtain sharp gradient estimates. Finally, we also address Schauder-type estimates for convex/concave operators and suitable Hölder data.