Pointwise arbitrarily high-order interior estimates for mixed local and nonlocal elliptic equations

In this paper, we mainly focus on pointwise, arbitrarily high-order interior estimates for the mixed local-nonlocal elliptic equation \begin{equation*} (-\Delta)^{s}u(x)-\Delta u(x)=f(x),\quad x\in B_r(0) \end{equation*} with $0<s<1$. The main challenges in this setting are the absence of an explicit Green function and the ineffectiveness of standard bootstrap arguments. Our approach overcomes these difficulties via the Campanato iteration method, which inductively constructs polynomials approximating the solution. Using the fact that all functions are locally $s$-harmonic up to a small error significantly reduces the computational complexity when studying the regularity of the fractional Laplacian acting on these polynomials. Finally, the regularity estimates are obtained from the analysis of a manageable recursive inequality system.