Space-only gradient estimates of Schrödinger equation with Neumann boundary condition under integral Ricci curvature bounds

Assume that M (M⊆N) is an n-dimensional compact Riemannian submanifold with boundary, satisfying the integral Ricci curvature assumption:D2supx∈N⁡(∮B(x,d)|Ric−|pdy)1p<κ for κ=κ(n,p) small enough, p>n/2, where diam(M)≤D. The boundary of M needs to satisfy the interior rolling R-ball condition. We prove a Hamilton type gradient estimate for positive solutions to the parabolic Schrödinger equation with Neumann boundary conditions, on a compact Riemannian submanifold M with boundary ∂M, satisfying the integral Ricci curvature assumption.