If $4$-convex vectors are closed in uniform norms then their second derivatives are also closed in weighted $L^2$-norm

Abstract

In this paper, we develop the weighted energy estimates for arbitrary 4-convex vectors and the vectors having both 4-convex and 4-concave functions as their arguments. To do this, we first develop these estimates for smooth 4-convex vectors and then, through mollification, extend the results for arbitrary 4-convex vectors. This type of estimates are valuable in problems of financial mathematics for the establishment of optimal investment strategies