In this paper, the study focuses on mixed Hessians within the class of m-convex functions in the space \(\mathbb {R}^n\) . Building on the foundational works of Trudinger and Wang, the authors extends the notion of mixed Hessians to vector-valued m-convex functions of the form \(\vec {u}(x)=(u_{1}(x),u_{2}(x),...,u_{k} (x))\) , where \(1\le k\le n-m+1\) . Mixed Hessians are introduced as Borel measures in the space of locally bounded functions, and their main analytic properties are rigorously developed. The paper establishes upper estimates for their integral averages and studies the weak convergence of mixed Hessians corresponding to decreasing sequences of m-convex vector functions.The results obtained form a connection between the theories of m-convex and m-subharmonic functions, thereby enhancing the potential theory framework and broadening the applications of Hessian measures in real convex geometry.
