Hardy-Poincaré Type Inequalities Related to k-Hessian Operator

Abstract

In this paper, we obtain some improved Hardy inequalities for Hessian integrals $$I_{p,k}[u,\Omega ]$$ by symmetrization method under conditions that $${\Omega }$$ is a bounded $$(k-1)$$ -convex starshaped domain of $${\mathbb {R}}^n$$ with $$1<p<n-k+1$$ , and $$u\in A_{k-1}({\Omega })$$ whose sub-level set $$\Omega _{t}=\{x\in \Omega \mid u(x)<t\}$$ is $$(k-1)$$ -convex starshaped , where $$A_{k-1}({\Omega })$$ is a particular class of function space whose sub-level domains satisfy some monotonicity property. Especially in the case of $$p=2$$ , the best contant for the remainder term is given.