Sharp conformally invariant Hardy-type inequalities with remainders

In the present paper we establish new Hardy-Maz'ya-type inequalities with remainders for all continuously differentiable functions with compact support in the half space $\mathbb{R}_+^n$. The weight functions depend on the distance to the boundary or on the distance to the origin. Also new sharp Avkhadiev-Hardy-type inequalities involving the distance to the boundary or the hyperbolic radius are proved. We consider Avkhadiev-Hardy-type inequalities in simply and doubly connected plain domains and in tube-domains.