Some Hardy-type inequalities with sharp constants via the divergence theorem

Abstract

Hardy’s inequality originated in the early twentieth century when G.H. Hardy introduced this fundamental result in real analysis to bound integral operators. Its elegant formulation and optimal constants spurred widespread interest, leading to numerous refinements. These developments laid the groundwork for further exploration and multidimensional extensions, deeply influencing harmonic analysis, partial differential equations, and mathematical physics. This historical evolution continues to inspire modern advancements in research. 
We discuss multidimensional generalizations of some improved Hardy inequalities based on the divergence theorem. The obtained  Hardy-type inequalities extend a recent version of the one-dimensional Hardy inequality with the best constant to multidimensional cases.