General Hardy’s inequalities for functions nonzero on the boundary

Abstract

Consider Hardy’s inequalities with general weight ϕ for functions nonzero on the boundary. By an integral identity in C¹($\overline \Omega $$), define Hilbert spaces H_k ¹ (Ω, ϕ) called Sobolev-Hardy spaces with weight ϕ. As a corollary of this identity, Hardy’s inequalities with weight ϕ in C¹ ($\overline \Omega $$) follow. At last, by Hardy’s inequalities with weight ϕ = 1, discuss the eigenvalue problem of the Laplace-Hardy operator with critical parameter (N − 2)²/4 in H¹ ₁ (Ω).

Keywords

Hardy’s inequality / embedding inequality / critical parameter

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