Erratum to 'L~p measure of growth and higher order Hardy-Sobolev-Morrey inequalities on R~N'

摘要

In the proof of part (i) of Theorem 3.2 of [1], the argument for the existence of a sequence r_n → ∞ such that lim ‖u(r_n,·) - u(r_n)||_(p,S~(n-1)) = 0 must be slightly modified. Indeed. ▽_(S~N-1)-u(r,σ) is not the orthogonal projection of ▽u(r,σ)) on the tangent space {σ}~⊥ to S~(N-1) at σ, but r times this projection. To account for the omitted factor r. the left-hand side of the inequality ∫_0~∞(1+r)~(-sp-N_rN-1)‖u(r,·)-u(r)‖_(p,S~(N-1)~p dr≤C‖ |▽u| ‖_L_(s~p)~P must be replaced with ∫_0~∞(1+r)~(-sp-N_rN-1-p)‖u(r,·) - u(r)‖_(p,S~(N-1))~p dr. Since s < -1 is assumed, the function (1 + r)~(-sp-N_rN-1-p) (equivalent to r~((s+1)p-1) for large r) is not integrable at infinity, which suffices to ensure the existence of the sequence r_n.