Optimality of constants in power-weighted Birman-Hardy-Rellich-Type inequalities with logarithmic refinements

Abstract

. The principal aim of this paper is to establish the optimality (i.e., sharpness) of the constants A ( m,α ) and B ( m,α ), m ∈ N , α ∈ R , of the form in the power-weighted Birman–Hardy–Rellich-type integral inequalities with logarithmic refinement terms recently proved in [41], namely, ˆ where sharpness is meant in the sense that A ( m,α ) as well as the N constants B ( m,α ) appearing in this inequality are optimal. Here the iterated logarithms are given by )) , j ∈ N , and the iterated exponentials are defined via e 0 = 0 , e j +1 = e e j , j ∈ N 0 = N ∪ { 0 } . Moreover, we prove the analogous sequence of inequalities on the exterior interval ( r, ∞ ) for f ∈ C ∞ 0 (( r, ∞ )), r ∈ (0 , ∞ ).