On Birman's sequence of Hardy–Rellich-type inequalities

Abstract

In 1961, Birman proved a sequence of inequalities {In}, for n∈N, valid for functions in C0n((0,∞))⊂L2((0,∞)). In particular, I1 is the classical (integral) Hardy inequality and I2 is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space Hn([0,∞)) of functions defined on [0,∞). Moreover, f∈Hn([0,∞)) implies f′∈Hn−1([0,∞)); as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite b>0, these inequalities hold on the standard Sobolev space H0n((0,b)). Furthermore, in all cases, the Birman constants [(2n−1)!!]2/22n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in L2((0,∞)) (resp., L2((0,b))). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.