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

Abstract

Dedicated with great pleasure to Eduard Tsekanovskii on the occasion of his 80th birthday.Abstract. In 1961, Birman proved a sequence of inequalities {In}, for n ∈ N, valid for functions in C n 0 ((0, ∞)) ⊂ L 2 ((0, ∞)). In particular, I 1 is the classical (integral) Hardy inequality and I 2 is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space; 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 H n 0 ((0, b)). Furthermore, in all cases, the Birman constants [(2n − 1)!!] 2 /2 2n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in L 2 ((0, ∞)) (resp., L 2 ((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.

A survey of some norm inequalities

“…The generalized version of weighted Hardy inequalities, especially, its integral inequality version (replacing f (x) by ´x a dt F (t) or ´b x dt F (t), and hence f ′ (x) by F (x), etc.) in L 2 ((a, b); dx) was established by Talenti [93] and Tomaselli [94] in 1969 and independently rediscovered by Chisholm and Everitt [15] in 1971 (see also [16] for a more general result in the conjugate index case 1/p + 1/q = 1, and [39], [40], and the references therein, for recent developments). In addition, a 1972 paper by Muckenhoupt [79] has further generalizations.…”

A Sequence of Weighted Birman-Hardy-Rellich Inequalities with Logarithmic Refinements

“…We start by stating a power-weighted extension of (1.1) for vector-valued functions, which is derived from the more general Hardy result [23, Example 1] by simple iteration (see also [45,Theorem 8.1] for the special case α = 0, a = 0, b = ∞). Inequality (4.1) will replace (1.1) in the base step of each induction proof.…”

“…In the present onedimensional context at hand, sharpness of (1.1) (and typically, it's power weighted version, the first line in (1.10)), are often proved in an integral form (rather than the currently presented differential form) where f (m) on the left-hand side is replaced by F and f on the right-hand side by m repeated integrals over F . For pertinent one-dimensional sources, we refer, for instance, to [14, p. 3-5], [22], [25, p. 104-105], [45,52,54], [55, p. 240-243], [64,Ch. 3], [65, p. 5-11], [68,77,86].…”

Factorizations and Hardy–Rellich-type inequalities

“…For a variant of (2.48) on the interval (0, 1) we refer to [16, p. 114]; the case of higher-order Hardy-type inequalities for general interval is also considered in [54]. We will reconsider this sequence of higher-order Hardy-type inequalities in [30].…”