On Birman's sequence of Hardy–Rellich-type inequalities (original) (raw)

2018, Journal of Differential Equations

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 Hn([0, ∞)) of functions defined on [0, ∞). Moreover, f ∈ Hn([0, ∞)) implies f ∈ H n−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 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. Contents 1. Introduction 1 2. An Integral Inequality 3 3. The Function Spaces Hn([0, ∞)) and Hn((0, ∞)) 4 4. A New Proof of Birman's Sequence of Hardy-Rellich-type Inequalities 9 5. Optimality of Constants 13 6. The Continuous Cesàro Operator T1 and its Generalizations Tn 15 7. The Birman Inequalities on the Finite Interval [0, b] 25 8. The Vector-Valued Case 27 References 31