Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8898994 | Journal of Differential Equations | 2018 | 41 Pages |
Abstract
In 1961, Birman proved a sequence of inequalities {In}, for nâN, valid for functions in C0n((0,â))âL2((0,â)). 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,â)) of functions defined on [0,â). Moreover, fâHn([0,â)) implies fâ²âHnâ1([0,â)); 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 [(2nâ1)!!]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,â)) (resp., L2((0,b))). We also show that these Birman constants are related to the norm of a generalized continuous Cesà ro averaging operator whose spectral properties we determine in detail.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Fritz Gesztesy, Lance L. Littlejohn, Isaac Michael, Richard Wellman,