On Birman--Hardy--Rellich-type inequalities.

Abstract

In 1961, Birman proved a sequence of inequalities on the space of m-times continuously di↵erentiable functions of compact support Cm 0 ((0, 1)) ⇢ L2((0, 1)), containing the classical (integral) Hardy inequality and the well-known Rellich inequality. In this dissertation, we give a proof of this sequence of inequalities on a certain Hilbert space Hm L ([0, 1)) as well as the standard Sobolev space Hm 0 ((0, b)) for 0 trivial function in L2((0, 1)) (resp., L2((0, b))). These Birman constants are closely related to the norm of generalized continuous Ces`aro averaging operators whose spectral properties we determine in detail. We then revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants, then discuss the infinite sequence of power weighted Birman–Hardy–Rellich-type inequalities and derive an operator-valued version thereof. We further improve this sequence of inequalities by adding recursive logarithmic refinement terms with unrestricted weight and logarithmic parameters. In the multidimensional setting, we derive variants of Hardy’s inequality involving power-type weights, radial derivatives and logarithmic refinements. Finally, we establish the multidimensional Birman–Hardy–Rellich-type inequalities with power-type weights and radial refinements.