Weak-type and end-point norm estimates for Hardy operators

Abstract

We explicitly calculate the best constants for weak-type and other end-point estimates for the Hardy operator and its adjoint. In particular, we find the right value for decreasing power weights, fixing some previously unclear results.

Weighted inequalities involving Hardy and Copson operators

“…Apart from the motivation described above, there seems to be a rapidly growing interest in finding optimal bounds for inequalities of the form (1.4), both for nonnegative functions and for non-increasing functions, motivated by some very interesting connections to Poincaré or Sobolev inequalities, rearrangement estimates of BMO functions, the conjecture of Iwaniec concerning the norm of the Beurling operator, and more. The interested reader is kindly referred to checking the papers [11,12,13,27,31,32,33,36,48,49], and also the references given therein.…”