Hardy's operator minus identity and power weights

Hardy’s inequality and its descendants: a probability approach

“…This set of connections deserves to be explored further. In particular we conjecture that many of the interesting properties of the classical Hardy operator H and the dual Hardy operator H * established in the series of papers by [37], [38], [12], [35], [13], [36], and [60] will have useful analogues for H F and H * F in the probability setting for Hardy's inequalities which we have considered here. On the other hand, the martingale connections of the operators L and R perhaps deserve to be better known in the world of classical Hardy type inequalities.…”

“…For more on this and connections to counting process martingales and survival analysis see [54], [22], and [7]. [60] studies I − H and I − H * as operators on L p (R + , λ) where λ denotes Lebesgue measure.…”

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.…”