A lower bound for $A_p$ exponents for some weighted weak-type inequalities

Abstract

We give a weak-type counterpart of the main result in [13] which allows to provide a lower bound for the exponent of the Ap constant in terms of the behaviour of the unweighted inequalities when p → ∞ and when p → 1 + . We also provide some applications to classical operators.

Sparse domination for the lattice Hardy–Littlewood maximal operator

“…Remark. In the particular case E = ℓ q , the dependence on the A p -characteristic is sharp both in the strong-type weighted estimate (5.3) (see [6]) and in the weak-type weighted estimate (5.5) (this follows from combining [6] and [32,Theorem 1]). In the general case that E is Banach lattice that is q-convex for some q ∈ (1, ∞), the exponent Ap holds for all q < q * and fails for all q > q * .…”