Sharp A1 Bounds for Calderón-Zygmund Operators and the Relationship with a Problem of Muckenhoupt and Wheeden

Weak Type Estimates for Singular Integrals Related to a Dual Problem of Muckenhoupt–Wheeden

“…We recall that w is an A 1 weight if there is a finite constant c such that M w ≤ c w a.e., and where w A 1 denotes the smallest of these c. In a recent paper [11] we proved the following related result: Theorem 1.3. Let ϕ(t) = t(1 + log + t)(1 + log + log + t).…”

$A_1$ bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden

“…The standard proofs applied to this concrete weight yield constants of exponential type C(p) ∼ 2 p . In [8], the growth of C(p) at infinity was improved to C(p) ∼ p log p. Lemma 1.2 represents the subsequent improvement to the best possible growth C(p) ∼ p. This can be seen by taking w ≡ 1 and recalling that M L p ≈ c n as p → ∞.…”

“…In the next section, we give a detailed proof of Lemma 1.2 along with some auxiliary statements. In the third section we outline briefly the main steps from [8] showing how this lemma leads to Theorem 1.1. In Section 4 we prove Corollaries 1.3 and 1.4.…”

Weighted L p estimates for the area integral associated to self-adjoint operators

“…Note that sharp weighted optimal bounds for singular integrals has been studied extensively, see for examples, [14,24,28,29,31] and the references therein. One then has the analogous statement as in Theorems 1.1, 1.2, 1.3 and 1.4 replacing s p , s h , S P , S H by g p , g h , G P , G H , respectively.…”