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

Abstract

Abstract. We obtain an L p (w) bound for Calderón-Zygmund operators T when w ∈ A 1 . This bound is sharp both with respect to w A 1 and with respect to p. As a result, we get a new L 1,∞ (w) estimate for T related to a problem of Muckenhoupt and Wheeden.

Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

“…We refer the reader to the work of Buckley [3], Petermichl and Volberg [27], Petermichl [25,26], Lacey, Moen, Pérez, and Torres [16], Lerner [19,20], Lerner, Ombrosi, and Pérez [21,22], Lacey, Petermichl, and Reguera [15] and Hytonen, Lacey, Reguera, and Vagharshakyan [14], etc.…”

Classical Fourier Analysis

“…However, Nazarov, Reznikov, Vasyunin, and Volberg [265] disproved the weaker inequality T L 1 (w)→L 1,∞ (w) ≤ C [w] A 1 log(e + [w] A 1 ) α for α < 1 5 . Lerner, Ombrosi, and Pérez [223] had previously shown that the preceding inequality holds with α = 1 for any Calderón-Zygmund operator T .…”

Extrapolation of weights revisited: New proofs and sharp bounds

“…Applying Corollary 4.2 we obtain (4.4). For Calderón-Zygmund operators, the estimate (4.3) was proved in [22] with N(t) = t. Then (4.4) holds linearly in [w] A q . 2…”

“…We consider this question in Section 4 in two ways, depending on whether the basic estimate is for p 0 > p or for p 0 < p. In particular, for Calderón-Zygmund operators, we extend to A q weights an estimate for A 1 weights due to Lerner, Ombrosi and Pérez [22].…”