New bounds for bilinear Calderón–Zygmund operators and applications

Abstract

In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calderón-Zygmund operators with Dini-continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hytönen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple A∞ constants inspired in the Fujii-Wilson and Hrusčěv classical constants. These estimates have many new applications including mixed bounds for multilinear Calderón-Zygmund operators and their commutators with BM O functions, square functions and multilinear Fourier multipliers.

Sparse Domination Theorem for Multilinear Singular Integral Operators with Lr-Hörmander Condition

“…We also remark that our assumption is weaker than the assumption (H2) used in [1] (see Proposition 3.3). It is also easy to see that our assumption is weaker than the Dini condition used in [9] (see Proposition 3.2). Now to state our main result, we need a multilinear analogue of grand maximal truncated operator.…”

“…For more background, see [16,25,26]. Now we shall briefly show that our conditions are weaker than Dini condition, which is used in [9] by Damián, Hormozi and the author. Recall that the Dini condition is defined by…”

“…We have For the proof, we refer the readers to [1]. One can also follow the maximal function trick used in [9] and then utilize the result in [22]. We can also obtain the A p -A ∞ type bounds, see [23,9] for details.…”

Ap-A∞ estimates for multilinear maximal and sparse operators

“…For m = 1 our FW characteristic coincides with (the q 1 -th power of) the A ∞ characteristic originating in the work of Fujii [Fuj77] and Wilson [Wil87], which is in turn smaller than the A ∞ characteristic based on the logarithmic maximal function originating in the work of García-Cuerva and Rubio de Francia [GCF85] and Hruščev [Hru84] (the latter observation has been first made in [HP13]). For m ≥ 2 our version of the FW characteristic is smaller than the version introduced in [CD13] and also smaller than the H ∞ characteristic in [DHL15] (the latter being a consequence of L 1 boundedness of the logarithmic maximal function, see [HP13, Lemma 2.1]). A key advantage of our FW characteristic is that, in the case of dependent weights, it can be estimated by a nice power of the multilinear Muckenhoupt characteristic: Lemma 1.6.…”

“…We apply to r ρ and r ρ the same index omission conventions as to r ρ . The sparse operators are known to dominate many multilinear singular integral operators, see [LMS14], [DHL15], [FSZK16]. The sparse operators ρ with ρ i = 0 dominate multilinear fractional integrals, see [LS16].…”