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Quadratic sparse domination and weighted estimates for non-integral square functions. (English) Zbl 1505.42016

After the development of the Calderón-Zygmund theory, researchers have encountered numerous examples of operators outside its scope but for which the Calderón-Zygmund theory still plays a guiding role; see for example [W. Hebisch, Colloq. Math. 60/61, No. 2, 659–664 (1990; Zbl 0779.35025)].
With the surge of activity around the \(A_2\) conjecture and the appearance of sparse domination, a wide family of operators, termed non-integrable singular operators, are now better understood [F. Bernicot et al., Anal. PDE 9, No. 5, 1079–1113 (2016; Zbl 1344.42009)]. Non-integrable operators lack good control over the size or regularity of their kernel, like the Riesz operator \(\nabla L^{-1/2}\) for \(L = \operatorname{div}(A\nabla\cdot)\), so different characterizing properties must be sought.
The paper under review builds upon the work of Bernicot et al. on linear operators [loc. cit.], and the authors investigate quadratic operators of the form \[ Sf(x) := \Bigl(\int_0^\infty \lvert \mathcal{Q}_tf(x)\rvert^2\frac{dt}{t}\Bigr)^\frac{1}{2}, \] where \(\{\mathcal{Q}_t\}_{t>0}\) are operators bounded in \(L^2(M,\mu)\), with \(M\) a doubling metric space. The authors assume that the operators \(\mathcal{Q}_t\) satisfy the off-diagonal estimates \[ \lVert\mathcal{Q}_t\rVert_{L^{p_0}(B_1) \to L^{q_0}(B_2)} \lesssim \lvert B_1\rvert^{-\frac{1}{p_0}}\lvert B_2\rvert^{\frac{1}{q_0}} \Bigl(1 + \frac{d(B_1, B_2)^2}{t}\Bigr)^{-(\nu + 1)}, \] where \(B_1\) and \(B_2\) are balls of radius \(\sqrt{t}\), and \(1\le p_0<2<q_0\le\infty\) – \(\nu\) is related to \((M, \mu)\). They also assume that the operators \(\mathcal{Q}_t\) satisfy a cancellation property with respect to another operator \(L\) (which satisfies additional hypotheses) and that a Cotlar-type inequality holds.
The main theorems are (see details in the paper) (i) sparse domination: \[ \int_M (Sf)^2g\,d\mu \le c\sum_{P\in\mathcal{S}}\Bigl(\diagup\!\!\!\!\!\!{\int}_{5P} \lvert f\rvert^{p_0}\,d\mu\Bigr)^{2/p_0}\Bigl(\diagup\!\!\!\!\!\!{\int}_{5P}\lvert g\rvert^{q_0^\ast}\,d\mu\Bigr)^{1/q_0^\ast}\lvert P\rvert, \tag{1} \] where \(p_0<2<q_0\), \(q_0^\ast = (q_0/2)^\prime\), and \(\mathcal{S}\) is a sparse family; and (ii) explicit constant: \[ \sum_{P\in\mathcal{S}}\Bigl(\diagup\!\!\!\!\!\!{\int}_{5P} \lvert f\rvert^{p_0}\,d\mu\Bigr)^{2/p_0}\Bigl(\diagup\!\!\!\!\!\!{\int}_{5P}\lvert g\rvert^{q_0^\ast}\,d\mu\Bigr)^{1/q_0^\ast}\lvert P\rvert \le C_0 \bigr([w]_{A_\frac{p}{p_0}}\cdot[w]_{RH_{\bigl(\frac{q_0}{p}\bigr)^\prime}}\bigr)^{2\gamma(p)}\lVert f\rVert^2_{L^p(w)}\lVert g\rVert_{L^{p^\ast}(w^{1-p^\ast})}, \tag{2} \] where \(w\) is a weight, and \(2<p<q_0\).
The preliminaries are well explained in Section 2, and Section 3 presents examples. Section 4 contains the main technical step, a proof that a grand maximal operator \(S^\ast\) is bounded in \(L^2\) and of weak-type \((p_0, p_0)\); the technique originated in [M. T. Lacey, Isr. J. Math. 217, 181–195 (2017; Zbl 1368.42016)].
In Section 5 they prove (1), where the control over \(S^\ast\) allows for the use of a good-\(\lambda\) technique; the region where \(f\) is large is handled by iteration. In Section 6 they prove (2). A sharpness result is included in Section 7.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B37 Harmonic analysis and PDEs
35J30 Higher-order elliptic equations
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References:

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