Abstract
We prove a (sharp) pointwise estimate for positive dyadic shifts of complexity m which is linear in the complexity. This can be used to give a pointwise estimate for Calderón-Zygmund operators and to answer a question originally posed by Lerner. Several applications to weighted estimates for both multilinear Calderón-Zygmund operators and square functions are discussed.
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Acknowledgments
The authors wish to thank Javier Parcet, Ignacio Uriarte-Tuero and Alexander Volberg for insightful discussions, and Andrei Lerner and Fedor Nazarov for sharing with us the details of their construction.
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J. M. Conde-Alonso was partially supported by the ERC StG-256997-CZOSQP, the Spanish Grant MTM2010-16518 and by ICMAT Severo Ochoa Grant SEV-2011-0087 (Spain).
Appendix: The weak-type estimate for multilinear m-shifts
Appendix: The weak-type estimate for multilinear m-shifts
Here we prove the weak-type estimate for k-linear m-shifts needed in Sect. 2. Notice that the only important point of the estimates below is the independenceof the constants from the parameter m. The proof could be more or less standard by now, but the authors have not been able to find it elsewhere. Therefore we include it for completeness.
Theorem 4.1
where \(C_W > 0\) only depends on k and d, and in particular is independent of m.
We will essentially follow Grafakos-Torres [6, 9]. We first prove an \(L^{2}\) bound and then apply a Calderón-Zygmund decomposition. For the \(L^{2}\) bound we will use a multilinear Carleson embedding theorem by Chen and Damián [2], from which we only need the unweighted result:
whenever
Now we can prove
Proposition 4.2
Proof
We begin by using duality and homogeneity to reduce to showing
assuming that \(\Vert f_i\Vert _{L^{2k}(P_0)} = \Vert g\Vert _{L^2(P_0)} = \Vert \alpha \Vert _{{\text {Car}}(P_0)} = 1\) and \(g\ge 0\). By definition and Cauchy-Schwarz, this is equivalent to
The second term can be estimated, using (4.2) in the linear case, by
For the first term observe that the sequence \(\beta _Q\) defined by
is a Carleson sequence adapted to \(P_0\) of the same constant. Indeed:
Therefore, we can write the first term as
which can also be estimated by (4.2) as follows:
Combining both terms we arrive at
which is what we wanted. \(\square \)
Now we can prove Theorem 4.1.
Proof
By homogeneity we can assume \(\Vert \alpha \Vert _{{\text {Car}}(P_0)} = \Vert f_i\Vert _{L^1(P_0)} = 1\). We now follow the classical scheme which uses the \(L^2\) bound and a standard Calderón-Zygmund decomposition, see for example Grafakos-Torres [6]. However, we need to be careful with the dependence on m, so we will adapt the proof in [9] to our operators.
Assume without loss of generality that \(f_i \ge 0\). Define
If \(\langle f_i \rangle _{P_0} > \lambda ^{1/k}\) then by the homogeneity assumption
and the estimate follows. Therefore, we can assume \(\langle f_i \rangle _{P_0} \le \lambda ^{1/k}\) for all \(1 \le i \le k\) and hence we can write \(\Omega _i\) as a union the cubes in a collection \(\mathcal {R}_i\) consisting of pairwise disjoint dyadic (strict) subcubes of \(P_0\) with the property
For each \(1 \le i \le k\) let \(b_i = \sum _{R \in \mathcal {R}_i} b_i^R\), where
We now let \(g_i = f_i-b_i\).
Observe that we have
as well as
Define \(\Omega = \cup _{i=1}^k \Omega _i\), then we have
To estimate the second term observe that
where the functions \(h_i^j\) are either \(g_i\) or \(b_i\) and, furthermore, for each \(1 \le j \le 2^k-1\) there is at least one \(1 \le i \le k\) such that \(h_i^j = b_i\). Fix j and let \(i_j\) be such that \(h_{i_j}^j = b_{i_j}\), then
So we deduce that \(\mathcal {A}_{P_0,\alpha }^m (h_1^j, \ldots , h_k^j)(x) = 0\) for all \(x \notin \Omega _{i_j}\). With this fact we can see that the second term in (4.3) is actually identical to
Now we can use the \(L^2\) bound as follows:
Putting both estimates together we arrive at
which yields the result with \(C_{W} = 2^{k(5+ d(2k-1))}\). \(\square \)
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Conde-Alonso, J.M., Rey, G. A pointwise estimate for positive dyadic shifts and some applications. Math. Ann. 365, 1111–1135 (2016). https://doi.org/10.1007/s00208-015-1320-y
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DOI: https://doi.org/10.1007/s00208-015-1320-y