A pointwise estimate for positive dyadic shifts and some applications

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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Notes

  1. 1.

    Since we uploaded this document to arXiv, two other articles have appeared: [7, 11], in which similar estimates are obtained.

  2. 2.

    We use \(\mathscr {D}_{\ge k}(P)\) to denote those cubes Q in \(\mathscr {D}(P)\) of generation at least k, so \(|Q| \le 2^{-dk}|P|\).

References

  1. 1.

    Bennett, C., Sharpley, R.: Interpolation of operators. Pure and Applied Mathematics, vol. 129. Academic Press Inc., Boston (1988)

  2. 2.

    Chen, W., Damián, W.: Weighted estimates for the multisublinear maximal function. Rend. Circ. Mat. Palermo (2), 62(3), 379–391 (2013)

  3. 3.

    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Sharp weighted estimates for approximating dyadic operators. Electron. Res. Announc. Math. Sci. 17, 12–19 (2010)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Sharp weighted estimates for classical operators. Adv. Math. 229(1), 408–441 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Damián, W., Lerner, A.K., Pérez, C.: Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators (2012). arXiv:1211.5115

  6. 6.

    Grafakos, L., Torres, R.H.: Multilinear Calderón-Zygmund theory. Adv. Math. 165(1), 124–164 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Hänninen, T.S.: Remark on dyadic pointwise domination and median oscillation decomposition (2015,ArXiv e-prints)

  8. 8.

    Hytönen, T.: \({A_2}\) theorem: remarks and complements (2012, preprint)

  9. 9.

    Hytönen, T.P.: The sharp weighted bound for general Calderón-Zygmund operators. Ann. of Math. (2). 175(3), 1473–1506 (2012)

  10. 10.

    Hytönen, T.P., Lacey, M.T., Pérez, C.: Sharp weighted bounds for the \(q\)-variation of singular integrals. Bull. Lond. Math. Soc. 45(3), 529–540 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Lacey, M.T.: Weighted weak type estimates for square functions (2015, ArXiv e-prints)

  12. 12.

    Lacey, M.T., Sawyer, E.T., Uriarte-Tuero, I.: Two weight inequalities for discrete positive operators (2009, ArXiv e-prints)

  13. 13.

    Lacey, M.T., Scurry, J.: Weighted weak type estimates for square functions (2012, ArXiv e-prints)

  14. 14.

    Lerner, A.K.: A pointwise estimate for the local sharp maximal function with applications to singular integrals. Bull. Lond. Math. Soc. 42(5), 843–856 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Lerner, A.K.: On an estimate of Calderón-Zygmund operators by dyadic positive operators. J. Anal. Math. 121, 141–161 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lerner, A.K.: A simple proof of the \(A_2\) conjecture. Int. Math. Res. Not. IMRN 90(14), 3159–3170 (2013)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Lerner, A.K.: On sharp aperture-weighted estimates for square functions. J. Fourier Anal. Appl. 20(4), 784–800 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Lerner, A.K., Nazarov, F.: Intuitive dyadic calculus: the basics (2015). arXiv:1508.05639

  19. 19.

    Lerner, A.K., Ombrosi, S., Pérez, C.: \(A_1\) bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden. Math. Res. Lett. 16(1), 149–156 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Li, K., Moen, K., Sun, W.: The sharp weighted bound for multilinear maximal functions and Calderón-Zygmund operators. J. Fourier Anal. Appl. 20(4), 751–765 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Nazarov, F., Reznikov, A., Vasyunin, V., Volberg, A.: \({A}_1\) conjecture: weak norm estimates of weighted singular integral operators and Bellman functions (2013, Preprint)

  23. 23.

    Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical \(A_p\) characteristic. Am. J. Math. 129(5), 1355–1375 (2007)

    MathSciNet  Article  MATH  Google Scholar 

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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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Correspondence to Guillermo Rey.

Additional information

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

$$\begin{aligned} \sup _{\lambda > 0} \lambda |\left\{ x\in P_0: \mathcal {A}_{P_0,\alpha }^m \vec {f}(x) > \lambda \right\} |^{k} \le C_{W} \Vert \alpha \Vert _{{\text {Car}}(P_0)} \prod _{i=1}^k \Vert f_i\Vert _{L^1(P_0)}, \end{aligned}$$
(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:

$$\begin{aligned} \left( \sum _{Q \in \mathscr {D}(P_0)} \alpha _Q \left( \prod _{i=1}^k \langle f_i \rangle _Q \right) ^p \right) ^{\frac{1}{p}} \le \Vert \alpha \Vert _{\text {Car}(P_0)} \prod _{i=1}^k p_i' \Vert f_i\Vert _{L^{p_i}(P_0)} \end{aligned}$$
(4.2)

whenever

$$\begin{aligned} \frac{1}{p} = \frac{1}{p_1} + \dots + \frac{1}{p_k}. \end{aligned}$$

Now we can prove

Proposition 4.2

$$\begin{aligned} \Vert \mathcal {A}_{P_0,\alpha }^m \vec {f}\Vert _{L^2(P_0)} \le 4\Vert \alpha \Vert _{{\text {Car}}(P_0)}\prod _{i=1}^k \Vert f_i\Vert _{L^{2k}(P_0)} \end{aligned}$$

Proof

We begin by using duality and homogeneity to reduce to showing

$$\begin{aligned} \int _{P_0} g(x) \mathcal {A}_{P_0,\alpha }^m \vec {f}(x) \, dx \le 4 \end{aligned}$$

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

$$\begin{aligned} \left( \sum _{Q \in \mathscr {D}_{\ge m}(P_0)} \alpha _Q \left( \prod _{i=1}^k \langle f_i \rangle _{Q^{(m)}} \right) ^2 |Q| \right) ^{1/2} \left( \sum _{Q \in \mathscr {D}_{\ge m}(P_0)} \alpha _Q \langle g \rangle _Q^2 |Q| \right) ^{1/2}. \end{aligned}$$

The second term can be estimated, using (4.2) in the linear case, by

$$\begin{aligned} \left( \sum _{Q \in \mathscr {D}_{\ge m}(P_0)} \alpha _Q \langle g \rangle _Q^2 |Q| \right) ^{1/2} \le 2. \end{aligned}$$

For the first term observe that the sequence \(\beta _Q\) defined by

$$\begin{aligned} \beta _Q = \frac{1}{2^{dm}} \sum _{R \in \mathscr {D}_m(Q)} \alpha _R \end{aligned}$$

is a Carleson sequence adapted to \(P_0\) of the same constant. Indeed:

$$\begin{aligned} \frac{1}{|Q|} \sum _{R \in \mathscr {D}(Q)} \beta _R|R|&= \frac{1}{|Q|}\sum _{R \in \mathscr {D}(Q)} |R| \frac{1}{2^{dm}} \sum _{T \in \mathscr {D}_m(R)} \alpha _T \\&= \frac{1}{|Q|}\sum _{R \in \mathscr {D}(Q)}\sum _{T \in \mathscr {D}_m(R)} \alpha _T |T| \\&= \frac{1}{|Q|} \sum _{R \in \mathscr {D}_{\ge m}(Q)} \alpha _R|R| \\&\le \Vert \alpha \Vert _{\text {Car}(I)} \\&=1. \end{aligned}$$

Therefore, we can write the first term as

$$\begin{aligned} \left( \sum _{Q \in \mathscr {D}(P_0)} \beta _Q \left( \sum _{i=1}^k \langle f_i \rangle _Q \right) ^2 |Q| \right) ^{1/2}, \end{aligned}$$

which can also be estimated by (4.2) as follows:

$$\begin{aligned} \left( \sum _{Q \in \mathscr {D}(P_0)} \beta _Q \left( \sum _{i=1}^k \langle f_i \rangle _Q \right) ^2 |Q| \right) ^{1/2} \le \left( \frac{2k}{2k-1} \right) ^k \le 2. \end{aligned}$$

Combining both terms we arrive at

$$\begin{aligned} \int _{P_0} g(x) \mathcal {A}_{P_0,\alpha }^m \vec {f}(x) \, dx \le 4 \end{aligned}$$

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

$$\begin{aligned} \Omega _i = \left\{ x\in P_0: \mathcal {M}^d f_i(x) > \lambda ^{1/k}\right\} . \end{aligned}$$

If \(\langle f_i \rangle _{P_0} > \lambda ^{1/k}\) then by the homogeneity assumption

$$\begin{aligned} |P_0| < \lambda ^{-1/k} \end{aligned}$$

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

$$\begin{aligned} \langle f_i \rangle _R > \lambda ^{1/k} \quad \text {and} \quad \langle f_i \rangle _{R^{(1)}} \le \lambda ^{1/k}. \end{aligned}$$

For each \(1 \le i \le k\) let \(b_i = \sum _{R \in \mathcal {R}_i} b_i^R\), where

$$\begin{aligned} b_i^R(x) := \left( f_i(x) - \langle f_i \rangle _R \right) \mathbbm {1}_R(x). \end{aligned}$$

We now let \(g_i = f_i-b_i\).

Observe that we have

$$\begin{aligned} |g_i(x)| \le 2^d \lambda ^{1/k}, \end{aligned}$$

as well as

$$\begin{aligned} |\Omega _i| = \sum _{R \in \mathcal {R}_i} |R| \le \lambda ^{-1/k}. \end{aligned}$$

Define \(\Omega = \cup _{i=1}^k \Omega _i\), then we have

$$\begin{aligned} |\{x \in P_0: \, \mathcal {A}_{P_0,\alpha }^m \vec {f}(x) > \lambda \}|&\le |\Omega | + |\{x\in P_0{\setminus } \Omega : \, \mathcal {A}_{P_0,\alpha }^m \vec {f}(x) > \lambda \}| \nonumber \\&\le k \lambda ^{-1/k} + |\{x\in P_0{\setminus } \Omega : \, \mathcal {A}_{P_0,\alpha }^m \vec {f}(x) > \lambda \}|. \end{aligned}$$
(4.3)

To estimate the second term observe that

$$\begin{aligned} \mathcal {A}_{P_0,\alpha }^m \vec {f}(x)&= \mathcal {A}_{P_0,\alpha }^m (\vec {g} + \vec {b})(x) \\&= \mathcal {A}_{P_0,\alpha }^m \vec {g}(x) + \sum _{j=1}^{2^k-1} \mathcal {A}_{P_0,\alpha }^m(h_1^j, \ldots , h_k^j)(x), \end{aligned}$$

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

$$\begin{aligned}&\mathcal {A}_{P_0}^m (h_1^j, h_2^j, \ldots , h_{i_j}^j, \ldots , h_k^j)(x)\\&\quad = \sum _{Q \in \mathscr {D}_{\ge m}(P_0)} \alpha _Q \left( \prod _{i=1}^k \langle h_i^j \rangle _{Q^{(m)}} \right) \mathbbm {1}_Q(x) \\&\quad = \sum _{Q \in \mathscr {D}_{\ge m}(P_0)} \alpha _Q \langle b_{i_j} \rangle _{Q^{(m)}}\left( \prod _{1 \le i \le k, \, i \ne i_j} \langle h_i^j \rangle _{Q^{(m)}} \right) \mathbbm {1}_Q(x) \\&\quad = \sum _{R \in \mathcal {R}_{i_j}} \sum _{Q \in \mathscr {D}_{\ge m}(P_0)} \alpha _Q \langle b_{i_j}^R \rangle _{Q^{(m)}}\left( \prod _{1 \le i \le k, \, i \ne i_j} \langle h_i^j \rangle _{Q^{(m)}} \right) \mathbbm {1}_Q(x) \\&\quad = \sum _{R \in \mathcal {R}_{i_j}} \sum _{Q \in \mathscr {D}_{> m}(R)} \alpha _Q \langle b_{i_j}^R \rangle _{Q^{(m)}}\left( \prod _{1 \le i \le k, \, i \ne i_j} \langle h_i^j \rangle _{Q^{(m)}} \right) \mathbbm {1}_Q(x). \end{aligned}$$

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

$$\begin{aligned} |\{x\in P_0{\setminus } \Omega : \, \mathcal {A}_{P_0,\alpha }^m \vec {g}(x) > \lambda \}|. \end{aligned}$$

Now we can use the \(L^2\) bound as follows:

$$\begin{aligned} |\{x\in P_0{\setminus } \Omega : \, \mathcal {A}_{P_0,\alpha }^m \vec {g}(x) > \lambda \}|&\le \frac{1}{\lambda ^2} \Vert \mathcal {A}_{P_0,\alpha }^m \vec {g}\Vert _{L^2(P_0)}^2 \\&\le \frac{16}{\lambda ^2} \prod _{i=1}^k \Vert g_i\Vert _{L^{2k}(P_0)}^{2} \\&\le \frac{16}{\lambda ^2} \prod _{i=1}^k \left( 2^d \lambda ^{1/k} \right) ^{\frac{2k-1}{k}} \Vert g_i\Vert _{L^1(P_0)}^{1/k} \\&= \frac{16}{\lambda ^2} 2^{d(2k-1)} \lambda ^{2-1/k} \\&= 2^{4+ d(2k-1)}\lambda ^{-1/k}. \end{aligned}$$

Putting both estimates together we arrive at

$$\begin{aligned} |\{x \in P_0: \, \mathcal {A}_{P_0,\alpha }^m \vec {f}(x) > \lambda \}| \le 2^{5+ d(2k-1)}\lambda ^{-1/k} \end{aligned}$$

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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