On the separated bumps conjecture for Calderón-Zygmund operators. (English) Zbl 1345.42015
The motivation of this paper is the long-standing question about which is the sharpest condition on a pair of weights \(w\) and \(\sigma\) so that the following inequality holds for all Calderón-Zygmund operators \(T\):
\[
\int |T(\sigma f)|^pw(dx)\leq C_{T,\sigma,w}^p\int|f|^p\sigma(dx), \tag{1}
\]
where \(\sigma(dx)=\sigma(x)dx\) and \(w(dx)=w(x)dx\). When \(w\) and \(\sigma\) are related via \(\sigma(x)=w(x)^{1-p'}\), the necessary and sufficient condition for (1) to hold is the so-called \(A_p\)-Muckenhoupt condition. Nevertheless, when \(w\) and \(\sigma\) do not have an explicit relationship, this is known as the two-weight problem.
Partial results towards the solution of the two-weight problem have been given thanks to the introduction of “bumped” \(A_p\) conditions (classical two-weighted \(A_p\) condition with the localized \(L^p\) and \(L^{p'}\) norms “bumped up” in the scale of Orlicz spaces), see [D. Cruz-Uribe and C. Pérez, Math. Res. Lett. 6, No. 3–4, 417–427 (1999; Zbl 0961.42013); Indiana Univ. Math. J. 49, No. 2, 697–721 (2000; Zbl 1033.42009)] and [D. Cruz-Uribe et al., Adv. Math. 255, 706–729 (2014; Zbl 1290.42033)].
Let \(A\) be a Young function. One says that \(A\in B_p\), \(1<p<\infty\) if, for some \(c>0\), \[ \int_c^{\infty} \frac{A(t)}{t^p}\frac{dt}{t}<\infty. \] In [F. Nazarov et al., J. Anal. Math. 121, 255–277 (2013; Zbl 1284.42037)], for \(p=2\), and in [A. K. Lerner, Int. Math. Res. Not. 2013, No. 14, 3159–3170 (2013; Zbl 1318.42018)], for all \(p\), it was proved that, letting \(A\in B_p\) and \(B\in B_{p'}\), a sufficient condition for the estimate (1) is that \[ \sup_Q\langle\sigma^{1/p'}\rangle_{\bar{A},Q}\langle w^{1/p}\rangle_{\bar{B},Q}<\infty, \] where \(\bar{A}, \bar{B}\) denote the dual Young functions of \(A\) and \(B\), and \(\langle f\rangle_{\bar{A}, Q}\) is the Luxemburg norm associated to \(A\).
Cruz-Uribe, Reznikov and Volberg suggested that “separated” bump conditions are sufficient for the two-weight problem, and they conjectured the following. Let \(A\in B_p\) and \(B\in B_{p'}\). The norm estimate (1) holds provided that \[ [\sigma, w]_{\bar{A},p'}+[w,\sigma]_{\bar{B},p}<\infty, \] where \[ [\sigma, w]_{\bar{A},p'}:=\sup_Q\langle\sigma^{1/p'}\rangle_{\bar{A},Q}\langle w\rangle_{Q}^{1/p}. \] Here, \(\langle w\rangle_{Q}\) is the average of \(w\). The word “separated” means that one is dealing with the sum of bumps instead of the product.
Now let again \(A\in B_p\) and \(B\in B_{p'}\) and let \(\varepsilon_p, \varepsilon_{p'}\) be two monotonic increasing functions on \((1,\infty)\) satisfying \(\int_1^{\infty}\varepsilon_p(t)^{-p'}(dt/t)=1\) and similarly for \(\varepsilon_{p'}\). Define \[ [\sigma,w]_{\bar{A},\varepsilon_p,p'}:=\sup_Q\varepsilon_p\Big(1+\frac{\langle\sigma^{1/p'}\rangle_{\bar{A},Q}}{\langle \sigma\rangle_Q^{1/p'}}\Big)\langle\sigma^{1/p'}\rangle_{\bar{A},Q}\langle w\rangle_Q^{1/p}. \] The author proves that a sufficient condition for (1) is \[ [\sigma, w]_{\bar{A},\varepsilon_p,p'}+[\sigma,w]_{\bar{B},\varepsilon_{p'},p}<\infty. \] The result above is a version of the conjecture by Cruz-Uribe et al., and it was already proved in the case \(p=2\) and under some additional condition in [F. Nazarov et al. “Bellman approach to the one-sided bumping for weighted estimates of Calderón-Zygmund operators”, Preprint (2013), arXiv:1306.2653].
In order to prove the main theorem, testing inequalities for sparse operators are verified, and this is done by means of a version of the parallel corona, based on the construction of stopping data.
Finally, the author illustrates the main result by giving examples of log bumps and log-log bumps which satisfy the hypotheses of the theorem.
Partial results towards the solution of the two-weight problem have been given thanks to the introduction of “bumped” \(A_p\) conditions (classical two-weighted \(A_p\) condition with the localized \(L^p\) and \(L^{p'}\) norms “bumped up” in the scale of Orlicz spaces), see [D. Cruz-Uribe and C. Pérez, Math. Res. Lett. 6, No. 3–4, 417–427 (1999; Zbl 0961.42013); Indiana Univ. Math. J. 49, No. 2, 697–721 (2000; Zbl 1033.42009)] and [D. Cruz-Uribe et al., Adv. Math. 255, 706–729 (2014; Zbl 1290.42033)].
Let \(A\) be a Young function. One says that \(A\in B_p\), \(1<p<\infty\) if, for some \(c>0\), \[ \int_c^{\infty} \frac{A(t)}{t^p}\frac{dt}{t}<\infty. \] In [F. Nazarov et al., J. Anal. Math. 121, 255–277 (2013; Zbl 1284.42037)], for \(p=2\), and in [A. K. Lerner, Int. Math. Res. Not. 2013, No. 14, 3159–3170 (2013; Zbl 1318.42018)], for all \(p\), it was proved that, letting \(A\in B_p\) and \(B\in B_{p'}\), a sufficient condition for the estimate (1) is that \[ \sup_Q\langle\sigma^{1/p'}\rangle_{\bar{A},Q}\langle w^{1/p}\rangle_{\bar{B},Q}<\infty, \] where \(\bar{A}, \bar{B}\) denote the dual Young functions of \(A\) and \(B\), and \(\langle f\rangle_{\bar{A}, Q}\) is the Luxemburg norm associated to \(A\).
Cruz-Uribe, Reznikov and Volberg suggested that “separated” bump conditions are sufficient for the two-weight problem, and they conjectured the following. Let \(A\in B_p\) and \(B\in B_{p'}\). The norm estimate (1) holds provided that \[ [\sigma, w]_{\bar{A},p'}+[w,\sigma]_{\bar{B},p}<\infty, \] where \[ [\sigma, w]_{\bar{A},p'}:=\sup_Q\langle\sigma^{1/p'}\rangle_{\bar{A},Q}\langle w\rangle_{Q}^{1/p}. \] Here, \(\langle w\rangle_{Q}\) is the average of \(w\). The word “separated” means that one is dealing with the sum of bumps instead of the product.
Now let again \(A\in B_p\) and \(B\in B_{p'}\) and let \(\varepsilon_p, \varepsilon_{p'}\) be two monotonic increasing functions on \((1,\infty)\) satisfying \(\int_1^{\infty}\varepsilon_p(t)^{-p'}(dt/t)=1\) and similarly for \(\varepsilon_{p'}\). Define \[ [\sigma,w]_{\bar{A},\varepsilon_p,p'}:=\sup_Q\varepsilon_p\Big(1+\frac{\langle\sigma^{1/p'}\rangle_{\bar{A},Q}}{\langle \sigma\rangle_Q^{1/p'}}\Big)\langle\sigma^{1/p'}\rangle_{\bar{A},Q}\langle w\rangle_Q^{1/p}. \] The author proves that a sufficient condition for (1) is \[ [\sigma, w]_{\bar{A},\varepsilon_p,p'}+[\sigma,w]_{\bar{B},\varepsilon_{p'},p}<\infty. \] The result above is a version of the conjecture by Cruz-Uribe et al., and it was already proved in the case \(p=2\) and under some additional condition in [F. Nazarov et al. “Bellman approach to the one-sided bumping for weighted estimates of Calderón-Zygmund operators”, Preprint (2013), arXiv:1306.2653].
In order to prove the main theorem, testing inequalities for sparse operators are verified, and this is done by means of a version of the parallel corona, based on the construction of stopping data.
Finally, the author illustrates the main result by giving examples of log bumps and log-log bumps which satisfy the hypotheses of the theorem.
Reviewer: Luz Roncal (Logroño)
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B35 | Function spaces arising in harmonic analysis |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
