Vector-valued operators, optimal weighted estimates and the \(C_p\) condition. (English) Zbl 1450.42011
This article contains a wide variety of results on optimal weighted estimates and sparse dominations.
The first part of the paper provides results concerning the \(C_p\) classes introduced by Muckenhoupt. The \(C_p\) classes contain the \(A_\infty\) class for all \( p > 0 \). B. Muckenhoupt [in: Functional analysis and approximation, Proc. Conf., Oberwolfach 1980, ISNM 60, 219–231 (1981; Zbl 0465.42014)] showed that if the Coifman-Fefferman estimate (labeled (1.1)) holds for the Hilbert transform with \(p>1\), then the weight involved must be a \(C_p\) weight. At the same time, he conjectured that the \(C_p\) condition is also sufficient for the Coifman-Fefferman estimate to hold.
In this context, the Coifman-Fefferman estimate is extended to the full expected range \(0 < p < \infty\) to operators that include Calderón-Zygmund operators and their commutators (including \(k\)-order, and vector-valued extensions), weakly strongly singular integral operators, pseudo-differential operators that belong to the Hörmander class, oscillatory integral operators and to some vector-valued operators. Furthermore, a suitable Coifman-Fefferman estimate for the fractional integral is deduced from one of the main proofs.
Also, a multilinear version of the Coifman-Fefferman estimate is extended to the full expected range \(0 < p < \infty \) for \(m\)-linear Calderón-Zygmund operators.
Additionally, a Coifman-Fefferman estimate where the strong norm is replaced by the weak norm, is obtained for \(w\)-Calderón-Zygmund operators with \(w\) satisfying a log-Dini condition, within the range \((1,\infty)\). Likewise, Coifman-Fefferman estimates are obtained for commutators which are completely new in both the scalar and the vector-valued cases.
On the other hand, sparse dominations are proved for vector-valued extensions of the maximal function, Calderón-Zygmund operators and their commutators, rough singular integral operator and Bochner-Riesz multiplier at the critical index. In addition, the paper contains an appendix where quantitative estimates for the operators mentioned above follow from the sparse domination results. These estimates include \(A_p\) weak and strong type estimates, endpoint estimates and Fefferman-Stein type inequalities. Also, quantitative unweighted estimates are given for Calderón-Zygmund operators satisfying the Dini condition and their vector-valued counterparts.
The first part of the paper provides results concerning the \(C_p\) classes introduced by Muckenhoupt. The \(C_p\) classes contain the \(A_\infty\) class for all \( p > 0 \). B. Muckenhoupt [in: Functional analysis and approximation, Proc. Conf., Oberwolfach 1980, ISNM 60, 219–231 (1981; Zbl 0465.42014)] showed that if the Coifman-Fefferman estimate (labeled (1.1)) holds for the Hilbert transform with \(p>1\), then the weight involved must be a \(C_p\) weight. At the same time, he conjectured that the \(C_p\) condition is also sufficient for the Coifman-Fefferman estimate to hold.
In this context, the Coifman-Fefferman estimate is extended to the full expected range \(0 < p < \infty\) to operators that include Calderón-Zygmund operators and their commutators (including \(k\)-order, and vector-valued extensions), weakly strongly singular integral operators, pseudo-differential operators that belong to the Hörmander class, oscillatory integral operators and to some vector-valued operators. Furthermore, a suitable Coifman-Fefferman estimate for the fractional integral is deduced from one of the main proofs.
Also, a multilinear version of the Coifman-Fefferman estimate is extended to the full expected range \(0 < p < \infty \) for \(m\)-linear Calderón-Zygmund operators.
Additionally, a Coifman-Fefferman estimate where the strong norm is replaced by the weak norm, is obtained for \(w\)-Calderón-Zygmund operators with \(w\) satisfying a log-Dini condition, within the range \((1,\infty)\). Likewise, Coifman-Fefferman estimates are obtained for commutators which are completely new in both the scalar and the vector-valued cases.
On the other hand, sparse dominations are proved for vector-valued extensions of the maximal function, Calderón-Zygmund operators and their commutators, rough singular integral operator and Bochner-Riesz multiplier at the critical index. In addition, the paper contains an appendix where quantitative estimates for the operators mentioned above follow from the sparse domination results. These estimates include \(A_p\) weak and strong type estimates, endpoint estimates and Fefferman-Stein type inequalities. Also, quantitative unweighted estimates are given for Calderón-Zygmund operators satisfying the Dini condition and their vector-valued counterparts.
Reviewer: Guillermo Flores (Córdoba)
MSC:
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B35 | Function spaces arising in harmonic analysis |
Keywords:
\(A_p\) weights; quantitative estimates; \(C_p\) estimates; vector-valued extensions; sparse domination; maximal functions; Calderón-Zygmund operators; commutatorsCitations:
Zbl 0465.42014References:
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