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WEIGHTED WEAK TYPE ENDPOINT ESTIMATES FOR THE COMPOSITIONS OF CALDERÓN–ZYGMUND OPERATORS

Published online by Cambridge University Press:  08 April 2019

GUOEN HU*
Affiliation:
School of Applied Mathematics, Beijing Normal University, Zhuhai519087, PR China
*

Abstract

Let $T_{1}$, $T_{2}$ be two Calderón–Zygmund operators and $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in \text{BMO}(\mathbb{R}^{n})$. In this paper, by establishing new bilinear sparse dominations and a new weighted estimate for bilinear sparse operators, we prove that the composite operator $T_{1}T_{2}$ satisfies the following estimate: for $\unicode[STIX]{x1D706}>0$ and weight $w\in A_{1}(\mathbb{R}^{n})$,

$$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log \bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx,\nonumber\end{eqnarray}$$
while the composite operator $T_{1,b}T_{2}$ satisfies
$$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1,b}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}^{2}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log ^{2}\bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx.\nonumber\end{eqnarray}$$

Type
Research Article
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by C. Meaney

The research was supported by the NNSF of China under grant no. 11871108.

References

Benea, C. and Bernicot, F., ‘Conservation de certaines propriétés á travers un contrôle épars d’un opérateur et applications au projecteur de Leray–Hopf’, Preprint, 2017, arXiv:1703:00228.Google Scholar
Buckley, S. M., ‘Estimates for operator norms on weighted spaces and reverse Jensen inequalities’, Trans. Amer. Math. Soc. 340 (1993), 253272.Google Scholar
Carozza, N. and Passarelli di Napoli, A., ‘Composition of maximal operators’, Publ. Mat. 40 (1996), 397409.Google Scholar
Chung, D., Pereyra, M. C. and Pérez, C., ‘Sharp bounds for general commutators on weighted Lebesgue spaces’, Trans. Amer. Math. Soc. 364 (2012), 11631177.Google Scholar
Coifman, R. R. and Meyer, Y., Wavelets: Calderón–Zygmund Operators and Multilinear Operators (Cambridge University Press, Cambridge, 1997).Google Scholar
Grafakos, L., Modern Fourier Analysis, 2nd edn, Graduate Texts in Mathematics, 250 (Springer, New York, 2008).Google Scholar
Hänninen, T., Hytönen, T. and Li, K., ‘Two-weight L p - L q bounds for positive dyadic operators: unified approach to pq and p > q ’, Potential Anal. 45 (2016), 579608.+q+’,+Potential+Anal.+45+(2016),+579–608.>Google Scholar
Hu, G., ‘Weighted vector-valued estimates for a non-standard Calderón–Zygmund operator’, Nonlinear Anal. 165 (2017), 143162.Google Scholar
Hu, G., ‘Quantitative weighted bounds for the composition of Calderón–Zygmund operators’, Banach J. Math. Anal. 13 (2019), 133150.Google Scholar
Hu, G. and Li, D., ‘A Cotlar type inequality for the multilinear singular integral operators and its applications’, J. Math. Anal. Appl. 290 (2004), 639653.Google Scholar
Hu, G. and Yang, D., ‘Weighted estimates for singular integral operators with nonsmooth kernels’, J. Aust. Math. Soc. 85 (2008), 377417.Google Scholar
Hytönen, T., ‘The sharp weighted bound for general Calderón–Zygmund operators’, Ann. of Math. (2) 175 (2012), 14731506.Google Scholar
Hytönen, T. and Lacey, M. T., ‘The A p - A inequality for general Calderón–Zygmund operators’, Indiana Univ. Math. J. 61 (2012), 20412052.Google Scholar
Hytönen, T., Lacey, M. T. and Pérez, C., ‘Sharp weighted bounds for the q-variation of singular integrals’, Bull. Lond. Math. Soc. 45 (2013), 529540.Google Scholar
Hytönen, T. and Pérez, C., ‘Sharp weighted bounds involving A ’, Anal. PDE 6 (2013), 777818.Google Scholar
Hytönen, T. and Pérez, C., ‘The L (logL)𝜖 endpoint estimate for maximal singular integral operators’, J. Math. Anal. Appl. 428 (2015), 605626.Google Scholar
Hytönen, T., Pérez, C. and Rela, E., ‘Sharp reverse Hölder property for A 1 weights on spaces of homogeneous type’, J. Funct. Anal. 263 (2012), 38833899.Google Scholar
Krantz, S. G. and Li, S., ‘Boundedness and compactness of integral operators on spaces of homogeneous type and applications’, J. Math. Anal. Appl. 258 (2001), 629641.Google Scholar
Lerner, A. K., ‘A simple proof of the A 2 conjecture’, Int. Math. Res. Not. 14 (2013), 31593170.Google Scholar
Lerner, A. K., ‘On pointwise estimate involving sparse operator’, New York J. Math. 22 (2016), 341349.Google Scholar
Lerner, A. K., ‘A weak type estimate for rough singular integrals’, Preprint, 2017,arXiv:1705:07397.Google Scholar
Lerner, A. K. and Nazarov, F., ‘Intuitive dyadic calculus: the basics’, Expo. Math. doi:10.1016/j.exmath.2018.01.001.Google Scholar
Lerner, A. K., Ombrosi, S. and Rivera-Rios, I., ‘On pointwise and weighted estimates for commutators of Calderón–Zygmund operators’, Adv. Math. 319 (2017), 153181.Google Scholar
Li, K., ‘Two weight inequalities for bilinear forms’, Collect. Math. 68 (2017), 129144.Google Scholar
Li, K., Pérez, C., Rivera-Rios, I. P. and Roncal, L., ‘Weighted norm inequalities for rough singular integral operators’, J. Geom. Anal. doi:10.1007/s12220-018-0085-4.Google Scholar
Rao, M. and Ren, Z., Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146 (Marcel Dekker, New York, 1991).Google Scholar
Sawyer, E., ‘Norm inequalities relating singular integrals and the maximal function’, Studia Math. 75 (1983), 253263.Google Scholar
Wilson, M. J., ‘Weighted inequalities for the dyadic square function without dyadic A ’, Duke Math. J. 55 (1987), 1950.Google Scholar