## Journal of the Mathematical Society of Japan

### Strongly singular bilinear Calderón–Zygmund operators and a class of bilinear pseudodifferential operators

#### Abstract

Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón–Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón–Zygmund operators satisfy pointwise estimates in terms of maximal operators, which imply their boundedness in weighted Lebesgue spaces.

#### Note

The first author is partially supported by a grant from the Simons Foundation (No. 246024). The third author is partially supported by the NSF under grant DMS 1500381.

#### Article information

Source
J. Math. Soc. Japan, Volume 71, Number 2 (2019), 569-587.

Dates
First available in Project Euclid: 4 March 2019

https://projecteuclid.org/euclid.jmsj/1551690077

Digital Object Identifier
doi:10.2969/jmsj/79327932

Mathematical Reviews number (MathSciNet)
MR3943451

Zentralblatt MATH identifier
07090056

#### Citation

BÉNYI, Árpád; CHAFFEE, Lucas; NAIBO, Virginia. Strongly singular bilinear Calderón–Zygmund operators and a class of bilinear pseudodifferential operators. J. Math. Soc. Japan 71 (2019), no. 2, 569--587. doi:10.2969/jmsj/79327932. https://projecteuclid.org/euclid.jmsj/1551690077

#### References

• [1] J. Alvarez and M. Milman, $H^p$ continuity properties of Calderón–Zygmund-type operators, J. Math. Anal. Appl., 118 (1986), 63–79.
• [2] Á. Bényi, F. Bernicot, D. Maldonado, V. Naibo and R. H. Torres, On the Hörmander classes of bilinear pseudodifferential operators II, Indiana Univ. Math. J., 62 (2013), 1733–1764.
• [3] Á. Bényi, D. Maldonado, V. Naibo and R. H. Torres, On the Hörmander classes of bilinear pseudodifferential operators, Integral Equations Operator Theory, 67 (2010), 341–364.
• [4] Á. Bényi and R. H. Torres, Symbolic calculus and the transposes of bilinear pseudodifferential operators, Comm. Partial Differential Equations, 28 (2003), 1161–1181.
• [5] Á. Bényi and R. H. Torres, Almost orthogonality and a class of bounded bilinear pseudodifferential operators, Math. Res. Lett., 11 (2004), 1–11.
• [6] J. Brummer and V. Naibo, Bilinear operators with homogeneous symbols, smooth molecules, and Kato–Ponce inequalities, Proc. Amer. Math. Soc., 146 (2018), 1217–1230.
• [7] R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque, 57, Société Mathématique de France, Paris, 1978. with an English summary.
• [8] D. Cruz-Uribe, J. M. Martell and C. Pérez, Extrapolation from $A_\infty$ weights and applications, J. Funct. Anal., 213 (2004), 412–439.
• [9] J. Duoandikoetxea, Fourier analysis, Graduate Studies in Mathematics, 29, American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe.
• [10] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math., 124 (1970), 9–36.
• [11] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137–193.
• [12] L. Grafakos and R. H. Torres, Multilinear Calderón–Zygmund theory, Adv. Math., 165 (2002), 124–164.
• [13] J. Herbert and V. Naibo, Bilinear pseudodifferential operators with symbols in Besov spaces, J. Pseudo-Differ. Oper. Appl., 5 (2014), 231–254.
• [14] J. Herbert and V. Naibo, Besov spaces, symbolic calculus, and boundedness of bilinear pseudodifferential operators, In: Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory, Vol. 1, Besov Spaces, Symbolic Calculus, and Boundedness of Bilinear Pseudodifferential Operators, Assoc. Women Math. Ser., Springer, 2016, 275–305.
• [15] K. Koezuka and N. Tomita, Bilinear pseudo-differential operators with symbols in $BS^m_{1,1}$ on Triebel–Lizorkin spaces, J. Fourier Anal. Appl., 24 (2018), 309–319.
• [16] N. Michalowski, D. Rule and W. Staubach, Multilinear pseudodifferential operators beyond Calderón–Zygmund theory, J. Math. Anal. Appl., 414 (2014), 149–165.
• [17] A. Miyachi and N. Tomita, Calderón–Vaillancourt-type theorem for bilinear operators, Indiana Univ. Math. J., 62 (2013), 1165–1201.
• [18] A. Miyachi and N. Tomita, Bilinear pseudo-differential operators with exotic symbols, Ann. Inst. Fourier (Grenoble), 2017.
• [19] A. Miyachi and N. Tomita, Bilinear pseudo-differential operators with exotic symbols, II, Pseudo-Differ. Oper. Appl., 2018.
• [20] V. Naibo, On the bilinear Hörmander classes in the scales of Triebel–Lizorkin and Besov spaces, J. Fourier Anal. Appl., 21 (2015), 1077–1104.
• [21] V. Naibo, On the $L^{\infty} \times L^{\infty} \rightarrow BMO$ mapping property for certain bilinear pseudodifferential operators, Proc. Amer. Math. Soc., 143 (2015), 5323–5336.
• [22] V. Naibo and A. Thomson, Bilinear Hörmander classes of critical order and Leibniz-type rules in Besov and local Hardy spaces, J. Math. Anal. Appl., 2019.
• [23] S. Rodríguez-López and W. Staubach, Estimates for rough Fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators, J. Funct. Anal., 264 (2013), 2356–2385.
• [24] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993. with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.