Abstract
The main purpose of this paper is to establish a Calderón–Vaillancourt-type theorem on
Funding source: National Science Foundation
Award Identifier / Grant number: DMS#1301595
Funding statement: Research of this work is partly supported by an NSF grant DMS#1301595 and a Simons Fellowship from the Simons Foundation.
Acknowledgements
The authors wish to thank the referee for his/her very careful reading of the paper and for having made some very valuable comments which improved the exposition of the paper and also helped us to simplify the proof of Theorem 3.2.
References
[1] Alvarez L. and Hounie J., Estimates for the kernel and continuity properties of pseudo-differential operators, Ark. Mat. 28 (1990), 1–22. 10.1007/BF02387364Search in Google Scholar
[2]
Bényi Á., Bernicot F., Maldonado D., Naibo V. and Torres H.,
On the H
[3] Calderón A. and Vaillancourt R., On the boundedness of pseudo-differential operators, J. Math Soc. Japan 23 (1971), 374–378. 10.2969/jmsj/02320374Search in Google Scholar
[4] Chen J. and Lu G., Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal. 101 (2014), 98–112. 10.1016/j.na.2014.01.005Search in Google Scholar
[5] Christ M. and Journé J-L., Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159 (1987), 51–80. 10.1007/BF02392554Search in Google Scholar
[6] Coifman R. R. and Meyer Y., On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315–331. 10.1090/S0002-9947-1975-0380244-8Search in Google Scholar
[7] Coifman R. R. and Meyer Y., Au delá des Opérateurs Pseudo-Differentiels, Astérisque 57, Société Mathématique, Paris, 1978. Search in Google Scholar
[8] Coifman R. R. and Meyer Y., Au delá des Operateurs Pseudo-Differentiels, 2nd ed., Astérisque 57, Société Mathématique de France, Paris, 1978. Search in Google Scholar
[9]
Dai W. and Lu G.,
[10]
Fefferman C.,
[11] Folland G. B., Lectures on Partial Differential Operators, Springer, Berlin, 1983. 10.1007/978-3-662-00729-7Search in Google Scholar
[12] Goldberg M., A local version of real Hardy space, Duke Math. J. 46 (1979), 27–41. 10.1215/S0012-7094-79-04603-9Search in Google Scholar
[13] Grafakos L. and Torres R. H., Multilinear Calderón–Zygmund theory, Adv. Math. 165 (2002), 124–164. 10.1006/aima.2001.2028Search in Google Scholar
[14] Hong Q. and Lu G., Symbolic calculus and boundedness for multi-parameter and multilinear pseudo-differential operators, Adv. Nonlinear Stud. 14 (2014), no. 4, 1055–1082. 10.1515/ans-2014-0413Search in Google Scholar
[15] Hörmander L., Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 501–517. 10.1007/978-3-540-49938-1_3Search in Google Scholar
[16] Hörmander L., Pseudo-differential operators and hypoelliptic equations, Proc. Sympos. Pure Math. 10 (1967), 138–183. 10.1090/pspum/010/0383152Search in Google Scholar
[17] Kato T. and Ponce G., Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907. 10.1002/cpa.3160410704Search in Google Scholar
[18] Kenig C. and Stein E. M., Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), 1–15. 10.4310/MRL.1999.v6.n1.a1Search in Google Scholar
[19] Kohn J. and Nirenberg L., An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269–305. 10.1002/cpa.3160180121Search in Google Scholar
[20] Kumano-go H., Pseudodifferential Operators, MIT Press, Cambridge, 1981. Search in Google Scholar
[21] Li K. and Sun W., Weighted estimates for multilinear pseudodifferential operators, Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 8, 1281–1288. 10.1007/s10114-014-3027-5Search in Google Scholar
[22] Lu G. and Zhang P., Multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications, Nonlinear Anal. 107 (2014), 92–117. 10.1016/j.na.2014.05.005Search in Google Scholar
[23] Maldonado D. and Naibo V., Weighted norm inequalities for paraproducts and bilinear pseudo-differential operators with mild regularity, J. Fourier Anal. Appl. 15 (2009), no. 2, 218–261. 10.1007/s00041-008-9029-xSearch in Google Scholar
[24] Miyachi A. and Tomita N., Calderón–Vaillancourt type theorem for bilinear pseudo-differential operators, Indiana Univ. Math. J. 62 (2013), no. 4, 1165–1201. 10.1512/iumj.2013.62.5059Search in Google Scholar
[25] Muscalu C., Pipher J., Tao T. and Thiele C., Bi-parameter paraproducts, Acta. Math. 193 (2004), 269–296. 10.1007/BF02392566Search in Google Scholar
[26]
Päivärinta L. and Somersalo E.,
A generalization of the Calderón–Vaillancourt theorem to
[27] Petersen B. E., Introduction to the Fourier Transform and Pseudo-Differential Operators, Pitman, Boston, 1983. Search in Google Scholar
[28] Sogge C., Fourier Integrals In Classical Analysis, Cambridge Tracts in Math. 105, Cambridge University Press, Cambridge, 1993. 10.1017/CBO9780511530029Search in Google Scholar
[29] Stein E. M., Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. Search in Google Scholar
[30] Taylor M., Pseudodifferential Operators, Princeton University Press, Princeton, 1981. 10.1515/9781400886104Search in Google Scholar
[31] Treves F., Introduction to pseudodifferential and Fourier Integral Operators. Vol. 1: Pseudodifferential Operators, Univ. Ser. Math., Plenum Press, New York, 1980. 10.1007/978-1-4684-8780-0Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Behavior of bounds of singular integrals for large dimension
- The twofold way of super holonomy
- On mod p singular modular forms
- Martin boundary of unbounded sets for purely discontinuous Feller processes
- Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order
- Nontrivial solutions of superlinear nonlocal problems
- Hopfian $\ell$-groups, MV-algebras and AF~{C}*-algebras
- On regularization of vector distributions on manifolds
- The Addition Theorem for algebraic entropies induced by non-discrete length functions
- When are Zariski chambers numerically determined?
- Convexity theorems for semisimple symmetric spaces
- On amenability of groups generated by homogeneous automorphisms and their cracks
Articles in the same Issue
- Frontmatter
- Behavior of bounds of singular integrals for large dimension
- The twofold way of super holonomy
- On mod p singular modular forms
- Martin boundary of unbounded sets for purely discontinuous Feller processes
- Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order
- Nontrivial solutions of superlinear nonlocal problems
- Hopfian $\ell$-groups, MV-algebras and AF~{C}*-algebras
- On regularization of vector distributions on manifolds
- The Addition Theorem for algebraic entropies induced by non-discrete length functions
- When are Zariski chambers numerically determined?
- Convexity theorems for semisimple symmetric spaces
- On amenability of groups generated by homogeneous automorphisms and their cracks