Weighted estimates for multilinear pseudodifferential operators

Springer Science and Business Media LLC - Tập 30 - Trang 1281-1288 - 2014
Kang Wei Li1, Wen Chang Sun1
1School of Mathematical Sciences and LPMC, Nankai University, Tianjin, P.R. China

Tóm tắt

In this paper, we study the weighted estimates for multilinear pseudodifferential operators. We show that a multilinear pseudodifferential operator is bounded with respect to multiple weights whenever its symbol satisfies some smoothness and decay conditions. Our result generalizes similar ones from the classical A p weights to multiple weights.

Tài liệu tham khảo

Bényi, Á., Maldonado, D., Naibo, V., et al.: On the Hörmander classes of bilinear pseudodifferential operators. Integr. Equ. Oper. Theory, 67, 341–364 (2010) Bényi, Á., Torres, R. H.: Almost orthogonality and a class of bounded bilinear pseudodifferential operators. Math. Res. Lett., 11, 1–11 (2004) Bernicot, F.: A bilinear pseudodifferential calculus. J. Geom. Anal., 20, 39–62 (2010) Bernicot, F., Torres, R. H.: Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus. Anal. PDE, 4, 551–571 (2011) Bui, T. A., Duong, X. T.: Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers. Bull. Sci. Math., 137, 63–75 (2013) Coifman, R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc., 212, 315–331 (1975) Coifman, R., Meyer, Y.: Commutateurs d’intégrales singulieres et opérateurs multi-linéaires. Ann. Inst. Fourier, Grenoble, 28, 177–202 (1978) Coifman, R., Meyer, Y.: Au delà des opérateurs pseudo-différentiels. Astérisque, 57, 1–185 (1978) Coifman, R., Meyer, Y.: Wavelets. Calderón-Zygmund andMultilinear Operators, Translated from the 1990 and 1991 French originals by David Salinger. Cambridge Studies in Advanced Mathematics 48, Cambridge University Press, Cambridge, 1997 Dai, W., Lu, G.: L p estimates for multi-linear and multi-parameter pseudo-differential operators, preprint, available at http://arxiv.org/abs/1308.4062 Ding, Y., Lu, S., Yang, D.: A criterion on weighted L p boundedness for rough multilinear oscillatory singular integrals. Proc. Amer. Math. Soc., 129, 1127–1136 (2001) Duong, X. T., Gong, R., Grafakos, L., et al.: Maximal operator for multilinear singular integrals with non-smooth kernels. Indiana Univ. Math. J., 58, 2517–2541 (2009) Duong, X. T., Grafakos, L., Yan, L.: Multilinear operators with non-smooth kernels and commutators of singular integrals. Trans. Amer. Math. Soc., 362, 2089–2113 (2010) Grafakos, L.: Classical Fourier Analysis, Second Edition, Springer-Verlag, New York, 2008 Grafakos, L., Liu, L., Yang, D.: Multiple weighted norm inequalities for maximal multilinear singular integrals with non-smooth kernels. Proc. Royal Soc. of Edinburgh, 141A, 755–775 (2011) Grafakos, L., Torres, R. H.: Multilinear Calderón-Zygmund theory. Adv. Math., 165, 124–164 (2002) Hu, G., Meng, Y., Yang, D.: Multilinear commutators of singular integrals with non doubling measures. Integral Equations Operator Theory, 51, 235–255 (2005) Hu, G., Yang, D.: Sharp function estimates and weighted norm inequalities for multilinear singular integral operators. Bull. London Math. Soc., 35, 759–769 (2003) Lacey, M., Thiele, C.: L p estimates on the bilinear Hilbert transform for 2 < p < ∞. Ann. of Math., 146, 693–724 (1997) Lacey, M., Thiele, C.: On Calderóns conjecture. Ann. of Math., 149, 475–496 (1999) Lerner, A. K., Ombrosi, S., Pérez, C., et al.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. in Math., 220, 1222–1264 (2009) Li, K., Sun, W.: Weighted estimates for multilinear Fourier multipliers. Forum Math., in press, http://dx.doi.org/10.1515/forum-2012-0128 Li, W., Xue, Q., Yabuta, K.: Weighted version of Carleson measure and multilinear Fourier multiplier. Forum Math., to appear Michalowski, N., Rule, D. J., Staubach, W.: Multilinear pseudodifferential operators beyond Calderón-Zygmund theory. J. Math. Anal. Appl., 414, 149–165 (2014) Muscalu, C., Pipher, J., Tao, T., et al.: Bi-parameter paraproducts. Acta Math., 193, 269–296 (2004) Muscalu, C., Pipher, J., Tao, T., et al.: Multi-parameter paraproducts. Rev. Mat. Iberoam., 22, 963–976 (2006) Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993 Yang, D., Yuan, W., Zhuo, C.: Fourier multipliers on Triebel-Lizorkin-type spaces. J. Funct. Spaces Appl., 2012, Article ID 431016, 37 pages (2012)