Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
pseudo-differential operator; reverse Hölder inequality; $A_p$ weight; commutator
pseudo-differential operator; reverse Hölder inequality; $A_p$ weight; commutator
Summary:
Let $-(n+1)<m\leq -(n+1)(1-\rho )$ and let $T_{a}\in \mathcal {L}^{m}_{\rho ,\delta }$ be pseudo-differential operators with symbols $a(x,\xi )\in \mathbb {R}^n\times \mathbb {R}^n$, where $0<\rho \leq 1$, $0\leq \delta <1$ and $\delta \leq \rho $. Let $\mu $, $\lambda $ be weights in Muckenhoupt classes $A_{p}$, $\nu =(\mu \lambda ^{-1})^{1/p}$ for some $1<p<\infty $. We establish a two-weight inequality for commutators generated by pseudo-differential operators $T_{a}$ with weighted BMO functions $b\in {\rm BMO}_{\nu }$, namely, the commutator $[b,T_{a}]$ is bounded from $L^{p}(\mu )$ into $L^{p}(\lambda )$. Furthermore, the range of $m$ can be extended to the whole $m\leq -(n+1)(1-\rho )$.
References:
[1] Alvarez, J., Hounie, J.: Estimates for the kernel and continuity properties of pseudo-differential operators. Ark. Mat. 28 (1990), 1-22. DOI 10.1007/BF02387364 | MR 1049640 | Zbl 0713.35106
[2] Auscher, P., Taylor, M. E.: Paradifferential operators and commutator estimates. Commun. Partial Differ. Equations 20 (1995), 1743-1775. DOI 10.1080/03605309508821150 | MR 1349230 | Zbl 0844.35149
[3] Bloom, S.: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 292 (1985), 103-122. DOI 10.1090/S0002-9947-1985-0805955-5 | MR 0805955 | Zbl 0578.42012
[4] Bui, T. A.: New weighted norm inequalities for pseudodifferential operators and their commutators. Int. J. Anal. 2013 (2013), Article ID 798528, 12 pages. DOI 10.1155/2013/798528 | MR 3079558 | Zbl 1268.47061
[5] Calderón, A. P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nati. Acad. Sci. USA 69 (1972), 1185-1187. DOI 10.1073/pnas.69.5.1185 | MR 0298480 | Zbl 0244.35074
[6] Chanillo, S.: Remarks on commutators of pseudo-differential operators. Multidimensional Complex Analysis and Partial Differential Equations Contemporary Mathematics 205. American Mathematical Society, Providence (1997), 33-37. DOI 10.1090/conm/205/02651 | MR 1447213 | Zbl 0898.47039
[7] Chanillo, S., Torchinsky, A.: Sharp function and weighted $L^p$ estimates for a class of pseudo-differential operators. Ark. Mat. 24 (1986), 1-25. DOI 10.1007/BF02384387 | MR 0852824 | Zbl 0609.35085
[8] Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103 (1976), 611-635. DOI 10.2307/1970954 | MR 0412721 | Zbl 0326.32011
[9] Fefferman, C.: $L^p$ bounds for pseudo-differential operators. Isr. J. Math. 14 (1973), 413-417. DOI 10.1007/BF02764718 | MR 0336453 | Zbl 0259.47045
[10] Fefferman, C., Stein, E. M.: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193. DOI 10.1007/BF02392215 | MR 0447953 | Zbl 0257.46078
[11] Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics 249. Springer, New York (2014). DOI 10.1007/978-1-4939-1194-3 | MR 3243734 | Zbl 1304.42001
[12] Holmes, I., Lacey, M. T., Wick, B. D.: Commutators in the two-weight setting. Math. Ann. 367 (2017), 51-80. DOI 10.1007/s00208-016-1378-1 | MR 3606434 | Zbl 1364.42017
[13] Hörmander, L.: Pseudo-differential operators and hypoelliptic equations. Singular Integrals Proceedings of Symposia in Pure Mathematics 10. American Mathematical Society, Providence (1967), 138-183. DOI 10.1090/pspum/010/0383152 | MR 0383152 | Zbl 0167.09603
[14] Hounie, J., Kapp, R. A. S.: Pseudodifferential operators on local Hardy spaces. J. Fourier Anal. Appl. 15 (2009), 153-178. DOI 10.1007/s00041-008-9021-5 | MR 2500920 | Zbl 1178.35396
[15] Hung, H. D., Ky, L. D.: An Hardy estimate for commutators of pseudo-differential operators. Taiwanese J. Math. 19 (2015), 1097-1109. DOI 10.11650/tjm.19.2015.5003 | MR 3384681 | Zbl 1357.47051
[16] Kohn, J. J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18 (1965), 269-305. DOI 10.1002/cpa.3160180121 | MR 0176362 | Zbl 0171.35101
[17] Laptev, A. A.: Spectral asymptotics of a certain class of Fourier integral operators. Tr. Mosk. Mat. O.-va 43 (1981), 92-115 Russian. MR 0651330 | Zbl 0503.47044
[18] Lerner, A. K.: On weighted estimates of non-increasing rearrangements. East J. Approx. 4 (1998), 277-290. MR 1638347 | Zbl 0947.42012
[19] Lin, Y.: Commutators of pseudo-differential operators. Sci. China, Ser. A 51 (2008), 453-460. DOI 10.1007/s11425-008-0035-x | MR 2395439 | Zbl 1213.42047
[20] Michalowski, N., Rule, D. J., Staubach, W.: Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes. J. Funct. Anal. 258 (2010), 4183-4209. DOI 10.1016/j.jfa.2010.03.013 | MR 2609542 | Zbl 1195.47033
[21] Michalowski, N., Rule, D. J., Staubach, W.: Weighted $L^p$ boundedness of pseudodifferential operators and applications. Can. Math. Bull. 55 (2012), 555-570. DOI 10.4153/CMB-2011-122-7 | MR 2957271 | Zbl 1252.42019
[22] Miller, N.: Weighted Sobolev spaces and pseudodifferential operators with smooth symbols. Trans. Am. Math. Soc. 269 (1982), 91-109. DOI 10.1090/S0002-9947-1982-0637030-4 | MR 0637030 | Zbl 0482.35082
[23] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165 (1972), 207-226. DOI 10.1090/S0002-9947-1972-0293384-6 | MR 0293384 | Zbl 0236.26016
[24] Muckenhoupt, B.: The equivalence of two conditions for weight functions. Studia Math. 49 (1974), 101-106. DOI 10.4064/sm-49-2-101-106 | MR 0350297 | Zbl 0243.44003
[25] Nishigaki, S.: Weighted norm inequalities for certain pseudo-differential operators. Tokyo J. Math. 7 (1984), 129-140. DOI 10.3836/tjm/1270152995 | MR 0752114 | Zbl 0555.35133
[26] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43. Princeton University Press, Princeton (1993). DOI 10.1515/9781400883929 | MR 1232192 | Zbl 0821.42001
[27] Tang, L.: Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators. J. Funct. Anal. 262 (2012), 1603-1629. DOI 10.1016/j.jfa.2011.11.016 | MR 2873852 | Zbl 1248.47048
[28] Yabuta, K.: Weighted norm inequalities for pseudo-differential operators. Osaka J. Math. 23 (1986), 703-723. DOI 10.18910/7950 | MR 0866272 | Zbl 0632.35079
[29] Yang, J., Wang, Y., Chen, W.: Endpoint estimates for the commutators of pseudo-differential operators. Acta Math. Sci., Ser. B, Engl. Ed. 34 (2014), 387-393. DOI 10.1016/S0252-9602(14)60013-8 | MR 3174086 | Zbl 1313.42080
Search
Partner of
