| Title:
|
Double weighted commutators theorem for pseudo-differential operators with smooth symbols (English) |
| Author:
|
Deng, Yu-long |
| Author:
|
Chen, Zhi-tian |
| Author:
|
Long, Shun-chao |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
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71 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
173-190 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $m\leq -(n+1)(1-\rho )$. (English) |
| Keyword:
|
pseudo-differential operator |
| Keyword:
|
reverse Hölder inequality |
| Keyword:
|
$A_p$ weight; commutator |
| MSC:
|
35S05 |
| MSC:
|
42B25 |
| MSC:
|
47G30 |
| idZBL:
|
07332711 |
| idMR:
|
MR4226476 |
| DOI:
|
10.21136/CMJ.2020.0246-19 |
| . |
| Date available:
|
2021-03-12T16:12:39Z |
| Last updated:
|
2023-04-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148734 |
| . |
| Reference:
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[1] Alvarez, J., Hounie, J.: Estimates for the kernel and continuity properties of pseudo-differential operators.Ark. Mat. 28 (1990), 1-22. Zbl 0713.35106, MR 1049640, 10.1007/BF02387364 |
| Reference:
|
[2] Auscher, P., Taylor, M. E.: Paradifferential operators and commutator estimates.Commun. Partial Differ. Equations 20 (1995), 1743-1775. Zbl 0844.35149, MR 1349230, 10.1080/03605309508821150 |
| Reference:
|
[3] Bloom, S.: A commutator theorem and weighted BMO.Trans. Am. Math. Soc. 292 (1985), 103-122. Zbl 0578.42012, MR 0805955, 10.1090/S0002-9947-1985-0805955-5 |
| Reference:
|
[4] Bui, T. A.: New weighted norm inequalities for pseudodifferential operators and their commutators.Int. J. Anal. 2013 (2013), Article ID 798528, 12 pages. Zbl 1268.47061, MR 3079558, 10.1155/2013/798528 |
| Reference:
|
[5] Calderón, A. P., Vaillancourt, R.: A class of bounded pseudo-differential operators.Proc. Nati. Acad. Sci. USA 69 (1972), 1185-1187. Zbl 0244.35074, MR 0298480, 10.1073/pnas.69.5.1185 |
| Reference:
|
[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. Zbl 0898.47039, MR 1447213, 10.1090/conm/205/02651 |
| Reference:
|
[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. Zbl 0609.35085, MR 0852824, 10.1007/BF02384387 |
| Reference:
|
[8] Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables.Ann. Math. (2) 103 (1976), 611-635. Zbl 0326.32011, MR 0412721, 10.2307/1970954 |
| Reference:
|
[9] Fefferman, C.: $L^p$ bounds for pseudo-differential operators.Isr. J. Math. 14 (1973), 413-417. Zbl 0259.47045, MR 0336453, 10.1007/BF02764718 |
| Reference:
|
[10] Fefferman, C., Stein, E. M.: $H^p$ spaces of several variables.Acta Math. 129 (1972), 137-193. Zbl 0257.46078, MR 0447953, 10.1007/BF02392215 |
| Reference:
|
[11] Grafakos, L.: Classical Fourier Analysis.Graduate Texts in Mathematics 249. Springer, New York (2014). Zbl 1304.42001, MR 3243734, 10.1007/978-1-4939-1194-3 |
| Reference:
|
[12] Holmes, I., Lacey, M. T., Wick, B. D.: Commutators in the two-weight setting.Math. Ann. 367 (2017), 51-80. Zbl 1364.42017, MR 3606434, 10.1007/s00208-016-1378-1 |
| Reference:
|
[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. Zbl 0167.09603, MR 0383152, 10.1090/pspum/010/0383152 |
| Reference:
|
[14] Hounie, J., Kapp, R. A. S.: Pseudodifferential operators on local Hardy spaces.J. Fourier Anal. Appl. 15 (2009), 153-178. Zbl 1178.35396, MR 2500920, 10.1007/s00041-008-9021-5 |
| Reference:
|
[15] Hung, H. D., Ky, L. D.: An Hardy estimate for commutators of pseudo-differential operators.Taiwanese J. Math. 19 (2015), 1097-1109. Zbl 1357.47051, MR 3384681, 10.11650/tjm.19.2015.5003 |
| Reference:
|
[16] Kohn, J. J., Nirenberg, L.: An algebra of pseudo-differential operators.Commun. Pure Appl. Math. 18 (1965), 269-305. Zbl 0171.35101, MR 0176362, 10.1002/cpa.3160180121 |
| Reference:
|
[17] Laptev, A. A.: Spectral asymptotics of a certain class of Fourier integral operators.Tr. Mosk. Mat. O.-va 43 (1981), 92-115 Russian. Zbl 0503.47044, MR 0651330 |
| Reference:
|
[18] Lerner, A. K.: On weighted estimates of non-increasing rearrangements.East J. Approx. 4 (1998), 277-290. Zbl 0947.42012, MR 1638347 |
| Reference:
|
[19] Lin, Y.: Commutators of pseudo-differential operators.Sci. China, Ser. A 51 (2008), 453-460. Zbl 1213.42047, MR 2395439, 10.1007/s11425-008-0035-x |
| Reference:
|
[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. Zbl 1195.47033, MR 2609542, 10.1016/j.jfa.2010.03.013 |
| Reference:
|
[21] Michalowski, N., Rule, D. J., Staubach, W.: Weighted $L^p$ boundedness of pseudodifferential operators and applications.Can. Math. Bull. 55 (2012), 555-570. Zbl 1252.42019, MR 2957271, 10.4153/CMB-2011-122-7 |
| Reference:
|
[22] Miller, N.: Weighted Sobolev spaces and pseudodifferential operators with smooth symbols.Trans. Am. Math. Soc. 269 (1982), 91-109. Zbl 0482.35082, MR 0637030, 10.1090/S0002-9947-1982-0637030-4 |
| Reference:
|
[23] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function.Trans. Am. Math. Soc. 165 (1972), 207-226. Zbl 0236.26016, MR 0293384, 10.1090/S0002-9947-1972-0293384-6 |
| Reference:
|
[24] Muckenhoupt, B.: The equivalence of two conditions for weight functions.Studia Math. 49 (1974), 101-106. Zbl 0243.44003, MR 0350297, 10.4064/sm-49-2-101-106 |
| Reference:
|
[25] Nishigaki, S.: Weighted norm inequalities for certain pseudo-differential operators.Tokyo J. Math. 7 (1984), 129-140. Zbl 0555.35133, MR 0752114, 10.3836/tjm/1270152995 |
| Reference:
|
[26] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.Princeton Mathematical Series 43. Princeton University Press, Princeton (1993). Zbl 0821.42001, MR 1232192, 10.1515/9781400883929 |
| Reference:
|
[27] Tang, L.: Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators.J. Funct. Anal. 262 (2012), 1603-1629. Zbl 1248.47048, MR 2873852, 10.1016/j.jfa.2011.11.016 |
| Reference:
|
[28] Yabuta, K.: Weighted norm inequalities for pseudo-differential operators.Osaka J. Math. 23 (1986), 703-723. Zbl 0632.35079, MR 0866272, 10.18910/7950 |
| Reference:
|
[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. Zbl 1313.42080, MR 3174086, 10.1016/S0252-9602(14)60013-8 |
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