Geodesic
Double weighted commutators theorem for pseudo-differential operators with smooth symbols
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 1, pp. 173-190
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $-(n+1)
Let $-(n+1)$ 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 $1$. 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 $m\leq -(n+1)(1-\rho )$.
DOI : 10.21136/CMJ.2020.0246-19
Classification : 35S05, 42B25, 47G30
Keywords: pseudo-differential operator; reverse Hölder inequality; $A_p$ weight; commutator 
@article{10_21136_CMJ_2020_0246_19,
     author = {Deng, Yu-long and Chen, Zhi-tian and Long, Shun-chao},
     title = {Double weighted commutators theorem for pseudo-differential operators with smooth symbols},
     journal = {Czechoslovak Mathematical Journal},
     pages = {173--190},
     year = {2021},
     volume = {71},
     number = {1},
     doi = {10.21136/CMJ.2020.0246-19},
     mrnumber = {4226476},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0246-19/}
}
TY  - JOUR
AU  - Deng, Yu-long
AU  - Chen, Zhi-tian
AU  - Long, Shun-chao
TI  - Double weighted commutators theorem for pseudo-differential operators with smooth symbols
JO  - Czechoslovak Mathematical Journal
PY  - 2021
SP  - 173
EP  - 190
VL  - 71
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0246-19/
DO  - 10.21136/CMJ.2020.0246-19
LA  - en
ID  - 10_21136_CMJ_2020_0246_19
ER  - 
%0 Journal Article
%A Deng, Yu-long
%A Chen, Zhi-tian
%A Long, Shun-chao
%T Double weighted commutators theorem for pseudo-differential operators with smooth symbols
%J Czechoslovak Mathematical Journal
%D 2021
%P 173-190
%V 71
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0246-19/
%R 10.21136/CMJ.2020.0246-19
%G en
%F 10_21136_CMJ_2020_0246_19
Deng, Yu-long; Chen, Zhi-tian; Long, Shun-chao. Double weighted commutators theorem for pseudo-differential operators with smooth symbols. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 1, pp. 173-190. doi: 10.21136/CMJ.2020.0246-19

[1] Alvarez, J., Hounie, J.: Estimates for the kernel and continuity properties of pseudo-differential operators. Ark. Mat. 28 (1990), 1-22. | DOI | MR | JFM

[2] Auscher, P., Taylor, M. E.: Paradifferential operators and commutator estimates. Commun. Partial Differ. Equations 20 (1995), 1743-1775. | DOI | MR | JFM

[3] Bloom, S.: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 292 (1985), 103-122. | DOI | MR | JFM

[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 | MR | JFM

[5] Calderón, A. P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nati. Acad. Sci. USA 69 (1972), 1185-1187. | DOI | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[8] Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103 (1976), 611-635. | DOI | MR | JFM

[9] Fefferman, C.: $L^p$ bounds for pseudo-differential operators. Isr. J. Math. 14 (1973), 413-417. | DOI | MR | JFM

[10] Fefferman, C., Stein, E. M.: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193. | DOI | MR | JFM

[11] Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics 249. Springer, New York (2014). | DOI | MR | JFM

[12] Holmes, I., Lacey, M. T., Wick, B. D.: Commutators in the two-weight setting. Math. Ann. 367 (2017), 51-80. | DOI | MR | JFM

[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 | MR | JFM

[14] Hounie, J., Kapp, R. A. S.: Pseudodifferential operators on local Hardy spaces. J. Fourier Anal. Appl. 15 (2009), 153-178. | DOI | MR | JFM

[15] Hung, H. D., Ky, L. D.: An Hardy estimate for commutators of pseudo-differential operators. Taiwanese J. Math. 19 (2015), 1097-1109. | DOI | MR | JFM

[16] Kohn, J. J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18 (1965), 269-305. | DOI | MR | JFM

[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 | JFM

[18] Lerner, A. K.: On weighted estimates of non-increasing rearrangements. East J. Approx. 4 (1998), 277-290. | MR | JFM

[19] Lin, Y.: Commutators of pseudo-differential operators. Sci. China, Ser. A 51 (2008), 453-460. | DOI | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[22] Miller, N.: Weighted Sobolev spaces and pseudodifferential operators with smooth symbols. Trans. Am. Math. Soc. 269 (1982), 91-109. | DOI | MR | JFM

[23] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165 (1972), 207-226. | DOI | MR | JFM

[24] Muckenhoupt, B.: The equivalence of two conditions for weight functions. Studia Math. 49 (1974), 101-106. | DOI | MR | JFM

[25] Nishigaki, S.: Weighted norm inequalities for certain pseudo-differential operators. Tokyo J. Math. 7 (1984), 129-140. | DOI | MR | JFM

[26] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43. Princeton University Press, Princeton (1993). | DOI | MR | JFM

[27] Tang, L.: Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators. J. Funct. Anal. 262 (2012), 1603-1629. | DOI | MR | JFM

[28] Yabuta, K.: Weighted norm inequalities for pseudo-differential operators. Osaka J. Math. 23 (1986), 703-723. | DOI | MR | JFM

[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 | MR | JFM

Cité par Sources :