Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
fractional operator; singular integral operator; commutator; weight
fractional operator; singular integral operator; commutator; weight
Summary:
We prove two-weighted norm estimates for higher order commutator of singular integral and fractional type operators between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of nontrivial weights in the optimal region satisfying the conditions required.
References:
[1] Bramanti, M., Cerutti, M. C.: $W_p^{1,2}$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients. Commun. Partial Differ. Equations 18 (1993), 1735-1763. DOI 10.1080/03605309308820991 | MR 1239929 | Zbl 0816.35045
[2] Bramanti, M., Cerutti, M. C.: Commutators of singular integrals and fractional integrals on homogeneous spaces. Harmonic Analysis and Operator Theory Contemporary Mathematics 189. AMS, Providence (1995), 81-94. DOI 10.1090/conm/189 | MR 1347007 | Zbl 0837.43013
[3] Bramanti, M., Cerutti, M. C.: Commutators of singular integrals on homogeneous spaces. Boll. Unione Mat. Ital., VII. Ser., B 10 (1996), 843-883. MR 1430157 | Zbl 0913.42013
[4] Bramanti, M., Cerutti, M. C., Manfredini, M.: $L^p$ estimates for some ultraparabolic operators with discontinuous coefficients. J. Math. Anal. Appl. 200 (1996), 332-354. DOI 10.1006/jmaa.1996.0209 | MR 1391154 | Zbl 0922.47039
[5] Chiarenza, F., Frasca, M., Longo, P.: Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients. Ric. Mat. 40 (1991), 149-168. MR 1191890 | Zbl 0772.35017
[6] Chiarenza, F., Frasca, M., Longo, P.: $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336 (1993), 841-853. DOI 10.1090/S0002-9947-1993-1088476-1 | MR 1088476 | Zbl 0818.35023
[7] Harboure, E., Salinas, O., Viviani, B.: Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces. Trans. Am. Math. Soc. 349 (1997), 235-255. DOI 10.1090/S0002-9947-97-01644-9 | MR 1357395 | Zbl 0865.42017
[8] Morvidone, M.: Weighted $\rm BMO_\phi$ spaces and the Hilbert transform. Rev. Unión Mat. Argent. 44 (2003), 1-16. MR 2051034 | Zbl 1075.42006
[9] Muckenhoupt, B., Wheeden, R. L.: Weighted bounded mean oscillation and the Hilbert transform. Stud. Math. 54 (1976), 221-237. DOI 10.4064/sm-54-3-221-237 | MR 0399741 | Zbl 0318.26014
[10] Pradolini, G.: A class of pairs of weights related to the boundedness of the fractional integral operator between $L^p$ and Lipschitz spaces. Commentat. Math. Univ. Carol. 42 (2001), 133-152. MR 1825378 | Zbl 1055.42015
[11] Pradolini, G.: Two-weighted norm inequalities for the fractional integral operator between $L^p$ and Lipschitz spaces. Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 41 (2001), 147-169. MR 1876717 | Zbl 0997.42010
[12] Pradolini, G., Ramos, W., Recchi, J.: On the optimal numerical parameters related with two weighted estimates for commutators of classical operators and extrapolation results. Collect. Math. 72 (2021), 229-259. DOI 10.1007/s13348-020-00287-1 | MR 4204587 | Zbl 1457.42029
[13] Rios, C.: The $L^p$ Dirichlet problem and nondivergence harmonic measure. Trans. Am. Math. Soc. 355 (2003), 665-687. DOI 10.1090/S0002-9947-02-03145-8 | MR 1932720 | Zbl 1081.35021
Search
Partner of
