Hardy Space Estimate for the Product of Singular Integrals

Miyachi, Akihiko

Canadian journal of mathematics, Tome 52 (2000) no. 2, pp. 381-411

Résumé

${{H}^{p}}$ estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given.

Détail

DOI : 10.4153/CJM-2000-018-2

Mots-clés : 42B30, Hp space, multilinear operator, singular integral, fractional integration, Sobolev space

Bibliographie

[BGS] [BGS] Burkholder, D. L., Gundy, R. F. and Silverstein, M. L., A maximal function characterization of the class Hp. Trans. Amer. Math. Soc. 157(1971), 137–153. Google Scholar

[CT] [CT] Calderón, A. P. and Torchinsky, A., Parabolic maximal functions associated with a distribution, II. Adv. in Math. 24(1977), 101–171. Google Scholar

[Ch] [Ch] Chanillo, S., A note on commutators. Indiana Univ. Math. J. 31(1982), 7–16. Google Scholar

[CG] [CG] Coifman, R. R. and Grafakos, L., Hardy space estimates for multilinear operators, I. Rev.Mat. Iberoamericana 8(1992), 45–67. Google Scholar

[CLMS] [CLMS] Coifman, R., Lions, P. L., Meyer, Y. and Semmes, S., Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72(1993), 247–286. Google Scholar

[CRW] [CRW] Coifman, R. R., Rochberg, R., and Weiss, G., Factorization theorems for Hardy spaces in several variables. Ann. of Math. 103(1976), 611–635. Google Scholar

[F] [F] Franke, J., On the spaces Fs p, q of Triebel-Lizorkin type: Pointwise multipliers and spaces on domains. Math. Nachr. 125(1986), 29–68. Google Scholar

[G] [G] Grafakos, L., Hardy space estimates for multilinear operators, II. Rev. Mat. Iberoamericana 8(1992), 69–92. Google Scholar

[K1] [K1] Komori, Y., The factorization of Hp and the commutators. Tokyo J. Math. 6(1983), 435–445. Google Scholar

[K2] [K2] Komori, Y., Hp estimate for multilinear singular integrals. (Japanese) Real Analysis Seminar 1994, Proceedings of a symposium held at Okayama Prefectural University, Okayama, Japan, 1994, 58–59. Google Scholar

[M1] [M1] Miyachi, A., Products of distributions in Hp spaces. Tôhoku Math. J. 35(1983), 483–498. Google Scholar

[M2] [M2] Miyachi, A., A factorization theorem for the real Hardy spaces. Proc. Analysis Conference (Singapore 1986), North-Holland Math. Stud. 150(1988), 167–175. Google Scholar

[M3] [M3] Miyachi, A., Multiplication and factorization of functions in Sobolev spaces and in Cαp spaces on general domains. Math. Nachr. 176(1995), 209–242. Google Scholar

[M4] [M4] Miyachi, A., Atomic decomposition for Sobolev spaces and for the Cpα spaces on the general domains. Tsukuba J. Math. 21(1997), 59–96. Google Scholar

[SiT] [SiT] Sickel, W. and Triebel, H., Hölder inequalities and sharp embeddings in function spaces of Bspq and Fspq type. Preprint. Google Scholar

[S] [S] Stein, E. M., Harmonic Analysis: Real-VariableMethods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, NJ, 1993. Google Scholar

[StT] [StT] Strömberg, J-O. and Torchinsky, A., Weighted Hardy Spaces. Lecture Notes in Math. 1381, Springer-Verlag, 1989. Google Scholar

[U] [U] Uchiyama, A., The factorization of Hp on the space of homogeneous type. Pacific. J. Math. 92(1981), 453–468. Google Scholar