Certain Operators with Rough Singular Kernels

Chen, Jiecheng ; Fan, Dashan ; Ying, Yiming

Canadian journal of mathematics, Tome 55 (2003) no. 3, pp. 504-532

Résumé

We study the singular integral operator $${{T}_{\Omega ,\alpha }}f\left( x \right)\,=\,\text{p}\text{.v}\text{.}\,{{\int }_{{{R}^{n}}}}\,b\left( \left| y \right| \right)\Omega \left( {{y}'} \right){{\left| y \right|}^{-n-\alpha }}\,f\left( x\,-\,y \right)\,dy,$$ defined on all test functions $f$ , where $b$ is a bounded function, $\alpha \ge 0,\,\Omega \left( {{y}'} \right)$ is an integrable function on the unit sphere ${{S}^{n-1}}$ satisfying certain cancellation conditions. We prove that, for $1\,<\,p\,<\infty$ , ${{T}_{\Omega ,\alpha }}$ extends to a bounded operator from the Sobolev space $L_{\alpha }^{p}$ to the Lebesgue space ${{L}^{p}}$ with $\Omega$ being a distribution in the Hardy space ${{H}^{q}}\left( {{S}^{n-1}} \right)$ where $q=\frac{n-1}{n-1+\alpha }$ . The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for ${{T}_{\Omega ,\alpha }}$ on the Hardy spaces, as well as the boundedness for the truncated maximal operator $T_{\Omega ,m}^{*}$ .

Détail

DOI : 10.4153/CJM-2003-021-4

Mots-clés : 42B20, 42B25, 42B15

Bibliographie

[BF] [BF] Blank, B. and Fan, D. Hardy spaces on compact Lie groups. Ann. Fac. Sci. Toulouse Math. 6(1997), 429–479. Google Scholar

[BL] [BL] Bergh, J. and Lofstrom, J., Interpolation Spaces, An introduction. Grundlehren der MathematischenWissenschafen 233, Spring-Verlag, Berlin, Heidelberg, New York, 1976. Google Scholar

[CDF] [CDF] Chen, J., Ding, Y. and Fan, D. A class of integral operators with variable kernels in Hardy spaces. Chinese Ann.Math. Ser. A 23(2002), 289–296. Google Scholar

[CF] [CF] Chen, J. and Fan, D., Maximal singular integrals on Sobolev spaces. Preprint. Google Scholar

[Ch] [Ch] Chen, L. On a singular integrals. Studia Math. TLXXXV(1987), 61–72. Google Scholar

[Co1] [Co1] Colzani, L., Hardy spaces on sphere. Ph.D. Thesis, Washington University, St Louis, Missouri, 1982. Google Scholar

[Co2] [Co2] Colzani, L. Hardy spaces on Unit Sphere. Boll. Un. Mat. Ital. C (6) Anal. Funz. Appl. IV–C(1985), 219–244. Google Scholar

[CTW] [CTW] Colzani, L., Taibleson, M. and G.Weiss Maximal estimates for Cesaro and Riesz means on sphere. Indiana Univ.Math. J. (6) 33(1984), 873–889. Google Scholar

[CZ1] [CZ1] Calderon, A. P. and Zygmund, A. On existence of certain singular integrals. Acta. Math. 88(1952), 85–139. Google Scholar

[CZ2] [CZ2] Calderon, A. P. and Zygmund, A. On singular integrals. Amer. J. Math. 18(1956), 289–309. Google Scholar

[CZ3] [CZ3] Calderon, A. P. and Zygmund, A. On singular integrals with variable kernels. Apl. Anal. 7(1978), 221–238. Google Scholar

[DL] [DL] Ding, Y. and Lu, S. Homogeneous fractional integrals on Hardy spaces. Tohoku Math. J. 52(2000), 153–162. Google Scholar

[DR] [DR] Duoandikoetxea, J. and Rubio, J. L. de Francia Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84(1986), 541–561. Google Scholar

[Fe] [Fe] Feferman, R. A note on singular integrals. Proc. Amer.Math. Soc. 74(1979), 266–270. Google Scholar

[FJW] [FJW] Frazier, M., Jawerth, B. and Weiss, G., Littlewood-Paley Theory and the Study of Function Spaces. AMS-CBMS Regional Conf. Ser., Conf. Board Math. Sci., Washington, D.C. Google Scholar

[FP1] [FP1] Fan, D. and Pan, Y. Singular integral operators with rough kernels supported by subvarieties. Amer. J. Math. 119(1997), 799–839. Google Scholar

[FP2] [FP2] Fan, D. and Pan, Y. L2 boundedness of a singular integral operators. Publ. Math. 41(1997), 317–333. Google Scholar

[FP3] [FP3] Fan, D. and Pan, Y. A singular integral operator with rough kernel. Proc. Amer.Math. Soc. 125(1997), 3695–3703. Google Scholar

[GS] [GS] Grafakos, L. and Stefanov, , Convolution Calderon-Zygmund singular integral operators with rough kernels. Analysis of divergence Orono, ME, 1997, 119-143, Appl. Numer. Anal., Birkhauser Boston, Massachusetts, 1999. Google Scholar

[HPW] [HPW] Han, S. Y., Paluszynski, M. and Weiss, G., A new atomic decomposition for the Triebel-Lizorkin spaces. Harmonic Analysis and operator Theory, Caracas, 1994, 235-249, Contemp. Math. , Amer. Math. Soc., Providence, Rhode Island, 1995. Google Scholar

[KW] [KW] Kurtz, D. and Wheeden, R. L. Results on weighted norm inequalities for multipliers. Trans. Amer. Math. Soc. 255(1979), 343–362. Google Scholar

[MW] [MW] Muckenhoupt, B. and Wheeden, R. L. Weighted norm inequalities for singular and fractional integrals. Trans. Amer. Math. Soc. 161(1971), 249–258. Google Scholar

[Na] [Na] Namazi, J., A singular integral. Ph. D. Thesis, Indiana University, Bloomington, 1984. Google Scholar

[RW] [RW] Ricci, F. and Weiss, G., A characterization of H1(∑). Proc. Sympos. Pure Math. , (eds., S. Wainger and G. Weiss), Amer. Math. Soc. Providence, Rhode Island, 289–294. Google Scholar

[S] [S] Sonine, Math. Ann. (1887), 157-161. Google Scholar

[Str] [Str] Strichartz, R. S. HP Sobolev spaces. Colloq. Math. LX/LXI(1990), 129–139. Google Scholar

[St] [St] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton, New Jersey, 1993. Google Scholar

[SW] [SW] Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton, New Jersey, 1971. Google Scholar

[T] [T] Torres, R., On the boundedness of certain operators with singular kernels on distribution spaces. Mem. Amer. Math. Soc. 442(1992), Amer.Math. Soc. Google Scholar

[Tr] [Tr] Triebel, H., Interpolation Theory, Function Spaces and Differential Operators, 2nd edition. Johann Ambrosius Barth Verlag, Heidlberg, Leipzig, 1995. Google Scholar