Rough singular integrals supported on submanifolds. (English) Zbl 1196.42016

Chen, Yanping; Ding, Yong; Liu, Honghai

J. Math. Anal. Appl. 368, No. 2, 677-691 (2010).

The authors show that the rough singular integral operator \(\mathcal T_{\Omega,\mathcal P}\) along polynomials \(\mathcal P = (p_1 , \dots , p_d)\) is bounded on the Triebel-Lizorkin spaces \(\dot{F}_\alpha^{p,q}(\mathbb R^d)\) and Besov spaces \(\dot{B}_\alpha^{p,q}(\mathbb R^d)\), where \(\mathcal T_{\Omega,\mathcal P}\) is defined by \[ \mathcal T_{\Omega,\mathcal P}f(x)=\text{ p.v. }\int_{\mathbb R^n}\frac{b(|y|)\Omega(y)}{|y|^n} f(x-\mathcal P(y))dy,\quad x\in\mathbb R^d, \] with \(\Omega\in H^1 (S^{n-1})\), \(\Omega(\lambda y) =\Omega (y)\) for \(\lambda > 0\), and \(\int_{S^{n-1}}\Omega (y') d\sigma(y') = 0\). Moreover, \(\mathcal P = (p_1 , \dots , p_d)\) is a mapping from \(\mathbb R^n\) into \(\mathbb R^d\) with \(p_j\) being polynomials in \(y\in\mathbb R^n\). The results obtained in the paper under review are extensions of D. Fan and Y. Pan’s result [Am. J. Math. 119, No. 4, 799–839 (1997; Zbl 0899.42002)].
To get the above results, the authors establish an \(l^q\)-valued \(L^p (\mathbb R^d)\)-boundedness of the maximal operator \[ \mathcal M_{\mathcal P}f(x):=\sup_{r>0}\frac{1}{r^n}\int_{\{y\in\mathbb R^n:|y|\leq r\}}|f(x-\mathcal P(y))|dy,\quad x\in\mathbb R^d, \] along polynomials \(\mathcal P = (p_1 , \dots, p_d)\), which was defined by E. M. Stein in 1986 [Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 196–221 (1987; Zbl 0718.42012)].

Reviewer: Chin-Cheng Lin (Chung-Li)

MSC:

42B25	Maximal functions, Littlewood-Paley theory
42B35	Function spaces arising in harmonic analysis

Keywords:

rough singular integrals; Triebel-Lizorkin spaces; maximal operators; vector-valued norm inequality

Citations:

Zbl 0899.42002; Zbl 0718.42012

References:

[1]	Chen, J.; Fan, D.; Ying, Y., Singular integral operators on function spaces, J. Math. Anal. Appl., 276, 691-708 (2002) · Zbl 1018.42009
[2]	Chen, J.; Zhang, C., Boundedness of rough singular integral on the Triebel-Lizorkin spaces, J. Math. Anal. Appl., 337, 1048-1052 (2008) · Zbl 1213.42034
[3]	Chen, Y.; Ding, Y., Rough singular integrals on Triebel-Lizorkin space and Besov space, J. Math. Anal. Appl., 347, 493-501 (2008) · Zbl 1257.42021
[5]	Cwikel, M.; Janson, S., Interpolation of analytic families of operators, Studia Math., 79, 61-71 (1984) · Zbl 0556.46041
[6]	Fan, D.; Pan, Y., Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math., 119, 799-839 (1997) · Zbl 0899.42002
[7]	Fefferman, C.; Stein, E. M., Some maximal inequalities, Amer. J. Math., 93, 107-115 (1971) · Zbl 0222.26019
[8]	Frazier, M.; Jawerth, B., A discrete transform and decompositions of distribution spaces, J. Funct. Anal., 93, 34-170 (1990) · Zbl 0716.46031
[9]	Frazier, M.; Jawerth, B.; Weiss, G., Littlewood-Paley Theory and the Study of Function Spaces, CBMS Reg. Conf. Ser., vol. 79 (1991), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0757.42006
[10]	Grafakos, L., Classical and Modern Fourier Analysis (2003), Prentice Hall: Prentice Hall Upper Saddle River, NJ
[11]	Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals (1993), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0821.42001
[13]	Stein, E. M.; Wainger, S., Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., 84, 1239-1295 (1978) · Zbl 0393.42010
[14]	Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0232.42007
[15]	Triebel, H., Theory of Function Spaces, Monogr. Math., vol. 78 (1983), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0546.46028
[16]	Triebel, H., Interpolation Theory, Function Spaces and Differential Operators (1995), Johann Ambrosius Barth Verlag: Johann Ambrosius Barth Verlag Heidelberg, Leipzig · Zbl 0830.46028