On the product theory of singular integrals.(English)Zbl 1057.42016

Rev. Mat. Iberoam. 20, No. 2, 531-561 (2004); corrigenda ibid. 21, No. 2, 693-694 (2005).
Let $$1<p<\infty$$. In this nice paper, the authors establish $$L^p$$-boundedness for a class of product singular integral operators on product spaces $$\widetilde{M}=M_1\times M_2\times\cdots \times M_n$$. Each factor space $$M_i$$ is a smooth manifold on which the basic geometry is given by a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. In particular, the authors focus their attention on the following two specific settings:
(A) Here $$M_i$$ is a compact connected $$C^\infty$$-manifold. Suppose that $$k$$ smooth real vector fields $$\{X_1,\,\cdots,\,X_k\}$$ on $$M_i$$ are given, which are of finite-type $$m$$ in the sense that these vector fields together with their commutators of order $$\leq m$$ span the tangent space to $$M_i$$ at each point.
(B) Here $$M_i$$ arises as the boundary of an unbounded model polynomial domain in $${\mathbb C}^2$$. Then, let $$\Omega=\{(z,\,w)\in{\mathbb C}^2| \,\text{ Im}[w]>P(z)\}$$, where $$P$$ is a real, subharmonic, non-harmonic polynomial of degree $$m$$. Then $$M_i=\partial\Omega$$ can be identified with $${\mathbb C}\times {\mathbb R}=\{(z,\,t):\;z\in{\mathbb C},\,t\in{\mathbb R}\}$$. The basic $$(0,1)$$ Levi vector field is then $$\bar{Z}=\frac{\partial}{\partial\bar{z}}-i\frac{\partial P}{\partial\bar{z}}\frac{\partial}{\partial t}$$, and let $$\bar{Z}=X_1+iX_2$$. The real vector fields $$\{X_1,\,X_2\}$$ and their commutators of orders $$\leq m$$ span the tangent space at each point. Thus, this $$M_i$$ is a special non-compact variant, with $$k=2$$, of the manifolds considered in (A).
The standard singular integrals on $$M_i$$ are non-isotropic smoothing operators of order zero. The authors prove that the boundedness of the product operators is then a consequence of a natural Littlewood-Paley theory on $$\widetilde{M}$$, which in turn is a consequence of a corresponding theory on each factor space. The square function for this theory is constructed from the heat kernel for the sub-Laplacian on each factor. The $$L^p$$ theory of product singular integrals established in this paper can be used in a number of different situations, and in particular for estimates of fundamental solutions of $$\square_b$$ on certain model domains in several complex variables.

MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 43A99 Abstract harmonic analysis
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References:

 [1] Chang, D.-C., Nagel, A. and Stein, E. M.: Estimates for the \? \partial -Neumann problem in pseudoconvex domains of finite type in C2. Acta Math. 169 (1992), 153-228. · Zbl 0821.32011 · doi:10.1007/BF02392760 [2] David, G. and Journé, J.-L.: A boundedness criterion for general- ized Calderón-Zygmund operators. Ann. of Math.(2) 120 (1984), 371-397. · Zbl 0567.47025 · doi:10.2307/2006946 [3] Fefferman, R. and Stein, E.M.: Singular integrals on product spaces. Adv. in Math. 45 (1982), 117-143. · Zbl 0517.42024 · doi:10.1016/S0001-8708(82)80001-7 [4] Jessen, B., Marcinkiewicz, J. and Zygmund, A.: Note on the differentiability of multiple integrals. Fund. Math. 25 (1935), 217-234. · Zbl 0012.05901 [5] Journé, J.L.: Calderón-Zygmund operators on product spaces. Rev. Mat. Iberoamericana 1 (1985), 55-91. · Zbl 0634.42015 · doi:10.4171/RMI/15 [6] Koenig, K.: On maximal Sobolev and Hölder estimates for the tan- gential Cauchy-Riemann operator and boundary Laplacian. Amer. J. Math. 124 (2002), 129-197. · Zbl 1014.32031 · doi:10.1353/ajm.2002.0003 [7] Müller, D., Ricci, F. and Stein, E.M.: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, I. Invent. Math. 119 (1995), 119-233. · Zbl 0857.43012 · doi:10.1007/BF01245180 [8] Nagel, A., Ricci, F. and Stein, E.M.: Singular integrals with flag kernels and analysis on quadratic CR manifolds. J. Funct. Anal. 181 (2001), 29-118. · Zbl 0974.22007 · doi:10.1006/jfan.2000.3714 [9] Nagel, A., Rosay, J.-P., Stein, E.M. and Wainger, S.: Esti- mates for the Bergman and Szegö kernels in C2. Ann. of Math. 129 (1989), 113-149. 561 · Zbl 0667.32016 · doi:10.2307/1971487 [10] Nagel, A. and Stein, E.M.: The b-Heat equation on pseudoconvex manifolds of finite type in C2. Math. Z. 238 (2001), 37-88. · Zbl 1039.32051 · doi:10.1007/s002090000245 [11] Nagel, A. and Stein, E.M.: Differentiable control metrics and scaled bump functions. J. Differential Geom. 57 (2001), 465-492. · Zbl 1041.58006 [12] Nagel, A. and Stein, E.M.: The \? \partial b-complex on decoupled bound- aries in Cn. Preprint. [13] Nagel, A., Stein, E.M. and Wainger. S.: Balls and metrics defined by vector fields I. Basic properties. Acta Math. 155 (1985), 103-147. · Zbl 0578.32044 · doi:10.1007/BF02392539 [14] Stein, E.M.: Topics in harmonic analysis related to the Littlewood- Paley theory. Annals of Mathematics Studies 63. Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.10502 [15] Stein, E.M.: Harmonic analysis: real-variable methods, orthogonal- ity, and oscillatory integrals. Princeton Mathematical Series 43. Mono- graphs in Harmonic Analysis, III. Princeton University Press, Prince- ton, NJ, 1993 · Zbl 0821.42001
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