By Andryan A. A.

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F. Lawden, Elliptic Functions and Applications, Applied Math. Sciences 80, Springer-Verlag, Berlin-Heidelberg-New York, 1989. G. Mikhlin, Compounding of double singular integrals, Doklady Akad. Nauk. S S S R , 2, (1936), 3-6. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, New Jersey, 1970. M. Stein, Harmonic Analysis: Real- Variable Method, Orthogonality, and Oscillatory Integrals, Princeton University Press, New Jersey, 1993. R. Strichartz, L P harmonic analysis and radon transform on the Heisenberg group, J.

22. 23. 24. 25. C. H. Dolley and C. Ratcliff, Fundamental solutions for the powers of the Heisenberg sub-Laplacian, Illinois J . , 37, (1993), 455-476. C. C. Chang and J. Tie, Laguerre Calculus and Its Applications i n the Heisenberg Group, AMS/IP series in advanced mathematics #22, International Press, Cambridge, Massachusetts, 2001. C. Chang and P. Greiner, Harmonic analysis and subRiemannian geometry on Heisenberg groups, too appear in Bull. Academia Sinica, 2002. C. Chang and P. Greiner, Analysis and geometry on Heisenberg groups, too appear in Proceedings of the 2nd International Congress of Chinese Mathematicians, ed.

E 12, z E Cd. We would like t o point out that, unlike the finite dimensional space Cd associated with Lebesgue measure which is of translation invariance, l2 is not associated with any measure of translation invariance. 2. Hardy space of infinite complex variables Given a complex Hilbert space E. For each n = 1 , 2 , ... we write E@" for the Hilbert tensor product of n copies of E . Eo is defined as the one-dimensional vector space C with its usual inner product. For a fixed vector z E E we will use the notation zn = z@" for the n-fold tensor product of copies of z ( z o E Eo is defined as the complex number 1).

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