Operator theory in harmonic analysis
The purpose of this paper is to give an illustration of how operator theoretic results in Hilbert space can be applied to obtain results in classical and abstract harmonic analysis. An inequality for integral operators will be used to give new proofs for the classical Hausdorff-Young Theorems on the unit circle (Fourier coefficients) and on the real line (Fourier integral). These proofs were discovered in the course of the author's investigations into Hausdorff-Young phenomena on non-commutative non-compact groups. The proof for the unit circle is short and will be given in §1. The proof for the real line is more involved and will be given in §2. In §3 a brief summary of the evolution of abstract harmonic analysis is given which includes the statement of the Hausdorff-Young Theorem for unimodular groups. In §4 the question of sharpness in the Hausdorff-Young Theorem in various settings is discussed and recent results of W. Beckner, J. Fournier and the author are described.
KeywordsFourier Coefficient Convolution Operator Compact Abelian Group Good Constant Fourier Integral
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- I.F. Riesz, Uber orthogonale Funktionensysteme, Nachr. Akad. Wiss. Gottingen Math. Phys. Kl. 1907, 116–122.Google Scholar
- 5.J. J. F. Fournier, Sharpness in Young's inequality for convolution, (to appear)Google Scholar
- 6.J. J. F. Fournier and B. Russo, Abstract interpolation and operator valued kernels, (to appear)Google Scholar
- 7.E. Hewitt and K. A. Ross, "Abstract Harmonic analysis", Springer-Verlag I (1963), II (1970).Google Scholar
- 8.A. Klein and B. Russo, Young's inequality for semi-direct products, in preparation.Google Scholar
- 11._____, The norm of the Lp-Fourier transform. II, Can. J. Math 28(1976)Google Scholar
- 12._____, On the Hausdorff-Young Theorem for integral operators, Pacific. J. Math. (to appear)Google Scholar
- 13._____, The norm of the Lp-Fourier transform III, (compact extensions), to appear.Google Scholar
- 14.A. Weil, L'integration dans les groupes topologiques et ses applications, Actualities Sci. et Ind. 869, 1145 Paris: Hermann & Cie. 1941 and 1951.Google Scholar