Operator theory in harmonic analysis

Abstract

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 CoefficientConvolution OperatorCompact Abelian GroupGood ConstantFourier IntegralThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.