NCMW Harmonic Analysis (2017)

School Convener(s)

Speakers and Syllabus 

Construction of eigenfunctions of the Laplace-Beltrami operator which are constant on horocycles. Definition of the Fourier transform. Fourier transform is well defined for Lp functions on the Poincar ́e disc. K−biinvariant functions and the spherical Fourier transform. The slice projection theorem for Radon, Fourier and the Abel transform.

The Plancherel theorem for L2 functions on G/K assuming the Plancherel theorem for K−biinvariant functions.The Abel inversion and the Fourier inversion for smooth, compactly supported K−biinvariant functions on G = SU (1, 1). Harish Chandra series for the elementary spherical functions. Harish Chandras c−function and asymptotics of the elementary spherical functions. Elementary cases of Kunze–Stein phenomena. Young’s convolution inequality for unimodular and non-unimodular groups. View a function on the hyperbolic space H 2 as a function on its isometry group which is unimodular and also as a function on the associated Iwasawa N A group which is non-unimodular and use this dichotomy to obtain a new convolution inequality known as Kunze–Stein phenomenon. Mapping properties of spectral projection, restriction and extension operators on the hyperbolic space H 2 and its connection to the convolution inequality mentioned above. Hardy- Littlewood maximal function (centered and uncentered), their mapping properties. Counter example to establish the sharpness of these properties and emphasize the difference between the centered and uncentered versions. Comments on the effect of polynomial versus exponential volume growth vis-a-vis the lack of ball doubling property of the metric-measure space H 2 on some results of analysis on this space. Discussion of some open questions.