Image: Íñigo de Amescua/ICMAT

What happens to a given function if we redefine its Fourier transform to be zero outside certain domain? This is in the root of foundational problems in harmonic analysis. Fefferman’s celebrated theorem for the ball (1971) shows that this transformation preserves Lp-integrability if and only if the boundary of such domain is piecewise flat. What if we truncate the Fourier transform on a curved space like a Lie group? What do we mean by “boundary flatness” in that case? And if we truncate the entries of a given matrix with Euclidean entries? How do we relate the shape of such domain with its effect on the singular numbers of this matrix?

In this paper, the authors fully characterize the local geometry of such Lp-preserving truncations for these (apparently unrelated) problems in terms of a surprising lax notion of boundary flatness. The matrix ones are all diffeomorphic variations of a fundamental map: the triangular projection or matrix Hilbert transform. The Lie group ones are all modeled on one of three fundamental examples: the classical Hilbert transform, and two new examples of Hilbert transforms over affine and universal covering groups. This vastly generalizes Fefferman’s theorem to nontrigonometric and noncommutative scenarios, and confirms that Schur multipliers (matrix transformations by entrywise multiplication) share profound similarities with Euclidean Fourier multipliers —even in the lack of a Fourier transform— and complete for Lie groups a longstanding search.

Reference

Parcet J, de la Salle M, Tablate E. The local geometry of idempotent Schur multipliers. Forum of Mathematics, Pi. 2025;13:e14. doi:10.1017/fmp.2025.6