What happens to an $L_p$ function when one truncates its Fourier transform to a domain? This is in the root of foundational problems in harmonic analysis. Fefferman’s celebrated theorem for the ball (1971) imposes that, to preserve $L_p$-integrability, the boundary of such domain must be flat. What if we truncate on a curved space like a Lie group? And if we truncate the entries of a given matrix? What happens with the singular numbers of it or with its Schatten $p$-norm? We fully characterize the local geometry of such $L_p$-preserving truncations for these (apparently unrelated) problems, in terms of a surprisingly lax notion of boundary flatness. The matrix ones are all diffeomorphic variations of a fundamental example: the triangular projection. The Lie group ones are all modeled on one of three fundamental examples: the classical Hilbert transform, and two new examples of Hilbert transforms that we call affine and projective. This vastly generalizes Fefferman’s theorem to nontrigonometric and noncommutative scenarios. It confirms the intuition that Schur multipliers share profound similarities with Euclidean Fourier multipliers – even in the lack of a Fourier transform connection – and complete, for Lie groups, a longstanding search of Fourier $L_p$-idempotents. Joint work with M. de la Salle and E. Tablate.

The video of this talk is available on the IISc Math Department channel.