2024/10/30 by Jacob Denson, Denson, Jacob
Medicine · #58J40 (Primary) 58C40 (Secondary) #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Medical Imaging Techniques and Applications #Ultrasound Imaging and Elastography
paper · pdf · doi:10.48550/arxiv.2410.23505
openalex publication_date 2024/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any bounded, regulated function m: [0,∞) → ℂ, consider the family of operators \ TR \ on the sphere Sd such that TR f = m(k/R) f for any spherical harmonic f of degree k. We completely characterize the compactly supported functions m for which the operators \ TR \ are uniformly bounded on Lp(Sd), in the range 1/(d-1) < |1/p - 1/2| < 1/2. We obtain analogous results in the more general setting of multiplier operators for eigenfunction expansions of an elliptic pseudodifferential operator P on a compact manifold M, under curvature assumptions on the principal symbol of P, and assuming the eigenvalues of P are contained in an arithmetic progression. One consequence of our result are new transference principles controlling the Lp boundedness of the multiplier operators associated with a function m, in terms of the Lp operator norm of the radial Fourier multiplier operator with symbol m(|⋅|): ℝd → ℂ. In order to prove these results, we obtain new quasi-orthogonality estimates for averages of solutions to the half-wave equation ∂t - i P = 0, via a connection between pseudodifferential operators satisfying an appropriate curvature condition and Finsler geometry.