$\gamma$-Valued Stein--Tomas Restriction for Schatten Classes

Abstract

Let $\Sigma \subset {\mathbb{R}}^3$ be a compact $C^2$ hypersurface with nonvanishing Gaussian curvature. We prove the sharp matrix-dimension-independent Schatten-valued Stein--Tomas range \[ R_Σ:L^p(\mathbb R^3;S^a)\longrightarrowγ(L^2(Σ),S^a), \qquad 1<a<\infty,\quad 1\le p\le\min\{a,4/3\}. \] The core endpoint is $E_{\Sigma }:\gamma (L^2(\Sigma ),S^4)\to L^4({\mathbb{R}}^3;S^4)$. After squaring the quartic norm, multiplication lands in the Hilbert--Schmidt class. A local coarea argument splits resonant quartets according to the shorter adjacent frequency chord, producing a dimension-free row/column estimate for elliptic and hyperbolic curvature. We also show that Gaussian type $4/3$ alone does not characterize the corresponding endpoint for arbitrary Banach spaces. Applications include matrix Herglotz waves, a two-dimensional Schrödinger $L^4$ estimate, and free spectral-measure bounds.