γ-Valued Stein--Tomas Restriction for Schatten Classes
Abstract
Let Σ⊂ℝ3 be a compact C2 hypersurface with nonvanishing Gaussian curvature. We prove the sharp matrix-dimension-independent Schatten-valued Stein--Tomas range RΣ:Lp(ℝ3;Sa)⟶γ(L2(Σ),Sa), 1<a<∞, 1≤ p≤min{a,4/3}. The core endpoint is EΣ:γ(L2(Σ),S4)→ L4(ℝ3;S4). 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 L4 estimate, and free spectral-measure bounds.
Related
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- The sharp Stein-Tomas inequality on the circle
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BibTeX
@misc{shahmurov2026γvalued,
title = {{$γ$-Valued Stein--Tomas Restriction for Schatten Classes}},
author = {R. Shahmurov and V. Shahmurov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40044},
eprint = {2610.40044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40044},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - γ-Valued Stein--Tomas Restriction for Schatten Classes
AU - R. Shahmurov
AU - V. Shahmurov
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40044
UR - https://arxiv.org/abs/2610.40044
AB - Let Σ⊂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(ℝ^3;S^a)⟶γ(L^2(Σ),S^a), 1<a<∞, 1≤ p≤min{a,4/3}. The core endpoint is E_Σ:γ(L^2(Σ),S^4)→ L^4(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.
ER -