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We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function f f from the homgeneous Sobolev space W ˙ d 1 ( R d ) \dot {W}^1_d(\mathbb {R}^d) on R d . \mathbb {R}^d. The asymptotic coefficient ‖ ∇ f ‖ L d ( R d ) \|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to the norm of \dj \mkern 1muf in the principal ideal L d , ∞ , \mathcal {L}_{d,\infty }, thus, providing a non-asymptotic, uniform bound on the spectrum of \dj \mkern 1muf. Our methods are based on the C ∗ C^{\ast } -algebraic notion of the principal symbol mapping on R d \mathbb {R}^d , as developed recently by the last two authors and collaborators.
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This content will become publicly available on September 1, 2026
Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains
Abstract We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary positive order. Moreover, when the underlying domain is convex, we obtain universal, non-asymptotic bounds that correctly reproduce the two leading terms in the asymptotics and depend on the domain only through simple geometric characteristics. Important ingredients in our proof are non-asymptotic versions of various Tauberian theorems.
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- Award ID(s):
- 1954995
- PAR ID:
- 10628073
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Inventiones mathematicae
- Volume:
- 241
- Issue:
- 3
- ISSN:
- 0020-9910
- Page Range / eLocation ID:
- 999 to 1079
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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