Spectral projections for the twisted Laplacian

• Autorzy: Herbert Koch , Fulvio Ricci
• Języki publikacji: EN

Abstrakty

EN

Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian
$L = -1/2 ∑_{j=1}^{n} [(∂_{x_{j}} + iy_{j})² + (∂_{y_{j}} - ix_{j})²]$
has the spectrum {n + 2k = λ²: k a nonnegative integer}. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate
$||P_{λ}u||_{L^{p}(ℝ^{d})} ≲ λ^{ϱ(p)} ||u||_{L²(ℝ^{d})}$
for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is
ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1),
ϱ(p) = (d-2)/2 - d/p if 2(d+1)/(d-1) ≤ p ≤ ∞.

Kategorie tematyczne

• 43A85: Analysis on homogeneous spaces
• 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 180
• Numer: 2
• Strony: 103-110

Daty

wydano

2007

Twórcy

autor

Herbert Koch

• Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany

autor

Fulvio Ricci

• Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Identyfikatory

DOI

10.4064/sm180-2-1