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Spectral projections for the twisted Laplacian

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Koch H., Ricci F.

Studia Mathematica

2007

tom 180

nr 2

103-110

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 ≤ ∞.

Ograniczanie wyników

1 Studia Mathematica

1 Koch H.

1 Ricci F.

1 2007

Wyniki wyszukiwania

Spectral projections for the twisted Laplacian