Spectral projections for the twisted Laplacian

Herbert Koch¹ ; Fulvio Ricci²

¹Mathematisches Institut Universität Bonn Beringstr. 1 53115 Bonn, Germany
²Scuola Normale Superiore Piazza dei Cavalieri 7 56126 Pisa, Italy

Studia Mathematica, Tome 180 (2007) no. 2, pp. 103-110

Résumé

Let $n \ge 1$, $d=2n$, and let $(x,y)\in \mathbb R^n\times \mathbb R^n$ be a generic point in $\mathbb R^{2n}$. The twisted Laplacian $$ L = -\frac12 \sum_{j=1}^n [ (\partial_{x_j} + iy_j)^2 + (\partial_{y_j} - i x_j)^2 ] $$ has the spectrum $ \{ n + 2k =\lambda^2: k \hbox{ a nonnegative integer}\}. $ Let $P_\lambda$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent $\varrho(p)$ in the estimate $$ \Vert P_\lambda u \Vert_{L^p(\mathbb R^d)} \lesssim \lambda^{\varrho(p)} \Vert u \Vert_{L^2(\mathbb R^d)} $$ for all $p\in[2,\infty]$, improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for $\varrho(p)$ is $$ \varrho(p) = \cases{ 1/p - 1/2 \hbox{if $ 2 \le p \le 2(d+1)/(d-1)$,} \cr (d-2)/2 - {d}/p \hbox{if $ 2(d+1)/(d-1) \le p \le \infty $.}\cr} $$

Détail

DOI : 10.4064/sm180-2-1

Keywords: mathbb times mathbb generic point mathbb twisted laplacian frac sum partial partial has spectrum lambda hbox nonnegative integer lambda spectral projection infinite dimensional eigenspace optimal exponent varrho estimate vert lambda vert mathbb lesssim lambda varrho vert vert mathbb infty improving previous partial results ratnakumar rawat thangavelu stempak zienkiewicz expression varrho varrho cases hbox d d hbox d infty