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Abstract

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 $$ \varrho(p) = \cases{ 1/p - 1/2 & \hbox{if $ 2 \le p \le 2(d+1)/(d-1)$,} \cr (d-2)/2 - {d}/p & \hbox{if $ 2(d+1)/(d-1) \le p \le \infty $.}\cr} $$