L^p Boundedness and Compactness of Localization Operators

Localization operators are special anti-Wick operators, which arise in many fields of pure and applied mathematics. We study in this paper some properties of two-wavelet localization operators, $i.e.,$ operators which depend on a symbol and two different windows. In the case when the symbol $F$ belongs to $L^p(\mathbb{R}^{2n})$, we give an extension of some results proved by Boggiatto and Wong. More precisely, we obtain the boundedness and compactness of such operators on $L^q(\mathbb {R}^n),\,\frac{2p}{p+1}\leq q\leq\frac{2p}{p-1}$, for every $p\in[1,\infty]$.