Eigenvalues of Hille–Tamarkin operators and geometry of Banach function spaces

Thomas Kühn¹ ; Mieczysław Mastyło²

¹ Fakultät für Mathematik und Informatik Mathematisches Institut Universität Leipzig Johannisgasse 26 D-04103 Leipzig, Germany
² Faculty of Mathematics and Computer Science Adam Mickiewicz University and Institute of Mathematics Polish Academy of Sciences (Poznań branch) Umultowska 87 61-614 Poznań, Poland

Studia Mathematica, Tome 207 (2011) no. 3, pp. 275-296

Résumé

We investigate how the asymptotic eigenvalue behaviour of Hille–Tamarkin operators in Banach function spaces depends on the geometry of the spaces involved. It turns out that the relevant properties are cotype $p$ and $p$-concavity. We prove some eigenvalue estimates for Hille–Tamarkin operators in general Banach function spaces which extend the classical results in Lebesgue spaces. We specialize our results to Lorentz, Orlicz and Zygmund spaces and give applications to Fourier analysis. We are also able to show the optimality of our eigenvalue estimates in the Lorentz spaces $L_{2,q}$ with $1\le q2$ and in Zygmund spaces $L_p(\log L)_a$ with $2\le p\infty$ and $a>0$.

Détail

DOI : 10.4064/sm207-3-4

Keywords: investigate asymptotic eigenvalue behaviour hille tamarkin operators banach function spaces depends geometry spaces involved turns out relevant properties cotype p concavity prove eigenvalue estimates hille tamarkin operators general banach function spaces which extend classical results lebesgue spaces specialize results lorentz orlicz zygmund spaces applications fourier analysis able optimality eigenvalue estimates lorentz spaces zygmund spaces log infty