Eigenvalues of Hille–Tamarkin operators and geometry of Banach function spaces
Tom 207 / 2011
Studia Mathematica 207 (2011), 275-296
MSC: Primary 47B06, 47G10; Secondary 47B10, 46E30.
DOI: 10.4064/sm207-3-4
Streszczenie
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 $1\le q<2$ and in Zygmund spaces $L_p(\log L)_a$ with $2\le p<\infty$ and $a>0$.
