Boundary multipliers of a family of Möbius invariant function spaces

Abstract

For $1<p<\infty$ and $0<s<1$, let ${\mathcal{Q}}_s^p(\mathbb{T})$ be the space of those functions $f$ which belong to $L^p(\mathbb{T})$ and satisfy \[ \sup_{I\subset \mathbb{T}}\frac{1}{|I|^s}\int_I\int_I\frac{|f(ζ)-f(η)|^p}{|ζ-η|^{2-s}}|dζ||dη|<\infty, \] where $|I|$ is the length of an arc $I$ of the unit circle $\mathbb{T}$ . In this paper, we give a complete description of multipliers between ${\mathcal{Q}}_s^p(\mathbb{T})$ spaces. The spectra of multiplication operators on ${\mathcal{Q}}_s^p(\mathbb{T})$ are also obtained.