Boundary multipliers of a family of Möbius invariant function spaces (Q2790816)

The authors give a complete description of the space of pointwise multipliers between BMO-type spaces \(Q^p_s(\mathbb{T})\) on the unit circle \(\mathbb{T}\).NEWLINENEWLINEFor \(1<p<\infty\) and \(s> 0\), the space \(Q^p_s(\mathbb{T})\) consists of all the functions \(f\in L^p(\mathbb{T})\) such that NEWLINE\[NEWLINE \|f\|_{Q^p_s}=\sup_{I\subset \mathbb{T}}\frac{1}{|I|^s}\int_I\int_I \frac{|f(\zeta)-f(\eta)|^p}{|\zeta-\eta|^{2-s}}\,|d\zeta|\,|d\eta|<\infty, NEWLINE\]NEWLINE where \(|I|\) denotes the length of the arc \(I\).NEWLINENEWLINEThe main result of this paper gives the following complete description of the multipliers between these spaces for the range \(1<p,q<\infty\) and \(0<s,r<1\): {\parindent=6mm \begin{itemize}\item[1)] If either \(p\leq q\) and \(s\leq r\), or \(p> q\), \(s\leq r\) and \((1-s)/p>(1-r)/q\), then \(f\) is a multiplier from \(Q^p_s(\mathbb{T})\) to \(Q^q_r(\mathbb{T})\) if and only if \(f\in L^\infty(\mathbb{T})\) and NEWLINE\[NEWLINE \sup_{I\subset \mathbb{T}} \frac{1}{|I|^r}\left(\log\frac{2}{|I|}\right)^q\int_I\int_I \frac{|f(\zeta)-f(\eta)|^q}{|\zeta-\eta|^{2-r}}\,|d\zeta|\,|d\eta|<\infty. NEWLINE\]NEWLINE \item[2)] In the remainder cases, that is, \(s>r\) or \(p> q\), \(s\leq r\) and \((1-s)/p\leq (1-r)/q\), the unique multiplier from \(Q^p_s(\mathbb{T})\) to \(Q^q_r(\mathbb{T})\) is \(f=0\). NEWLINENEWLINE\end{itemize}} As an application of this result, the authors show that, for \(1<p<\infty\) and \(0<s<1\), the spectrum of the multiplication operator \(M_f(g)=fg\) on \(Q^p_s(\mathbb{T})\) coincides with the set \(R(f)\) of all \(\lambda\in\mathbb{C}\) such that \(\{\zeta\in\mathbb{T}:\,|f(\zeta)-\lambda|<\varepsilon\}\) has positive measure for every \(\varepsilon>0\).NEWLINENEWLINEThe proofs of these results require a study of some properties of the space \(Q^p_s(\mathbb{T})\), which is interesting by itself. This includes its relation with Möbius invariant holomorphic Besov spaces on the unit disk, and also an analytic version of the above mentioned results.