Session 33. Spaces of analytic functions

The logarithmic-\(BMOA\) space, multipliers, and spaces of Dirichlet type

Daniel Girela, Universidad de Málaga, Spain
If \(X\) and \(Y\) are two spaces of analytic functions in the unit disc \(\mathbb D \) which are continuously contained in \({\mathcal Hol} (\mathbb D )\), \(\mathcal M(X, Y)\) denotes the space of multipliers from \(X\) to \(Y\), \(\mathcal M (X, Y)=\{g\in{\mathcal Hol}(\mathbb D): fg\in Y,\quad\text{for all \(f\in X\)}\}.\) The space of multipliers from \(X\) to itself will be simply denoted by \(\mathcal M(X)\).

The spaces \(\mathcal M (X, Y)\) have been studied for a big number of spaces \(X, Y\). In this talk we shall concentrate our attention in the case where \(X\) and \(Y\) are spaces related with the spaces of Dirichlet type \({\mathcal D^p_{\alpha}}\) (\(0<p<\infty\), \(\alpha >-1\)), \(BMOA\) and the Bloch space \(\mathcal B\).

Let us remark that the spaces \(M(\mathcal B)\) and \(\mathcal M (BMOA)\) are known:

  • \(M(\mathcal B)=H^\infty \cap \mathcal B _{\log}\), where \(\mathcal B_{\log}\) is the logarithmic Bloch space which consists of those \(f\in {\mathcal Hol} (\mathbb D )\) with \(\sup_{z\in\mathbb D}(1-|z|^2)|f'(z)|\log\frac{1}{1-|z|^2}<\infty \).
  • \(\mathcal M (BMOA)=H^\infty \cap BMOA_{\log}\), where the logarithmic-\(BMOA\) space \(BMOA_{\log}\) consists of those \(f\in {\mathcal Hol} (\mathbb D )\) for which the Borel measure \(\mu _f\) in \(\mathbb D \) defined by \(d\mu_f(z)=(1-\vert z\vert^2)\vert f^\prime (z)\vert ^2 dA(z)\) is a \(2\)-logarithmic Carleson measure.

Our starting point is the fact that whenever \(p\neq q\), the only multiplier from \({\mathcal D^p_{p-1}} \) to \({\mathcal D^q_{q-1}}\) is the trivial one. It is easy to see that if \(0<p<q<\infty \) then \(\mathcal B\cap {\mathcal D^p_{p-1}}\subset\mathcal B\cap {\mathcal D^q_{q-1}}\). This clearly implies the following: "If \(X\) is a subspace of the Bloch space and \(0<p<q<\infty \), then the space of multipliers \(\mathcal M(X\cap {\mathcal D^p_{p-1}}, X\cap {\mathcal D^q_{q-1}})\) is non trivial". Then the question of characterizing the space \(\mathcal M(X\cap {\mathcal D^p_{p-1}}, X\cap {\mathcal D^q_{q-1}})\) for classical subspaces of the Bloch space such as \(H^\infty \), \(BMOA\) or \(\mathcal B\) arises naturally. In this talk we shall consider the case \(X=BMOA\). We shall present a number of results on the space \(BMOA_{\log}\) and we shall use them to study the spaces \(\mathcal M(BMOA\cap {\mathcal D^p_{p-1}}, BMOA\cap {\mathcal D^q_{q-1}})\), \(0<p, q<\infty \).

This talk is based on several recent works in collaboration with several colleagues such as C. Chatzifountas, R. Hernández, P. Galanopoulos, M. J. Martín, and José Ángel Peláez.

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