From Zygmund space to Bergman-Zygmund space (Q6613287)

Given a positive measure \(\mu\) on the unit disk \(\mathbb D\) and given parameters \(p\in(0,\infty)\) and \(\beta\in\mathbb R\), the Zygmund space \(L^{p,\beta}(d\mu)\) consists of the measurable functions \(f\) in \(\mathbb D\) such that\N\[\N\int_{\mathbb D} |f|^p \log^\beta (e+|f|)^\beta d\mu< \infty.\N\]\NFor \(\beta>0\) these are the usual Lorentz-Zygmund spaces \(L^p(\log^+L)^\beta(d\mu)\).\N\NA classical interpolation result of Zygmund proves that if a linear operator \(T\) is of weak types \((L^1, L^1)\) and \((L^\infty, L^\infty)\), then it is bounded from \(L^p\) to \(L^p\), \(1<p<\infty\). The first result of the paper, Theorem A, provides interpolation results for the Zygmund spaces for various combinations of the parameters \(p\) and \(\beta\). Their common feature of these results is that the assumption is a weak-type estimate between Zygmund spaces at the endpoints of the parameters range.\N\NGiven \(p,\beta_1, \beta_2\) as before, and given \(\alpha>-1\) consider the space \(A^{p, \beta_1}_{\alpha,\beta_2}\) of analytic functions \(f\) such that\N\[\N\int_{\mathbb D} |f(z)|^p \log^{\beta_1}(e+|f(z)|)\, \log^{\beta_2}\Bigl(\frac e{1-|z|}\Bigr) \, (1-|z|)^\alpha dA(z)<+\infty.\N\]\NWhen \(\beta_1=\beta_2=0\) these are the usual weighted Bergman spaces, while \(A^{p, \beta_1}_{0,0}\) coincides with a Nevanlinna area class.\N\NThe second result of the paper, Theorem B, proves that the parameters \(\beta_1,\beta_2\) are interchangeable, in the sense that\N\[\NA^{p, \beta_1}_{\alpha,\beta_2}= A^{p, 0}_{\alpha,\beta_1+\beta_2}=A^{p, \beta_1+\beta_2}_{\alpha,0}.\N\]\NThe third and last result, Theorem C, characterizes the Carleson measures for the Bergman-Zygmund space. Given \(\gamma\in\mathbb R\), a finite positive Borel measures \(\mu\) on \(\mathbb D\) is \(\gamma\)-Carleson for \(A_\alpha^{p,\beta}=A_{\alpha,0}^{p,\beta}\) is there exists \(C>0\) such that for all \(f\in A_\alpha^{p,\beta}\)\N\[\N\int_{\mathbb D}|f|^p\log^\gamma(e+|f|)\, d\mu\leq C \int_{\mathbb D} |f|^p \log^\beta(e+|f|)\, (1-|z|)^\alpha dA.\N\]\NAccording to Theorem C, \(\mu\) is \(\gamma\)-Carleson for \(A_\alpha^{p,\beta}\) if and only if there exists a constant \(C>0\) such that for all \(z\in\mathbb D\)\N\[\N\mu(D\bigl(z,r(1-|z|)\bigr))\leq C \left(\int_{A_\alpha^{p,\beta}}(1-|z|)^\alpha dA\right)\, \log^{\beta-\gamma}\Bigl(\frac e{1-|z|}\Bigr).\N\]\NThe parameter \(r\) can have any value in \((0,1)\), and the constant \(C\) depends on \(r\).