On the angular distribution of mass by Besov functions (Q2381141)

For \(p>1\), let \(B_p\) be the Besov space of holomorphic functions on the unit disk \(D\) such that \[ \| f\| _{B_p}^p=\int_D(1-| z| ^2)^{p-2}| f'(z)| ^p\,dA(z)<\infty, \] where \(dA(z)\) is the two-dimensional Lebesgue measure on \(D\). Denote \(\Sigma_{\varepsilon}=\{w\in\mathbb C: | \arg w| <\varepsilon\}\). The authors prove the Main Theorem: Given \(p>1\) and \(\varepsilon>0\), there exists a constant \(K>0\) depending only on \(p\) such that \[ \int_{f^{-1}(\Sigma_{\varepsilon})}(1-| z| ^2)^{p-2}| f'(z)| ^p \,dA(z)\geq K\varepsilon\frac{| f'(0)| ^{p+4}}{\| f\| _{B_p}^4}, \] for any nonzero function \(f\in B_p\) with \(f(0)=0\). This Theorem extends the results of D. Marshall and W. Smith for univalent functions and F. Pérez-González and J. Ramos for functions of the widest class of weighted Bergman spaces. In addition to the Main Theorem the authors prove the result for conformal mappings in Theorem 2: Given \(\varepsilon>0\) and a conformal map \(h\) on \(B_p\), \(p>1\), with \(h(0)=0\), there exists a constant \(K>0\) depending on \(p\) such that \[ \int_{h^{-1}(\Sigma_{\varepsilon})}(1-| z| ^2)^{p-2}| h'(z)| ^p \,dA(z)\geq K\varepsilon| h'(0)| ^p. \]