Generalized Hölder spaces of holomorphic functions in domains in the complex plane
The paper is concerned with the study of some nonstandard Banach spaces of analytic functions that enjoy a kind of smoothness property up to the boundary. Let \(\omega : [0,d]\to \mathbb{R}^+\) satisfies the following conditions: {\parindent=6mm \begin{itemize}\item[1)] \(\omega\) is continuous and \(\omega(0) = 0\), \item[2)] \(\omega\) is almost increasing, in the sense that there is a constant \(C\geq 1\) such that \(x \leq y\) implies \(\omega(x) \leq C\,\omega(y)\), \item[3)] \(\omega(h)/h\) is almost decreasing (there exists \(C\geq 1\) such that \(x \leq y\) implies \(\omega (y) \leq C\, \omega (x)\)). \end{itemize}} Such a function \(\omega\) is called a modulus of continuity. Now, let \(\omega : [0,2]\to \mathbb{R}^+\) be a modulus of continuity, and define \(A^\omega(D)\) as the space of all analytic functions in the unit disk \(D\) such that \[ |f(z)-f(w)|\leq C\, \omega (|z-w|),\quad z,w \in D, \] where \(C\) is a constant. The norm of a function \(f \in A^\omega(D)\) is given by \[ \| f\|_{A^\omega(D)}= \sup_{ z,w\in D}\frac{ |f(z)-f(w)|}{ \omega (|z-w|)} +\| f\| _{L^\infty (D)}. \] Similarly, define \(B^\omega(D)\) as the space of all analytic functions in the unit disk that satisfy \[ |f^\prime(z)|\leq C\frac{\omega (1-|z|)}{ 1-|z|},\quad z\in D, \] where \(C\) is a constant independent of \(z\) and \(w\). The norm of \(f\in B^\omega (D)\) is given by \[ \| f\| _{B^\omega(D)} = \sup_{ z\in D}\frac{ (1-|z|)|f^\prime (z)|}{\omega (1-|z|)} +\|f\|_{L^\infty(D)}. \] The first main result of the paper under review says that if \(\omega\) satisfies \[ \int_t^d \frac{\omega (s)}{s^2} ds \leq C \frac{\omega(t)}{t},\quad 0 < t < d, \] then \(A^\omega(D)\) is continuously embedded in \(B^\omega(D)\); that is, the inclusion \(A^\omega(D) \hookrightarrow B^\omega(D)\) is continuous. On the other hand, if \(\omega\) satisfies \[ \int_0^t \frac{\omega (s)}{s} ds \leq C \omega(t),\quad 0 < t < d, \] then the imbedding \(B^\omega(D) \hookrightarrow A^\omega(D)\) is continuous. The authors then take up the variable exponent Hölder space \(A^{\lambda (.)}(D)\) of analytic functions in the unit disk for which \[ \| f\|_{A^{\lambda (.)}(D)} := \sup_{ z,w\in D}\frac{ |f(z)-f(w)|}{ |z-w|^{\lambda (z)}} +\| f\|_{L^\infty (D)} \] is finite, where \(\lambda : D\to [0,1]\) is a continuous function that satisfies the following log-condition: \[ |\lambda(z)-\lambda(w)|\leq \frac{C}{ \log \frac{1}{|z-w|}},\quad z,w \in D,\, |z-w| < 1/2. \] The space \(B^{\lambda (.)}(D)\) consists of the analytic functions in the unit disk that satisfy \[ |f^\prime (z)|\leq C(1-|z|)^{\lambda (z)-1}, \] with the norm \[ \sup_{ z\in D}|f^\prime (z)|(1-|z|)^{1-\lambda(z)} +\| f\|_{L^\infty (D)}. \] It is proved that, if \(\sup_{z\in D} \lambda(z)< 1\), then the imbedding \(A^{\lambda(.)}(D) \hookrightarrow B^{\lambda(.)}(D)\) is continuous. On the other hand, if \(\inf_{z\in D} \lambda(z)>0\), then \(B^{\lambda(.)}(D) \hookrightarrow A^{\lambda(.)}(D)\) is continuous. Similar results in the upper-half plane, instead of the unit disk, are established as well.
scientific article; zbMATH DE number 6997105