Diagonal mapping and problems of representation in anisotropic spaces of holomorphic functions in the polydisk (Q757646)

Let \(\omega_ j\) be non-negative functions of class \(L^ 1(0,1)\), \(1\leq j\leq n\), \(0<p<+\infty.\) By \(H^ p(\omega_ 1,...,\omega_ n)\) we denote the class of functions holomorphic in the polydisk \(U^ n\), for which \[ \| f\|_{H^ p(\omega_ 1,...,\omega_ n)}= \] \[ (\int_{U^ n}| f(z_ 1,...,z_ n)|^ p\omega_ 1(1-| z_ 1|)...\omega_ n(1-| z_ n|)dm_{2n}(z_ 1,...,z_ n))^{1/p}<+\infty, \] where \(dm_{2n}-2n\)-dimensional Lebesgue measure on \(U^ n.\) In {\S} 2 of this paper with some restrictions on \(\omega_ j\), \(1\leq j\leq n\), the author gives complete characterization of those functions g, holomorphic in U, for which the following representation is valid: \[ g(z)=Df(z)=^{def}f(z,z,...,z),\quad z\in U,\quad f\in H^ p(\omega_ 1,...,\omega_ n). \] In {\S} 3 the author obtains the integral representation of functions from the classes \(H^ p(\omega_ 1,...,\omega_ n)\), \(0<p<+\infty\). Further the author gives a complete description of linear continuous functionals on indicated spaces.