On an integral-type operator between $H^2$ space and weighted Bergman spaces

Abstract

Let $H(\mathbb B)$ denote the space of all holomorphic functions on the unit ball $\mathbb B$ of $\mathbb C^n$ and $\Re h(z)=\sum_{j=1}^nz_j\frac{\pt h}{\pt z_j}(z)$ the radial derivative of $h.$ Motivated by recent results by S. Li and S. Stević , in this paper we study the boundedness and compactness of the following integral operator $$ L_gf(z)= \int_0^1 \Re f(tz) g(tz)\frac{dt}{t},\quad z\in \mathbb B, $$ between the Hardy space $H^2$ and weighted Bergman spaces.