Embedding Bergman spaces into tent spaces

Abstract

Let \(A^p_\omega \) denote the Bergman space in the unit disc \(\mathbb {D}\) of the complex plane induced by a radial weight \(\omega \) with the doubling property \(\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds\). The tent space \(T^q_s(\nu ,\omega )\) consists of functions such that $$\begin{aligned} \begin{aligned} \Vert f\Vert _{T^q_s(\nu ,\omega )}^q =\int _\mathbb {D}\left( \int _{\varGamma (\zeta )}|f(z)|^s\,d\nu (z)\right) ^\frac{q}{s}\omega (\zeta )\,dA(\zeta ) <\infty ,\quad 0<q, \; s<\infty . \end{aligned} \end{aligned}$$ Here \(\varGamma (\zeta )\) is a non-tangential approach region with vertex \(\zeta \) in the punctured unit disc \(\mathbb {D}{\setminus }\{0\}\). We characterize the positive Borel measures \(\nu \) such that \(A^p_\omega \) is embedded into the tent space \(T^q_s(\nu ,\omega )\), where \(1+\frac{s}{p}-\frac{s}{q}>0\), by considering a generalized area operator. The results are provided in terms of Carleson measures for \(A^p_\omega \).