Imbedding potentials in tent spaces (Q1868675)

Let \(H_\alpha^p\) be the space of holomorphic functions on the unit ball \(B_n\) in \(\mathbb{C}^n\) such that \[ \left(\int\limits_{D(\zeta)}\left|(I+R)^k f(z)\right|^2 (1-|z|^2)^{2(k-\alpha)-(n+1)} dV(z) \right)^\frac{1}{2}\in L^p(S^n) \] for some \(k>\alpha\), where \(R\) is the operator of the radial differentiation, \(S^n\) is the unit sphere in \(\mathbb{C}^n\) and \[ D(\zeta)=\left\{z\in B^n: |1-z\zeta|<\tfrac{\beta}{2}(1-|z|^2)\right\}, \quad \beta >1. \] The authors treat the popular idea of tent spaces (which goes back to papers by \textit{R. R. Coifman, Y. Meyer} and \textit{E. M. Stein}, 1982-1985). The corresponding tent space \(T_s^q(d\mu)\) is defined as the space of measurable functions on \(B^n\) such that \[ \|f\|_{T_s^q(d\mu)}=\left\{\int_{S^n}\left(\int_{D(\zeta)} |f(z)|^s\frac{d\mu(z)}{(1-|z|^2)^n}\right)^\frac{q}{s} d\sigma(\zeta)\right\}\frac{1}{q}< + \infty. \] The authors obtain a characterization of positive Borel measures on \(B^n\) for which the potential space \(K_\alpha[L^p(d\sigma)]\), where \[ K_\alpha[d\nu](z)=\int\limits_{S^n}\frac{d\nu(\zeta)}{|1-z\zeta|^{n-\alpha}}, \] is imbedded into the tent space \(T_2^q(d\mu) .\) From this characterization some description of pointwise multipliers of the space \(H_\alpha^p\) is also derived.