On Dirichlet type spaces, \(\alpha\)-Bloch spaces and \(Q_p\) spaces on the unit ball of \(\mathbb{C}^n\) (Q2718028)

Let \(f\) be in the Dirichlet type space \(D_\tau(\tau\in\mathbb{R})\) on the unit ball \(B\) of \(\mathbb{C}^n\). NEWLINENEWLINENEWLINEThe author obtains that the following conditions are equivalent. For the gradient \(\nabla f\), invariant gradient \(\widetilde{\nabla}f\), radial derivative \(Rf\), NEWLINE\[NEWLINE\|f\|_{D_\tau}^{2} < \infty,\tag{i}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\int_B |\widetilde{\nabla}f(z)|^2 (1-|z|^2)^{-1-\tau} d\nu(z) < \infty\;\text{if} \tau < 1,\tag{ii}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\int_B |\nabla f(z)|^2 (1-|z|^2)^{1-\tau} d\nu(z) < \infty\text{ if }\tau < 2,\tag{iii}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\int_B |Rf(z)|^2 (1-|z|^2)^{1-\tau} d\nu(z) < \infty\text{ if }\tau < 2,\tag{iv}NEWLINE\]NEWLINE where \(d\nu\) is the normalized Lebesgue measure on \(B\). Moreover, in the case of \(n > 1\), the inclusion relations among \(D_\tau, B^\alpha,\) and \(Q_p\) are studied and examples are given to show that all these inclusions are strict and best possible.