Let \(H\) be the space of holomorphic functions on the open unit ball \(\mathbb{B}\) in \(\mathbb{C}^n\). For \(g\in H\), let \(J_g:H\to H\) be the integral operator defined by \[ J_g(f)(z)=\int_0^1 f(tz)(Rg)(tz)\frac{dt}{t}, \] where \(R\) denotes the radial derivative.
The author obtains a complete characterization of the symbols \(g\) for which the operator \(J_g\) between Hardy spaces \(H^p\) is bounded, compact or in the Schatten class, respectively. Although these characterizations coincide with the well-known ones for the case \(n=1\), their proofs require different techniques in order to avoid certain specific tools used in the one-dimensional case as, for example, the factorization of Hardy spaces.
The main results proved by the author are the following:

1)

\(J_g:H^p\to H^p\), \(0<p<\infty\), is bounded (resp. compact) if and only if \(g\in \mathrm{BMOA}\) (resp. \(g\in \mathrm{VMOA}\)).

2)

Let \(0<p<q<\infty\) and \(\alpha=n/p-n/q\). If \(\alpha< 1\), \(J_g:H^p\to H^q\) is bounded (resp. compact) if and only if \(g\in \Lambda_\alpha\) (resp. \(g\in \lambda_\alpha\)), where \(\Lambda_\alpha\) is the Lipschitz space of order \(\alpha\) (resp. \(\lambda_\alpha\) is the little Lipschitz space of order \(\alpha\)). If \(\alpha=1\), then \(J_g\) is bounded if and only if \(Rg\in H^\infty\), and it is compact if and only if \(J_g=0\), that is, \(g\) is constant. If \(\alpha>1\), \(J_g\) is bounded if and only if \(J_g=0\).

3)

Let \(0<q<p<\infty\) and \(1/r=1/q-1/p\). The operator \(J_g:H^p\to H^q\) is bounded if and only if \(g\in H^r\), and it is compact if and only if it is bounded.

4)

For \(0<p<\infty\), let \(S_p(H^2)\) be the \(p\)-Schatten class of operators on \(H^2\). If \(p>n\), then \(J_g\in S_p(H^2)\) if and only if \(g\) is in the Besov space
\[ B_p=\Big\{f\in H\;:\; \big\|(1-|z|^2)^{1-(n+1)/p}Rf(z)\big\|_{L^p(\mathbb{B})}<\infty\Big\}. \] If \(p\leq n\), then \(J_g\in S_p(H^2)\) if and only if \(J_g=0\).