Leibenson divisors for Hardy classes in the convex case and in several variables (Q1849627)

Let \(\Omega\) be a convex open subset of \(\mathbb C^n\) with \(\mathcal C^2\)-boundary. The equivalent norms \(\|f\|_{q,b}= \sup_{0\leq r<1}\left(\int_{\partial\Omega}|f(b+r(z-b))|^q d\sigma(z)\right)^{\frac{1}{q}}<\infty\), \(b\in\Omega,\) make the Hardy class \(H^q(\Omega)\), \(q\in [1, +\infty[\), a Banach space. The author proves that for a holomorphic function \(f(z)\) belonging to the Hardy class \(H^q(\Omega)\), the Leibenson's divisors at the point \(a\in\Omega\) \[ \mathcal L_jf(z)=\int_0^1\frac{\partial f}{\partial z_j}(a+t(z-a)) dt,\quad j=1,2,\dots,n, \] are the continuous endomorphisms of the Banach space \(H^q(\Omega)\), i.e., for all \(f\in H^q(\Omega)\), there exists a constant \(C(n,\Omega,a, q)\) such that \[ \sup_{0\leq r<1}\int_{\partial\Omega}\left\|f'(a+tr(z-a)) dt\right\|^q_\ast d\sigma(z)\leq C(n,\Omega,a, q)\|f\|^q_{q,a}, \] where the norm from the left is the dual norm of the Euclidian space \(\mathbb C^n\). The case of the Bergman spaces is also considered.