It is well-known that if \(f\) is an analytic function on the open unit ball \(\mathbb{B}\) in \(\mathbb{C}^n\) , then \[ \sup_{0\le r<1}\|f_r\|_{L^p(\mathbb{S})}<\infty \quad\text{if and only if}\quad \int_{\mathbb{B}} \Delta\,(|f(z)|^p) (1-|z|^2)dV(z)<\infty, \] for any \(0 < p < \infty\). Here \(\mathbb{S}\) denotes the unit sphere and \(f_r(\zeta)=f(r\zeta)\), for \(0 < r < 1\) and \(\zeta\in \mathbb{S}\). Observe that this equivalence gives a characterization of the functions \(f\) in the Hardy space \(H^p(\mathbb{B})\).
In this paper the authors prove that if \(f\in C^2(\mathbb{B})\) satisfies \(\operatorname{Re}(f\overline{\Delta f})\ge 0\), then the above equivalence holds for any \(p\ge 2\). In particular, if \(p\ge 2\), the solutions \(f\) of the non-homogeneous Yukawa PDE \(\Delta f=\varphi f+\psi\operatorname{Re} f\), where \(\varphi\) and \( \psi\) are nonnegative continuous functions on \(\mathbb{B}\), satisfy the equivalence.
It is also well known that there exist analogous characterizations for analytic functions \(f\) in weighted Bergman spaces \(A^p_\alpha(\mathbb{B})\) (with respect to the weights \(\omega_\alpha(z)=(1-|z|^2)^\alpha\), \(\alpha>-1\)) or in Dirichlet-type spaces. Similarly to the Hardy case, the authors extend the corresponding equivalences to spaces of \(C^k\) functions on \(\mathbb{B}\). Such spaces are defined in terms of the above condition \(\operatorname{Re}(f\overline{\Delta f})\ge 0\) and/or of the Heinz’s nonlinear differential inequality \[ |\Delta f(z)| \le \varphi(z)\, \big(\|\partial f(z)\|^2+\|\overline{\partial} f(z)\|^2\big)^{1/2} + \psi(z)\, |f(z)|+\phi(z), \] where \(\varphi, \psi\) and \(\phi\) are nonnegative continuous functions on \(\mathbb{B}\), among others.