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On the Littlewood-Paley inequalities for subharmonic functions on domains in \(\mathbb R^n\). (English) Zbl 1276.31003

Bilyk, Dmitriy (ed.) et al., Recent advances in harmonic analysis and applications. In honor of Konstantin Oskolkov on the occasion of his 65th birthday. Based on the AMS sectional meeting, Georgia Southern University, Statesboro, GA, USA, March 11–13, 2011. Berlin: Springer (ISBN 978-1-4614-4564-7/hbk; 978-1-4614-4565-4/ebook). Springer Proceedings in Mathematics & Statistics 25, 357-383 (2013).
The Littlewood-Paley inequality reads \[ \int_{D}(1-|z|)^{p-1}|\nabla h(z)|^{p}dxdy\leq C\left(|h(0)|^p+\sup _{0<r<1}\int_{0}^{2\pi}|h(re^{i\theta})|^pd\theta\right). \] Here \(D\) is the unit disc in the complex plane, \(h\) is a harmonic function on \(D\), \(C\) is some constant independent of \(h\) and \(p\) is some number \(\geq 2\) (for \(1<p \leq2\) the inequality is in the opposite direction).
The author proves the following generalizations of this inequality:
Let \(\Omega\) be a domain in \(\mathbb R^{n}\), \(n\geq 2\). Assume \(f\) to be a nonnegative subharmonic function on this domain. If \(\Delta f\) has subharmonic behavior on \(\Omega\) (a notion explained in the paper, more or less this means that \(f\) satisfies some integral means assumptions), then \[ \int_{\Omega}\delta(x)^{\gamma}(\Delta f(x))^{p}dx\leq C\int_{\Omega}\delta(x)^{\gamma-2p+2\frac{p}{q}}(\Delta f(x)^q)^{\frac{p}{q}}dx, \] for all \(1\leq q\leq p <\infty\), \(\gamma \in \mathbb R\), and \(C\) depending only on \(p,q,\gamma\) and the domain \(\Omega\). The function \(\delta\) is the distance to the boundary. Moreover, if \(f^p\) is \(\mathcal C^{2}\) smooth and subharmonic, as well as \(\Delta f^{q}\) has subharmonic behavior, then \[ \int_{\Omega}\delta(x)^{\gamma}\Delta f(x)^pdx\leq C\int_{\Omega}\delta(x)^{\gamma-2+2\frac{p}{q}}(\Delta f(x)^q)^{\frac{p}{q}}dx \] (this time \(0<p\leq q<\infty\)).
Putting \(\gamma=2p-1\) and \(q=1\) with some additional assumptions on the Green function of \(\Omega\) as well as on the function \(f\), the author obtains \[ \int_{\Omega}\delta(x)^{2p-1}(\Delta f(x))^{p}dx\leq CN_{p}^{p}(f), \] where \(N_p(f)\) is the least harmonic majorant of \(f^p\) at a given point in \(\Omega\). This is a generalization of the aforementioned Littlewood-Paley inequality. A reverse inequality holds when \(0<p\leq 1\).
Similarly, for \(q\geq2\) one has \[ \int_{\Omega}\delta(x)^{q-1}|\nabla g(x)|^{q}dx\leq CN_{q}^{q}(f), \] when \(g\) is a nonnegative \(\mathcal C^2\) subharmonic function for which \(|\nabla g|\) has subharmonic behavior.
Similar inequalities involving the non-tangential maximal function of a function which has (together with its gradient) subharmonic behavior.
Also when the domain is bounded and Lipschitz, a version of the inequality is proved where the exponent over \(\delta(x)\) depends on the Lipschitz constant.
For the entire collection see [Zbl 1253.00015].

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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