Monday Analysis Seminar: On Generalizations of the Littlewood-Paley Inequalities for Subharmonic Functions on Domains in $\mathbb R^n\, (n\ge 2)$

2011-12-9 14:45 - 2011-12-9 16:15
Faculty of Science Buliding #3 Room 210
Manfred Stoll (University of South Carolina)
For the unit disc $\mathbb D$ in $\mathbb C$, the classical
Littlewood-Paley Inequalities (1936) are as follows: Let $h$ be harmonic on
$\mathbb D$. There
exists a positive constant $C$, independent of $h$, such that for all
$p,\, 1<p\le 2$,
\sup_{0<r<1} \int_0^{2\pi}|h(re^{i\theta})|^pd\theta \le C_1 \left[|h(0)|^p +
\iint_{\mathbb D}(1-|z|)^{p-1}|\nabla h(z)|^pdx\,dy\right],
with the reverse inequality valid for all $p \ge 2$.
By a 1956 result of T. M. Flett it follows that (\ref{eq1}) is also valid for all $p,\, 0<p\le 1$.

In the talk we will explain why \eqref{eq1} is not the correct formulation of the Littlewood--Paley inequality for harmonic functions when $0<p\le 1$, and present the correct formulation of the result for bounded domains in $\mathbb R^n,\, n\ge 2$.  Specifically, we will prove the following.

Theorem. Let $\Omega\subset\mathbb R^n$ be a bounded domain with $C^{1,1} $ boundary, and let $f $ be a $C^1$ function on $\Omega$ for which $|f|$ and $|\grad f|$ have subharmonic behavior on
$\Omega$. Then for $0<p\le 1$,  $\alpha >1$, and $t_o\in\Omega$ fixed, there exists a constant $C_\alpha$, independent of $f$, such that
\int_{\partial\Omega} (M_{\alpha}f)^p(\zeta)\, ds(\zeta)  \le C_\alpha \left[ |f(t_o)|^p + \int_{\Omega} \delta(y)^{p-1} |\grad f(y)|^p dy \right]
where $M_{\alpha} f$ is the non--tangential maximal function of $f$.
We will also consider extensions of the Littlewood--Paley inequalities for non--negative subharmonic functions $f$ for which  $\Delta f$  is  subharmonic or has subharmonic behavior.