Fischer decompositions for entire functions and the Dirichlet problem for parabolas

Abstract

Let $P_{2k}$ be a homogeneous polynomial of degree 2k and assume that there exist $C>0$, $D>0$ and $\alpha \ge 0$ such that

$$\begin{aligned} \left\langle P_{2k}f_{m},f_{m}\right\rangle _{L^2(\mathbb {S}^{d-1})}\ge \frac{1}{C\left( m+D\right) ^{\alpha }}\left\langle f_{m},f_{m}\right\rangle _{\mathbb {S}^{d-1}} \end{aligned}$$

for all homogeneous polynomials $f_{m}$ of degree m. Assume that $P_{j}$ for $j=0, \dots ,\beta <2k$ are homogeneous polynomials of degree j. The main result of the paper states that for any entire function f of order $\rho <\left( 2k-\beta \right) /\alpha$ there exist entire functions q and h of order bounded by $\rho$ such that

$$\begin{aligned} f=\left( P_{2k}-P_{\beta }- \dots -P_{0}\right) q+h\text { and }\Delta ^k r=0. \end{aligned}$$

This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem for parabola-shaped domains on the plane, with data given by entire functions of order smaller than $\frac{1}{2}$.