Fischer decompositions for entire functions and the Dirichlet problem for parabolas

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{equation*} \left\langle P_{2k}f_{m},f_{m}\right\rangle_{L^2($\mathbb{S}$^{d-1})}\geq \frac{1}{C\left( m+D\right) ^{α}}\left\langle f_{m},f_{m}\right\rangle_{$\mathbb{S}$^{d-1}} \end{equation*} for all homogeneous polynomials $f_m$ of degree $m\mathrm{.}$ 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 $$ there exist entire functions $q$ and $h$ of order bounded by $\rho$ such that \begin{equation*} f=\left( P_{2k}-P_{β}- \dots -P_{0}\right) q+h$\text{ and }$Δ^{h}r=0. \end{equation*} 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 $\frac{1}{2}$.