Counting dimensions of L-harmonic functions with exponential growth

Abstract

Let $$\Omega \subset {\mathbb {R}}^{n-1}$$ be a bounded open set, $$X=\Omega \times {\mathbb {R}}\subseteq {\mathbb {R}}^{n}$$ be the infinite strip. Let L be a second order uniformly elliptic operator of divergence form acting on a function $$f\in W_{\text {loc}}^{1,2}(X)$$ given by $$Lf=\sum _{i,j=1}^{n}\frac{\partial }{\partial x_{i}}\bigl (a^{ij}(x)\frac{\partial f}{\partial x_{j}}\bigr )$$. It is natural to consider the solutions of $$Lu=0$$ with boundary value $$u|_{\partial \Omega \times {\mathbb {R}}}=0$$ and exponential growth at most d: $$|u(x',x_{n})|\le {\tilde{C}}e^{d|x_{n}|}$$ for some $${\tilde{C}}>0$$. Denote by $${\mathcal {A}}_{d}$$ the solution space. In (Acta Math Sin (Engl Ser)15:525–534, 1999), Hang and Lin proved that $$\text {dim}{\mathcal {A}}_{d}\le Cd^{n-1}$$. The power $$n-1$$ is sharp, but one may wonder whether there are more precise estimates for the constant C. In this note, we consider some natural subspaces of $${\mathcal {A}}_{d}$$ and obtain some estimates of dimensions of these subspaces. Compared with the case $$L=\Delta _{X}$$, when d is sufficiently large, the estimates obtained in this note are sharp both on the power $$n-1$$ and the constant C.