Let Ω⊂Rn-1\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {R}}^{n-1}$$\end{document} be a bounded open set, X=Ω×R⊆Rn\documentclass[12pt]{minimal}
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\begin{document}$$X=\Omega \times {\mathbb {R}}\subseteq {\mathbb {R}}^{n}$$\end{document} be the infinite strip. Let L be a second order uniformly elliptic operator of divergence form acting on a function f∈Wloc1,2(X)\documentclass[12pt]{minimal}
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\begin{document}$$f\in W_{\text {loc}}^{1,2}(X)$$\end{document} given by Lf=∑i,j=1n∂∂xi(aij(x)∂f∂xj)\documentclass[12pt]{minimal}
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\begin{document}$$Lf=\sum _{i,j=1}^{n}\frac{\partial }{\partial x_{i}}\bigl (a^{ij}(x)\frac{\partial f}{\partial x_{j}}\bigr )$$\end{document}. It is natural to consider the solutions of Lu=0\documentclass[12pt]{minimal}
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\begin{document}$$Lu=0$$\end{document} with boundary value u|∂Ω×R=0\documentclass[12pt]{minimal}
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\begin{document}$$u|_{\partial \Omega \times {\mathbb {R}}}=0$$\end{document} and exponential growth at most d: |u(x′,xn)|≤C~ed|xn|\documentclass[12pt]{minimal}
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\begin{document}$$|u(x',x_{n})|\le {\tilde{C}}e^{d|x_{n}|}$$\end{document} for some C~>0\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{C}}>0$$\end{document}. Denote by Ad\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}_{d}$$\end{document} the solution space. In (Acta Math Sin (Engl Ser)15:525–534, 1999), Hang and Lin proved that dimAd≤Cdn-1\documentclass[12pt]{minimal}
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\begin{document}$$\text {dim}{\mathcal {A}}_{d}\le Cd^{n-1}$$\end{document}. The power n-1\documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document} 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 Ad\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}_{d}$$\end{document} and obtain some estimates of dimensions of these subspaces. Compared with the case L=ΔX\documentclass[12pt]{minimal}
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\begin{document}$$L=\Delta _{X}$$\end{document}, when d is sufficiently large, the estimates obtained in this note are sharp both on the power n-1\documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document} and the constant C.
