An asymptotic Cauchy problem for the Laplace equation

Abstract

The Cauchy problem for the Laplace operator $\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$ is modified by replacing the Laplace equation by an asymptotic estimate of the form $\begin{gathered} \Delta u(x,y) = 0, \hfill \\ u(x,0) = f(x),\frac{{\partial u}}{{\partial y}}(x,0) = g(x) \hfill \\ \end{gathered} $ with a given majorant h, satisfying h(+0)=0. This asymptotic Cauchy problem only requires that the Laplacian decay to zero at the initial submanifold. It turns out that this problem has a solution for smooth enough Cauchy data f, g, and this smoothness is strictly controlled by h. This gives a new approach to the study of smooth function spaces and harmonic functions with growth restrictions. As an application, a Levinson-type normality theorem for harmonic functions is proved.