An asymptotic Cauchy problem for the Laplace equation

The Cauchy problem for the Laplace operator $$sumlimits_{k = 1}^infty {frac{{left| {hat f(n_k )} right|}}{k}} leqslant constleft| f right|1$$ 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 majoranth, satisfyingh(+0)=0. Thisasymptotic 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 dataf, g, and this smoothness is strictly controlled byh. 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.