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.

Given positive integers$n$_{1}<$n$_{2}<... we show that the Hardy-type inequality $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ holds true for all$f$∈$H$^{1}, provided that the$n$_{$k$}'s, satisfy an appropriate (and indispensable) regularity condition. On the other hand, we exhibit inifinite-dimensional subspaces of$H$^{1}for whose elements the above inequality is always valid, no additional hypotheses being imposed. In conclusion, we extend a result of Douglas, Shapiro and Shields on the cyclicity of lacunary series for the backward shift operator.

Let$f$be an entire function of order at least 1/2,$M(r)$=max_{|}$z$|=$r$|$f(z)$|, and$n(r, a)$the number of zeros of$f(z)-a$in |$z$|≤$r$. It is shown that lim sup_{$r$→∞}$n(r, a)$/log$M (r)$≥1/2π for all except possibly one$a$∈C.

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a “two circles” Morera type theorem (earlier announced by Agranovskii).

A new elementary approach to uniform resolvent estimates of the Carleman-type is developed. Schatten-von Neumann's $$\mathfrak{S}_p $$ perturbations of self-adjoint and unitary operators are considered. Examples of typical growth are provided.