Equivalent norms in some spaces of analytic functions and the uncertainty principle

The main object of this work is to describe such weight functions w(t) that for all elements ${\displaystyle f\in L_{p,\Omega }}$ the estimate ${\displaystyle \parallel wf{\parallel }_p\ge K(\Omega )\parallel f{\parallel }_p}$ is valid with a constant K(Ω), which does not depend on f and it grows to infinity when the domain Ω shrinks, i.e. deforms into a lower dimensional convex set ${\displaystyle {\Omega }_{\infty }}$. In one-dimensional case means that ${\displaystyle K(\sigma ):=K\left({\Omega }_{\sigma }\right)\to \infty }$ as σ → 0. It should be noted that in the framework of the signal transmission problem such estimates describe a signal's behavior under the influence of detection and amplification. This work contains some results of the above-mentioned type which I presented at the Banach Centre in the Summer of 1994. Some of these results had been published earlier, some are new ones.