Beurling's theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations (Q1946419)

The authors consider functions from homogeneous spaces of functions defined on a locally compact Abelian group. The notion of Beurling spectrum, or essential spectrum, of functions is introduced. If a continuous unitary character is an essential point of the spectrum of a function, then it is the c-limit of a linear combination of shifts of the function in question. The notion of a slowly varying function at infinity is introduced, and properties of such functions are considered. The authors consider the Cauchy problem \[ \frac{\partial x}{\partial t}=\Delta x, \qquad x(0,s)=x_0(s), \;s\in {\mathbb R}^n, \] with initial function \(x_0\) from a homogeneous space. It is proved that the weak solution as a function of the first argument is a slowly varying function at infinity.