Fourier series and delta-subharmonic functions on open semi-annulus

Abstract

We consider a class $SK(R)$ of subharmonic functions on the unbounded open semi--annulus
$$D_+(R)=\{z:\,|z|>R,\ Im\,z>0\},$$ which on each semi--annulus
$$D_+(R_1,R_2)=\{z:\, R<R_1<|z|<R_2<\infty,\ Im\,z>0\}$$
possesses a positive harmonic majorant. We introduce a class $JS(R)$ of subharmonic functions on $D_+(R)$, whose boundary values on the real boundary $D_+(R)$ are non--positive. We obtain some properties of functions in the classes $SK(R)$ and $JS(R)$. The class $\delta S(R)$ of delta--subharmonic functions on $D_+(R)$ is defined as the difference of classes $SK(R)$ or $JS(R)$:
$$\delta S(R)=SK(R)-SK(R)=JS(R)-JS(R).$$ For functions $v\in \delta S(R)$ we introduce a growth characteristic $T_R(r,v)$, which differs from the characteristics used for the functions defined on the upper half-plane. It determines the growth of function in the vicinity of semi-circumference $L_R=\{Re^{i\theta}:0\leq\theta\leq\pi\}$. For an arbitrary function of growth $\gamma$ (unbounded non--decreasing positive function defined on the real semi--axis $R_+=\{r:r>0\}$) we define the class $\delta S_{L_R}(R,\gamma)\subset\delta S(R)$ of delta--subharmonic functions $v$ of finite $\gamma$--type on $D_+(R)$ in the vicinity of circumference $L_R$ as
$$\displaystyle T_R(r,v)\leq A\gamma\left(\frac{B}{r-R}\right)$$ for all $R<r<2R$ and some positive $A$ and $B$ depending on $v$, but independent of $r$. We obtain the criterions of belonging of function $v$ to the class $\delta S_{L_R}(R,\gamma)$ in terms of its Fourier coefficients.