Complex Analysis in Several Variables, MTH 3108
Home Assignement 3
Professor M.Shubin
Fall 1997


Textbook:

Introduction to Complex Analysis in Several Variables, by Lars Hörmander. 3rd edition, North Holland, 1990


1. Find out whether the following domain $\Omega\subset\hbox{{\bbb C}}^2$is a domain of holomorphy or not:

\begin{displaymath} \Omega=\{(z_1,z_2)\vert\;z_1=x_1+iy_1,\ z_2=x_2+iy_2,\ \vert x_1\vert<1\ \hbox{or}\ \vert y_2\vert<1 \}\;.\end{displaymath}


2. Find out whether the following domain $\Omega\subset\hbox{{\bbb C}}^2$is pseudoconvex or not:

\begin{displaymath} \Omega=\{(z_1,z_2)\vert\;z_1=x_1+iy_1,\ z_2=x_2+iy_2,\ \vert... ...\vert^2+\vert z_2\vert^2<1\ \hbox{or}\ \vert y_1\vert<1/2 \}\;.\end{displaymath}


3. In $\hbox{{\bbb C}}$ calculate ${\partial\over\partial\bar{z}}\bigl({1\over z}\bigr)$ in the sense of distributions.


4. For any function $f\in L^1(S^1)$ prove that its Fourier series converges to this function in the sense of distributions.


5. Let $T: {\cal H}\to {\cal H}$ be a closed densely defined operator in a Hilbert space ${\cal H}$. Prove that the operator A=T*T is self-adjoint i.e. it is densely defined and A*=A. Here the equality A*=A means that the operators A and A* have the same domain (i.e. DA=DA*) and they coincide on their common domain.


 
Sergey Bratus
11/8/1997