4
$\begingroup$

Let $D$ be a (real) domain in $\mathbb R^n=\mathbb R^n+i\lbrace 0 \rbrace\subset \mathbb C^n$. Then, due to P. Lelong, there exists a maximal (complex) domain $\tilde D\subset\mathbb C^n$, $D=\tilde D\cap(\mathbb R^n+i\lbrace 0 \rbrace )$, such that for every harmonic function $h$ on $D$ there exists a holomorphic function $\tilde h$ on $\tilde D$ such that $\tilde h|_D=h$. $\tilde D$ is called the harmonic envelope of holomorphy for $D$. If $D=\mathbb B\subset\mathbb R^n$, $\mathbb B$ the real unit ball, then $\tilde D$ is known to be the Lie-ball (see the works of M. Jarnicki and J. Siciak). Now fix $p=(p_1,\dots,p_n)\in(0,\infty)^n$ and let $E(p):= \lbrace x\in\mathbb R^n:|x_1|^{2p_1}+\cdots +|x_n|^{2p_n}<1 \rbrace$. Note that if $(1,\dots,1)=:p^\ast$, then $E(p^\ast)=\mathbb B$.

Question: is there an explicit description of $\widetilde{E(p)}$ ?

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.