We know that if $f:\mathbb{C}\to \mathbb{C}$ is a holomorphic function and $f$ vanishes on a set $E\subset \mathbb{R} \subset \mathbb{C}$ such that $E$ has a limit point in $\mathbb{R}$ then $f$ is identically zero.
The same is not true for multivariable case. E.g. $f(z_1,\dots z_n)=z_1$
For single variable case, the result remains true if we consider the set $E\subset \mathbb{R}$ with positive Lebesgue measure in $\mathbb{R}$ as a positive Lebesgue measure set is always uncountable and hence has a limit point.
My question: Is the result remains true for multivariable case if we consider $E\subset \mathbb{R} ^n\subset \mathbb{C}^n$ with positive Lebesgue measure in $\mathbb{R}^n$ i.e. if $f: \mathbb{C}^n\to \mathbb{C}$ is a holomorphic function vanishing on $E\subset \mathbb{R} ^n\subset \mathbb{C}^n$ with positive Lebesgue measure in $\mathbb{R}^n$, then is $f$ identically zero?
Best Answer
It's true and you can prove it easily by induction on the number of variables. Indeed, for sake of simplicity assume that we have only two variables. Let $f$ be a holomorphic function defined on $\mathbb{C}^2$. You can write \begin{equation*} \forall z, w \in \mathbb{C},\ f(z, w) = \sum_{j=0}^{\infty} a_j(z) w^j \end{equation*} where $a_j$ are entire functions. By using the Fubini theorem, we know there exists a set $F$ of $\mathbb{C}$ such that \begin{equation*} \lambda_1(F) > 0\ \mbox{ and }\ \forall z \in F,\ \lambda_1(\{ w \in \mathbb{C} \mid (z, w) \in E\}) > 0, \end{equation*} where $\lambda_1$ denotes the one dimensional Lebesgue measure. Thus for each $z \in F$, the power series $f_z(w) = f(z, w)$ vanishes on a set of positive measure. So, $f_z$ is identically zero for all $z \in F$ and its coefficients too. But its coefficients are holomorphic functions and vanish identically on $F$.
If you are interested in sets on which holomorphic functions of several variables can be identically zero, you have to check out the notion of capacity in non linear potential theory. Siciak, Bedford and Taylor defined a capacity $\mathrm{cap}$ in this context and proved that a set is of zero capacity if and only if there exists locally a plurisubharmonic function whose poles are exactly this set. Since $\log \|f\|$ is a plurisubharmonic function for any holomorphic function $f$ and its poles are the zeros of $f$, you deduce that $\mathrm{cap}(E) > 0$ implies $f = 0$.