Polar set

The polar set of analytic function $f(z)$ of the complex variable $z=(z_1,\dots,z_n)$, $n\geq1$, is the set $P$ of points in some domain $D$ of the complex space $\mathbf C^n$ for which: a) $f(z)$ is holomorphic everywhere in $D\setminus P$; b) $f(z)$ cannot be analytically continued to any point of $P$; and c) for every point $a\in P$ there are a neighbourhood $U_a$ and a function $q_a(z)\not\equiv0$, holomorphic in $U_a$, for which the function $p_a(z)=q_a(z)f(z)$, which is holomorphic in $D\cap\{U_a\setminus P\}$, can be holomorphically continued to $U_a$. At every point $a\in P$ one has $q_a(a)=0$. The polar set $P$ consists of the poles (cf. Pole (of a function)) $a\in P$ of $f(z)$, for which $p_a(a)\neq0$, and the points $a\in P$ of indeterminacy of $f(z)$, for which $p_a(a)=0$ (it is assumed that $p_a(z)$ and $q_a(z)$ have no common factors that are holomorphic and vanish at $a$). Every polar set is a complex analytic variety (by which we mean the set of common zeros of a finite set of holomorphic functions) of complex dimension $n-1$.

A polar set in potential theory is a set $E$ of points of the Euclidean space $\mathbf R^n$, $n\geq2$, for which there exists a potential $U_\mu(x)$, $x\in\mathbf R^n$, for some Borel measure $\mu$, that takes the value $+\infty$ at the points of $E$ and only at those points.

In the case of the logarithmic potential for $n=2$ and the Newton potential for $n\geq3$, for a bounded set $E$ to be a polar set it is necessary and sufficient that $E$ is a set of type $G_\delta$ and has zero outer capacity. Here, in the definition of a polar set, one can replace "potential" by "superharmonic function". The main properties of polar sets in this case are: a) the set $\{a\}$ which consists of a single point $a\in\mathbf R^n$ is a polar set; b) a countable union of polar sets is a polar set; c) any polar set has Lebesgue measure zero in $\mathbf R^n$; and d) under a conformal mapping a polar set goes to a polar set.

For a local criterion for being polar see Thinness of a set.

References

[1]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[2]	N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[3]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)

Comments

A set $E$ as described under 2) is usually called a complete polar set. A (not necessarily complete) polar set is defined as a subset of a complete polar set. A bounded set is polar if and only if it has zero outer capacity.

The sets described under 1) are also called pole sets, or sets of poles, which avoids confusion, cf. [a4] and Meromorphic function.

In parabolic potential theory, a set $A$ is polar if and only if there exists an open covering $\mathcal W$ of $A$ and, for any $V\in\mathcal W$, a positive supercaloric function $u_V$ on $V$ such that $u_V=\infty$ on $A\cap V$, see [a3]. Again, points are polar and a countable union of polar sets is polar. Any polar set is totally thin but, in contrast with classical potential theory, not every totally thin set is polar. A similar theory of polarity holds in harmonic spaces, see [a2], or in the more general case of balayage spaces, cf. [a1]. In probabilistic potential theory, a Borel set is polar if its first hitting time $T_A$ satisfies $T_A=\infty$ a.s.

References

[a1]	J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a2]	C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
[a3]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1983)
[a4]	H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) (Translated from German)
[a5]	H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Chapt. 8

Comment

For polar sets in convex geometry, see polar body.