Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is below the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the boundary of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also inside the ball.
Superharmonic functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions.
- 1 Formal definition
- 2 Properties
- 3 Examples
- 4 Riesz Representation Theorem
- 5 Subharmonic functions in the complex plane
- 6 Subharmonic functions on Riemannian manifolds
- 7 See also
- 8 Notes
- 9 References
Formally, the definition can be stated as follows. Let be a subset of the Euclidean space and let
be an upper semi-continuous function. Then, is called subharmonic if for any closed ball of center and radius contained in and every real-valued continuous function on that is harmonic in and satisfies for all on the boundary of we have for all
Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.
A function is called superharmonic if is subharmonic.
- A function is harmonic if and only if it is both subharmonic and superharmonic.
- If is C2 (twice continuously differentiable) on an open set in , then is subharmonic if and only if one has on , where is the Laplacian.
- The maximum of a subharmonic function cannot be achieved in the interior of its domain unless the function is constant, this is the so-called maximum principle. However, the minimum of a subharmonic function can be achieved in the interior of its domain.
- Subharmonic functions make a convex cone, that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic.
- The pointwise maximum of two subharmonic functions is subharmonic.
- The limit of a decreasing sequence of subharmonic functions is subharmonic (or identically equal to ).
- Subharmonic functions are not necessarily continuous in the usual topology, however one can introduce the fine topology which makes them continuous.
If is analytic then is subharmonic. More examples can be constructed by using the properties listed above, by taking maxima, convex combinations and limits. In dimension 1, all subharmonic functions can be obtained in this way.
Riesz Representation Theorem
If is subharmonic in a region , in Euclidean space of dimension , is harmonic in , and , then is called a harmonic majorant of . If a harmonic majorant exists, then there exists the least harmonic majorant, and
while in dimension 2,
Subharmonic functions in the complex plane
One can show that a real-valued, continuous function of a complex variable (that is, of two real variables) defined on a set is subharmonic if and only if for any closed disc of center and radius one has
If is a holomorphic function, then
is a subharmonic function if we define the value of at the zeros of to be −∞. It follows that
is subharmonic for every α > 0. This observation plays a role in the theory of Hardy spaces, especially for the study of Hp when 0 < p < 1.
In the context of the complex plane, the connection to the convex functions can be realized as well by the fact that a subharmonic function on a domain that is constant in the imaginary direction is convex in the real direction and vice versa.
Harmonic majorants of subharmonic functions
Subharmonic functions in the unit disc. Radial maximal function
Let φ be subharmonic, continuous and non-negative in an open subset Ω of the complex plane containing the closed unit disc D(0, 1). The radial maximal function for the function φ (restricted to the unit disc) is defined on the unit circle by
If Pr denotes the Poisson kernel, it follows from the subharmonicity that
It can be shown that the last integral is less than the value at e iθ of the Hardy–Littlewood maximal function φ∗ of the restriction of φ to the unit circle T,
so that 0 ≤ M φ ≤ φ∗. It is known that the Hardy–Littlewood operator is bounded on Lp(T) when 1 < p < ∞. It follows that for some universal constant C,
If f is a function holomorphic in Ω and 0 < p < ∞, then the preceding inequality applies to φ = |f | p/2. It can be deduced from these facts that any function F in the classical Hardy space Hp satisfies
With more work, it can be shown that F has radial limits F(e iθ) almost everywhere on the unit circle, and (by the dominated convergence theorem) that Fr, defined by Fr(e iθ) = F(r e iθ) tends to F in Lp(T).
Subharmonic functions on Riemannian manifolds
Subharmonic functions can be defined on an arbitrary Riemannian manifold.
Definition: Let M be a Riemannian manifold, and an upper semicontinuous function. Assume that for any open subset , and any harmonic function f1 on U, such that on the boundary of U, the inequality holds on all U. Then f is called subharmonic.
- Rosenblum, Marvin; Rovnyak, James (1994), p.35 (see References)
- Greene, R. E.; Wu, H. (1974). "Integrals of subharmonic functions on manifolds of nonnegative curvature". Inventiones Mathematicae. 27 (4): 265–298. doi:10.1007/BF01425500., MR0382723
- Conway, John B. (1978). Functions of one complex variable. New York: Springer-Verlag. ISBN 0-387-90328-3.
- Krantz, Steven G. (1992). Function Theory of Several Complex Variables. Providence, Rhode Island: AMS Chelsea Publishing. ISBN 0-8218-2724-3.
- Doob, Joseph Leo (1984). Classical Potential Theory and Its Probabilistic Counterpart. Berlin Heidelberg New York: Springer-Verlag. ISBN 3-540-41206-9.
- Rosenblum, Marvin; Rovnyak, James (1994). Topics in Hardy classes and univalent functions. Birkhauser Advanced Texts: Basel Textbooks. Basel: Birkhauser Verlag.