Hartogs's extension theorem

From testwiki
Jump to navigation Jump to search

Template:Short description Template:Redirect

In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be compact, therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an isolated singularity is always a removable singularity for any analytic function of Template:Math complex variables. A first version of this theorem was proved by Friedrich Hartogs,[1] and as such it is known also as Hartogs's lemma and Hartogs's principle: in earlier Soviet literature,[2] it is also called the Osgood–Brown theorem, acknowledging later work by Arthur Barton Brown and William Fogg Osgood.[3] This property of holomorphic functions of several variables is also called Hartogs's phenomenon: however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of systems of partial differential or convolution equations satisfying Hartogs-type theorems.[4]

Historical note

The original proof was given by Friedrich Hartogs in 1906, using Cauchy's integral formula for functions of several complex variables.[1] Today, usual proofs rely on either the Bochner–Martinelli–Koppelman formula or the solution of the inhomogeneous Cauchy–Riemann equations with compact support. The latter approach is due to Leon Ehrenpreis who initiated it in the paper Template:Harv. Yet another very simple proof of this result was given by Gaetano Fichera in the paper Template:Harv, by using his solution of the Dirichlet problem for holomorphic functions of several variables and the related concept of CR-function:[5] later he extended the theorem to a certain class of partial differential operators in the paper Template:Harv, and his ideas were later further explored by Giuliano Bratti.[6] Also the Japanese school of the theory of partial differential operators worked much on this topic, with notable contributions by Akira Kaneko.[7] Their approach is to use Ehrenpreis's fundamental principle.

Hartogs's phenomenon

For example, in two variables, consider the interior domain

Hε={z=(z1,z2)∈Δ2:|z1|<ε  or  1−ε<|z2|}

in the two-dimensional polydisk Δ2={z∈ℂ2;|z1|<1,|z2|<1} where 0<ε<1.

Theorem Template:Harvtxt: Any holomorphic function f on Hε can be analytically continued to Δ2. Namely, there is a holomorphic function F on Δ2 such that F=f on Hε.

Such a phenomenon is called Hartogs's phenomenon, which lead to the notion of this Hartogs's extension theorem and the domain of holomorphy.

Formal statement and proof

Let Template:Mvar be a holomorphic function on a set Template:Math, where Template:Mvar is an open subset of Template:Math (Template:Math) and Template:Mvar is a compact subset of Template:Mvar. If the complement Template:Math is connected, then Template:Mvar can be extended to a unique holomorphic function Template:Mvar on Template:Mvar.Template:Sfnm

Ehrenpreis' proof is based on the existence of smooth bump functions, unique continuation of holomorphic functions, and the Poincaré lemma — the last in the form that for any smooth and compactly supported differential (0,1)-form Template:Mvar on Template:Math with Template:Math, there exists a smooth and compactly supported function Template:Mvar on Template:Math with Template:Math. The crucial assumption Template:Math is required for the validity of this Poincaré lemma; if Template:Math then it is generally impossible for Template:Mvar to be compactly supported.Template:Sfnm

The ansatz for Template:Mvar is Template:Math for smooth functions Template:Mvar and Template:Mvar on Template:Mvar; such an expression is meaningful provided that Template:Mvar is identically equal to zero where Template:Mvar is undefined (namely on Template:Mvar). Furthermore, given any holomorphic function on Template:Mvar which is equal to Template:Mvar on some open set, unique continuation (based on connectedness of Template:Math) shows that it is equal to Template:Mvar on all of Template:Math.

The holomorphicity of this function is identical to the condition Template:Math. For any smooth function Template:Mvar, the differential (0,1)-form Template:Math is Template:Math-closed. Choosing Template:Mvar to be a smooth function which is identically equal to zero on Template:Mvar and identically equal to one on the complement of some compact subset Template:Mvar of Template:Mvar, this (0,1)-form additionally has compact support, so that the Poincaré lemma identifies an appropriate Template:Mvar of compact support. This defines Template:Mvar as a holomorphic function on Template:Mvar; it only remains to show (following the above comments) that it coincides with Template:Mvar on some open set.

On the set Template:Math, Template:Mvar is holomorphic since Template:Mvar is identically constant. Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of Template:Math.[8] Thus, on this open subset, Template:Mvar equals Template:Mvar and the existence part of Hartog's theorem is proved. Uniqueness is automatic from unique continuation, based on connectedness of Template:Mvar.

Counterexamples in dimension one

The theorem does not hold when Template:Math. To see this, it suffices to consider the function Template:Math, which is clearly holomorphic in Template:Math but cannot be continued as a holomorphic function on the whole of Template:Math. Therefore, the Hartogs's phenomenon is an elementary phenomenon that highlights the difference between the theory of functions of one and several complex variables.

Notes

Template:Reflist

References

Template:Refbegin

Historical references

Scientific references

Template:Refend

  1. ↑ 1.0 1.1 See the original paper of Template:Harvtxt and its description in various historical surveys by Template:Harvtxt, Template:Harvtxt and Template:Harvtxt. In particular, in this last reference on p. 132, the Author explicitly writes :-"As it is pointed out in the title of Template:Harv, and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula".
  2. ↑ See for example Template:Harvtxt, which refers the reader to the book of Template:Harvtxt for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).
  3. ↑ See Template:Harvtxt and Template:Harvtxt.
  4. ↑ See Template:Harvtxt and Template:Harvtxt Template:Harv.
  5. ↑ Fichera's proof as well as his epoch making paper Template:Harv seem to have been overlooked by many specialists of the theory of functions of several complex variables: see Template:Harvtxt for the correct attribution of many important theorems in this field.
  6. ↑ See Template:Harvtxt Template:Harv.
  7. ↑ See his paper Template:Harv and the references therein.
  8. ↑ Any connected component of Template:Math must intersect Template:Math in a nonempty open set. To see the nonemptiness, connect an arbitrary point Template:Mvar of Template:Math to some point of Template:Mvar via a line. The intersection of the line with Template:Math may have many connected components, but the component containing Template:Mvar gives a continuous path from Template:Mvar into Template:Math.
Retrieved from "https://en.wiki.beta.math.wmflabs.org/w/index.php?title=Hartogs%27s_extension_theorem&oldid=28843"