§ 2. Cauchy’s Integral Formulas
39
C
∞ functions with compact support in G. Like on R (Chapter V, § 10), it
can be used to define distributions on G. For all r ∈ N, define a semi-norm
on D(G) (Appendix of Chap. III, end of no 8)
N r (ϕ) =
p,q≤r
sup
z∈G
|D
p
1 D
q
2 ϕ(z)| =
D
p
1 D
q
2 ϕ G ,
where D 1 and D 2 are the differential operators with respect to the real coordinates x, y of z. Writing D(G, K) for the subspace of the functions ϕ ∈ D(G)
which vanish outside a given compact set K ⊂ G, a distribution of G is then
a linear functional ϕ → T (ϕ) in D(G), continuous in the following sense : for
any compact set K ⊂ G, there exists r ∈ N and a constant M K (T ) ≥ 0 such
that
|T (ϕ)| ≤ M K (T )N r (ϕ) for all ϕ ∈ D(G, K) .
Like on R, the successive derivatives of the distributions can be defined by
iterating the formulas defining the first derivatives
D 1 T : ϕ −→ −T (D 1 ϕ) , D 2 T : ϕ −→ −T (D 2 ϕ)
of T . If T is defined by a function f of class C
∞ on G, i.e. if
T (ϕ) =
ϕ(z)f (z)dm(z) ,
it can be immediately checked that D i T is defined by a function D i f : like over
R, integrate by parts
29 with respect to x or y. The notion of a holomorphic
function can then be generalized by saying that a distribution T on G is
holomorphic if it satisfies Cauchy’s condition
D 2 T = iD 1 T
or, equivalently, if T vanishes for all functions of the form ∂ϕ/∂ ¯
z.
Having set this, (i) any holomorphic distribution is defined by a holomorphic function, in other words this is not a generalization; (ii) if a sequence T n
of holomorphic distributions converges to a distribution T , i.e. if
lim T n (ϕ) = T (ϕ) for all ϕ ∈ D(G) ,
then T is holomorphic (obvious from the definition); (iii) if f n and f are
holomorphic functions defining T n and T , then f n (z) converges uniformly to
f (z) on any compact subset of G.
Regardingholomorphic functions, any definition of convergence more restrictive than of convergence in the sense of distributions implies, therefore,
29 As ϕ is zero outside a compact subset K ⊂ G, the function under the
sign
is the restriction to G of a C
∞ function on R
2 and vanishes outside K, hence
outside a square I × I, where I is a compact interval of R.
39
C
∞ functions with compact support in G. Like on R (Chapter V, § 10), it
can be used to define distributions on G. For all r ∈ N, define a semi-norm
on D(G) (Appendix of Chap. III, end of no 8)
N r (ϕ) =
p,q≤r
sup
z∈G
|D
p
1 D
q
2 ϕ(z)| =
D
p
1 D
q
2 ϕ G ,
where D 1 and D 2 are the differential operators with respect to the real coordinates x, y of z. Writing D(G, K) for the subspace of the functions ϕ ∈ D(G)
which vanish outside a given compact set K ⊂ G, a distribution of G is then
a linear functional ϕ → T (ϕ) in D(G), continuous in the following sense : for
any compact set K ⊂ G, there exists r ∈ N and a constant M K (T ) ≥ 0 such
that
|T (ϕ)| ≤ M K (T )N r (ϕ) for all ϕ ∈ D(G, K) .
Like on R, the successive derivatives of the distributions can be defined by
iterating the formulas defining the first derivatives
D 1 T : ϕ −→ −T (D 1 ϕ) , D 2 T : ϕ −→ −T (D 2 ϕ)
of T . If T is defined by a function f of class C
∞ on G, i.e. if
T (ϕ) =
ϕ(z)f (z)dm(z) ,
it can be immediately checked that D i T is defined by a function D i f : like over
R, integrate by parts
29 with respect to x or y. The notion of a holomorphic
function can then be generalized by saying that a distribution T on G is
holomorphic if it satisfies Cauchy’s condition
D 2 T = iD 1 T
or, equivalently, if T vanishes for all functions of the form ∂ϕ/∂ ¯
z.
Having set this, (i) any holomorphic distribution is defined by a holomorphic function, in other words this is not a generalization; (ii) if a sequence T n
of holomorphic distributions converges to a distribution T , i.e. if
lim T n (ϕ) = T (ϕ) for all ϕ ∈ D(G) ,
then T is holomorphic (obvious from the definition); (iii) if f n and f are
holomorphic functions defining T n and T , then f n (z) converges uniformly to
f (z) on any compact subset of G.
Regardingholomorphic functions, any definition of convergence more restrictive than of convergence in the sense of distributions implies, therefore,
29 As ϕ is zero outside a compact subset K ⊂ G, the function under the
sign
is the restriction to G of a C
∞ function on R
2 and vanishes outside K, hence
outside a square I × I, where I is a compact interval of R.