In mathematics, the Dirichlet space on the domain Ω ⊆ C , D ( Ω ) (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space H 2 ( Ω ) , for which the Dirichlet integral, defined by
D ( f ) := 1 π ∬ Ω | f ′ ( z ) | 2 d A = 1 4 π ∬ Ω | ∂ x f | 2 + | ∂ y f | 2 d x d y is finite (here dA denotes the area Lebesgue measure on the complex plane C ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on D ( Ω ) . It is not a norm in general, since D ( f ) = 0 whenever f is a constant function.
For f , g ∈ D ( Ω ) , we define
D ( f , g ) := 1 π ∬ Ω f ′ ( z ) g ′ ( z ) ¯ d A ( z ) . This is a semi-inner product, and clearly D ( f , f ) = D ( f ) . We may equip D ( Ω ) with an inner product given by
⟨ f , g ⟩ D ( Ω ) := ⟨ f , g ⟩ H 2 ( Ω ) + D ( f , g ) ( f , g ∈ D ( Ω ) ) , where ⟨ ⋅ , ⋅ ⟩ H 2 ( Ω ) is the usual inner product on H 2 ( Ω ) . The corresponding norm ∥ ⋅ ∥ D ( Ω ) is given by
∥ f ∥ D ( Ω ) 2 := ∥ f ∥ H 2 ( Ω ) 2 + D ( f ) ( f ∈ D ( Ω ) ) . Note that this definition is not unique, another common choice is to take ∥ f ∥ 2 = | f ( c ) | 2 + D ( f ) , for some fixed c ∈ Ω .
The Dirichlet space is not an algebra, but the space D ( Ω ) ∩ H ∞ ( Ω ) is a Banach algebra, with respect to the norm
∥ f ∥ D ( Ω ) ∩ H ∞ ( Ω ) := ∥ f ∥ H ∞ ( Ω ) + D ( f ) 1 / 2 ( f ∈ D ( Ω ) ∩ H ∞ ( Ω ) ) .
We usually have Ω = D (the unit disk of the complex plane C ), in that case D ( D ) := D , and if
f ( z ) = ∑ n ≥ 0 a n z n ( f ∈ D ) , then
D ( f ) = ∑ n ≥ 1 n | a n | 2 , and
∥ f ∥ D 2 = ∑ n ≥ 0 ( n + 1 ) | a n | 2 . Clearly, D contains all the polynomials and, more generally, all functions f , holomorphic on D such that f ′ is bounded on D .
The reproducing kernel of D at w ∈ C ∖ { 0 } is given by
k w ( z ) = 1 z w ¯ log ( 1 1 − z w ¯ ) ( z ∈ C ∖ { 0 } ) . 