In mathematics , the Dirichlet space on the domain Ω ⊆ C , D ( Ω ) {\displaystyle \Omega \subseteq \mathbb {C} ,\,{\mathcal {D}}(\Omega )} (named after Peter Gustav Lejeune Dirichlet ), is the reproducing kernel Hilbert space of holomorphic functions , contained within the Hardy space H 2 ( Ω ) {\displaystyle H^{2}(\Omega )} , 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 {\displaystyle {\mathcal {D}}(f):={1 \over \pi }\iint _{\Omega }|f^{\prime }(z)|^{2}\,dA={1 \over 4\pi }\iint _{\Omega }|\partial _{x}f|^{2}+|\partial _{y}f|^{2}\,dx\,dy} is finite (here dA denotes the area Lebesgue measure on the complex plane C {\displaystyle \mathbb {C} } ). The latter is the integral occurring in Dirichlet's principle for harmonic functions . The Dirichlet integral defines a seminorm on D ( Ω ) {\displaystyle {\mathcal {D}}(\Omega )} . It is not a norm in general, since D ( f ) = 0 {\displaystyle {\mathcal {D}}(f)=0} whenever f is a constant function .
For f , g ∈ D ( Ω ) {\displaystyle f,\,g\in {\mathcal {D}}(\Omega )} , we define
D ( f , g ) := 1 π ∬ Ω f ′ ( z ) g ′ ( z ) ¯ d A ( z ) . {\displaystyle {\mathcal {D}}(f,\,g):={1 \over \pi }\iint _{\Omega }f'(z){\overline {g'(z)}}\,dA(z).} This is a semi-inner product, and clearly D ( f , f ) = D ( f ) {\displaystyle {\mathcal {D}}(f,\,f)={\mathcal {D}}(f)} . We may equip D ( Ω ) {\displaystyle {\mathcal {D}}(\Omega )} with an inner product given by
⟨ f , g ⟩ D ( Ω ) := ⟨ f , g ⟩ H 2 ( Ω ) + D ( f , g ) ( f , g ∈ D ( Ω ) ) , {\displaystyle \langle f,g\rangle _{{\mathcal {D}}(\Omega )}:=\langle f,\,g\rangle _{H^{2}(\Omega )}+{\mathcal {D}}(f,\,g)\;\;\;\;\;(f,\,g\in {\mathcal {D}}(\Omega )),} where ⟨ ⋅ , ⋅ ⟩ H 2 ( Ω ) {\displaystyle \langle \cdot ,\,\cdot \rangle _{H^{2}(\Omega )}} is the usual inner product on H 2 ( Ω ) . {\displaystyle H^{2}(\Omega ).} The corresponding norm ‖ ⋅ ‖ D ( Ω ) {\displaystyle \|\cdot \|_{{\mathcal {D}}(\Omega )}} is given by
‖ f ‖ D ( Ω ) 2 := ‖ f ‖ H 2 ( Ω ) 2 + D ( f ) ( f ∈ D ( Ω ) ) . {\displaystyle \|f\|_{{\mathcal {D}}(\Omega )}^{2}:=\|f\|_{H^{2}(\Omega )}^{2}+{\mathcal {D}}(f)\;\;\;\;\;(f\in {\mathcal {D}}(\Omega )).} Note that this definition is not unique, another common choice is to take ‖ f ‖ 2 = | f ( c ) | 2 + D ( f ) {\displaystyle \|f\|^{2}=|f(c)|^{2}+{\mathcal {D}}(f)} , for some fixed c ∈ Ω {\displaystyle c\in \Omega } .
The Dirichlet space is not an algebra , but the space D ( Ω ) ∩ H ∞ ( Ω ) {\displaystyle {\mathcal {D}}(\Omega )\cap H^{\infty }(\Omega )} 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 ) ) . {\displaystyle \|f\|_{{\mathcal {D}}(\Omega )\cap H^{\infty }(\Omega )}:=\|f\|_{H^{\infty }(\Omega )}+{\mathcal {D}}(f)^{1/2}\;\;\;\;\;(f\in {\mathcal {D}}(\Omega )\cap H^{\infty }(\Omega )).} We usually have Ω = D {\displaystyle \Omega =\mathbb {D} } (the unit disk of the complex plane C {\displaystyle \mathbb {C} } ), in that case D ( D ) := D {\displaystyle {\mathcal {D}}(\mathbb {D} ):={\mathcal {D}}} , and if
f ( z ) = ∑ n ≥ 0 a n z n ( f ∈ D ) , {\displaystyle f(z)=\sum _{n\geq 0}a_{n}z^{n}\;\;\;\;\;(f\in {\mathcal {D}}),} then
D ( f ) = ∑ n ≥ 1 n | a n | 2 , {\displaystyle D(f)=\sum _{n\geq 1}n|a_{n}|^{2},} and
‖ f ‖ D 2 = ∑ n ≥ 0 ( n + 1 ) | a n | 2 . {\displaystyle \|f\|_{\mathcal {D}}^{2}=\sum _{n\geq 0}(n+1)|a_{n}|^{2}.} Clearly, D {\displaystyle {\mathcal {D}}} contains all the polynomials and, more generally, all functions f {\displaystyle f} , holomorphic on D {\displaystyle \mathbb {D} } such that f ′ {\displaystyle f'} is bounded on D {\displaystyle \mathbb {D} } .
The reproducing kernel of D {\displaystyle {\mathcal {D}}} at w ∈ C ∖ { 0 } {\displaystyle w\in \mathbb {C} \setminus \{0\}} is given by
k w ( z ) = 1 z w ¯ log ( 1 1 − z w ¯ ) ( z ∈ C ∖ { 0 } ) . {\displaystyle k_{w}(z)={\frac {1}{z{\overline {w}}}}\log \left({\frac {1}{1-z{\overline {w}}}}\right)\;\;\;\;\;(z\in \mathbb {C} \setminus \{0\}).} See also [ edit ] References [ edit ] Arcozzi, Nicola; Rochberg, Richard; Sawyer, Eric T.; Wick, Brett D. (2011), "The Dirichlet space: a survey" (PDF) , New York J. Math. , 17a : 45–86 El-Fallah, Omar; Kellay, Karim; Mashreghi, Javad; Ransford, Thomas (2014). A primer on the Dirichlet space . Cambridge, UK: Cambridge University Press. ISBN 978-1-107-04752-5 .