Dirichlet space

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 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 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,,gin {mathcal {D}}(Omega )} , we define 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 where ⟨ ⋅ , ⋅ ⟩ H 2 ( Ω ) {displaystyle langle cdot ,,cdot angle _{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 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 cin 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 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