- Publication venue
- 'Pleiades Publishing Ltd'
- Publication date
- Field of study
- Publication venue
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Publication date
- 01/01/1991
- Field of study
In 1945 the first author introduced the classes Hp(α), 1 †p -1, of holomorphic functions in the unit disk with finite integral
(1) âŹDââŁf(ζ)âŁp(1ââŁÎ¶âŁ2)αdΟdη<â(ζ=Ο+iη)
and established the following integral formula for fâHp(α):
(2) f(z) = (α+1)/Ï âŹ_\mathbb{D} f(ζ) ((1-|ζ|ÂČ)^α)//((1-zζÌ
)^{2+α}) dΟdη, zâ \mathbb{D} .
We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes Lp(Ω;[K(w)]αdm(w)), where:
1) Ω=w=(w1â,...,wnâ)âCn:Imw1â>âk=2nââŁwkââŁ2, K(w)=Imw1âââk=2nââŁwkââŁ2;
2) Ω is the matrix domain consisting of those complex m Ă n matrices W for which I(m)âWâ
Wâ is positive-definite, and K(W)=det[I(m)âWâ
Wâ];
3) Ω is the matrix domain consisting of those complex n Ă n matrices W for which ImW=(2i)â1(WâWâ) is positive-definite, and K(W) = det[Im W].
Here dm is Lebesgue measure in the corresponding domain, I(m) denotes the unit m Ă m matrix and W* is the Hermitian conjugate of the matrix W - Publication venue
- 'Allerton Press'
- Publication date
- Field of study
- Publication venue
- 'Springer Science and Business Media LLC'
- Publication date
- 01/01/1993
- Field of study
- Publication venue
- 'Allerton Press'
- Publication date
- Field of study
- Publication venue
- Publication date
- 01/01/2007
- Field of study