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On some properties of a differential operator on the polydisk. (English) Zbl 1166.32002

The paper is devoted to the study of relations between the following differential operators in the space H(Δ n ), H p (Δ n ) of holomorphic functions in the unit polydisc Δ n :

R s f= h 1 ,,h n 0 (k 1 ++k n +1) s a k 1 k n z 1 k 1 y n k n ,D α f= k 1 ,,k n 0 (k 1 +1) α (k n +1) α a k 1 k n z 1 k 1 z n k ·

Here, f(z)= k 1 ,,k+n0 a k 1 k n z 1 k 1 z n k n belongs to H(Δ n ) or H p (Δ n ) for some 0<p, s, α. The authors goal is to reduce the study of R s to a study of D α , which were studied by many authors. An example of a result is the following.

Theorem 2.7. Let 0<p<, α>-1, s and fH(Δ n ). If γ>α+2 p-2 for p1 or γ>α+1 p+1 n(1-1 p) for p>1 and v=sp+αn-γpn+n-1, then

Δ n |D γ f(z)| p (1-|z| 2 ) α dm 2n (z)C 0 1 (Δ) n |R s f(w)| p (1-|w| 2 ) v dm n (ζ)d|w|,

where w=|w|ζ.

From here some new embedding theorems for various quasinorms, where the operators R s are participating, are obtained.

MSC:
32A18Bloch functions, normal functions
32A36Bergman spaces