Corrections to the paper “Sharp estimates of the growth of the Poisson-Stieltjes integral in the polydisc” by I.E.Chyzhykov, O.A.Zolota

Математичнi Студiї. Т.37, №2 Matematychni Studii. V.37, No.2

УДК 517.55

I. E. Chyzhykov, O. A. Zolota

Corrections to the paper “Sharp estimates of the growth of the Poisson-Stieltjes integral in the polydisc” by I.E.Chyzhykov, O.A.Zolota

The proof of the sufficiency of the theorem in [1] contains a gap, because the estimate of the Poisson kernel on the set Qk =Ek\Ek−1 (p.194) does not hold. The authors do not

know whether the sufficiency statement is true in the general case, but here we give the proof under the stronger assumption 0< βj <2, j ∈ {1, . . . , n} instead of _β1₁ +· · ·+ _β1_n > 1₂ as it

was stated in [1].

Theorem. Let µ be a finite Borel measure on Tn_{, n ∈ N, 0< β}

j <2, 1 ≤ j ≤n. In order

that

Z

Tn

P(z, w)dµ(w) ≤Cδ

1−1

β∗, 0< δ <1, |z_j|= 1−δ

1

βj_{, (1)}

hold for some positive constant C, it is sufficient, and for nonnegative µit is necessary that µ∈H(β1,...,βn)_.

Proof. Sufficiency. For fixed δ ∈(0,1) we set rj = 1−δ

1

βj_{, and let z}

j = rjeiϕj, 1≤ j ≤ n.

For1≤j ≤n we denote

Ek=

n

eiθ1

, . . . , eiθn_:|θ

j−ϕj| ≤ 2k·δ

_βj1

,1≤j ≤no, 0≤k≤Nj,

where Nj =

h

log₂πβj δ

i

+ 1, and

Rm1...mn =

(

eiθ1_{, . . . , e}iθn_∈Tn _{: (2}mj−1_δ) 1 βj _{≤ |θ}

j−ϕj| ≤(2mjδ)

1

βj_{, if m}

j >0,

|θj −ϕj| ≤δ

1

βj _{, if m}

j = 0,

)

.

Then Tn =

N1

S

m1=0 . . .

Nn

S

mn=0

Rm1...mn. As in [2] (proof of Theorem 3.1) we have for fixed k

that:

Z

Rm1...mn

P(z, w)dµ(w)

≤ |µ|(Rm1...mn)·

n

Y

j=1

P(|z|,w),˜

where z ∈Un_{, w˜ = ( ˜w}

1, . . . ,w˜n), w˜j =ei

˜

θj_{, θ}˜

j = 2k−1·δ

1

βj _{,1≤j ≤n.} The following estimates are known ([2])

P0 reiϕ, eiθ

≤ π

2

(ϕ−θ)2 (1−r), P0 re

iϕ_{, e}iθ

≤ 2

1−r. (2)

2010Mathematics Subject Classification:31B05, 31B10, 32A22, 32A25, 32A26, 32A35.

c

224

Applying the first estimate, we get

P0(|zj|,w˜j)≤

π2 ˜

θ2_j (1−rj) =

π2

(2k−1_·δ)

2 βj

(1−rj), k ≥1.

Using the second inequality in (2), we obtain

Z

R0...0

P(z, w)dµ(w)

≤ |µ|(R0...0)·

2n

(1−r1)·. . .·(1−rn)

.

Using the definition of the class H(β1,...,βn) _{we have that |µ|(R}

0...0) ≤Cδ. The equalities

1−rj =δ

1

βj_{, 1≤j ≤n imply}

Z

R0...0

P(z, w)dµ(w) ≤Cδ

1−1 β1+...+

1 βn

.

Then we estimate the integral over the setsRm1...mn. Since,Rm1...mn ⊂Emax{m1,...,mn}, we get that |µ|(Rm1...mn)≤C·2

m1+...+mn_δ.

Therefore, for N = maxNj, 1≤j ≤n

N

X

m1=1 . . .

N

X

mn=1

Z

Rm1...mn

P(z, w)dµ(w) ≤

N

X

m1=1 . . .

N

X

mn=1

|µ|(Rm1...mn)·

n

Y

j=1

π2(1−rj)

(2mj−1δ) 2 βj

≤

≤C

N

X

m1=1 . . .

N

X

mn=1 2m1

·. . .·2mn_·δ·

n

Y

j=1

δ 1 βj

2 2_mj

βj _·δ 2 βj

≤

≤Cδ1−

1 β1+...+

1 βn

N

X

m1=1

21− 2 β1

m1 . . .

N

X

mn=1

21−βn2

mn

< Cδ1−β1∗.

REFERENCES

1. Chyzhykov I.E., Zolota O.A. Sharp estimates of the growth of the Poisson-Stieltjes integral in the poly-disc// Mat. Stud. – 2010. – V.34, №2. – P. 193–196.

2. Rudin W. Function theory in polydiscs. – New York-Amsterdam: Math. Lecture Note Series, 1969. – 162p.

Lviv Ivan Franko National University [email protected]

Drohobych Ivan Franko State Pedagogical University [email protected]