# Blaschke products with zeros outside a Stolz angle, in [Q.sub.p] spaces.

1. Introduction

Given a [zeta] [member of] D = {z [member of] C, |z| < 1}. We denote by g(., [zeta]) the Green function on D with pole at [zeta], that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For 0 < p < 1 the space [Q.sub.p] is defined as the class of all analytic functions in D with sup

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Functions in [Q.sub.p] spaces have been studied in the last years by several authors. (see [1], [3], [8], [10]).

For a Blaschke product ([4], [6])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(where we assume without loss of generality 0 < |zn| < |zn+1|, n. N). Essen and Xiao proved in [7] that for p. (0, 1) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

In [5] the authors study the condition (1.1) using the notion of asymptotically concentrated sequences.

Definition 1.1. Let {[a.sub.n]}be a sequence with [a.sub.n] > 0, [a.sub.n] > [a.sub.n+1], n [member of] N and lim n [right arrow] [infinity] [a.sub.n] = 0 We call {[a.sub.n]} asymptotically concentrated if for every k [member of] N there is an infinite sequence {[n.sub.p]} [subset] N depending on k and such that [a.sub.np]/[a.sub.np+k] [right arrow] 1, p [right arrow] [infinity]. If this condition is not satisfied, we say that {an} is not asymptotically concentrated.

In most cases it is relatively easy to see if a sequence {[a.sub.n]} is asymptotically concentrated or not. In a plausible way we would say that {[a.sub.n]} is asymptotically concentrated if it contains families of arbitrarily many asymptotically equal terms. We have a characterization theorem for these sequences:

Theorem 1.2. Let {[a.sub.n]} be a sequence with [a.sub.n] > 0, [a.sub.n] > [a.sub.n+1], n [member of] N and lim n [right arrow] [infinity] [a.sub.n] = 0

Then the sequences [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], n [member of] N are both N are both bounded if and only if {an} is not asymptotically concentrated.

Using the above Definition and Theorem 1.2, the authors prove the following:

Theorem 1.3. Let [z.sub.n] be a Blaschke sequence in D, with [a.sub.n] = 1-|[z.sub.n]|. If the sequence an is not asymptotically concentrated, then for the Blaschke product B(z) = B(z, {[z.sub.n]}) we have B [member of] [intersection] [Q.sub.p] 0<p<1. [Q.sub.p].

The opposite in general is not true. But the next theorem gives a 0-1 lawfor sequences lying in angular domains:

Theorem 1.4. Let {[z.sub.n]} be a Blaschke sequence in D, with [a.sub.n] = 1 - |[z.sub.n]|. We suppose also that all zn lie in an angular domain in D with vertex on a D, i.e. that there exist a [zeta] [member of] [partial derivative] D and a positive constant M so that |[zeta] - zn|/1-|Zn|<M fof all n.

If the sequence {an} is asymptotically concentrated, then for the Blaschke product B(z) = B(z, {zn}) we have B [??] [union] [Q.sub.p] 0<p<1 [Q.sub.p].

The Blaschke product with zeros [z.sub.n] = 1/2+1/2 [[e.sup.i[theta]n, where [theta.sub.n] = acroos (1-1/[n.sup.4], n [member of] N proves that the condition of the angular domain in Theorem C can not be eliminated.

2. Main Results

We give the characterization for the Blaschke products in [Q.sub.p] spaces in the terms of [alpha] curves. Let [gamma] : [0, [pi]) [right arrow] [0, 1) be a continuous function with [GAMMA] (0) = 0 and [gamma] (x) > 0 for x [not equal to] = 0. We define, the Jordan curve [GAMMA] on D as follows:

z = [[re.sup.i[theta]] [member of] [GAMMA], such that 1 - r = [gamma] (|[theta]|), |[theta]| < n.

[GAMMA] is symmetric with respect to the real ax and lies on D, except of the intersection point H with the a D. Let a [member of] N. We will call [GAMMA] an [alpha]-curve if [gamma] ([theta]/[tehta][alpha] has finite and non-zero limit as [theta] [right arrow] [0.sup.+]. Specially, in the case that that [gamma]([theta])/[theta] [right arrow]0 as [theta] [right arrow] [0.sup.+] we say that [GAMMA] tends in the boundary tangentially.

The following characterization of [alpha]-curves is known [9]:

Lemma 2.1. Let [alpha] [greater than or equal to] 1. [GAMMA] is [alpha]-curve if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is bounded and non-zero.

Trying to answer the question for which p0, for a given Blaschke product B(z, {zn}), we have

B(z) [member of] [Q.sub.p] for all [p.sub.0] < p < 1 and B(z) [??] [Q.sub.p] for all 0 < p < [p.sub.0] we obtained the following theorems:

Theorem 2.2. Let {[z.sub.n]} be a Blaschke sequence in D, with [alpha.sub.n] = 1 - |[z.sub.n]|, n [member of] N. We suppose that {[z.sub.n]} is asymptotically concentrated and all [z.sub.n] lie on an [alpha]-curve.

If the Blaschke product B(z) = B(z, {zn}) [member of] [Q.sub.p], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the constant c [member of] R does not depend on m.

Proof. Let the sequence {zn} in D satisfy the hypothesis of the Theorem, and let B(z) = B(z, {zn}). [Q.sub.p].

Let m [member of] N and [omega] m = 1 - |1 - [z.sub.m]|. From (1.1) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where the constant c does not depend on m. From the other side

|1- [bar.z.sub.n][omega]m| [less than or equal to] |1-[z.sub.n]|+|[z.sub.n] - [bar.z.sub.n][omega]m| [less than or equal to] |1-[z.sub.n]|+|1-[omega]m| = |1-zn|+|1-[z.sub.m]|. (2.2)

From our hypothesis {[z.sub.n]} are located on an [alpha]-curve. From the Lemma A there exist [c.sub.1] and [c.sub.2] constants, which depend only on [alpha], such that:

c1(1 - |[z.sub.n]|) [less than or equal to] |1 - zn|[alpha] [less than or equal to] = [c.sub.2](1 - |[z.sub.n]|), n [member of] N. (2.3)

The sequence {1-|zn|}, n [member of] N is decreasing, so there exists a constant [c.sub.3] which depends only on [alpha], such that |1 - [z.sub.n]| < [c.sub.3]|1 - [z.sub.m]| [for all] n > m, and because of (2.2) we get

|1 - [bar.zn[omega]m|.sup.2] [less than or equal to] [c.sub.4]|1 - [z.sub.m]|.sup.2] = [c.sub.4]|1 - [omega]m|.sup.2]], [for all]n > m, (2.4)

where c4 = (1 + c3)2. Now using (2.4) in the inequality (2.1) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

So combining the result above with (2.3) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where [c.sub.0] depends on p and [alpha] only.

Trying to see if the condition in the Theorem 2.2 above is also sufficient we give the following:

Theorem 2.3. Let {[z.sub.n]} be a Blaschke sequence in D, with an = 1 - |[z.sub.n]|, n [member of] N. Let A([theta], [member of]) = {z [member of] D : |z - exp i[theta]| < [member of] [member of]}. If there is an absolute constant c such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

then the Blaschke product B(z) = B(z, {[z.sub.n]}) [member of] [Q.sub.p].

Proof. This proof is based on an idea which apears in [2]We will show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If |[less than or equal to] | = 3/4, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where A([theta] 2) = D, [for all][theta] [member of] [0, 2[pi]).

Now, if |[omega]| [greater than or equal to] 3/4 let us define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with [E.sub.0] = 0

We observe that [for all] z [member of] D, En [member of] N such that z [member of] [E.sub.n]. We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let us consider now z [menber of] [E.sub.n] \ [E.sub.n-1]. We get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and as a conclusion we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So B(z) = B(z, {[z.sub.n]}). [Q.sub.p].

We observe that the condition in Theorem 2.2 is more easy to be used than the one in Theorem 2.3. So, it is natural to ask if we can have the opposite implication of

Theorem 2.2.

References

[1] R. Aulaskari,Y. He J. Ristoja, and R. Zhao, [Q.sub.p] spaces on riemann surfaces, Canadian J.Math., 50:449-464, 1998.

[2] R. Aulaskari, D. Stegenga, and J. Xiao, Some subclasses of BMOA and their characterisation in terms of carleson measures, Rocky Mountain J. Math., 26:485- 506, 1996.

[3] R. Aulaskari, J. Xiao, and R. Zhao, On subspaces and subsets of BMOA and UBC, Analysis, 15:101-121, 1995.

[4] P. Colwell, Blaschke Products (Bounded Analytic functions), The University of Michigan, 1985.

[5] N. Danikas and C. Mouratides, Blaschke products in[Q.sub.p] spaces, ComplexVariables, 43:109-209, 2000.

[6] P. Duren, Theory of Hp spaces, volume 38 of Pure and Applied Mathematics, Academic Press, 1970.

[7] M. Essen and J. Xiao, Some results on [Q.sub.p] spaces, 0 < p < 1, J. reine angew. Math., 485:173-195, 1997.

[8] M. Essen and J. Xiao, [Q.sub.p] spaces - a survey, U.U.D.M. Report, 4, 1998.

[9] J. Shapiro, Composition Operators and classical functions theory, SpringerVerlag, 1993.

[10] J. Xiao, Holomorphic Q classes, Springer Verlag, 2002.

G. Majchrowska

Aristotle University of Thessaloniki,

Computer Science Department

E-mail: grazyna@csd.auth.gr

Chr. Mouratides

Technological Institute of West Macedonia,

Koila, 50100, Kozani, Greece

E-mail: cmourati@kozani.teikoz.gr

Given a [zeta] [member of] D = {z [member of] C, |z| < 1}. We denote by g(., [zeta]) the Green function on D with pole at [zeta], that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For 0 < p < 1 the space [Q.sub.p] is defined as the class of all analytic functions in D with sup

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Functions in [Q.sub.p] spaces have been studied in the last years by several authors. (see [1], [3], [8], [10]).

For a Blaschke product ([4], [6])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(where we assume without loss of generality 0 < |zn| < |zn+1|, n. N). Essen and Xiao proved in [7] that for p. (0, 1) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

In [5] the authors study the condition (1.1) using the notion of asymptotically concentrated sequences.

Definition 1.1. Let {[a.sub.n]}be a sequence with [a.sub.n] > 0, [a.sub.n] > [a.sub.n+1], n [member of] N and lim n [right arrow] [infinity] [a.sub.n] = 0 We call {[a.sub.n]} asymptotically concentrated if for every k [member of] N there is an infinite sequence {[n.sub.p]} [subset] N depending on k and such that [a.sub.np]/[a.sub.np+k] [right arrow] 1, p [right arrow] [infinity]. If this condition is not satisfied, we say that {an} is not asymptotically concentrated.

In most cases it is relatively easy to see if a sequence {[a.sub.n]} is asymptotically concentrated or not. In a plausible way we would say that {[a.sub.n]} is asymptotically concentrated if it contains families of arbitrarily many asymptotically equal terms. We have a characterization theorem for these sequences:

Theorem 1.2. Let {[a.sub.n]} be a sequence with [a.sub.n] > 0, [a.sub.n] > [a.sub.n+1], n [member of] N and lim n [right arrow] [infinity] [a.sub.n] = 0

Then the sequences [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], n [member of] N are both N are both bounded if and only if {an} is not asymptotically concentrated.

Using the above Definition and Theorem 1.2, the authors prove the following:

Theorem 1.3. Let [z.sub.n] be a Blaschke sequence in D, with [a.sub.n] = 1-|[z.sub.n]|. If the sequence an is not asymptotically concentrated, then for the Blaschke product B(z) = B(z, {[z.sub.n]}) we have B [member of] [intersection] [Q.sub.p] 0<p<1. [Q.sub.p].

The opposite in general is not true. But the next theorem gives a 0-1 lawfor sequences lying in angular domains:

Theorem 1.4. Let {[z.sub.n]} be a Blaschke sequence in D, with [a.sub.n] = 1 - |[z.sub.n]|. We suppose also that all zn lie in an angular domain in D with vertex on a D, i.e. that there exist a [zeta] [member of] [partial derivative] D and a positive constant M so that |[zeta] - zn|/1-|Zn|<M fof all n.

If the sequence {an} is asymptotically concentrated, then for the Blaschke product B(z) = B(z, {zn}) we have B [??] [union] [Q.sub.p] 0<p<1 [Q.sub.p].

The Blaschke product with zeros [z.sub.n] = 1/2+1/2 [[e.sup.i[theta]n, where [theta.sub.n] = acroos (1-1/[n.sup.4], n [member of] N proves that the condition of the angular domain in Theorem C can not be eliminated.

2. Main Results

We give the characterization for the Blaschke products in [Q.sub.p] spaces in the terms of [alpha] curves. Let [gamma] : [0, [pi]) [right arrow] [0, 1) be a continuous function with [GAMMA] (0) = 0 and [gamma] (x) > 0 for x [not equal to] = 0. We define, the Jordan curve [GAMMA] on D as follows:

z = [[re.sup.i[theta]] [member of] [GAMMA], such that 1 - r = [gamma] (|[theta]|), |[theta]| < n.

[GAMMA] is symmetric with respect to the real ax and lies on D, except of the intersection point H with the a D. Let a [member of] N. We will call [GAMMA] an [alpha]-curve if [gamma] ([theta]/[tehta][alpha] has finite and non-zero limit as [theta] [right arrow] [0.sup.+]. Specially, in the case that that [gamma]([theta])/[theta] [right arrow]0 as [theta] [right arrow] [0.sup.+] we say that [GAMMA] tends in the boundary tangentially.

The following characterization of [alpha]-curves is known [9]:

Lemma 2.1. Let [alpha] [greater than or equal to] 1. [GAMMA] is [alpha]-curve if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is bounded and non-zero.

Trying to answer the question for which p0, for a given Blaschke product B(z, {zn}), we have

B(z) [member of] [Q.sub.p] for all [p.sub.0] < p < 1 and B(z) [??] [Q.sub.p] for all 0 < p < [p.sub.0] we obtained the following theorems:

Theorem 2.2. Let {[z.sub.n]} be a Blaschke sequence in D, with [alpha.sub.n] = 1 - |[z.sub.n]|, n [member of] N. We suppose that {[z.sub.n]} is asymptotically concentrated and all [z.sub.n] lie on an [alpha]-curve.

If the Blaschke product B(z) = B(z, {zn}) [member of] [Q.sub.p], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the constant c [member of] R does not depend on m.

Proof. Let the sequence {zn} in D satisfy the hypothesis of the Theorem, and let B(z) = B(z, {zn}). [Q.sub.p].

Let m [member of] N and [omega] m = 1 - |1 - [z.sub.m]|. From (1.1) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where the constant c does not depend on m. From the other side

|1- [bar.z.sub.n][omega]m| [less than or equal to] |1-[z.sub.n]|+|[z.sub.n] - [bar.z.sub.n][omega]m| [less than or equal to] |1-[z.sub.n]|+|1-[omega]m| = |1-zn|+|1-[z.sub.m]|. (2.2)

From our hypothesis {[z.sub.n]} are located on an [alpha]-curve. From the Lemma A there exist [c.sub.1] and [c.sub.2] constants, which depend only on [alpha], such that:

c1(1 - |[z.sub.n]|) [less than or equal to] |1 - zn|[alpha] [less than or equal to] = [c.sub.2](1 - |[z.sub.n]|), n [member of] N. (2.3)

The sequence {1-|zn|}, n [member of] N is decreasing, so there exists a constant [c.sub.3] which depends only on [alpha], such that |1 - [z.sub.n]| < [c.sub.3]|1 - [z.sub.m]| [for all] n > m, and because of (2.2) we get

|1 - [bar.zn[omega]m|.sup.2] [less than or equal to] [c.sub.4]|1 - [z.sub.m]|.sup.2] = [c.sub.4]|1 - [omega]m|.sup.2]], [for all]n > m, (2.4)

where c4 = (1 + c3)2. Now using (2.4) in the inequality (2.1) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

So combining the result above with (2.3) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where [c.sub.0] depends on p and [alpha] only.

Trying to see if the condition in the Theorem 2.2 above is also sufficient we give the following:

Theorem 2.3. Let {[z.sub.n]} be a Blaschke sequence in D, with an = 1 - |[z.sub.n]|, n [member of] N. Let A([theta], [member of]) = {z [member of] D : |z - exp i[theta]| < [member of] [member of]}. If there is an absolute constant c such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

then the Blaschke product B(z) = B(z, {[z.sub.n]}) [member of] [Q.sub.p].

Proof. This proof is based on an idea which apears in [2]We will show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If |[less than or equal to] | = 3/4, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where A([theta] 2) = D, [for all][theta] [member of] [0, 2[pi]).

Now, if |[omega]| [greater than or equal to] 3/4 let us define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with [E.sub.0] = 0

We observe that [for all] z [member of] D, En [member of] N such that z [member of] [E.sub.n]. We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let us consider now z [menber of] [E.sub.n] \ [E.sub.n-1]. We get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and as a conclusion we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So B(z) = B(z, {[z.sub.n]}). [Q.sub.p].

We observe that the condition in Theorem 2.2 is more easy to be used than the one in Theorem 2.3. So, it is natural to ask if we can have the opposite implication of

Theorem 2.2.

References

[1] R. Aulaskari,Y. He J. Ristoja, and R. Zhao, [Q.sub.p] spaces on riemann surfaces, Canadian J.Math., 50:449-464, 1998.

[2] R. Aulaskari, D. Stegenga, and J. Xiao, Some subclasses of BMOA and their characterisation in terms of carleson measures, Rocky Mountain J. Math., 26:485- 506, 1996.

[3] R. Aulaskari, J. Xiao, and R. Zhao, On subspaces and subsets of BMOA and UBC, Analysis, 15:101-121, 1995.

[4] P. Colwell, Blaschke Products (Bounded Analytic functions), The University of Michigan, 1985.

[5] N. Danikas and C. Mouratides, Blaschke products in[Q.sub.p] spaces, ComplexVariables, 43:109-209, 2000.

[6] P. Duren, Theory of Hp spaces, volume 38 of Pure and Applied Mathematics, Academic Press, 1970.

[7] M. Essen and J. Xiao, Some results on [Q.sub.p] spaces, 0 < p < 1, J. reine angew. Math., 485:173-195, 1997.

[8] M. Essen and J. Xiao, [Q.sub.p] spaces - a survey, U.U.D.M. Report, 4, 1998.

[9] J. Shapiro, Composition Operators and classical functions theory, SpringerVerlag, 1993.

[10] J. Xiao, Holomorphic Q classes, Springer Verlag, 2002.

G. Majchrowska

Aristotle University of Thessaloniki,

Computer Science Department

E-mail: grazyna@csd.auth.gr

Chr. Mouratides

Technological Institute of West Macedonia,

Koila, 50100, Kozani, Greece

E-mail: cmourati@kozani.teikoz.gr

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Author: | Majchrowska, G.; Mouratides, Chr. |
---|---|

Publication: | Global Journal of Pure and Applied Mathematics |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Apr 1, 2008 |

Words: | 1718 |

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