2.6.
THE INTEGRABILITY
OF THE DERIVATIVE
OF A CONFORMAL MAPPING*
Let ~ be a simply connected domain having at least ...

Author:
Brennan J.

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2.6.

THE INTEGRABILITY

OF THE DERIVATIVE

OF A CONFORMAL MAPPING*

Let ~ be a simply connected domain having at least two boundary points in the extended complex plane and let ~ be a conformal mapping of ~ onto the open unit disk m . In this note we pose the following question: For which numbers p is

n For p = 2 the integral is equal to the area of the disk and is therefore finite. In general, it is known to converge for 4/3 < p < 3 and if ~ is the plane slit along the negative real axis then it obviously diverges for p = 4/3 and p = 4. These facts are consequences of the Koebe ,distortion theorem and were first discovered by Gehring and Hayman (unpublished) for p < 2 and by Metzger [I] for p > 2. Recently, the author has succeeded in proving that the upper bound 3 can be increased. The following theorem summarizes the known results. THEOREM

I.

There exists a number

T, T > 0, not depending

on ~, such t h a t

ffl'iPgw&

<+oo,

if 4/3 < p < 3 + T. For a wide class of regions, is the correct

upper bound

(cf.

including

"starlike"

[2], Theorem 2).

p < 4 in all cases but, unfortunately,

and "close-to-convex"

Quite

the argument

likely, ~ - l ~ ' I P ~ < + o o

d~ r is harmonic

the Koebe distortion if and only if

Thus,

Theorem

integral

I.

on the curve

theorem that

I~I=~

}~(%)I~K ~

for 4/3 <

relative near

to x 0.

~o~, ~(~0)=0, and It is easy to see,

Moreover,

and, consequently,

it follows

consequence

of the following

from

[[l~IP~

I is now an immediate

~ ( ~ ) ~

LEMMA

measure

p = 4

in [2] will not give this result.

Here is a sketch of the proof of Theorem I. We shall assume that we shall denote by 6(z) the Euclidean distance from the point z to ~s using polar coordinates, that

where

domains,

a

lemma on the growth of the

as r ยง i.

There exists

a constant

p, p > 0, such that if h >

MI=~ ~

(l-~) s

I/2 then

"

Of course, if we could prove the lemma for all p, p < I, then we could prove Theorem 1 for 4/3 < p < 4. So far, however, this has still not been done. The proof of the lemma is based on an idea of Carleson [3], which he expressed in connection with another problem. The question is the following: On a Jordan curve is harmonic measure absolutely continuous with *J. BRENNAN.

University

of Kentucky,

Lexington,

Kentucky

40506.

2211

respect to ~-dimensional Hausdorff measure for every ~, ~ < I? On the one hand, according to the Beurling projection theorem (cf. [4], p. 72), the question can be answered affirmatively if ~ ~ I/2. On the other hand, Lavrent'ev [5], McMillan and Piranian [6], and Carleson [3] have shown by means of counterexamples that absolute continuity does not always occur if ~ = I. In addition, Carleson was able to show that the upper bound I/2 in Beurling's theorem can be increased. It is interesting to speculate on the extent to which it is possible to observe a similarity between the two problems. For example, it is well known (cf. [7], p. 44) that harmonic measure is absolutely continuous with respect to l-dimensional Hausdorff measure if there are no points ~ on ~ for which

(i)

~,

(2) The question arises: if this condition is satisfied must

~l~'lPgm~~ <+oo

~

for 4 /3 < p < 4?

At this time the answer is not known. Before proceeding to the solution of the general problem it apparently remains to answer this more modest question. To the best of my knowledge, the question about the integrability of the derivative of a conformal mapping arose in connection with several problems in approximation theory. We shall mention only one of these and then indicate an application of Theorem I. Our problem was first posed by Keldy~ in 1939 (cf. [8] and [9], p~ i0) and he obtained the first results in this direction. Further progress has been achieved in the works of D~rbaw [10], ~aginjan [ I], Maz'ja and Havin [12, 13], and the author [14, 15, 2]. A complete discussion of the results obtained up to 1975 can be found in the surveys of Mergeljan [9], Mel'nikov and Sinanjan [16]. Let us assume that D, U are two Jordan domains in the complex plane, U c D, and let = Int (D \U). We shall denote by HP(~), p k I, the closure of the set of all polynomials in the space LP(~, dxdy) and we shall denote by L~(~) the subspace consisting of those functions f, f ~ LP(~), which are analytic in ~. Clearly, H p ~ L~. An interesting question concerns the possibility of equality in this inclusion. It is well known that in order for H p and L~ to coincide the determining factor is the "thinness" of the region ~ near multiple boundary points (i.e., near points of ~D n ~U). Here is a result which gives a quantitative description of that dependence. The proof is based in part on Theorem I (cf. [2] and [5], pp. 143-148). THEOREM 2. Let ~(z) be the distance from z to C x D and let d~ be harmonic measure on 8U relative to the domain U. There exists an absolute constant T, T > 0, not depending on ~, such that if

then HP(~)

= L~(~)

for all p, p < 3 + T.

The question remains: is p = 4 the upper bound or is the theorem true for all p, p < +~? LITERATURE CITED T. A. Metzger, "On polynomial approximation in Aq(D)," Proc. Am. Math. Soc., 37, 468470 (1973). J. Brennan, " T h e i n t e g r a b i l i t y of the derivative in conformal mapping," J. London Math. Soc., 18, 261-272 (1978). L. Carleson, "On the distortion of sets on a Jordan curve under conformal mapping," Duke Math. J., 40, 547-559 (1973). J. E. McMillan, "Boundary behavior under conformal mapping," Proc. of the N.R.L. Conference on Classical Function Theory, Washington D.C. 1970, pp. 59-76. M. A. Lavrent'ev, "On some boundary problems in the theory of univalent functions," Mat. Sb., No. I, 815-844 (1936). J. E. McMillan and G. Piranian, "Compression and expansion of boundary sets," Duke Math. J., 40, 599-605 (1973).

2212

7~ 8.

9, 10. 11. 12. 13. 14. 15. 16.

J. E. McMillan, "Boundary behavior of a conformal mapping," Acta Math., 123, 43-67 (1969). M. V Keldy~, "Sur l'approximation en moyenne quadratique des fonctions analytiques," Mat. Sb., 47, No. 5, 391-402 (1939). S. N Mergeljan, "On the completeness of systems of analytic functions," Usp. Mat. Nauk, 8, No. 4, 3-63 (1953). M. M D~rba~jan, "Metric theorems on completeness and the representation of analytic functions," Doctoral Dissertation, Erevan (1948). A. L Saginjan, "On a criterion for the incompleteness of a system of analytic functions," Dokl. Akad. Nauk Arm. SSR, y, No. 4, 97-100 (1946). V. G Maz'ja and V. P. Havin, "On approximation in the mean by analytic functions," Vestn. Leningr. Univ., Ser. Mat. Mekh. Astron., No. 13, 62-74 (1968). V. G Maz'ja and V. P. Havin, "Applications of (p, l)- capacity to some problems in the theory of exceptional sets," Mat. Sb., 90, No. 4, 558-591 (1973)., J. Brennan, "Invariant subspaces and weighted polynomial approximation," Ark. Mat., 11, 167-189 (1973). J. Brennan, "Approximation in the mean by polynomials on non-Carathe~dory domains," Ark. Mat., I__5, 117-168 (1977). M. S. Mel'nikov and S. O. Sinanjan, "Questions in the theory of approximation of functions of one complex variable," in: Contemporary Problems of Mathematics, Vol. 4, Itogi Nauki i Tekhniki, VINITI, Moscow (1975), pp. 143-250.

2213

THE INTEGRABILITY

OF THE DERIVATIVE

OF A CONFORMAL MAPPING*

Let ~ be a simply connected domain having at least two boundary points in the extended complex plane and let ~ be a conformal mapping of ~ onto the open unit disk m . In this note we pose the following question: For which numbers p is

n For p = 2 the integral is equal to the area of the disk and is therefore finite. In general, it is known to converge for 4/3 < p < 3 and if ~ is the plane slit along the negative real axis then it obviously diverges for p = 4/3 and p = 4. These facts are consequences of the Koebe ,distortion theorem and were first discovered by Gehring and Hayman (unpublished) for p < 2 and by Metzger [I] for p > 2. Recently, the author has succeeded in proving that the upper bound 3 can be increased. The following theorem summarizes the known results. THEOREM

I.

There exists a number

T, T > 0, not depending

on ~, such t h a t

ffl'iPgw&

<+oo,

if 4/3 < p < 3 + T. For a wide class of regions, is the correct

upper bound

(cf.

including

"starlike"

[2], Theorem 2).

p < 4 in all cases but, unfortunately,

and "close-to-convex"

Quite

the argument

likely, ~ - l ~ ' I P ~ < + o o

d~ r is harmonic

the Koebe distortion if and only if

Thus,

Theorem

integral

I.

on the curve

theorem that

I~I=~

}~(%)I~K ~

for 4/3 <

relative near

to x 0.

~o~, ~(~0)=0, and It is easy to see,

Moreover,

and, consequently,

it follows

consequence

of the following

from

[[l~IP~

I is now an immediate

~ ( ~ ) ~

LEMMA

measure

p = 4

in [2] will not give this result.

Here is a sketch of the proof of Theorem I. We shall assume that we shall denote by 6(z) the Euclidean distance from the point z to ~s using polar coordinates, that

where

domains,

a

lemma on the growth of the

as r ยง i.

There exists

a constant

p, p > 0, such that if h >

MI=~ ~

(l-~) s

I/2 then

"

Of course, if we could prove the lemma for all p, p < I, then we could prove Theorem 1 for 4/3 < p < 4. So far, however, this has still not been done. The proof of the lemma is based on an idea of Carleson [3], which he expressed in connection with another problem. The question is the following: On a Jordan curve is harmonic measure absolutely continuous with *J. BRENNAN.

University

of Kentucky,

Lexington,

Kentucky

40506.

2211

respect to ~-dimensional Hausdorff measure for every ~, ~ < I? On the one hand, according to the Beurling projection theorem (cf. [4], p. 72), the question can be answered affirmatively if ~ ~ I/2. On the other hand, Lavrent'ev [5], McMillan and Piranian [6], and Carleson [3] have shown by means of counterexamples that absolute continuity does not always occur if ~ = I. In addition, Carleson was able to show that the upper bound I/2 in Beurling's theorem can be increased. It is interesting to speculate on the extent to which it is possible to observe a similarity between the two problems. For example, it is well known (cf. [7], p. 44) that harmonic measure is absolutely continuous with respect to l-dimensional Hausdorff measure if there are no points ~ on ~ for which

(i)

~,

(2) The question arises: if this condition is satisfied must

~l~'lPgm~~ <+oo

~

for 4 /3 < p < 4?

At this time the answer is not known. Before proceeding to the solution of the general problem it apparently remains to answer this more modest question. To the best of my knowledge, the question about the integrability of the derivative of a conformal mapping arose in connection with several problems in approximation theory. We shall mention only one of these and then indicate an application of Theorem I. Our problem was first posed by Keldy~ in 1939 (cf. [8] and [9], p~ i0) and he obtained the first results in this direction. Further progress has been achieved in the works of D~rbaw [10], ~aginjan [ I], Maz'ja and Havin [12, 13], and the author [14, 15, 2]. A complete discussion of the results obtained up to 1975 can be found in the surveys of Mergeljan [9], Mel'nikov and Sinanjan [16]. Let us assume that D, U are two Jordan domains in the complex plane, U c D, and let = Int (D \U). We shall denote by HP(~), p k I, the closure of the set of all polynomials in the space LP(~, dxdy) and we shall denote by L~(~) the subspace consisting of those functions f, f ~ LP(~), which are analytic in ~. Clearly, H p ~ L~. An interesting question concerns the possibility of equality in this inclusion. It is well known that in order for H p and L~ to coincide the determining factor is the "thinness" of the region ~ near multiple boundary points (i.e., near points of ~D n ~U). Here is a result which gives a quantitative description of that dependence. The proof is based in part on Theorem I (cf. [2] and [5], pp. 143-148). THEOREM 2. Let ~(z) be the distance from z to C x D and let d~ be harmonic measure on 8U relative to the domain U. There exists an absolute constant T, T > 0, not depending on ~, such that if

then HP(~)

= L~(~)

for all p, p < 3 + T.

The question remains: is p = 4 the upper bound or is the theorem true for all p, p < +~? LITERATURE CITED T. A. Metzger, "On polynomial approximation in Aq(D)," Proc. Am. Math. Soc., 37, 468470 (1973). J. Brennan, " T h e i n t e g r a b i l i t y of the derivative in conformal mapping," J. London Math. Soc., 18, 261-272 (1978). L. Carleson, "On the distortion of sets on a Jordan curve under conformal mapping," Duke Math. J., 40, 547-559 (1973). J. E. McMillan, "Boundary behavior under conformal mapping," Proc. of the N.R.L. Conference on Classical Function Theory, Washington D.C. 1970, pp. 59-76. M. A. Lavrent'ev, "On some boundary problems in the theory of univalent functions," Mat. Sb., No. I, 815-844 (1936). J. E. McMillan and G. Piranian, "Compression and expansion of boundary sets," Duke Math. J., 40, 599-605 (1973).

2212

7~ 8.

9, 10. 11. 12. 13. 14. 15. 16.

J. E. McMillan, "Boundary behavior of a conformal mapping," Acta Math., 123, 43-67 (1969). M. V Keldy~, "Sur l'approximation en moyenne quadratique des fonctions analytiques," Mat. Sb., 47, No. 5, 391-402 (1939). S. N Mergeljan, "On the completeness of systems of analytic functions," Usp. Mat. Nauk, 8, No. 4, 3-63 (1953). M. M D~rba~jan, "Metric theorems on completeness and the representation of analytic functions," Doctoral Dissertation, Erevan (1948). A. L Saginjan, "On a criterion for the incompleteness of a system of analytic functions," Dokl. Akad. Nauk Arm. SSR, y, No. 4, 97-100 (1946). V. G Maz'ja and V. P. Havin, "On approximation in the mean by analytic functions," Vestn. Leningr. Univ., Ser. Mat. Mekh. Astron., No. 13, 62-74 (1968). V. G Maz'ja and V. P. Havin, "Applications of (p, l)- capacity to some problems in the theory of exceptional sets," Mat. Sb., 90, No. 4, 558-591 (1973)., J. Brennan, "Invariant subspaces and weighted polynomial approximation," Ark. Mat., 11, 167-189 (1973). J. Brennan, "Approximation in the mean by polynomials on non-Carathe~dory domains," Ark. Mat., I__5, 117-168 (1977). M. S. Mel'nikov and S. O. Sinanjan, "Questions in the theory of approximation of functions of one complex variable," in: Contemporary Problems of Mathematics, Vol. 4, Itogi Nauki i Tekhniki, VINITI, Moscow (1975), pp. 143-250.

2213

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