Printer Friendly

Continuity of the restriction maps on Smirnov classes.

1. Introduction

As usual, we define the Hardy space [H.sup.2] = [H.sup.2]([DELTA]) as the space of all functions f : z [right arrow] [[summation].sup.[infinity].sub.n=0] [a.sub.n][z.sup.n] for which the norm [([parallel]f[parallel] = [[summation].sup.[infinity].sub.n=0] [[absolute value of [a.sub.n]].sup.2]).sup.1/2] is finite. Here, [DELTA] is the open unit disc. For a more general simply connected domain D in the sphere or extended plane [bar.C] = C [union] ([infinity]) with at least two boundary points, and a conformal mapping [phi] from D onto [DELTA] (i.e., a Riemann mapping function, abbreviation is RMF), a function g analytic in D is said to belong to the Smirnov class [E.sup.2](D) if and only if g = (f [omicron] [phi])[[phi]'.sup.1/2] for some f [member of] [H.sup.2]([DELTA]) where [[phi]'.sup.1/2] is an analytic branch of the square root of [phi]'. The reader is referred to [1-7] and references therein for the basic properties of these spaces.

Let C = ([C.sub.1], [C.sub.2], [C.sub.3], ..., [C.sub.N]) be an N-tuple of closed distinct curves on the sphere [bar.C] and suppose that, for each i, 1 [less than or equal to] i [less than or equal to] N, [C.sub.i] is a circle, a line [union]{[infinity]}, an ellipse, a parabola [union] {[infinity]}, or a branch of a hyperbola [union]{[infinity]}. Let [D.sub.i] be the complementary domain of [C.sub.i]. Recall that a complementary domain of a closed F [subset or equal to] [bar.C] is a maximal connected subset of [bar.C] - F, which must be a domain. For 1 [less than or equal to] i [less than or equal to] N, suppose that [[phi].sub.i] : [D.sub.i] [right arrow] [DELTA] is a conformal equivalence (i.e., RMF) and let [[psi].sub.i] : [DELTA] [right arrow] [D.sub.i] be its inverse. For 1 [less than or equal to] i [less than or equal to] N, let us keep the notations of [C.sub.i], [D.sub.i], [[phi].sub.i], [[psi].sub.i] fixed until the end of the paper.

In this paper we prove the following.

Theorem 1. Let 1 [less than or equal to] i, j [less than or equal to] N. Suppose that [GAMMA] is an open subarc of [C.sub.j] and suppose also that [GAMMA] [subset or equal to] [D.sub.i] if i [not equal to] j. Then the restriction f [right arrow] f|[sub.[GAMMA]] defines a continuous linear operator mapping [E.sup.2]([D.sub.i]) into [L.sup.2]([GAMMA]).

For similar work regarding restriction maps, see [8, 9]. Our conjecture is that Theorem 1 is valid if, for each j, 1 [less than or equal to] j [less than or equal to] N, [C.sub.j] is a [sigma]-rectifiable analytic Jordan curve.

There are some similar results for rectifiable curves in Havin's paper [10]. Also the Cauchy projection operator from [L.sup.p] to [E.sup.p] is bounded on all Carleson regular curves; compare the papers of David, starting with [11].

We need the following Theorem to simplify the proof of Theorem 1.

Theorem 2 (Theorem 1 in [12]). Let D be a complementary domain of [[union].sup.N.sub.i=1] [C.sub.i] and suppose that D is simply connected so that [D.sub.i] is the complementary domain of [C.sub.i] which contains D. Then

(i) [partial derivative]D is a [sigma]-rectifiable closed curve and every f [member of] [E.sup.2](D) has a nontangential limit function [??] [member of] [L.sup.2]([partial derivative]D);

(ii) (Parseval's identity) the map f [right arrow] [??] ([E.sup.2](D) [right arrow] [L.sup.2]([partial derivative]D)) is an isometric isomorphism onto a closed subspace [E.sup.2]([partial derivative]D) of [L.sup.2]([partial derivative]D), so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

If [GAMMA] [subset or equal to] [C.sub.i] is an open subarc, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

because Parseval's identity is true for the trivial chain ([C.sub.i]) of curves. Hence Theorem 1 will be proved if the following theorem can be proved.

Theorem 3. Let 1 [less than or equal to] i [not equal to] j [less than or equal to] N. Suppose that [GAMMA] is an open subarc of [C.sub.j] and that [GAMMA] [subset or equal to] [D.sub.i]. Then the restriction f [right arrow] f|[sub.[GAMMA]] defines a continuous linear operator mapping [E.sup.2]([D.sub.i]) into [L.sup.2](T).

2. Preliminaries for the Proof of Theorem 3

Let us keep the notation of Theorem 3 fixed for the rest of the paper and let us also agree to use I for arc-length measure.

An arc or closed curve [gamma] is called [alpha]-rectifiable if and only if it is a countable union of rectifiable arcs in C, together with ([infinity]) in the case when [infinity] [member of] [gamma]. For instance, a parabola without [infinity] is [sigma]-rectifiable arc, and a parabola with [infinity] is [sigma]-rectifiable Jordan curve. The following definition will simplify the language.

Definition 4. Let [gamma] [subset opr equal to] C be a simple [sigma]-rectifiable arc contained in a simply connected domain G [subset or equal to] [bar.C]. We say that [gamma] has the restriction property in G if and only if the map g [right arrow] g|[sub.[GAMMA]] defines a continuous linear operator mapping [E.sup.2](G) into [L.sup.2]([gamma]).

Thus, the last sentence of Theorem 3 reads "[GAMMA] has the restriction property in [D".sub.i].

Lemma 5_(Invariance Lemma (Lemma 4 in [9])). Let [G.sub.1], [G.sub.2] [subset or equal to] [bar.C] be simply connected domains and suppose that [[gamma].sub.1] [subset or equal to] [G.sub.1] [intersection] C, [[gamma].sub.2] [subset or equal to] [G.sub.2] [union] C are simple [sigma]-rectifiable arcs. If [chi] : [G.sub.1] [right arrow] [G.sub.2] is a conformal equivalence onto [G.sub.2] and [chi]([[gamma].sub.1]) = [[gamma].sub.2], then [[gamma].sub.1] has the restriction property in [G.sub.1] if and only if [y.sub.2] has the restriction property in [G.sub.2].

Corollary 6. Theorem 3 is true; that is, [GAMMA] has the restriction property in [D.sub.i], if and only if [[phi].sub.i]([GAMMA]) has the restriction property in [DELTA], for some RMF [[phi].sub.i] : [D.sub.i] [right arrow] [DELTA].

A subarc [gamma] of [GAMMA] has the restriction property in [D.sub.i] if and only if [[phi].sub.i]([gamma]) has the restriction property in [DELTA]. Corollary 6 will be used in the following way. [GAMMA] will be written as the union of finitely many subarcs and we will show that each of these subarcs has the restriction property in [D.sub.i]; it will then follow that r itself has the required restriction property. Three different kinds of subarc will be considered.

Definition 7. A subarc [gamma] [subset or equal to] [GAMMA] is said to be of type I if and only if [bar.[gamma]] [subset or equal to] [D.sub.i] (i.e., both of its end-points a, b belong to [D.sub.i]).

Lemma 8 (Lemma 6 in [9]). Let [gamma] be a subarc of [GAMMA] and suppose that [[phi].sub.i], [[theta].sub.i] are Riemann mapping functions for [D.sub.i].

(i) [[phi].sub.i]([gamma]) has the restriction property in [DELTA] if and only if [[theta].sub.i]([gamma]) has the restriction property in [DELTA];

(ii) [[phi].sub.i]([gamma]) is rectifiable if and only if [[theta].sub.i]([gamma]) is rectifiable;

(iii) if [gamma] is of type I, then [bar.[[phi].sub.i]([gamma])] [subset or equal to] [DELTA] and [[phi].sub.i]([gamma]) is rectifiable;

(iv) if [gamma] is of type I, it has the restriction property in [D.sub.i].

We can now "ignore" subarcs of r whose closure (in C) is contained in [D.sub.i]. We will now restrict our attention to subarcs of r with a single end-point a [member of] [partial derivative][D.sub.i], the other being in [D.sub.i]. There are two types, depending on whether a [member of] C or a = [infinity].

Definition 9. (i) An open subarc [gamma] of [GAMMA] is of type II if and only if it has an end-point a [member of] [partial derivative][D.sub.i] [intersection] C and [bar.[gamma]] -(a) [subset or equal to] [D.sub.i] [intersection] C.

(ii) In the case where [C.sub.i] is unbounded (so that [infinity] [member of] [partial derivative][D.sub.i]) an open subarc [gamma] [subset or equal to] [GAMMA] is of type III if and only if [infinity] is an endpoint of [gamma] and [bar.[gamma]] - ([infinity]) [subset or equal to] [D.sub.i].

Modulo a finite subset of [D.sub.i], T is the union of at most three open subarcs, each of which is of type I, II, or III; see Figure 1.

If [gamma] is a type II or type III subarc of [GAMMA] then [[phi].sub.i]([gamma]) is a simple open analytic arc in [DELTA] with one end-point on the circle T and the other in [DELTA]. We will show that [[phi].sub.i]([gamma]) has the restriction property in [DELTA] using the powerful Carleson theorem (Theorem 11 below).

Definition 10 (see [1, p.157]). For 0 < h < 1 and 0 [less than or equal to] [theta] < 2[pi], let [C.sub.[theta]h] = {z [member of] C : 1-h [less than or equal to] [absolute value of z] [less than or equal to] 1, 0 < arg z [less than or equal to] [theta] + h}. A positive regular Borel measure [mu] on [DELTA] is called a Carleson measure if there exists a positive constant M such that [mu]([C.sub.[theta]h]) [less than or equal to] Mh, for every h and every [theta].

Theorem 11 (see [1, p. 157, Theorem 9.3] or see [13, p. 37]). Let [mu] be a finite positive regular Borel measure on [DELTA]. In order that there exists a constant C > 0 such that

[[integral].sub.[DELTA]] [[absolute value of (z)].sup.2] d[mu](Z) [less than or equal to] C[[parallel]f[parallel].sup.2], [for all]f [member of] [H.sup.2] ([DELTA]), (3)

it is necessary and sufficient that [mu] be a Carleson measure.

To complete the proof of Theorem 3 it is sufficient to show that arc-length measure on [[phi].sub.i] ([gamma]) is a Carleson measure whenever [gamma] is of type II or III.

It will be useful to use arc-length to parametrize [gamma] and [[phi].sub.i]([gamma]). Recall that [sigma] compact arc a is called smooth if there exists some parametrization g : [a, b] [right arrow] [sigma] such that g [member of] [C.sup.1] [a, b] and g'(t) [not equal to] 0, [for all]t [member of] [a, b]. Note that if a is smooth, then it is rectifiable; that is,

l([sigma]) = [[integral].sup.b.sub.a] [absolute value of g' (t)] dt < [infinity]. (4)

To define the arc-length parametrization of [sigma] put s = s(t) = [[integral].sup.t.sub.a] [absolute value of g'(u)] du for a [less than or equal to] t [less than or equal to] b so that 0 [less than or equal to] s [less than or equal to] l([sigma]). Then s'(t) = [absolute value of g'(t)] and t [right arrow] s(t) ([a, b] [right arrow] [0, l]) is [C.sup.1] with strictly positive derivative. Hence also its inverse s [right arrow] t(s) ([0,l] [right arrow] [a, b]) is [C.sup.1] with strictly positive derivative. Recall that the arc-length parametrization of the smooth arc [sigma] is the map h : [0, l] [right arrow] [sigma] satisfying h(s) = {the point on [sigma] length s from the initial point (g(a))}; that is, h(s) = g(t(s)) 0 [less than or equal to] s [less than or equal to] l.

Since h'(s) = g'(t(s))t'(s), h [member of] [C.sup.1] [0,l], with nonzero derivative, necessarily [absolute value of h'(s)] = 1 since

h'(s(t)) = g'(t)t' (s) = g'(t)/s'(t) = g'(t)/[absolute value of g'(t)]. (5)

We need the following lemma.

Lemma 12 (Theorem 1 in [14]). Let [sigma] [subset or equal to] [bar.[DELTA]] be a smooth simple arc with arc-length parametrization g [member of] [C.sup.1] [0, l]. Suppose that [absolute value of g(0)] = 1, [absolute value of g(s)] < 1 for 0 < s [less than or equal to] l. Then arc-length measure on [sigma] [intersection] [DELTA] is a Carleson measure; hence [sigma] [intersection] [DELTA] has the restriction property in [DELTA].

3. Type II Subarcs

The following lemma gives the continuity of the restriction map for finite end-points.

Lemma 13. A type II arc [gamma] [subset or equal to] [GAMMA] [subset or equal to] [D.sub.i] has the restriction property in [D.sub.i].

Proof. By Lemmas 12 and 5 it is sufficient to show that [absolute value of [[phi].sub.i]([gamma])] is a smooth arc in [bar.[delta]]. Suppose that [gamma] has end-points a [member of] [partial derivative][D.sub.i] [intersection] C and b [member of] [D.sub.i] [intersection] C, so that [bar.[gamma]] = [gamma] [union] (a) [union] (b). Clearly [bar.[gamma]] is a smooth arc. Because [C.sub.i] is an open analytic arc, [[phi].sub.i] can be continued analytically into a neighbourhood U of a so as to be conformal in [D.sub.i] [union] U. This means that [[phi].sub.i] is conformal in a neighbourhood of [bar.[gamma]] and so [bar.[[phi].sub.i]([gamma])] = [[phi].sub.i]([bar.[gamma]]) is a smooth arc in [bar.[DELTA]] with [bar.[[phi].sub.i](a)] = 1 and <[[phi].sub.i]([bar.[gamma]] - (a)) [subset equal to] A. The result now follows from Lemmas 12 and 5.

We have now made a good deal of progress because of the following.

Lemma 14. Theorem 3 is true if [C.sub.i] is a circle or an ellipse.

Proof. In this case T isa finite union of type I and type II arcs only, so the result follows by Lemma 8(iv) and Lemma 13.

4. Type III Subarcs

The proof of Theorem 3 will be completed by showing that every type III arc in [D.sub.i] has the restriction property in [D.sub.i]. We have an open subarc [gamma] of an open subarc [GAMMA] of [C.sub.j] and [GAMMA] [subset or equal or] [D.sub.i]. In this case [infinity] is an end-point of [gamma] and [infinity] [member of] [partial derivative][D.sub.i], so both [C.sub.i] and [C.sub.j] are unbounded. We will use the same strategy we used for type II arcs in Lemma 13; we show that [sigma] = [bar.[[phi].sub.i]([gamma])] is a smooth arc in [DELTA] as in Lemma 12, so that [[phi].sub.i]([gamma]) has the restriction property in [DELTA] and so [gamma] has the restriction property in [D.sub.i]. The proof is more complicated because conformality of [[phi].sub.i] at [infinity] cannot necessarily be used. Instead we make use of the fact that as z [right arrow] [infinity] along [gamma], the unit tangent vector of [gamma] at z tends to a limit. The following two Lemmas help us exploit this fact.

Lemma 15. Let g [member of] [C.sup.1] [0, [infinity]) with g'(t) [not equal to] 0 (t [greater than or equal to] 0). Suppose that c [member of] C and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

exist. Define [sigma] = g([0, [infinity])) [union] (c). Then

(i) [sigma] is a compact arc,

(ii) [sigma] is rectifiable,

(iii) [sigma] is smooth.

Proof. (i) Define f on [0, 1] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Then f [member of] C[0,1] is a continuous parametrization of [sigma].

(ii) To prove that [sigma] is rectifiable, it suffices to show that, for some T > 0, [[integral].sup.[infinity].sub.T] [absolute value of g'(u)] du < [infinity]. Let [epsilon](t) = [omega] - (g' (t)/ [absolute value of g'(t)]). So [epsilon](t) [right arrow] 0 as t [right arrow] [infinity]. Choose T [greater than or equal to] 0 such that [absolute value of [epsilon](t)] [less than or equal to] 1/2 for t [greater than or equal to] T. Then, for t [greater than or equal to] T,

[absolute value of g'(t)] (1 - [bar.[omega]][epsilon](t)) = [bar.[omega]]g' (t). (8)

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

and hence

[[integral].sup.[infinity].sub.T] [absolute value of g'(u)] du < [infinity], (11)

which establishes the rectifiability of [sigma].

(iii) Let h : [0,1] [right arrow] [sigma] be the arc-length parametrization of [sigma]. Then h [member of] C[0, l], h(s) = g(t) where [[integral].sup.t.sub.0] [absolute value of g'(u)] du = s and s'(t) = [absolute value of g'(t)]. Therefore the map t [right arrow] s ([0, [infinity]) [right arrow] [0, l)) is [C.sup.1] with strictly positive derivative. So the inverse map s [right arrow] t ([0, l) [right arrow] [0, [infinity])) is [C.sup.1]. Since t(s(t)) [equivalent to] t and t'(s) = 1/s'(t) where 0 [less than or equal to] t [less than or equal to] [infinity] and 0 [less than or equal to] s [less than or equal to] l, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Hence h' is continuous and so h [member of] [C.sup.1] [0, l].

Lemma 16. Let k [member of] [C.sup.1] [0, [infinity]) with k'(t) [not equal to] 0 (t [greater than or equal to] 0) and suppose that k(t) [right arrow] [infinity] as t [right arrow] + [infinity]. Then, if [absolute value of [omega]] = 1,

k'(t)/[absolute value of k'(t)] [right arrow] [omega] [??] k(t)/[absolute value of k(t)] [right arrow] [omega].

Proof. Write [omega] = [e.sup.i[alpha]]. Choose T' such that t [greater than or equal to] T' [??] [Ree.sup.-i[alpha]](k'(t)/ [absolute value of k'(t)]) > 0. Then using [??] to denote the principal value of arg we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

is a branch of arg(k'/[absolute value of k']) and hence also of arg k' on [T', [infinity]) which tends to a as t [right arrow] [infinity]. We will find a branch [??] of arg k which also tends to a as t [right arrow] [infinity].

Let [epsilon] > 0. Choose T such that t [greater than or equal to] T [greater than or equal to] T' [??] [alpha] - [epsilon]/2 [less than or equal to] [theta] [less than or equal to] [alpha] + [epsilon]/2. Now k(t) - k(T) = [[integral].sup.t.sub.T] k'(u)du is a limit of Riemann sums [summation]([t.sub.i+1] - [t.sub.i])k' ([[xi].sub.i]).

The sector S (see Figure 2) is closed under addition and multiplication by positive scalars; therefore

k(t) - k(T) [member of] S for t [greater than or equal to] T. (15)

So there is an argument [mu](t) of k(t) - k(T) satisfying

[alpha] - [epsilon]/2 [less than or equal to] [mu] (t) [less than or equal to] [alpha] + [epsilon]/2 (t [greater than or equal to] T). (16)

Now k(t)/(k(t) - k(T)) [right arrow] 1 as t [right arrow] [infinity]. So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

If we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

then [??](t) is an argument of k(t) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]]. (19)

Hence also

[absolute value of (k(t)/[absolute value of k(t)] - [omega])] = [absolute value of ([e.sup.i[??](t)] - [e.sup.i[alpha]])] < [epsilon]. (20)

Consequently,

k(t)/[absolute value of k(t)] [right arrow] [omega] = [e.sup.i[alpha]], (21)

and our Lemma is proved.

There are now four cases to prove depending on the geometry of [C.sub.i] and [D.sub.i].

4.1. Case 1: [D.sub.i] Is a Half-Plane. The following lemma will be needed here and in Case 2.

Lemma 17. Let G be the open right half-plane Re z > 0 and let [theta](z) = (z - 1)/(z + 1) so that [theta] is a Riemann mapping function for G. Let k: [0, [infinity]) [right arrow] G be an injective [C.sup.1] function such that k'(t) [not equal to] 0, for all t [greater than or equal to] 0, and [lim.sub.t[right arrow][infinity]](t) = [infinity]. Let [rho] be the (simple) arc parametrized by k. If [lim.sub.t[right arrow][infinity]](k'(t)/[absolute value of k'(t)]) = to (with [absolute value of [omega]] = 1), then [sigma] = [bar.[theta]([rho])] satisfies the hypothesis of Lemma 12 and, hence, [rho] has the restriction property in G.

Proof. Put g = [theta] [omicron] k, so that g [member of] [C.sup.1] [0, [infinity]) parametrizes [theta]([rho]). Clearly g(t) [right arrow] 1 as t [right arrow] [infinity]. Now g satisfies the hypothesis of Lemma 15, for we can show that g'(t)/[absolute value of g'(t)] [right arrow] [[omega].sup.-1] as t [right arrow] [infinity]. Since [theta]'(z) = 2/[(z + 1).sup.2] it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

using Lemma 16.

So [sigma] = g[0, [infinity]) [union] ([[omega].sup.-1]) satisfies Lemma 12; hence g[0, [infinity]) has the restriction property in [DELTA]. But g[0, [infinity]) = [theta]([rho]) and, therefore, by Lemma 5, [rho] has the restriction property in G.

Now suppose that [C.sub.i] is a line and [D.sub.i] is a half-plane. By Invariance Lemma 5 with a linear equivalence [chi](z) = [alpha]z + [beta] ([alpha] = 0) we can assume that [C.sub.i] is the imaginary axis and that [D.sub.i] = G, the open right half-plane, as above. If [gamma] [subset or equal to] [D.sub.i] is a type III arc, it is a subarc of a line, parabola, or hyperbola component. Obviously [gamma] has a parametrization k as in Lemma 17. Hence [gamma] has the restriction property in [D.sub.i].

4.2. Case 2: [D.sub.i] Is the Concave Complementary Domain of a Parabola. Any two parabolas are conformally equivalent via a linear equivalence: [mu](z) = az + b (a, b [member of] C, a [not equal to] 0). So assume that [C.sub.i] is the parabola

[y.sup.2] = 4(1 - x) (23)

and that [D.sub.i] is the complementary domain to the "right" of [C.sub.i]. The function

w - [right arrow] [(1 + w).sup.2] (24)

maps the open right half-plane G conformally onto [D.sub.i] and the imaginary axis onto [C.sub.i]. Its inverse is the function

I(z) = [z.sup.1/2] - 1, (z [member of] [D.sub.i]), (25)

where [z.sup.1/2] is the principal square-root of z (here and throughout all standard multivalued functions will take their principal values).

Now let [gamma] [subset or equal to] [D.sub.i] be a type III arc. Because G is conformally equivalent to [D.sub.i] via [??] it will be sufficient to show that the arc I([gamma]) [subset or equal to] G has a parametric function k as in Lemma 17. Letting h be the arc-length parametrization of [gamma], then h [member of] [C.sup.1][0, [infinity]), [absolute value of h'(t)] = 1 and h(t) [right arrow] [infinity] as t [right arrow] [infinity], and h is injective.

Now [gamma] is a subarc of a line, parabola, or hyperbola component. Hence as z [right arrow] [right arrow] along [gamma] the unit tangent vector at z tends to a limit [omega] ([absolute value of [omega]] = 1). Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

and therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

by Lemma 16.

Put k = [??] [omicron] h. Then k is an injective parametric function for [??]([gamma]). Clearly k [member of] [C.sup.1][0, [infinity]), k(t) [right arrow] [infinity] as t [right arrow] [infinity], and

k'(t) = [??]' (h (t)) h'(t) [not equal to] 0, [for all]t [greater than or equal to] 0. (28)

Moreover,

k'(t)/[absolute value of k'(t)] = [[[absolute value of h(t)].sup.1/2]/h[(t).sup.1/2]] h'(t)/[absolute value of h'(t)] [right arrow] [[omega].sup.1/2]

So k is as in Lemma 17, which shows that [gamma] has the restriction property in [D.sub.i].

Remark 18. The notation [[omega].sup.1/2] is ambiguous when [omega] = -1 ([gamma] could be part of another parabola). But, because type I arcs can be ignored, we can assume that either [gamma] is contained entirely in the upper half-plane, in which case [(-1).sup.1/2] = i, or else [gamma] is in the lower half-plane and [(-1).sup.l/2] = - i.

4.3. Case 3: [D.sub.i] Is the Convex Complementary Domain of a Parabola. In this case the parabola

[[gamma].sup.2] = 4 [([pi]/4).sup.2] ([([pi]/4).sup.2] - x) (30)

will be chosen for [C.sub.i], and [D.sub.i] will be the complementary domain to the "left" of [C.sub.i]. This choice is made because then we have the relatively simple Riemann mapping function

[[phi].sub.i] (z) = [tan.sup.2] ([z.sup.1/2]), (z [member of] [D.sub.i]). (31)

This function maps the real interval (-[infinity], [([pi]/4).sup.2]) in an increasing fashion onto (-1, 1), and so it maps the upper/lower half of [D.sub.i] onto the upper/lower half of [DELTA]. The formula for [[phi].sub.i] is indeterminate on (-[infinity], 0], but these singularities are removable and the formula

[[phi].sub.i] (x) = -[tanh.sup.2][(-x).sup.1/2] (32)

can be used to define [[phi].sub.i](x), for negative x. This mapping will be examined in detail in a moment, but first we dispose of a trivial case and make some simple observations.

Let [gamma] [subset or equal to] [D.sub.i] be a type III arc. If [gamma] is a real interval (-[infinity], a), with a < [([pi]/4).sup.2], then [[phi].sub.i]([gamma]) is a subinterval of (-1, 1) which obviously has the restriction property in [DELTA]. So this case is trivial and needs no more attention.

The following observations are elementary.

(i) If [gamma] is part of another line, then it must be parallel to R and certainly disjoint from (-[infinity], 0].

(ii) If [gamma] is part of another parabola [C.sub.j], then [C.sub.j] must be symmetric about R and have an equation of the form

[y.sup.2] = 4a(b - x), (33)

where 0 < a [less than or equal to] [([pi]/4).sup.2], b [less than or equal to] [([pi]/4).sup.2].

(iii) If [gamma] is part of a hyperbola, then its asymptote must be parallel to R.

(iv) In all (nontrivial) cases [gamma] intersects (-[infinity], 0] in at most two points. So, because type I arcs can be ignored there is no loss of generality in assuming that Im z has constant sign on [gamma] and that Re z < 0 on [gamma].

(v) Hence, for definiteness, we can assume that [gamma] is contained in the open second quadrant.

(vi) In all cases [y.sup.2]/x tends to a limit as z [right arrow] [infinity] along [gamma]. If [gamma] is part of a line or hyperbola, the limit is 0, and if [gamma] is part of the parabola in (ii) above the limit is-4a. For future reference let us note that

0 [less than or equal to] lim [[y.sup.2]/4[absolute value of x]] [less than or equal to] [([pi]/4).sup.2]. (34)

(vii) Because the lim in (34) exists and because type I arcs can be ignored, we can assume that

[y.sup.2]/[x.sup.2] < 1, on [gamma]. (35)

Now let [gamma] be type III arc in [D.sub.i] as in (v) and (vi). We will show that [[phi].sub.i]([gamma]) has the restriction property in [DELTA]. To elucidate [[phi].sub.i]([gamma]) it is convenient to work backwards, examining the mapping properties of the square map (z [right arrow] [z.sup.2]), then tan, and then the principal square root.

Lemma 19. Let [[DELTA].sup.+] be the open semidisc

{[DELTA].sup.+] = [z [member of] C : [absolute value of z] < 1, x > 0}. (36)

If [sigma]' is a smooth simple arc in [bar.[[DELTA].sup.+]], if i is an end-point of [sigma]', and if [sigma]' - {i} [subset or equal to] [[DELTA].sup.+], then the arc

[sigma] = {[z.sup.2] : z [member of] [sigma]'} (37)

is a smooth simple arc in [bar.[DELTA]] satisfying the hypothesis of Lemma 12, so that [sigma]-[-1] has the restriction property in [DELTA].

Proof. This is clear: the square map z [right arrow] [z.sup.2] is conformal in a neighbourhood of [sigma]'.

Now let S be the open strip

S = {z [member of] C : 0 < x < [pi]/4}. (38)

It is well known that tan maps S conformally onto [[DELTA].sup.+]. The imaginary axis is mapped to the vertical part of [partial derivative][[DELTA].sup.+], and the line [pi]/4 + iR is mapped to the semicircular part of [partial derivative][[DELTA].sup.+]. Moreover, if z tends to infinity in S in such a way that y [right arrow] + [infinity], then tan z [right arrow] i.

Lemma 20. Let k [member of] [C.sup.1][0, [infinity]) be injective and satisfy k'(t) = 0, for t [greater than or equal to] 0. Suppose also that

(i) k(t) [member of] S for all t [greater than or equal to] 0,

(ii) Im k(t) [right arrow] + [infinity] as t [right arrow] + [infinity],

(iii) [lim.sub.t[right arrow][infinity]] Re k(t) = [x.sub.0] exists (0 [less than or equal to] [x.sub.0] [less than or equal to] [pi]/4),

(iv) [lim.sub.t[right arrow][infinity]] (k'(t)/ [absolute value of k'(t)]) = i.

If [gamma]' is the arc parametrized by k, then [sigma]' = (tan [gamma]') [union] {i} satisfies the hypothesis of Lemma 19, so that [tan.sup.2][gamma]' has the restriction property in [DELTA].

Proof. Let g = tan [omicron] k, so that g parametrizes [gamma]' and tan [gamma]' = g[0, [infinity]). Now g [member of] [C.sup.1][0, [infinity]), g'(t) [not equal to] 0, for all t [greater than or equal to] 0, and g(t) [right arrow] i as t [right arrow] + [infinity]. Lemma 15 will be used to show that [sigma]' = g[0, [infinity]) [union] (i) satisfies the hypothesis of Lemma 19. For all t [greater than or equal to] 0,

g'(t)/[absolute value of g'(t)] = [[[absolute value of cos k(t)].sup.2]/[(cos k (t)).sup.2]] [k'(t)/[absolute value of k'(t)]] (39)

Let k(t) = x(t) + iy(t). Since x(t) [right arrow] [x.sub.0] and y(t) [right arrow] + [infinity], as t [right arrow] + [infinity], and because cos x, cosh y > 0 on [gamma],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

So [lim.sub.t[right arrow][infinity]] (g(t)/[absolute value of g(t)]) exists.

The function

[??](z) = [z.sup.1/2] (41)

maps [D.sub.i] - (-[infinity], 0] conformally onto the vertical strip S as above. The limiting values of [??] from above and below a point x on (-[infinity], 0] are at [+ or-]i[(-x).sup.1/2], respectively. Now tan maps S conformally onto [[DELTA].sup.+] and tan [+ or-]i[(-x).sup.1/2] = [+ or -] i tanh [(-x).sup.1/2]. Finally the square function maps [[DELTA].sup.+] conformally onto [DELTA] -((-1, 0]), and it maps both of [+ or -]i tanh[(-x).sup.1/2] and - [tanh.sup.2] [(-x).sup.1/2]. Thus the cut made by I is repaired by the square function (by Schwarz's Reflection Principle): [[phi].sub.i] is continuous at all points of (-[infinity], 0] and therefore analytic on [D.sub.i]. Because [[phi].sub.i](z) [member of] (-1, 0] if and only if z [member of] (-[infinity], 0] the injectivity of [[phi].sub.i] on [D.sub.i] is clear.

Let [gamma] [subset or equal to] [D.sub.i] be a type III arc. Assume that y > 0 and x < 0 when z = x + iy [member of] y. Let [gamma]' = [??]([gamma]) so that [gamma]' [subset or equal to] S. We show that [gamma] is as in Lemma 20 so that [tan.sup.2][gamma]' has the restriction property in [DELTA] and, hence, [gamma] has the restriction property in [D.sub.i].

Let z = x + iy be an arbitrary point of [gamma] and write

[z.sup.1/2] = u + iv, (42)

for the corresponding point I(z) [member of] [gamma]'; then

x + iy = [u.sup.2] - [v.sup.2] + 2iuv. (43)

Eliminating v, and remembering that x < 0, we see that

[u.sup.2] = 1/2 (x + [([x.sup.2] + [y.sup.2]).sup.1/2]) = [absolute value of x]/2 ([(1 + [y.sup.2]/[x.sup.2]).sup.1/2] - 1) (44)

Since [y.sup.2]/[x.sup.2] < 1 (observation (vii)), the binomial series implies that

[u.sup.2] = [y.sup.2]/4[absolute value of x] - [1/16] [[y.sup.4]/[[absolute value of x].sup.3]] + ... ~ [y.sup.2]/4[absolute value of x], (45)

as z tends to x along [gamma]. It follows from (34) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

Now let h be the arc-length parametrization of [gamma] and write h(t) = x(t) + iy(t). Let k = [??] [omicron]h = [h.sup.1/2] so that k parametrizes [gamma]'. Write k(t) = u(t) + iv(t). (i), (ii), (iii), and (iv) of Lemma 20 can now be verified.

Obviously k(t) [member of] S, for all t [greater than or equal to] 0, so (i) is true. As t [right arrow] [infinity], [absolute value of k(t)] = [[absolute value of h(t)].sup.1/2] [right arrow] [infinity], but since 0 [less than or equal to] u(t) [less than or equal to] [pi]/4 we must have v(t) [right arrow] + [infinity], so that (ii) is true. Item (iii) follows from (46). Now h(t) [right arrow] [infinity] as t [right arrow] [infinity], [absolute value of h'(t)] [equivalent to] 1, and h'(t) [right arrow] -1 as t [right arrow] [infinity]. So, by Lemma 16,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

So (iv) is true and we have now completed the proof.

4.4. Case 4: [C.sub.i] Is a Hyperbola Component. We can deal simultaneously with the convex and concave complementary domains of a hyperbola component as follows. Let -[pi]/2 < [alpha] < [pi]/2 and let [C.sub.i] = sin([alpha] + iR). If [alpha] < 0, [C.sub.i] is the arc

[C.sub.i] = {z = x + iy [member of] C : x < 0, [[x.sup.2]/[sin.sup.2][alpha]] - [[y.sup.2]/[cos.sup.2][alpha]] = 1}, (48)

and if [alpha] > 0, [C.sub.i] is the arc

[C.sub.i] = {z = x + iy [member of] C : x > 0, [[x.sup.2]/[sin.sup.2][alpha]] - [[y.sup.2]/[cos.sup.2][alpha]] = 1}. (49)

Let [D.sub.i] be the complementary domain to the "left" of [C.sub.i] - then [D.sub.i] is convex when [alpha] < 0 and concave when [alpha] > 0. Linear equivalence will be used as before to reduce the general case to this one.

The function [sin.sup.-1] maps the double cut plane C - {(-[infinity], -1] [union] [1, [infinity])} conformally onto the vertical strip [absolute value of x] < [pi]/2, mapping the upper/lower parts of the first domain onto the upper/lower parts of the second. The upper and lower limits of [sin.sup.-1] at a point -x [member of] (-[infinity], -1] are -[pi]/2 [+ or -] i[cosh.sup.-1] x. The arc [C.sub.i] = sin([alpha] + iR) is mapped to the line Re z = [alpha]. Therefore [sin.sup.-1] maps [D.sub.i]- (-[infinity], -1] conformally onto the strip

[D.sub.[alpha]] = {z = x + iy [member of] C : - [pi]/2 < x < [alpha]}. (50)

If

[lambda](z) = [[pi]/4] [[z + ([pi]/2)]/[[alpha] + ([pi]/2)], (51)

then [lambda] maps [D.sub.[alpha]] conformally onto the strip

S = {z = x + iy [member of] C : 0 < x < [pi]/4}. (52)

Therefore

[[phi].sub.i] (z) = [tan.sup.2]A ([sin.sup.-1]z) (53)

is a Riemann mapping function for [D.sub.i]. Now let [gamma] be a type III arc in [D.sub.i]. As in Case 3 the case [gamma] [subset or equal to] R is trivial, so we can assume that [gamma] lies entirely in the upper half-plane. It will be sufficient for us to show that [lambda]([sin.sup.-1][gamma]) has a parametric function k as in Lemma 20.

Let z = x + iy be arbitrary point of [gamma] and write [sin.sup.-1]z = u + iv for the corresponding point of [sin.sup.-1]y. Clearly, by (50),

u + iv [member of] [D.sub.[alpha]]. (54)

Now

z = x + iy = sin (u + iv) = sin u cosh v + i cos u sinh v, (55)

so that

[[absolute value of z].sup.2] = [sin.sup.2]u [cosh.sup.2]v + [cos.sup.2]u [sinh.sup.2]v = [sin.sup.2]u + [sinh.sup.2]v. (56)

As z [right arrow] [infinity] along [gamma], [[absolute value of z].sup.2] [right arrow] + [infinity] and [sin.sup.2]u remains bounded; therefore

v - [right arrow] + [infinity] as z - [right arrow] [infinity] along [gamma]. (57)

It now follows from (56) and 57) that

sin u = x/[absolute value of z] [([tanh.sup.2]v + | [[sin.sup.2]u/[cosh.sup.2]v]).sup.1/2] ~ x/[absolute value of z] as z - [right arrow] [infinity]. (58)

Let h be the arc-length parametrization of [gamma]. As z [right arrow] [infinity] along [gamma] its unit tangent vector has a limit [e.sup.i[theta]], say. The asymptotes of [C.sub.i] are the rays arg z = [+ or -]([pi]/2 - [alpha]). Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (59)

So, by (57) and Lemma 16,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (60)

Now g = [sin.sup.-1] [omicron] h is a parametric function for [sin.sup.-1] [gamma]. By

(54) it follows that

(i) g(t) [member of] [D.sub.[alpha]] (t [greater than or equal to] 0), and (57) shows that

(ii) Im g(t) [right arrow] + [infinity] as t [right arrow] [infinity].

Equation (60) shows that

(iii) [lim.sub.t[right arrow][infinity]] Re g(t) = [sin.sup.-1] cos [theta] = ([pi]/2) - [theta] and we notice that - [pi]/2 [less than or equal to] ([pi]/2) - [theta] [less than or equal to] [alpha], by (59).

Finally observe that

g'(t)/[absolute value of g'(t)] = [[1-h[(t).sup.2].sup.1/2]/[(1 - h[(t).sup.2]).sup.1/2]] [h'(t)/[absolute value of h'(t)]]. (61)

Now in the upper half-plane [(1 - [w.sup.2]).sup.1/2] ~ -iw, as w [right arrow] [infinity]. So, as t [right arrow] [infinity],

g'(t)/[absolute value of g'(t)] ~ [[absolute value of h(t)]/-ih(t)] [[h'(t)]/[absolute value of h'(t)]], (62)

and therefore

(iv) [lim.sub.t[right arrow][infinity]] (g'(t)/[absolute value of g'(t)]) = i.

It follows easily that k = [lambda] [omicron] g satisfies the hypothesis of Lemma 20, and therefore <p;(y) has the restriction property in [DELTA].

http://dx.doi.org/10.1155/2014/102169

The author declares that there is no conflict of interests regarding the publication of this paper.

References

[1] P. L. Duren, Theory of [H.sup.p] Spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, NY, USA, 1970.

[2] P. L. Duren, "Smirnov domains," Journal of Mathematical Sciences, vol. 63, no. 2, pp. 167-170, 1993.

[3] G. M. Goluzin, Functions of a Complex Variable, American Mathematical Society, Providence, RI, USA, 1969.

[4] D. Khavinson, "Factorization theorems for different classes of analytic functions in multiply connected domains," Pacific Journal of Mathematics, vol. 108, no. 2, pp. 295-318, 1983.

[5] I. I. Privalov, Boundary Properties of Analytic Functions, GITTL, Moscow, Russia, 1950, (Russian; German translation: Deutscher, Berlin, Germany, 1956).

[6] A. M. Ruben and P. Rosenthal, An Introduction to Operators on the Hardy-Hilbert Space, Springer, New York, NY, USA, 2007.

[7 ] Y. Soykan, "On equivalent characterisation of elements of Hardy and Smirnov spaces," International Mathematical Forum, vol. 2, no. 24, pp. 1185-1191, 2007

[8] Y. Soykan, "On the continuity of restriction maps," International Journal of Mathematical Analysis, vol. 2, no. 17-20, pp. 823-826, 2008.

[9] Y. Soykan, "Restriction maps," International Journal of Mathematical Analysis, vol. 5, no. 29-32, pp. 1491-1496, 2011.

[10] V. P. Havin, "Boundary properties of integrals of Cauchy type and of conjugate harmonic functions in regions with rectifiable boundary," Matematicheskii Sbornik, vol. 68, no. 110, pp. 499-517, 1965.

[11] G. David, "Singular integral operators over certain curves in the complex plane," Annales Scientifiques de l'Ecole Normale Superieure. Quatrieme Serie, vol. 17, no. 1, pp. 157-189, 1984.

[12] Y. Soykan, "Boundary behaviour of analytic curves," International Journal of Mathematical Analysis, vol. 1, no. 1-4, pp. 133-157, 2007.

[13] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, vol. 20 of Studies in Advanced Mathematics, CRC Press, New York, NY, USA, 1995.

[14] Y. Soykan, "Restriction maps on the Hardy spaces of the unit disc," International Journal of Mathematical Analysis, vol. 7, no. 49-52, pp. 2407-2411, 2013.

Yuksel Soykan

Department of Mathematics, Art and Science Faculty, Biilent Ecevit University, 67100 Zonguldak, Turkey

Correspondence should be addressed to Yuksel Soykan; yukseLsoykan@hotmail.com

Received 2 June 2014; Accepted 19 August 2014; Published 11 September 2014

Academic Editor: Dashan Fan

Conflict of Interests
COPYRIGHT 2014 Hindawi Limited
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2014 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Title Annotation:Research Article
Author:Soykan, Yuksel
Publication:Abstract and Applied Analysis
Article Type:Report
Date:Jan 1, 2014
Words:7096
Previous Article:On lacunary mean ideal convergence in generalized random n-Normed spaces.
Next Article:A reproducing Kernel Hilbert space method for solving systems of fractional integrodifferential equations.
Topics:

Terms of use | Privacy policy | Copyright © 2022 Farlex, Inc. | Feedback | For webmasters |