A new characterization of holomorphic functions in the unit disk Текст научной статьи по специальности «Математика»

134

Probl. Anal. Issues Anal. Vol. 7(25), No. 1, 2018, pp. 134-147

DOI: 10.15393/j3.art.2018.4690

UDC 517.444

V. V. VüLCHKQv, Vit. V. Vqlchkqv

A NEW CHARACTERIZATION OF HOLOMORPHIC FUNCTIONS IN THE UNIT DISK

Abstract. We study the conditions under which a function satisfying a weighted Morera property for all hyperbolic circles of a fixed radius is holomorphic. We show that one of such conditions is the restriction on a speed of decrease of the difference between the function and its Cauchy type integral.

Key words: Cauchy integral formula, holomorphy tests, Legend-re functions, hyperbolic plane

2010 Mathematical Subject Classification: 43A45

1. Introduction and the statement of the main result. Let

C be the complex plane, D = {z E C : \z\ < 1}, T = {z E C : \z\ = 1}, D = D U T. By the classical Cauchy theorem, the necessary and sufficient condition of a function f E C(D) to be holomorphic in D is

Other holomorphy tests which are based on the Cauchi integral formula can be found in [3]. Another characterization of holomorphic functions is related with the well-known Morera property which has been studied in many contexts and generality (we refer the reader to [1], [7]-[9], [11], [12] for an account of considerable amount of research).

Throughout what follows G is the conformal automorphism group of the disk D. We denote by gA the image of a set A C D under the map g E G. For q E (0,1) we set jg = {z E C : \z\ = q}.

Let f E C(D) and let E C (0,1) be a given set. Assume that

for all ze D.

(1)

T

©Petrozavodsk State University, 2018

for all g £ G, q £ E. For which E does this imply that f is holomorphic? One of the results in [2] states that f is holomorphic if and only if the equations

P~' =0 (Q £ E>

have no common solution z £ C. Here and below PV is the associated Legendre function of the first kind (see [4, Ch. 3, Sect. 3.2]). Also, there is a nonconstant radial real analytic (so nonholomorphic) function on D satisfying (2) for one fixed q £ (0,1) and all g £ G.

In this paper we present a new aspect: we study the case when an assumption of type (1) is replaced by an upper bound of the difference

f (z) - £ / «

T

as \z\ ^ 1, and assumption (2) holds for one fixed q £ (0,1) and all g £ G. Our main result is as follows.

Theorem 1. Let q £ (0,1) be fixed. Then

(i) If f £ C(D) satisfies (2) for all g £ G and

f (z) - ^ / f-zdZ = 0 (^—zi) as \z\ ^ 1, (3)

T

then f is holomorphic on D.

(ii) There exists a nonholomorphic function f £ C(D) such that condition (2) is fulfilled for all g £ G, and

f (z) - 2-jfzzdZ = O (Tr—M) as |z|^ 1. (4)

T

The proof of Theorem 1 is based on the development of the method proposed by the authors in [10]. We introduce a transmutation operator which establishes a homeomorphism between the space of smooth radial functions in D and the space of even functions in C^(R). In a certain general sense it commutes with the generalized convolution operator; this allows us to reduce the problem to the one-dimensional case. Finally,

we use some results of the theory of convolution equations in R (see [7, Part 3]).

2. Notation. It is known that for each g £ G there exist uniquely defined parameters t, z £ C such that \t\ = 1, \z\ < 1, and

w — z

gW = t -__

1 — zw

for each w £ D. The group G parametrized by pairs (t, z) is the motion group in the Poincare model of the hyperbolic plane H2 which is realized as the disk D and has the corresponding Riemannian structure (see [5, Introduction, § 4]). The hyperbolic distance d between the points zi, z2 £ £ H2 in this model is defined by

-¡( , 1, \1 — ziz2\ + \z2 — zi\ d(z1, z2) = - ln

2 \1 — ziz2\ — \z2 — zi\ In particular,

1 1 + Izl

d(z, 0) = -ln--= arth\z\ and \z\ = th d(z, 0), z £ H2.

2 1 — \ z\

The hyperbolic measure dfx on H2 has the form

, , , i dz A dz dfx(z) = --

2(1 -|z|2 )2

The distance d and the measure dfx are invariant with respect to the group G.

For r > 0, the symbol Br denotes the open hyperbolic disk of radius r centered at the origin of H2, i.e.,

Br = {z e H2 : d(0, z) < r}.

Let Bx = H2. For r > 0, we set

Br = {z e H2 : d(0, z) < r}, dBr = {z e H2 : d(0, z) = r}.

Furthermore, let xr be the characteristic function (the indicator) of the disk Br. We denote by L(H2) and Lloc (H2) the classes of functions integrable and locally integrable on H2 with respect to the measure df ,

respectively. Let dg be the Haar measure on G normalized so that the relation

^(g°)dg = ^(z) dKz)

(5)

G

H2

is valid for each function ^ E L(H2) (see [5, Introduction, § 4, Section 3]).

Let D(H2) (or D(R)) be the set of functions with compact supports in C^(H2) (in C^(R), respectively) endowed with the standard topology (for instance, see [5, Ch. 2, § 2.2]). We denote the spaces of radial functions in L(H2), C^(H2) and D(H2) with the induced topology by L(H2), C ^ (H2) and D^ (H2). In a similar way, we let Cf° (R) and D(R) denote the spaces of even functions in C^(R) and D(R), respectively.

Let fi,f2 be radial functions in the class Lloc(H2). Assume that at least one of the functions f1 and f2 has compact support. Then we define the generalized hyperbolic convolution f 1 o f2 by

(fi o f2)(g°)= f fi(z)f2 (g-1z)T^i4 d^(z),

H2

(1 - z • g°)2

g E G. (6)

Equality (6) shows that f1 o f2 is a radial function in the class Lloc(H2). It follows from (6) and (5) that

(fi o f2)(Z)= fi (g°)f2(g-1 Z)

(1 - \g°\2)2

G

(1 - Z • g°)2

dg, Z E D,

(7)

and

fi O f2 = f2 O fi.

If fi,f2 E Cb°°(H2) then, in view of (7),

L(fi of2) = fi oLf2 = (Lfi )of2,

(8)

where

22

L = 4(1 - \z\2)

d 2 d a^z -8(1 -z2)za?

(9)

In addition, if f1,f2, f3 E Lloc(H2) are radial functions, and at least two of the functions f1, f2, f3 have compact supports, then

(fi o f2) o f3 = fi o (f2 o f3).

Let h £ L(R). Its Fourier transform h is defined by

h(A) = J h(t)e—ixtdt, A £ R. Assume that A £ C and

iA — 1 v = v (A) = .

Now define

Ux(z) = (1 — \z\2)V+i F (v + 2, v + 1; 2; \z\2) , z £ D, (10)

where F is the Gauss hyperheometric function. Also let

Hx(z) = (1 — \z\2)V F (v + 2, v; 1; \z\2) , A £ C, z £ D. (11)

We can conclude from (11) and the expansion of F in a hypergeometric series that Ux is an entire function of A. It is even because

Hx(z) = (1 — z\2)—2 F (^, ; ^;izz-т) (12)

(see [4, formula 2.9 (3)]) and F(a, b; c; z) is symmetric in a and b. It follows from (9), (11) and the hypergeometric differential equation that

(LUx)(z) = —(A2 + 1) Ux(z). (13)

Suppose that the function T £ L^ (H2) is compactly supported. For

A £ C, let

F(T)(A) = JT(z) Hx(z) (1 — \z\2)2 df(z).

H2

Equality (12) shows that the function F(T) is an even entire function of A. If fi and f2 are functions with compact support in L^ (H2) then

F (fiof2)= F (fi) F (f2) (14)

(see (8), (13) and the proof of Lemma 8 in [10]).

3. Auxiliary results. We need the following lemmas in the proof of Theorem 1.

Lemma 1. The following equality holds d

— (zUx(z)) = 2Ux(z), A E C, z E D, (15)

dz

where the function Ux is defined by (1°).

Proof. First assume that z = °. Relation (1°) yields

dz (-zUx(z)) = 1 d (P2(1 - P2)V+1p(v + 2, V + 1; 2; p2)) .

Using now [4, formula 2.8 (25)] and (11) we arrive at (15). Now continuous extension to the point z = ° completes the proof. □

For a, 3, A E C, we set (a)(3) (a + 3 +1 - iA a + 3 + 1 + iA 2 \

(r)=f (—2—' —2—;a+1 ).

Lemma 2. For each A E C,

F (xr) (A) = * sh2 r ^(x'1](r). (16)

Proof. For brevity we set

2 1 + iA 1 - iA

z = — sh r, a =-, b =-.

22

The expansion of p(x'(\r) in a hypergeometric series shows that

lim ^q'1 (r) - 1 = ^ a (a +1) ... (a + n - 1) b(b +1) ... (b + n - 1) ^ = a—^i r(a + 1) n!r(n) ^

^ a (a + 1) ... (a + n) b(b + 1) ... (b + n) „ b + 1 + 1 2 )

= z ^-^T—, 7y\-z = abzF(a + 1,b +1;2;z) =

z—1 n! l(n + 2)

n=0 v '

= - 1(A2 + 1) sh2 r ^(r).

Now, by the definition of the transform F and [8, Proposition 7.2 (ii)] we see that

/r 4 (®>i) ( ) 1

sh(2t)(t)dt = — AI'+T «-—i V+1— •

0

This gives, by the equality above, the desired result. □

Lemma 3. For each r > 0, the following assertions hold.

(i) F (xr) (0) = 0.

(ii) The function F (xr) (A) has infinitely many zeros, all of which are real, simple and lie symmetrically relative to A = 0.

(iii) If F (xr) (A) = 0 then \F(xr)'(A)\ > c\A\—3/2, where the constant c > 0 is independent of A.

Proof. Using (16) and [4, formula 3.2 (7)] we obtain

F (xr)= *thrP(—x-i)/2(ch2r).

Now the assertions of Lemma 3 are partial cases of Lemmas 2.4, 2.5 and Corollary 2.2 in [7, Part 2, Ch. 2]. □

The next statements are analogs of the Paley-Wiener theorem and the inversion formula for the transform F.

Lemma 4.

(i) An even entire function w is the F-transform of a function in ^(H2) with support in Br if and only if for each N £ Z+ there exists a positive constant cN > 0 such that

er|imx|

\w(A)\< cN (1 + \A\)N , A £ C

(ii) Assume that f £ L Pi C)(H2) and

cc>

J A \F(f )(A)\ dA <

0

Then

CO

f (z) = n6 f F(f )(A) Hx(z) \c(A)\—2dA + F(f)(i) Hi(z),

where

23—iX r(iA)

c(A) =

r (X=!) r (X+3)

and the integral is absolutely convergent for each z £ C. Proof. To prove (i), let 0 <£ <t. We set

K(t,£) = ^(cht)2(ch 2t — ch 2£) —1/2F (2, —2± ^^f—f1) . For A £ C, z £ D, we have

t

Hx(z)= i K(t, 0 cos A^,

where t = arth \z\ (see [8, Proposition 7.3]). Owing to (13), the rest of the proof is identical to that of Proposition 1 in [10].

In view of Theorem 2.3 in [6], assertion (ii) can be proved in the same way as Proposition 2 in [10]. □

For f £ Dn(H2), t £ R1, let

C

1 CK f 1

A(f)(t) = ^ J F(f)(A)\c(A)\-2 cos(At)dA + - F(f )(i) cos(it).

0

Using Lemma 4 and Stirling's formula, it is easy to see that A(f) £ £ C^ (R1 ).

Suppose that T £ L^ (H2) has compact support. If F(T) £ L2 (R1 ) then, by the classical Paley-Wiener theorem, there exists a function A(T) £

£ L2(R1) with compact support such that A(T) = F(T).

The main properties of the map f ^ A(f) are contained in the following lemma.

Lemma 5.

(i) The transformation A can be extended to a linear homeomorphism of the spaces C^ (H2) and CC(R1).

(ii) Let f £ C^ (H2), r> 0. Then f = 0 in Br if and only if A(f) = 0 in (— r, r) .

(iii) Assume that T G L^ (H2) has compact support and let F(T) G G L2(R!). Then

A(f o T)(t) = A(f) * A(T) (17)

for each f G C(H2).

(iv) Let X G C. Then

A(HX )(t)=cos Xt. (18)

Proof. This lemma can be proved in the same way as Theorems 2 and 3 in [10] taking into account Lemma 4 and (14). □

4. Proof of the main result. We now proceed to the proof of Theorem 1. To prove (i), we remark that we may confine our attention to the case f G C(D) fi C^(D) (otherwise we can use the standard hyperbolic regularization, see [5, Ch. 1, the proof of Theorem 4.2]). Let r > 0 and

Jf(z) = r=W/ f(r^)*' c G D- (19)

dBr

We set

z + Z

gcz = _ ' z G D. 1 + Cz

By Green's formula, we have

T />n f d , „, ,, dz A dz n f df , , (1 — |z|2)2 , , N

Jf(z) =—J m(f gz)) —^ =2i7 dzgz) WoF Mz).

Br Br

Since gz G G and the measure d^(z) is G-invariant,

t (z) 2- i df ( ) (1 — g-1 w|2)2 d ( ) Jf(() =2i I dz(w)-—. d^(w)-

This implies easily that

Jf«)=2 j |—wg2

gz Br

g^Br dz (1 + Zg-1 w)2

Because of (6) we can write

: 2i _ dz

Jf = 2if o xr in H2. (20)

Next, let g £ G, z £ D, and

gz = , where a,b £ C, \a\2 — \b\2 = 1. (21)

bz + a

Putting Z = g0, we see from (19) and (31) that

Jf (g0) = \a\2 j f ( piN dz.

dBr ^ + a z'

a

The change of variable z = = w in the integral enables us to write

a

Jf (g0)= a2 J f (gz) dz Vg £ G. (22)

dBr

Equalities (20) and (22) show that the function f satisfies (2) for all g £ G if and only if

f o xr = 0 in H2, (23)

dz

where r = arth p. Introduce the following auxiliary function

u(z) = i / f (Zz) dZ, z £ D. (24)

IC|=i

It follows from (24) that u £ CC (D). We then find that u satisfies (2) and, if z = 0,

u(z) = v(\z\)e—i arg 2, where v(\z\) = — if (\z\e^) e^ dif. (25)

2n J

— n

A straightforward calculation shows that

dU = v'(W) + (26)

du

In particular, — £ CCC(H2). In addition, it follows from (25) that

dz N

v(0) = 0. (27)

Relation (23) leads to du

^^ o xr = 0 in H2, where r = arth p. (28)

dz

Then we find from (16) and the asymptotic expansion for fX1'1)(r) (see [7, Part 2, Corollary 2.2]) that F(xr) £ L2(R1). By the Paley-Wiener theorem, there exists a function Ar £ L2 (R1) with compact support such that Ar = F(xr). Owing to Lemma 5 and (17),

* Ar = 0 on R-

!)

Using now Lemma 3 and [7, Part 3, Theorem 1.3], we conclude that

A ( — j (t) = £ Cx cos (At), t E R, (29)

^ z' xeN(r)

where N(r) = {A > 0 : F(xr) = 0}, cx E C, the series converges in the space C^(R), and

\cx\ = O ((1 + A)-a) as A ^ for each fixed a > 0. According to Lemma 5 the series

y^ cx Hx

xeN (r)

converges in C^(H2) to some function w E (H2). Relations (29) and (18) yield

*( ^ = A(w),

du

whence — = w in D. By the definition of w this shows that dz

v(\z\)= \2 £ cx Ux(\z\) (30)

xeN (r)

(see (26), (27) and Lemma 1). Next, it follows from (25) and (3) that

v(|z|)= o( y/1 — lz|) as |z|^ 1. (31)

In addition, for each e > 0, X G N(r) we have

U\(z) = (a(X)elXt + a(—X)e~lXt)e-t + O(e-2t(32)

where

( r(iX/2)

a(X) = vnr((iX + 3)/2)' (33)

|z| = th t > e and the constant in the symbol O depends only on e (see [5, Introduction, Theorem 4.15]). Applying now (31) we see from (30) and (32) that

cx (a(X)elXt + a(—X)e-lXt) ^ 0 as t ^

xeN (r)

Together with (33) this implies that cx = 0 for all X G N(r) (see, for instance, [7, Part 3, Theorem 1.6]). Owing to (30) we obtain v = 0. In view of (25) and (24) this means that

J f (z) dz = 0

YR

for each R G (0,1).

Assume now that h G G. Writing (2) with f (hz) instead of f and using (3) with hz instead of z, we obtain

i f (hz) dz = 0

YR

for all R G (0,1). Then

df 2

o Xr = 0 in H for each r > 0.

dz

Now, from the arbitrariness of r > 0 it follows that f is holomorphic.

To prove (ii) consider the function f (z) = zUx(z), where X G N(r) and r = arth p. Owing to Lemma 1, we infer that

f = 2 ^,

dz

whence f is nonholomorphic in D. Next, the proof of (i) shows that f satisfies (28) and (2) for all g G G. Finally, it follows from [5, Introduction, Theorem 4.15] that relation (4) is fulfilled. This completes the proof of Theorem 1. □

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Received December 22, 2017. In revised form, March 11, 2018. Accepted March 11, 2018. Published online April 24, 2018.

Valerii V. Volchkov Donetsk National University 24, Universitetskaya str., Donetsk 83001 E-mail: [email protected]

Vitalii V. Volchkov Donetsk National University 24, Universitetskaya str., Donetsk 83001 E-mail: [email protected]