Yamashita, Shinji

English

Yamashita Shinji

University of Szeged

Acta Sei. Math., 43 (1981), 141—108 The hyperbolic M. Riesz theorem SHINJI YAMASHITA , 1. Introduction. We shall prove the non-Euclidean hyperbolic versions of the theorems of M . R I E S Z [7] and of L . F E J E R and F . R I E S Z [4] (see [ 8 , Theorem VIII.45, p. 339 and Theorem VIII.46, p. 340], [3, p. 46]) and show a property of conformal mappings in terms of the non-Euclidean geometry in the unit disk. Let D = {|z|<l}, T=\0,2n), and K= {eu\t£T}. Let , ; <T(Z, W) = tanh _ 1 ( |z — w|/|l —zw|) be the non-Euclidean hyperbolic distance between z and w in D, and let a(z) = ff(z,0) = ( l /2)log[(l + |z | ) / ( l - |z | )] , z£D. Let B be the family of functions / , holomorphic and bounded, | / | < 1 , in D. Then <x(/) for f<zB, like | / | , hais the property that log o ( f ) is subharmonic in D, so that ( r ( / ) p =exp [p log a(f)] is subharmonic in D for all />>0; see [10]. Let be the family of f £ B such that fo(f)>>(re")dt T is bounded for 1 (0</?<°=). The class Hpa is the hyperbolic counterpart of the (parabolic) Hardy class HF in D [3, p. 2], and is called the hyperbolic Hardy class. (Recently, it is observed that an "elliptic" analogue of Hp (0<p<°o) , namely, a meromorphic Hardy class yields no new family [11, Theorem 1].) Each f £ B has the radial limit f*(t)= l im o f ( re ' ' ) at e" for a.e. t^T, and as will be seen, CT(/*) is of class LP(T) for all f£Hpa The hyperbolic M. Riesz theorem is : T h e o r e m . 1. Let C be a rectifiable, curve with the initial point a and the terminal point b.(possibly, a = b) in the complex plane. Suppose that. / ' CCI D JK 'and CPl/Cc {a, &}. '-Received October 23, 1980. 142 S. Yamashita Set for tíT, with eu$CC\K, V(t)= J arg (e* — w)| (w£C). c Then, for each feHra (0</><°°), fo(f)p(z)\dz\ S /o(f*)'(t)V(t)dt. C T If C is the diameter {jce i s |-1 1} {s£T), then V(t)=n/2, so that the hyperbolic Fejér and F. Riesz theorem is T h e o r e m 2. For each f£Hp (0<^<«>) and each s£T, / *(/)> (*«*) dx ^ (1/2) / a ( f * Y ( t ) dt. -1 T The Fejér and F. Riesz theorem has the obvious application to conformal mappings from D onto a Jordan domain with the rectifiable boundary [4, Satz IV]; see [8, Corollary, p. 341] and [3, Corollary, p. 47]. The hyperbolic version is not so apparent as in the cited case; namely, the following theorem does not appear to be a direct consequence of Theorem 2. There is no relation between o( f ) and i / ' l / o - l / l 2 ) like that between \f \ and \f'\. T h e o r e m 3. Let y be a Jordan curve in D with finite non-Euclidean length L. Let fbe a one-to-one conformal mapping from D onto the interior of y. Then the non-Euclidean length of the image of each diameter by f is not greater than L/2. The constant 2 in L/2 cannot be replaced by any larger constant. For the proofs of Theorems 1 and 3, the principal idea is to obtain the M. Riesz theorem for subharmonic functions of class PL in the sense of E. F. B E C K E N B A C H and T. R A D Ó [2] (see also [6, p. 9]); see Theorem 4 in Section 2. 2. Subharmonic functions of class PL. A function u defined in D is called of. class PL in D if h § 0 (possibly, u=0) and log« is subharmonic in Ö; we regard — °° as a subharmonic function. The family of all functions of class PL in D is denoted by PL again. All members of PL are subharmonic in D, and if u^PL, then up£PL for each />>0. I f / i s holomorphic in D, then \f\£PL, and further, if f£B, then o(f)dPL. Let PLP be the family of all u£PL such that up has a harmonic majorant in D (0<p<«>). Here, a function v subharmonic in D is said to have a harmonic majorant h in D if h is harmonic and vsh in D. The class H" is the family of / holomorphic in D such that \f\£PL", while is the family of fdB such that a(f)ePLp (0</><oo). ..I The hyperbolic M. Riesz theorem 143 T h e o r e m 4. Let C and V be as in Theorem I. Then, each u£PLp (0<p<«=) has the radial limit u*{t) at e" for a.e. t£T, and f up(z) \dz\ S n'1 f (u*)'(t)V(i) dt. C T Earlier, a special case of Theorem 4, where p=1, C is an arbitrary diameter, u is continuous on DUK, and u£PL, was established by BECKENBACH [1, Theo-rem 2]. It is now an easy exercise to extend a geometric theorem of BECKENBACH [1, Theorem 3] with the aid of Theorem 4. Theorem 1 (and consequently, Theorem 2) now follows from Theorem 4, applied to a(f)£PLp; note that CT(/)*=CT(/*). The theory of subharmonic func-tions of class PL thus serves for the differential geometry, as originated by Becken-bach and Rado, as well as for the hyperbolic Hardy classes. For the proof of Theorem 4 we shall make use of L e m m a 1 [5, Theorem]. Let 0 be a function convex and increasing on ( — a n d suppose that q>(t)/t-* + oo as r— + °=>. Set ( p ( ~ l i m (pit), and let v be a subharmonic function in D such that <p(v), t—-co again subharmonic, has a harmonic majorant in D. Then the radial limit v* (i) exists at e" for a.e. t£T, and is of L1(T), such that Furthermore, tp(v*)€V-(T). In effect, v admits a positive harmonic majorant in D (see [9, p. 65]), so that v=vA —q, where # £ 0 is a Green's potential and is the least harmonic majorant of v in D, expressed by the Poisson integral of a signed measure dn(f) = v*(t)dt + dMs(t) on T, where dps is singular with respect to dt. Now, [5, Theorem] asserts that i//is(i) = 0 on T a n d -<p(v*)£V-(T). P r o o f of T h e o r e m 4. Since u*ePLx with (up)* = (u*)p, it suffices to prove the theorem in the case p= 1. Set <p(t) = e' and v = \og u and consider Lemma 1. Since <p(v) has a harmonic majorant, v has the harmonic majorant h ( z ) = ( 2 K ) - I / ¿ ¿ ^ v*(t) dt ( Z € D ) . T I I Furthermore, h*=v*—log u*. Since u*=(p(v*)£L1(T) by Lemma 1, it follows 144 S. Yamashita from Jensen's inequality that eh^g, where g is the Poisson integral of <p(v*)=u*. Thus, ff1, where A: is a harmonic conjugate of h in D. Therefore \f*\ — \f\tr=eh* = u* and u = e" ^ eh = | / | in D. We now apply M. Riesz's cited theorem to / o f Hardy class H1 to obtain the follow-ing chain of estimates: /u(z) \dz\ ^ f | /(z)| \dz\ *n~vf |/*(/)| m dt = C C T = 71-! Ju*(t)V(t) dt, T whence follows Theorem 4. 3. Conformal mappings. We remember that if / is holomorphic in D and if f'^H1, then / is continuous on DO K and /(e") is absolutely continuous as a function of /€ T with (3-1) = i e " ( f W • for a.e. tCT, where (/ ')*(*) is again the radial limit o f f at e"; see [3, Theorem 3.11, p. 42]. For f £ B we denote /#00 = |/'00|/(l-|/(z)|2), ziD, and for the proof of Theorem 3 we shall make use of L e m m a 2. Let f£B and f'^H1, and assume that | / ( e " ) | < l for all t£T. Then f* f PL1 and d z m C/*)*(>) = for a.e. t£ T. df / ( l - . l / ( ^ ' ) l2 ) P r o o f . A calculation yields that A l o g / * = 4 ( / # ) 2 = - 0 except for the zeros o f / ' , so that f*£PL. Since | / | is bounded by a constant c < l in D, it follows f r o m / # = i | / ' | / ( 1 - c 2 ) in D with f'ZH1 that f*ePL\ Since ( / * ) + = [ ( / ' f |/(1 - | / | 2 ) a.e. on T, the second assertion follows from (3.1). P r o o f of T h e o r e m 3. Since y is rectifiable (in the Euclidean sense), it fol-lows from [3, Theorem 3.12, p. 44] that f'€H\ By Lemma 2, f*£PD. Since •• J 1 — |/(e")l2 4 l-\f(euW\ The hyperbolic M. Riesz theorem 145 it further follows f rom Lemma 2 that L= f (/*)*<!) dt. T The first assertion in Theorem 3 now follows from Theorem 4 applied to each diameter C and f*£PL\ It remains to prove the sharpness of 2 in L/2. For simplicity we consider the half-plane /?={w|Re w=-0} with the non-Euclidean metric \dw\j\2 Re w] in the differential form. The non-Euclidean length of a curve f in R is denoted by X(r). Let £ > 0 and let 0 C o n s i d e r the Euclidean rectangle Q with the vertices z 1 = a + 6 / , z2=a—ei, zs=b—ei, and zt=b+Bi. Let fc be a one-to-one conformai mapping f rom D onto Q such that fc maps the diameter [ — 1,1] onto the segment ab on the real axis. If we show that (3.2) l{ft{K))IX(ab) - 2 as e 0, then the function ( / e —1)/( / £ +1) serves as an example for the sharpness. Let ZiZj, zaz3 , z3z4, and z4zx be the four sides o f f E (K) . A calculation yields that X(zizi) = ;.(z2z3) = (1/2) log (b /a ) = A(ab), and as e—0, A(z1z2) = e/a — 0 and J.(z3z4) — e/b — 0. Therefore (3.2) holds. Appendix. Tsuji's proof of M. Riesz's theorem contains an obscure point. There is a gap between (5) and (6) in [8, p. 341] ; the meanings of d/dx in (5) and (6) are different. Since M. Riesz did not raise his result explicitly as in [8, Theorem VIII.46, p. 340], we must avoid this difficulty. The principal point is to prove that, for / ho lomorph ic on D U AT, (A) / |/(vv)| \dw\ - i f | /(e")| V(t) dt, C T where C and F a r e the same as in Theorem 1. Choose points w0=a, wlt ...,w„_1, w„=fc on C in this order. Then V(t) = lim 2 l a r g ( e " - w , ) - a r g ( e " - w i t _ 1 ) | k= 1 as max \wk—M^-J— 0, where arg (eu—w) is a fixed branch in D; V(t) is Lebesgue measurable on T. Now it follows from [7, (3), p. 54] (a careful reading shows that the cited point is true even if A or B lies on K) that the following estimate of the integral 10 146 S. Yamashita: The hyperbolic M. Riesz theorem on the rectilinear segment w k_ 1w k holds : / l / M I \dw\ s TI—1 / | / ( e " ) | | a r g ( e ' ' - w 4 ) - a r g ( e " - w i _ 1 ) | d t , lSJfcSH. S u m m i n g u p b o t h sides f r o m & = 1 t o n, a n d lett ing ^ a x ^ — we obta in (A). R e m a r k . I t migh t be m o r e appropr ia te to call [8, T h e o r e m VIII.46] the F . Car l -son a n d M . Riesz theorem. References [1| E. F. BECKENBACH, On a theorem of Fejér and Riesz, J. London Math. Soc., 13 (1938), 82—86. [2J E. F. BECKENBACH and T. RADÓ, Subharmonic functions and minimal surfaces, Trans. Amer. Math. Soc., 35 (1933), 648—661. [3] P. L . DÜREN, Theory of Hp spaces, Pure and Applied Mathematics, vol. 38 , Academic Press (New York and London, 1970), [4] L . FEJÉR and F . RIESZ, Über einige funktionentheoretische Ungleichungen, Math. Z., 1 1 (1921), 305—314. [5] L . GÄRDING and L . HÖRMANDER, Strongly subharmonic functions. Math. Scand., 1 5 (1964) , 93—96; Correction, ibid., 18 (1966), 183. [6] T. RADÓ, Subharmonic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete, 5. Band, Springer-Verlag (Berlin—Göttingen—Heidelberg, 1937). [7] M . RIESZ, Remarque sur les fonctions holomorphes, Acta Sei. Math. 12 (1950) , 5 3 — 5 6 . [8] M. TSUJI, Potential theory in modern function theory, Maruzen Co., Ltd., (Tokyo, 1959). [9] S. YAMASHITA, On some families of analytic functions on Riemann surfaces, Nagoya Math. J., 31 (1968), 57—68. ;[10] S. YAMASHITA, Hyperbolic.Hardy class H\ Math. Scand., 45 (1979), 261—266. [11] S. YAMASHITA, The meromorphic Hardy class is the Nevanlinna class, J. Math. Anal. Appl., tó be published. DEPARTMENT O F MATHEMATICS TOKYO METROPOLITAN UNIVERSITY FUKAZAWA, SETAGAYA-KU, TOKYO, 158 JAPAN 

The hyperbolic M. Riesz theorem

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The hyperbolic M. Riesz theorem

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