A CHARACTERIZATION OF REAL ANALYTIC FUNCTIONS

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 43, 208, A CHARACTERIZATION OF REAL ANALYTIC FUNCTIONS Grzegorz Łysik Jan Kochanowski University, Faculty of Mathematics and Natural Science Kielce, Poland; lysik@iman.l Abstract. We give a characterization of real analytic functions in terms of integral means. The characterization justifies the introduction of a definition of analytic functions on metric measure saces.. Introduction It is well-known that harmonic functions, i.e., solutions to the Lalace equation u 0, where n i, can be characterized by the mean value roerty. 2 x 2 i Namely, a function u continuous on an oen set Ω R n is harmonic on Ω if, and only if, for any closed ball B(x,R Ω the value of u at the center of the ball is equal to the integral mean of u over the ball. In the revious aer [7] we roved that olyharmonic functions on Ω can be characterized as those continuous functions on Ω for which integral mean over balls of radius R is exressed as an even olynomial of R with coefficients continuous on Ω. Here we extend the above characterization to the case of real analytic functions. Namely we rove The main theorem. Let u be a continuous, comlex valued function on an oen set Ω R n. Then u is real analytic on Ω if, and only if, there exist functions u 2l C 0 (Ω,C for l N 0 and ǫ C 0 (Ω,R + such that ( B(x, R B(x,R u(ydy locally uniformly in {(x,r: x Ω,0 R < ǫ(x}. u 2l (xr 2l In fact, the necessity of the exansion ( for the real analyticity of u is well known, see Theorem below. The novelty of the main theorem is that the sufficient condition for real analyticity of u assumes only continuity of the functions u and u 2l in the exansion (. Thus it justifies the introduction of a definition of analytic functions on metric measure saces. 2. Preliminary results Throughout the aer Ω stands for an oen set. If Ω R n, x Ω and 0 < R < dist(x, Ω solid and sherical means of a continuous, comlex valued function u C 0 (Ω are defined by (2a (2b M(u; x,r N(u; x,r σ(nr n nσ(nr n B(x,R u(ydy u(x+rzdz, σ(n B(0, S(x,R u(yds(y nσ(n S(0, u(x+rzds(z, htts://doi.org/0.586/aasfm Mathematics Subject Classification: Primary 26E05, 32A05; Secondary 32K99, 3E05. Key words: Real analytic function, integral mean, metric measure sace.

2 476 Grzegorz Łysik where σ(n π n/2 /Γ(n/2 + (with Γ the Euler Γ-function is the volume of the unit ball B(0, in R n and ds denotes the surface measure on the shere S(x,R. Note that by the second formulas in (2a and (2b for a fixed x Ω the functions M and N are defined for R < dist(x, Ω and they are even continuous functions of R satisfying M(u; x,0 N(u; x,0 u(x. Comuting M(u; x,r in sherical coordinates one obtains the following differential relation between the functions M(u; x,r and N(u; x,r (see [5, Lemma ], ( R (3 n R + M(u; x,r N(u; x,r for any x Ω and R < dist(x, Ω. If u is a real analytic function, then its mean value functions M(u;, and N(u;, are exressed by the so called Pizzetti s serii. Theorem. [8, 6] (Mean-value roerty If u is a real analytic function on Ω R n, then for any x Ω and R small enough we have k u(x (4 M(u; x,r 4 k( n + R2k k! 2 k and (5 N(u; x,r k u(x 4 k( n 2 k k! R2k, where (a k a(a+ (a+k is the Pochhammer symbol. The exansions (4 and (5 are uniform on comact subsets of Ω. Conversely real analytic functions are characterized as those smooth ones for which the Pizzetti s serii converge. Theorem 2. [6, Theorem 3.2] (Converse to the mean-value roerty Let Ω R n, u C (Ω and ǫ C 0 (Ω,R +. If the series on the right hand side of (4 or (5 is locally uniformly convergent in {(x, R: x Ω, R < ǫ(x}, then u is real analytic on Ω. 3. A characterization of real analyticity In this section we shall rove the main theorem. Lets start with Lemma. Let Ω be a domain in R n and u C 0 (Ω. Assume that there exist functions v 2l C 0 (Ω for l N 0 and ǫ C 0 (Ω,R + such that (6 N(u; x,r v 2l (xr 2l locally uniformly in {(x,r: x Ω, R < ǫ(x}. Then u C (Ω and v 2l C (Ω for l N 0. Proof. Let η(r be a smooth function on [0, suorted by [0,] and constant forr close to zero. Assume also that nσ(n 0 η(rrn dr. Then η ε (y n η ( y ε ε is a radially symmetric mollifier suorted by B(0,ε. Integrating in sherical coordinates we get η ε (ydy nσ(n η(rr n dr. B(0,ε 0

3 A characterization of real analytic functions 477 Alying the Lalace oerator to η ε we have η ε (y ( y ε n+2 η + ( n y η : L ε ε n+ ε ( η( y. y ε For l N 0 ut m 2l (η B(0, η (yy 2l dy. For x from a comact subset of Ω and sufficiently small ε > 0 let us comute acting on the convolution η ε u. Using sherical coordinates, (2b and (6 we derive ( η ε u (x ( η ε u(x ε ( u(ζ ds(ζ L ε ( η(rdr 0 S(x,r (2b (6 ε 0 nσ(nn(u;x,rl ε ( η(rr n dr ε nσ(nv 2l (x L ε ( η(rr 2l+n dr 0 2l 2 v 2l (x nσ(nε L ( η(tt 2l+n dt 2l 2 v 2l (x nσ(nε 2l 2 v 2l (x ε l B(0, 0 B(0, v 2l+2 (x ε 2l m 2l+2 ( η L ( η( y y 2l dy η (yy 2l dy since by the Green formula B(0, η η (ydy (yds(y 0. Analogously S(0, n for k N 0 we obtain k( η ε u (7 (x v 2l+2k (x ε 2l m 2l+2k ( k η. Since k (η ε u is convergent in the distributional sense as ε 0 we get m 2k ( k η v 2k lim k (η ε u k( limη ε u k v 0 D (Ω. ε 0 ε 0 Since v 2k C(Ω alying the Weyl lemma [9, Lemma 2] we conclude that u v 0 C 2k (Ω. Note that for 0 l k we have k v 0 k l ( l v 0 m 2l ( l η k l v 2l C 0 (Ω. So v 2l C 2k 2l (Ω since by (8 below, m 2l ( l η 4 l l! ( n m 2 l 0(η and m 0 (η. Since k is arbitrary big we conclude that v 2l C (Ω for l N 0. Now the main theorem is a consequence of Theorem and the following. Theorem 3. Let Ω be a domain in R n, ǫ C 0 (Ω,R + and u C 0 (Ω. If there exist functions u k C 0 (Ω for k N 0 and ǫ C 0 (Ω,R + such that M(u; x,r u k (xr k locally uniformly in {(x,r: x Ω, R < ǫ(x}, then u is real analytic on Ω, u k 0 if k is odd and u k [ 4 l( n 2 + l l!] l u if k 2l with l N 0.

4 478 Grzegorz Łysik Proof. Since, by the second formula in (2a, M(u; x, R is an even function of R we have u k 0 if k is odd. Next alying (3 we get for x Ω and R < ǫ(x, ( R ( N(u; x,r n R + ( 2l u 2l (xr 2l n + u 2l (xr 2l. Hence the assumtions of Lemma are satisfied with v 2l ( 2l + u n 2l and so u 2l C (Ω for l N 0. By the Green formula we get for l N 0, m 2l ( η η (yy 2l dy η (y ( y 2l dy B(0, 2l(n+2l 2 B(0, B(0, η (yy 2l 2 dy { 0 if l 0, 2l(n+2l 2m 2l 2 (η if l. So for k N 0 we obtain { (8 m 2l ( k η 0 if l 0,...,k, 4 k (l k + k (l k + n 2 km 2l 2k (η if l k. By (7 for k N 0 we get k( η ε u (x v 2l+2k (x ε 2l m 2l+2k ( k η Taking the limit as ε 0 we get ( 2l+2k n + u 2l+2k (x ε 2l 4 k (l + k (l+ n 2 km 2l (η. k u 0 (x lim ε 0 k (η ε u(x ( 2k n + u 2k (x 4 k k!( n 2 km 0 (η. Since u 0 u and m 0 (η we conclude that 4 k k! ( n + u 2 k 2k k u for k N 0. Hence k u(x M(u; x,r 4 k( n + 2 k k!r2k locally uniformly in {(x,r: x Ω, R < ǫ(x} and Theorem 2 imlies analyticity of u in Ω. Remark. Analogues of Lemma and Theorem 3 hold true if one only assumes that u,u l,v 2l L loc (Ω for l N 0. Also the condition of continuity of ǫ can be relaced by its uniform ositivity on comact subsets of Ω. 4. Analytic functions on metric measure saces It is well known that the mean value characterization of harmonic functions can be used to define harmonic functions on metric measure saces (MMS, see [3, Definition 2.]. If the measure is continuous with resect to the metric, then harmonic functions on MMS satisfy the maximum rincile, the Harnack tye inequality and the Weierstrass and Montel convergence theorem, see [3]. On the other hand Alabern, Mateu and Verdera obtained in [] a characterization of Sobolev saces on R n only

5 A characterization of real analytic functions 479 in terms of the Euclidean metric and the Lebesgue measure which allowed them to define higher order Sobolev saces on MMS. Here we roose a definition of analytic functions on MMS. Definition. Let (X,ρ,µ be a metric measure sace with a metric ρ and a regular Borel measure µ which is ositive on oen sets and finite on bounded sets. Let Ω be an oen subset of X. For any x Ω and 0 < R < dist(x, Ω define the solid mean of a continuous, comlex valued function u C 0 (Ω by M X (u; x,r u(ydµ(y, µ(b ρ (x,r B ρ(x,r where B ρ (x,r is a ball with resect to the metric ρ of center at x and radius R. We also ut M X (u; x,0 : lim R 0 M X (u; x,r u(x. Definition 2. We say that a function u C 0 (Ω is (X,ρ,µ-analytic on Ω if there exist functions u k C 0 (Ω for k N 0 and ǫ C 0 (Ω,R + such that M X (u; x,r u k (xr k locally uniformly in {(x,r: x Ω,0 R < ǫ(x}. Since an (X,ρ,µ-analytic function u on Ω uniquely determines M X (u; x,r and vice versa the toology on the sace A X (Ω,ρ,µ of (X,ρ,µ-analytic functions on Ω can be defined by where E(K,ǫ { A X (Ω,ρ,µ rojlim K Ω indlim ǫ>0 E(K,ǫ, k F C 0 (K;C F(x,R R ( ǫ,ǫ : F K,ǫ su k R0 ǫ k k N 0,x K k! By Theorem 3 and the Pringsheim tye theorem [4, Theorem] we get < Corollary. Let X R n with the Euclidean metric ρ and the Lebesgue measure λ. Let Ω X. Then A X (Ω,ρ,λ A(Ω. In other words a function u continuous on Ω is (X,ρ,λ-analytic on Ω if, and only if, it is real analytic on Ω. The following definition gives an imortant class of MMS for which metricmeasure theoretic roerties of analytic functions are the same as the ones of real analytic functions on R n. Definition 3. The metric measure sace (X,ρ,µ is called analytizable if for any x X there exist oen sets U R n, x Ω X and a homeomorhism Φ: U Ω such that Φ ( B(Φ (y,r B ρ (y,r for y Ω and R small enough and µ(a Φ (A for Borel sets A Ω. Theorem 4. Under the notations of Definition 3 let u: Ω C be a continuous function. Then u is (X,ρ,µ-analytic on Ω if, and only if, u Φ is real analytic on Φ (Ω. }.

6 480 Grzegorz Łysik Proof. For x Ω and R small enough we have M X (u; x,r u(y dµ(y µ(b ρ (x,r B ρ(x,r u(φ(zdz Φ (B ρ (x,r Φ (B ρ(x,r u(φ(zdz B(Φ (x,r which roves the theorem. M(u Φ; Φ (x,r, B(Φ (x,r 5. L -mean value exansions The following theorem establishes the mean value exansion of a real analytic function with resect to the L metric on R n and the Lebesgue measure λ. Theorem 5. (Mean value roerty Let, X (R n,,λ, Ω R n and u A(Ω. Then for x Ω and R small enough we have (9 M X (u; x,r D(n,,k D(,κ 2k u(x R 2k x 2κ κ N n 0, κ k where (0 D(n,,k Γ ( n + Γ n ( + Γ (, D(,κ n+2k + The exansion (9 is locally uniform with resect to x Ω. Γ ( 2κ ( ++ Γ 2κn++. (2κ +! (2κ n +! Proof. Exending u into Taylor series we comute M X (u;x,r u(y dy B (x,r B (x,r l u(x (y x l dy B (x,r B (x,r l! x l l N n 0 l u(x (Rz l dz. B (0, l! x l B (0, l N n 0 Note that if at least one of the exonents l,...,l n is odd, then the integral of z l z l zn ln over B (0, vanishes. Next using [2, formula 676, 5] for k N 0 and κ N n 0 with κ k we have ( z 2κ dz 2n Γ 2κ ( + Γ 2κn+ B (0, n Γ ( 2k+n + ( 2 n 2κ + Γ ( 2k+n + Γ + Γ ( 2κ n+ +. (2κ + (2κ n + In articular, ( B (0, 2n Γ ( Γ ( n Γ ( + 2nΓn n + Γ ( n +.

7 Hence M X (u;x,r B (0, A characterization of real analytic functions 48 κ N n 0 (2κ! Γ ( n + Γ n( ( Γ n+2k + 2k u(x R2k z 2κ dz x2κ B (0, κ k Γ ( 2κ + ( Γ 2κn+ (2κ! 2k u(x x 2κ R 2k, which imlies (9. To rove the uniform convergence of the series (9 on a comact subset K Ω note that by the Cauchy inequalities we can find (R,...,R n R n + such that for any x K and any l (l,...,l n N n 0, l u(x C l x l R l! l n!. Rn ln Since for any ε > 0 one can find C ε such that ( s ( s s (+εs ( Γ(s+ C ε for s 0 e e we derive that the series (9 converges for R < min(r,...,r n. Theorem 6. (Converse to the mean-value roerty Let, X (R n,,λ, Ω R n, ǫ C 0 (Ω,R + and u C (Ω. If the series M X (x,r D(n,,k D(,κ 2k u(x R 2k, x 2κ κ N n 0, κ k where D(n,,k and D(n,κ are given by (0, is locally uniformly convergent in {(x,r: x Ω, R < ǫ(x}, then u A(Ω and M X (u; x,r M X (x,r for x Ω and 0 < R < min ( ǫ(x,dist(x, Ω. Proof. Fix a comact set K Ω and set ǫ inf x K ǫ(x > 0. Then the assumtion imlies that ( ( 2k Γ 2κ + + Γ ( 2κ n+ + (2k! 2κ (2κ + (2κ n +Γ ( u(x n+2k + 2k R 2k 0 x 2κ su x K κ N n 0, κ k as k uniformly on K { R ǫ } with any ǫ < ǫ. Hence for any ǫ < ǫ there exists a constant C(ǫ < such that for k N 0, ( ( 2k Γ 2κ + + Γ ( 2κ n+ + C(ǫ (2k!. 2κ x 2κ κ N n 0, κ k Γ ( n+2k + 2k u(x Since by ( the coefficient of 2k u(x in the above exression is not less then δ k( k x 2κ κ with some δ > 0 we conclude that for any comact set K Ω one can find C < and L < such that su k u(x CL 2k (2k! for k N 0. x K But by [4, Theorem] this inequality imlies that u A(Ω. Finally, by Theorem 5 we get M X (x,r M X (u; x,r. ǫ 2k

8 482 Grzegorz Łysik Finally let us remark that Theorems 5 and 6 remain valid also for 0 < <. In fact Definitions and 2 make sense if ρ is only a (non necessary symmetric seudometric. Added in roof. The uniqueness roerty for the analytic functions on MMS will be studied in a subsequent aer. References [] Alabern, R., J. Mateu, and J. Verdera: A new characterization of Sobolev saces on R n. - Math. Ann. 354, 202, [2] Fichtenholz, G. M.: Differential- und Integralrechnung, vol. III. - Hochschulbücher für Mathematik 63, Johann Ambrosius Barth Verlag GmbH, Leizig, 992. [3] Gaczkowski, M., and P. Górka: Harmonic functions on metric measure saces: convergence and comactness. - Potential Anal. 3, 2009, [4] Komatsu, H.: A characterization of real analytic functions. - Proc. Jaan Acad. 36, 960, [5] Łysik, G.: On the mean-value roerty for olyharmonic functions. - Acta Math. Hungar. 33, 20, [6] Łysik, G.: Mean-value roerties of real analytic functions. - Arch. Math. (Basel 98, 202, [7] Łysik, G.: A characterization of olyharmonic functions. - Acta Math. Hungar. 47, 205, [8] Nicolesco, M.: Les fonctions olyharmoniques. - Hermann & C ie Éditeurs, Paris, 936. [9] Weyl, H.: The method of orthogonal rojections in otential theory. - Duke Math. J. 7, 940, Received 3 January 207 Acceted 5 Setember 207