A lot of “counterexamples” to Liouville's theorem

TITLE: A LOT OF “COUNTEREXAMPLES” TO LIOUVILLE’S THEOREM. AUTHOR: LUIS BERNAL-GONZ´ ALEZ. AFFILIATION: DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. FOOTNOTES TO THE TITLE: *This work is supported in part by DGICYT grant PB93-0926. A.M.S. Subject Classification: Primary 30D15. Secondary 30E10, 44A05. Key words and phrases: Liouville’s theorem, entire functions, dense linear manifold, Radon transform, Arakelian set, strips and sectors, growth index. 1 ABREVIATED TITLE: LIOUVILLE’S THEOREM. NAME AND MAILING ADDRESS OF THE AUTHOR TO WHOM PROOFS SHOULD BE SENT: LUIS BERNAL-GONZ´ ALEZ. DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. 2 A LOT OF “COUNTEREXAMPLES” TO LIOUVILLE’S THEOREM by LUIS BERNAL–GONZ´ ALEZ* Abstract. We prove in this paper that, given α∈(0,1/2), there exists a linear manifold Mof entire functions satisfying that Mis dense in the space of all entire functions and, in addition, limz→∞ exp(|z|α) f(j)(z) = 0 on any plane strip for every f∈Mand for every derivation index j. Moreover, it is shown the existence of an entire function with infinite growth index satisfying the latter property. 1. INTRODUCTION AND NOTATION One of the most elementary, surprising and beautiful results in Complex Analysis is Liouville’s theorem: each bounded entire function is constant. Nevertheless, if boundedness condition is slightly weakened (for instance, by allowing boundedness *This work is supported in part by DGICYT grant PB93-0926. 1991 Mathematics Subject Classification: Primary 30D15. Secondary 30E10, 44A05. Key words and phrases: Liouville’s theorem, entire functions, dense linear manifold, Radon transform, Arakelian set, strips and sectors, growth index. 3 on every line), then nonconstant entire functions can be obtained. For this, see for instance [7, pp. 9-10], where it is even shown a nonconstant entire function having limit zero on any line. This is achieved by using a result of A. Roth about approximation on closed sets. But several sharper results have been obtained. Recently, D.H. Armitage [3] has constructed a nonconstant entire function fsuch that each derivative f(n)is integrable on every line lwith respect to length measure sand ∫l f(n)ds = 0 for every n∈N0, where N0=N∪ {0}and Nis the set of positive integers. Moreover, fsatisfies lim z→∞ z∈l f(n)(z) = 0 for every n∈N0and every line l(therefore each derivative is bounded on each line). This is obtained by an elementary pole-pushing technique. In the same paper it is noted that if fis a continuous function on the complex plane C, integrable on 4 Cwith respect to plane Lebesgue measure, and if ∫l f ds = 0 (1) for every line l, then fis identically zero (see [2] for instance). We remark here that the mapping ˆ f(l) = ∫l f ds (l∈L) is known as the Radon transform of f. We have denoted by Lthe set of all straight lines in the plane. Thus, we are dealing with the injectivity of the linear transform f7→ ˆ fin R2. The corresponding problem for hyperplanes in Rnis extensively studied in [8] for certain function spaces. L. Zalcman [16] in 1982 had indicated that it actually suffices for (1) to hold only for almost every line belonging to a dense set of directions in order that f≡0. In the same paper, he constructed –by using a result (Theorem 1 below) due to Arakelian concerning tangential approximation (see [1, p. 1189]and [6, pp. 160-162])– a nonconstant entire function satisfying (1) on every line l. Nonnull entire functions tending to zero on every line had already been constructed by Mittag-Leffler [10, pp. 290-294], and D.J. Newman [11] provided an explicit nonconstant entire function which is bounded on every line through the origin. Zalcman also pointed out that, if nearly nothing is assumed for f, then there exist nonnull functions for 5 which (1) holds: take f=χE, where Eis Sierpinski’s nonmeasurable set with the property that any line intersects Ein at most two points (see [15]). All these results can be carried into the setting of harmonic functions on Rn(n≥2) (see [4] and [5]). In this paper we improve all above assertions about entire functions and provide a dense linear manifold of entire functions “violating” Liouville’s theorem. All nonnull functions in this manifold have infinite growth order. We can get even for at least one of these functions to have extremely fast growth. We will need a bit of notation and a preliminary result. If r > 0, Brdenotes the closed ball {z:|z| ≤ r}. A strip is the plane region lying between two parallel straight lines. If β∈(0,2π), the sector sβis the set sβ={z: 0 ≤arg z≤β}(here arg zis evidently being allowed to be outside the “principal interval” (−π, π] used by some authors) . Σ will stand for the family consisting of all strips in Cand all sectors sβ(β∈(0,2π)). H(C) is the space of all entire functions, endowed with the compact-open topology. The extended plane C∞is the one-point compactification of C. If F⊂Cis a closed set, then A(F) is the space of all continuous functions 6 on Fwhich are holomorphic in the interior of F. A closed subset F⊂Cis said to be an Arakelian set [13] whenever C∞\Fis both connected and locally connected at infinity. We will use the following (above mentioned) theorem due to Arakelian. THEOREM 1. Assume that F⊂Cis an Arakelian set and that ε(t)is continuous and positive for t≥0. In addition, suppose that ∫∞ 1 t−3/2log ε(t)dt > −∞.(2) Then for every g∈A(F)there exists an entire function fsuch that |f(z)−g(z)|< ε(|z|)∀z∈F. The statement does not remain valid for every Fif (2) is violated. Note that, for instance, ε(t) = exp(−t1/2) does not satisfy (2), but ε(t) = exp(−tα) does for α < 1/2. Finally, let fbe an entire function. If r > 0, we define exp1r= exp r, expk+1 r= exp(expkr) (k∈N), and µ(r) = max{|f(z)|:|z|=r}. For r > 0 7 large enough, we denote log1r= log r, logk+1 r= log(logkr) (k∈N). The following definitions can be found in [12] and [14]. The growth k-order ρk=ρk(f) of f is ρk= lim sup r→∞ logk+1 µ(r) log r. If k= 1, then ρkis called the order of f, and we shall denote it by ρ(f). The growth index i(f) of fis i(f) = min{k∈N:ρk(f)<∞}, where we set i(f) = ∞when ρk(f) = ∞for all k. 2. RESULTS Recall that H(C) is separable: the family {Pn}∞ 1of holomorphic polynomials having coefficients with rational real and imaginary parts is an example of a countable dense subset of H(C). With this in mind, we are now ready to state the following theorem. THEOREM 2. Assume that α∈(0,1/2). Then there is a linear manifold M⊂H(C)which is dense in H(C)and satisfying lim z→∞ z∈S exp(|z|α)f(j)(z) = 0 (3) 8 ∀S∈Σ,∀f∈Mand ∀j∈N0. PROOF. Consider the sequence {Pn}∞ 1mentioned at the beginning of the paragraph and fix α∈(0,1/2). Fix also a number β∈(α, 1/2). For every n∈N, the function ε(t) = εn(t)≡min{1/n, exp(−tβ)} is positive and continuous for t≥0 and satisfies (2). Let Fn=Bn∪En, where En={z∈C:|z| ≥ n+ 1 and dist(z, P )≥1} and Pis the parabolic curve P={x−ix1/2:x≥0}. Then Fnis closed. In addition, a glance over its complement reveals that Fnis an Arakelian set. Define the function gn:Fn→Cby gn(z) = {Pn(z) if z∈Bn 0 if z∈En. Since Bn∩En=∅,gnis well-defined. Trivially, gn∈A(En). By Theorem 1, there exists an entire function fnsuch that |fn(z)−gn(z)|< ε(|z|)∀z∈Fn, 9 then the set T(V) = {f∈H(C) : fis bounded on V} is of the first category, because T(V) = ∞ ∪ 1 Cnwith Cn={f∈H(C) : |f(z)| ≤ non V}and each Cnis closed and has empty interior (for this, note that Cn∩ {nonconstant polynomials}=∅and that the second set in the intersection is dense in H(C)). Finally, observe that AL ⊂∩{T(r) : ris a ray from the origin}. So AL is, in this sense, very small. REFERENCES 1. N.V. ARAKELIAN, Uniform approximation on closed sets by entire functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1187-1206. 2. A.S. BESICOVITCH, A uniqueness theorem and a problem of integration, J. London Math. Soc. 33 (1958), 82-84. 3. D.H. ARMITAGE, A non-constant continuous function on the plane whose integral on every line is zero, Amer. Math. Monthly 101 (1994), 892-894. 4. D.H. ARMITAGE and M. GOLDSTEIN, Better than uniform approximation 16 on closed sets by harmonic functions with singularities, Proc. London Math. Soc. 60 (1990), 319-343. 5. D.H. ARMITAGE and M. GOLDSTEIN, Nonuniqueness for the Radon transform, Proc. Amer. Math. Soc. 117 (1993), 175-178. 6. D. GAIER, Lectures on complex approximation, Birkh¨auser, Boston, 1987. 7. P.M. GAUTHIER, Uniform approximation, 1-37, in Complex Potential Theory, Kluwer Academic Publishers, 1994. 8. S. HELGASON, The Radon transform, Birkh¨auser, Boston, 1980. 9. A.S.B. HOLLAND, Introduction to the theory of entire functions, Academic Press, New York and London, 1973. 10. G. MITTAG-LEFFLER, Sur la repr´esentation analytique d’une branche uniforme d’une fonction monog`ene (Sixi`eme note), Acta Math. 42 (1920), 285-308. 11. D.J. NEWMAN, An entire function bounded in every direction, Amer. Math. Monthly 83 (1976), 192-193. 12. A.R. REDDY, On entire Dirichlet series of infinite order, Rev. Mat. Hisp.- Amer. 27 (1967), 120-131. 13. J.P. ROSAY and W. RUDIN, Arakelian’s approximation theorem, Amer. Math. Monthly 96 (1989), 432-434. 14. D. SATO, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc. 69 (1963), 410-414. 15. W. SIERPINSKI, Sur un probl`eme concernant les ensembles mesurables super- 17 ficiellement, Fund. Math. 1(1920), 112-115. 16. L. ZALCMAN, Uniqueness and nonuniqueness for the Radon transform, Bull. London Math. Soc. 14 (1982), 241-245. DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO FACULTAD DE MATEM´ ATICAS AVENIDA REINA MERCEDES APARTADO 1160 41080 SEVILLA, SPAIN E-MAIL: lb[email protected] 18