Rate of Growth of Distributionally Chaotic Functions
aa r X i v : . [ m a t h . F A ] J u l RATE OF GROWTH OF DISTRIBUTIONALLY CHAOTICFUNCTIONS
CLIFFORD GILMORE, F´ELIX MART´INEZ-GIM´ENEZ, AND ALFRED PERIS
Abstract.
We investigate the permissible growth rates of functionsthat are distributionally chaotic with respect to differentiation operat-ors. We improve on the known growth estimates for D -distributionallychaotic entire functions, where growth is in terms of average L p -normson spheres of radius r > r → ∞ , for 1 ≤ p ≤ ∞ . We computegrowth estimates of ∂/∂x k -distributionally chaotic harmonic functionsin terms of the average L -norm on spheres of radius r > r → ∞ .We also calculate sup-norm growth estimates of distributionally chaoticharmonic functions in the case of the partial differentiation operators D α . Introduction
The term chaos first appeared in mathematical literature in an article byLi and Yorke [28], where they studied the dynamical behaviour of intervalmaps with period three. Schweizer and Sm´ıtal [33] subsequently introducedthe stronger notion of distributional chaos for self-maps of a compact inter-val.A continuous map g : Y → Y on a metric space ( Y, d ) is said to be
Li-Yorke chaotic if there exists an uncountable set Γ ⊂ Y such that for everypair ( x, y ) ∈ Γ × Γ of distinct points, we havelim inf n →∞ d ( g n ( x ) , g n ( y )) = 0 and lim sup n →∞ d ( g n ( x ) , g n ( y )) > . In this case, Γ is called a scrambled set for g and each such pair ( x, y ) iscalled a Li-Yorke pair for g . This definition captures the behaviour of orbitswhich are proximal without being asymptotic.Our setting will be the Fr´echet space X endowed with an increasing se-quence ( k · k k ) k ∈ N of seminorms that define the metric d ( x, y ) := ∞ X k =1 − k k x − y k k k x − y k k under which X is complete, where x, y ∈ X . We let T denote a continuouslinear operator on X . Mathematics Subject Classification.
Key words and phrases.
Distributional chaos, distributionally irregular vectors, growthrates, entire functions, harmonic functions, differentiation operator, partial differentiationoperators.C. Gilmore was supported by the Magnus Ehrnrooth Foundation and the Irish Re-search Council via a Government of Ireland Postdoctoral Fellowship. F. Mart´ınez-Gim´enezand A. Peris were supported by MICINN and FEDER, Projects MTM2016-75963-P andPID2019-105011GB-I00, and by Generalitat Valenciana, Project PROMETEO/2017/102.
The connection between Li-Yorke chaos and the linear dynamical notionof irregularity was identified in [10]. We say that x ∈ X is an irregular vector for T if there exist m ∈ N and increasing sequences ( j k ) and ( n k ) of positiveintegers such thatlim k →∞ T j k x = 0 and lim k →∞ k T n k x k m = ∞ . This notion was introduced by Beauzamy [9] for Banach spaces to describethe local aspects of the dynamics of pairs of vectors and it was generalisedto the Fr´echet space setting in [14].We recall if there exists x ∈ X such that its T -orbit is dense in X , that is { T n x : n ≥ } = X, then T is said to be hypercyclic and such an x ∈ X is known as a hypercyclicvector . It is well known that hypercyclic vectors are irregular. Compre-hensive introductions to the topic of hypercyclicity can be found in themonographs [7] and [25].A natural strengthening of Li-Yorke chaos was introduced by Schweizerand Sm´ıtal [33] with the notion of distributional chaos for interval maps.We first recall that the upper and lower densities of a set A ⊂ N are defined,respectively, as dens( A ) := lim sup n →∞ | A ∩ { , , . . . , n }| n , dens( A ) := lim inf n →∞ | A ∩ { , , . . . , n }| n . For a continuous map g : Y → Y on a metric space ( Y, d ), points x, y ∈ Y and δ >
0, we define F x,y ( δ ) := dens( { n ∈ N : d ( g n ( x ) , g n ( y )) < δ } )and F ∗ x,y ( δ ) := dens( { n ∈ N : d ( g n ( x ) , g n ( y )) < δ } ) . If the pair ( x, y ) satisfy F ∗ x,y ≡ F x,y ( ε ) = 0 for some ε >
0, then ( x, y ) iscalled a distributionally chaotic pair . The map g is said to be distributionallychaotic if there exists an uncountable set Γ ⊂ Y such that every distinctpair ( x, y ) ∈ Γ × Γ is a distributionally chaotic pair for g .The study of distributional chaos in the linear dynamical setting wasinitiated in [30]. The T -orbit of x ∈ X is said to be distributionally near to0 if there exists A ⊂ N with dens( A ) = 1 such thatlim n ∈ A T n x = 0 . We say x ∈ X has a distributionally unbounded orbit if there exist m ∈ N and B ⊂ N with dens( B ) = 1 such thatlim n ∈ B k T n x k m = ∞ . Combining these properties, x ∈ X is defined to be a distributionally irreg-ular vector for T if its orbit is both distributionally unbounded and distri-butionally near to 0. This strengthening of irregularity was introduced in[10]. ATE OF GROWTH OF DISTRIBUTIONALLY CHAOTIC FUNCTIONS 3
In the Fr´echet space setting, it follows from results in [10] and [13] that T admits a distributionally irregular vector if and only if T is distributionallychaotic. Hence, in the sequel our study focuses on distributionally irregularvectors. Distributional chaos has been investigated from many aspects, forinstance in [1, 8, 14, 15, 19, 31, 35, 36, 37, 38, 12].An example of a map that admits a distributionally irregular vectoris the differentiation operator D : f f ′ , acting on the space H ( C ) ofentire holomorphic functions on C (this follows from [13, Corollary 17]).Bernal and Bonilla [11] computed growth estimates for D -irregular and D -distributionally irregular entire functions, where growth is in terms of aver-age L p -norms, for 1 ≤ p ≤ ∞ , on spheres of radius r > r → ∞ .We note that permissible growth rates of D -hypercyclic and D -frequentlyhypercyclic entire functions have previously been investigated in [16, 17,20, 21, 24, 29, 32, 34]. We recall the notion of frequent hypercyclicity wasintroduced by Bayart and Grivaux [6], where they defined T : X → X tobe frequently hypercyclic if there exists x ∈ X such that for any nonemptyopen subset U ⊂ X it holds thatdens ( { n : T n x ∈ U } ) > . Such an x ∈ X is called a frequently hypercyclic vector for T .In the setting of the space H ( R N ) of harmonic functions on R N , for N ≥ ∂∂x k : H ( R N ) → H ( R N ) , where 1 ≤ k ≤ N . They identified sharp L -growth rates, on spheres ofradius r > r → ∞ , of harmonic functions that are universal (and hencehypercyclic) with respect to ∂/∂x k . Growth estimates in the frequentlyhypercyclic case were computed by Blasco et al. [16] and sharp growth rateswere subsequently identified in [23].Growth estimates with respect to the sup-norm, on spheres of radius r > r → ∞ , were computed by Aldred and Armitage in [3] for harmonicfunctions that are universal for general partial differentiation operators D α = ∂ | α | ∂x α · · · ∂x α N N where α = ( α , . . . , α N ) ∈ N N and | α | = α + · · · + α N . The frequentlyhypercyclic case was subsequently investigated by Blasco et al. [16].In this article we compute permissible growth rates of irregular and distri-butionally irregular functions. In Section 2 we improve the growth estimatesfrom [11] for distributionally irregular entire functions and we also providelower estimates. In Section 3 we investigate average L -growth estimatesof irregular and distributionally irregular harmonic functions with respectto partial differentiation operators ∂/∂x k . Then in Section 4 we computesup-norm growth rates of distributionally irregular harmonic functions inthe case of the partial differentiation operators D α .We will use the fact that absolutely Ces`aro bounded operators cannot bedistributionally irregular. We recall for a Banach space X , the continuous C. GILMORE, F. MART´INEZ-GIM´ENEZ, AND A. PERIS linear operator T : X → X is said to be absolutely Ces`aro bounded if thereexists a constant C > N ∈ N N N X j =1 (cid:13)(cid:13) T j x (cid:13)(cid:13) ≤ C k x k for all x ∈ X . If the orbit of x ∈ X is distributionally unbounded, then itwas proven in [15, Proposition 20] thatlim sup N →∞ N N X j =0 (cid:13)(cid:13) T j x (cid:13)(cid:13) = ∞ . We will need a technical result whose proof follows the argument of [16,Theorem 2.4].
Lemma 1.1.
Let X be a set, f n : X → R + , n ∈ N , a sequence of non-negative functions defined on X such that there exist α, β, C > with (1.1) ∞ X n =1 f n ( x ) R αn + β e − αR n ! α n β − α/ / ≤ C, ∀ x ∈ X, ∀ R > . Then there exists
B > such that m m X n =1 f n ( x ) ≤ B, ∀ x ∈ X, ∀ m ∈ N . Proof.
We consider the functions g n ( R ) := m R αn + β e − αR n ! α n β − α/ / , R > , n ∈ N . The function g n attains its maximum at a n := n + β/α . Moreover byStirling’s formula g n ( a n ) := ( n + β/α ) αn + β e − αR n ! α n β n − α/ n / ∼ √ n . The function g n has an inflection point at b n := a n + p n/α + β/α and, sincelim R →∞ g n ( R ) = 0, we have that g n ( R ) ≥ h n ( R ) for each R ∈ I n = [ a n , b n ],where h n is the affine map such that h n ( a n ) = g n ( a n ) and h n ( b n ) = 0. Wefix m ∈ N such that m < a n = n + β/α < b n = a n + p n/α + β/α < m for m ≥ m and for n = m, . . . , m . By (1.1)2 mC ≥ Z [ m, m ] m X n = m f n ( x ) g n ( R ) ! dR ≥ m X n = m f n ( x ) Z I n g n ( R ) dR ≥ m X n = m f n ( x ) Z I n h n ( R ) dR ≥ C ′ m X n = m f n ( x ) p n/α + β/α √ n for m ≥ m , where C ′ > m and x . Thereforewe find B > m m X n = m f n ( x ) ≤ B , ∀ m ≥ m , ∀ x ∈ X. ATE OF GROWTH OF DISTRIBUTIONALLY CHAOTIC FUNCTIONS 5
In particular, and again using (1.1), there is
B > m m X n =1 f n ( x ) ≤ B, ∀ m ∈ N , ∀ x ∈ X. (cid:3) Throughout this article we will need the following slight improvement ofa result which is a consequence of Proposition 7 and Theorems 15 and 19in [13]. For the convenience of the reader we outline the main steps of theproof.
Theorem 1.2 (Bernardes et al. [13]) . Let Y be a separable Fr´echet space, let T : Y → Y be a continuous linear operator and assume that X is a separableFr´echet space that is continuously embedded in Y . Suppose that:(a) There exists a dense subset X of X with T x ∈ X and lim n →∞ T n x = 0 in X for all x ∈ X .(b) There exist a subset Y of X , a mapping S : Y → Y with T Sy = y on Y , and a vector z ∈ Y \ { } such that P ∞ n =1 T n z and P ∞ n =1 S n z converge unconditionally in Y and X , respectively.Then there exists a dense subset of vectors in X which are distributionallyirregular for T .Proof. We denote by ( k · k k ) k ∈ N the increasing sequence of seminorms underwhich Y is complete.Arguing as in [13, Theorem 19], we define w k := P ∞ n =1 T k n z + z + P ∞ n =1 S k n z , where w k = 0 if k is sufficiently large and T k w k = w k . Let y k := P ∞ n = k S k n z . Then y k → T k j y k = j − k X n =1 T k n z + z + ∞ X n =1 S k n z → w k as j → ∞ . For 0 ≤ ℓ < k we havelim j →∞ T ℓ + k j y k = T ℓ w k and hence { T ℓ w k : 0 ≤ ℓ < k } are accumulation points of the orbit of y k .Next we define ε := 12 min n(cid:13)(cid:13)(cid:13) T ℓ w k (cid:13)(cid:13)(cid:13) m > ≤ ℓ < k , m ∈ N o . Then there exists an increasing sequence ( N k ) of positive integers such thatlim k →∞ N k (cid:12)(cid:12)(cid:8) ≤ j ≤ N k : (cid:13)(cid:13) T j y k (cid:13)(cid:13) m > ε (cid:9)(cid:12)(cid:12) = 1and it follows from [13, Proposition 7] that there exists y ∈ Y with a dis-tributionally m -unbounded orbit with respect to T . The result then followsfrom [13, Theorem 15] which gives that T admits a dense subset of distri-butionally irregular vectors in X . (cid:3) C. GILMORE, F. MART´INEZ-GIM´ENEZ, AND A. PERIS Growth of Distributionally Irregular Entire Functions
We consider the average L p -norm growth, for 1 ≤ p ≤ ∞ , of entire func-tions that are distributionally irregular with respect to the differentiationoperator D : H ( C ) → H ( C ). Initial growth estimates were obtained byBernal and Bonilla [11] and here we improve the permissible growth estim-ates while also computing estimates for lower rates of growth.We first introduce the pertinent notation and some background results.For an entire function f ∈ H ( C ) and 1 ≤ p < ∞ , the average L p -norm isdefined as M p ( f, r ) = (cid:18) π Z π | f ( re it ) | p dt (cid:19) /p and we denote the sup-norm by M ∞ ( f, r ) = sup | z | = r | f ( z ) | for r > Theorem 2.1.
Let ≤ p ≤ ∞ .(i) Let a = 1 / (2 max { , p } ) . For any ϕ : R + → R + with ϕ ( r ) → ∞ as r → ∞ , there exists a D -distributionally irregular entire function f with M p ( f, r ) ≤ ϕ ( r ) e r r a for r > sufficiently large.(ii) Let a = 1 / (2 min { , p } ) . There does not exist a D -distributionallyirregular entire function f that satisfies (2.1) M p ( f, r ) ≤ c e r r a where c > is a constant and for r > sufficiently large.Proof. (i) Since M p ( f, r ) ≤ M ( f, r )for 1 ≤ p < p ≥ ≤ p ≤ ∞ and we assume without loss of generality that inf r> ϕ ( r ) >
0. We consider X := ( f ∈ H ( C ) : k f k X := sup r> M p ( f, r ) r / p ϕ ( r ) e r < ∞ ) and we note that ( X, k · k X ) is Banach space which is continuously embeddedin H ( C ). Further note that the functions f ∈ X satisfy the desired growthcondition.Our strategy is to apply Theorem 1.2, so we let X = Y be the space ofpolynomials. Note that X is dense in X and it follows immediately thatTheorem 1.2 (a) is satisfied.For part (b), we define the mapping S : Y → Y for g ∈ Y as Sg ( z ) = Z z g ( ξ ) d ξ. ATE OF GROWTH OF DISTRIBUTIONALLY CHAOTIC FUNCTIONS 7
For all g ∈ Y we have that DSg = g , and since P ∞ n =1 D n g is a finite sum itconverges unconditionally.We also know that P ∞ n =1 S n g converges unconditionally in X (cf. [16,Theorem 2.3]) for any polynomial g ∈ X , and thus by Theorem 1.2 theresult follows.(ii) For p = 1 the result follows from [11, Theorem 7]. Moreover since M ( f, r ) ≤ M p ( f, r )for 2 < p ≤ ∞ , it suffices to prove the result for p ≤
2. So we assume1 < p ≤ f ∈ H ( C ) satisfies (2.1).Let B ( r ) denote the open ball of radius r which is centred at the origin.We define the translation f a ( z ) := f ( z + a ), for a ∈ C , which is an entirefunction with f a (0) = f ( a ).We let R > r and it follows from the Hausdorff-Young inequality (see[26]) and (2.1) that for a ∈ B ( r ) ∞ X n =0 (cid:18) | D n f ( a ) | n ! R n (cid:19) q ! /q = ∞ X n =0 (cid:18) | D n f a (0) | n ! R n (cid:19) q ! /q ≤ M p ( f a , R ) ≤ M p ( f, R + r ) ≤ c e R + r ( R + r ) / p where q is the conjugate exponent of p with 1 /p + 1 /q = 1.So it follows that(2.2) ∞ X n =0 | D n f ( a ) | q R qn + q/ p (1 + r/R ) q/ p e − qR c q n ! q e qr ≤ . We now apply Lemma 1.1 with α = q and β = q/ p to obtain that thereexists C > r such that1 m m X n =0 | D n f ( a ) | q ≤ C. Next let S ( r ) = { z ∈ C : | z | = r } denote the sphere of radius r centeredat the origin, k ∈ N arbitrary, and let a , . . . , a k ∈ S ( r ). We take theaverages 1 k k X j =1 m m X n =0 | D n f ( a j ) | q ≤ C and hence 1 m m X n =0 k k X j =1 | D n f ( a j ) | q ≤ C. Note that, if the a , . . . , a k ∈ S ( r ) are uniformly distributed,lim k →∞ k k X j =1 | D n f ( a j ) | q = c r M qq ( D n f, r ) C. GILMORE, F. MART´INEZ-GIM´ENEZ, AND A. PERIS where c r is a constant depending only on r . So it follows that the average1 m m X n =0 M qq ( D n f, r )is bounded and thus f cannot be a distributionally unbounded function forthe differentiation operator. (cid:3) Growth of Distributionally Irregular Harmonic Functions
In this section we compute the average L -norm growth of harmonic func-tions that are irregular and distributionally irregular for the partial differ-entiation operators ∂/∂x k , for 1 ≤ k ≤ N . We begin by introducing thenotation and background from [2, 4, 5] required for our investigation.The space H ( R N ) of harmonic functions on R N , for N ≥
2, is a Fr´echetspace when equipped with the complete metric d ( g, h ) := ∞ X n =1 − n | g − h | S ( n ) | g − h | S ( n ) for g, h ∈ H ( R N ), and it corresponds to the topology of local uniform con-vergence. Above we set | f | S ( n ) = sup | x | = n | f ( x ) | for f ∈ H ( R N ).We denote by B ( x, r ) and S ( x, r ), respectively, the open ball and thesphere of radius r (in the euclidean metric) with centre at x ∈ R N . Whenthey are centred at the origin of R N we simply write B ( r ) and S ( r ).The sup-norm of h ∈ H ( R N ) on S ( r ) is defined as M ∞ ( h, r ) = sup k x k = r | h ( x ) | . Let σ r denote the normalised ( N − S ( r ), so that σ r ( S ( r )) = 1. The average L -norm on S ( r ) of h ∈ H ( R N ) is given by M ( h, r ) = Z S ( r ) | h | d σ r ! / where r >
0. The corresponding inner product is defined by h g, h i r = Z S ( r ) gh d σ r for g, h ∈ H ( R N ).We denote by H m ( R N ) the space of homogeneous harmonic polynomialson R N of homogeneity degree m ≥
0. The harmonic analogue of the standardpower series representation of holomorphic functions states that any h ∈H ( R N ) has a unique expansion of the form(3.1) h = ∞ X m =0 H m where H m ∈ H m ( R N ) for each m ≥ d , see [5, Corollary 5.34]. Moreover, h H j , H k i r = 0 when j = k , so ATE OF GROWTH OF DISTRIBUTIONALLY CHAOTIC FUNCTIONS 9 by orthogonality one has for any r > L -average of (3.1) is M ( h, r ) = ∞ X m =0 M ( H m , r ) . We denote the dimension of H m ( R N ) by d m = d m ( N ) and it can be shown [5,Proposition 5.8] that d = 1 for N = 2 and(3.2) d m = N + 2 m − N + m − (cid:18) N + m − m (cid:19) for N + m ≥
3. It follows easily from (3.2) that(3.3) d m = O ( m N − )as m → ∞ (cf. [5, p.107]). Moreover, for α ∈ N N with | α | = m and H ∈ H m ( R N ), D α H is constant and it follows from [2, Lemma 1] that(3.4) | D α H | ≤ m ! p d m r − m M ( H, r )for r >
0. Further details on the spaces H m ( R N ) can be found in [5, Chapter5] and [4, Chapter 2].We also require an antiderivative for the partial differentiation operators ∂/∂x k on H ( R N ), where 1 ≤ k ≤ N . Suitable linear maps were definedby Aldred and Armitage [2] by using a specific orthogonal representationof harmonic polynomials constructed by Kuran [27]. We denote the n th antiderivative, with respect to the coordinate x k , by the linear map(3.5) P n,k : H m ( R N ) → H m + n ( R N )for m, n ≥
0. For our purposes we do not require to explicitly define the maps P n,k , however we will utilise the pertinent properties which are contained inthe following fundamental lemma taken from [2, Lemma 4]. Lemma 3.1.
Let m, n ≥ , N ≥ and ≤ k ≤ N . If H ∈ H m ( R N ) then P n,k ( H ) ∈ H m + n ( R N ) , ∂ n ∂x nk P n,k ( H ) = H and (3.6) M ( P n,k ( H ) , ≤ c n,m,N M ( H, where c n,m,N = ( N + 2 m − n !( N + 2 m + n − N + 2 m + 2 n − . Similar to line (4.2) in [16], for fixed m we will use the simpler estimate(3.7) c n,m,N ≤ c m ( n + m )! ( n + m + 1) N − for n ∈ N , where(3.8) c m = c m ( N ) = ( N + 2 m − P n,k are mutually compatible since for H ∈ H m ( R N ) and ℓ, n ≥ P ℓ + n,k ( H ) = P ℓ,k ( P n,k ( H )) . A proof of this fact can be found in [23, Lemma 3.3]. In particular it holdsthat ∂ n ∂x nk P ℓ,k ( H ) = P ℓ − n,k ( H ) for ℓ > n .We also need the following lemma on inequalities between L -norms ofharmonic functions with respect to N -spheres with different centres. Thisis essentially known but for completeness we include a proof. This requiresthe Poisson integral and Harnack’s inequality, which we recall below and fulldetails can be found in [4, Sections 1.3 and 1.4].We let σ N = σ ( S (1)) be the surface area of the N -sphere S (1), where σ denotes the (unnormalised) surface area measure. The Poisson kernel ofthe ball B ( x , r ) is given by the function K x ,r ( x, y ) := 1 σ N r r − k x − x k k x − y k N for y ∈ S ( x , r ) and x ∈ R N \ { y } .For a function h continuous on S ( x , r ), the Poisson integral is defined as I h,x ,r ( x ) := Z S ( x ,r ) K x ,r ( x, y ) h ( y ) d σ ( y )for x ∈ B ( x , r ). It is a fundamental result of potential theory that I h,x ,r defines a harmonic function on the ball B ( x , r ) with boundary values on S ( x , r ) given by h .We recall that Harnack’s inequality states if h is a positive harmonicfunction on B ( x , r ), then h ( x ) ≤ ( r + k x − x k ) r N − ( r − k x − x k ) N − h ( x )for each x ∈ B ( x , r ). Lemma 3.2.
Let N ≥ . Given h ∈ H ( R N ) , r > , R > r , and a ∈ R N with k a k ≤ r , we consider the translated harmonic function h a defined by h a ( x ) = h ( a + x ) . We then have M ( h a , R ) ≤ C N M ( h, r + R ) , where C N > is a constant that only depends on N .Proof. For brevity we denote the Poisson integrals of the subharmonic func-tion h on the spheres S ( a, R ) and S ( r + R ), respectively, by I a,R and I ,r + R .First observe that M ( h, r + R ) = Z S ( r + R ) | h ( y ) | d σ ( r + R ) ( y )= 1 σ N ( r + R ) N − Z S ( r + R ) h ( y ) d σ ( y ) = I ,r + R (0)and similarly M ( h a , R ) = Z S ( R ) | h a ( y ) | d σ R ( y ) = Z S ( a,R ) | h ( y ) | d σ R ( y ) = I a,R ( a ) . Next notice by the subharmonicity of h that h ≤ I ,r + R in the ball B ( r + R ). Since I a,R = h on the sphere S ( a, R ) ⊂ B ( r + R ), it holds that ATE OF GROWTH OF DISTRIBUTIONALLY CHAOTIC FUNCTIONS 11 I a,R = h ≤ I ,r + R on S ( a, R ). By the maximum principle it follows that I a,R ≤ I ,r + R on B ( a, R ) and in particular we have that I a,R ( a ) ≤ I ,r + R ( a ) . Since I ,r + R is a positive harmonic function on B ( r + R ), we have byHarnack’s inequality and the facts that k a k ≤ r and r < RI ,r + R ( a ) ≤ ( r + R + k a k ) ( r + R ) N − ( r + R − k a k ) N − I ,r + R (0) ≤ (2 r + R ) ( r + R ) N − R N − I ,r + R (0)= (cid:18) rR + 1 (cid:19) (cid:16) rR + 1 (cid:17) N − I ,r + R (0) = C N I ,r + R (0)where the constant C N depends only on N and the result follows. (cid:3) We are now ready to identify the average L -growth of harmonic functionsthat are irregular with respect to partial differentiation. Theorem 3.3.
Let ≤ k ≤ N .(i) Let ϕ : R + → R + be any function with ϕ ( r ) → ∞ as r → ∞ . Thenthere exists a ∂/∂x k -irregular harmonic function h ∈ H ( R N ) with M ( h, r ) ≤ ϕ ( r ) e r r ( N − / for r > sufficiently large.(ii) Let α ∈ N N . There does not exist a D α -irregular harmonic function h ∈ H ( R N ) that satisfies (3.9) M ( h, r ) ≤ C e r r ( N − / for r > and any constant C > .Proof. (i) This follows from [2, Theorem 1] since hypercyclic vectors areirregular.(ii) Let h ∈ H ( R N ). We recall that the translation h a ( x ) := h ( x + a )preserves harmonicity and we further note that h a (0) = h ( a ), where a ∈ R N .Furthermore it follows from (3.1) that h a has a unique representation of theform h a = ∞ X j =0 H a,j where H a,j ∈ H j ( R N ).For n ∈ N and α ∈ N N we may differentiate under the summation sign toobtain(3.10) D nα h ( a ) = D nα h a (0) = ∞ X j =0 ( D nα H a,j ) (0) = (cid:0) D nα H a,n | α | (cid:1) (0)where we use the convention that D nα = ( D α ) n . Fix r > a ∈ B ( r ). For R > r it follows from (3.10) and (3.4)that | D nα h ( a ) | ≤ ( n | α | )! q d n | α | R − n | α | M ( H a,n | α | , R ) ≤ ( n | α | )! q d n | α | R − n | α | M ( h a , R ) . Applying Lemma 3.2 we get that M ∞ ( D nα h, r ) ≤ c N ( n | α | )! q d n | α | R − n | α | M ( h, r + R ) . Next suppose that (3.9) holds. By (3.3) we know that d n | α | = O (( n | α | ) N − )as n → ∞ and hence there exists a constant C , independent of n and r ,such that M ∞ ( D nα h, r ) ≤ C ( n | α | )!( n | α | ) ( N − / e r + R R n | α | ( r + R ) ( N − / . Applying Stirling’s formula and choosing R = n | α | + ( N − / M ∞ ( D nα h, r ) ≤ C ( n | α | ) n | α | +( N − / e r + n | α | +( N − / ( n | α | + ( N − / n | α | +( N − / e n | α | (cid:16) rn | α | +( N − / (cid:17) ( N − / ≤ Ce r +( N − / (cid:18) N − n | α | (cid:19) − n | α | −→ e r as n → ∞ . So we get that the sequence { M ∞ ( D nα h, r ) } n is bounded andsince h does not have an unbounded orbit it cannot be irregular. (cid:3) Next we compute growth rates for distributionally irregular entire func-tions.
Theorem 3.4.
Let ≤ k ≤ N .(i) Let ϕ : R + → R + be any function with ϕ ( r ) → ∞ as r → ∞ . Thenthere exists a harmonic function h ∈ H ( R N ) which is distributionallyirregular with respect to the partial differentiation operator ∂/∂x k , suchthat (3.11) M ( h, r ) ≤ ϕ ( r ) e r r N/ − / for r > sufficiently large.(ii) There does not exist a ∂/∂x k -distributionally irregular h ∈ H ( R N ) satisfying (3.12) M ( h, r ) ≤ c e r r N/ − / where c > is constant and for r > sufficiently large. ATE OF GROWTH OF DISTRIBUTIONALLY CHAOTIC FUNCTIONS 13
Proof. (i) Fix 1 ≤ k ≤ N and we assume without loss of generality thatinf r> ϕ ( r ) >
0. We consider the space X := ( h ∈ H ( R N ) : k h k X := sup r> M ( h, r ) r N/ − / ϕ ( r ) e r < ∞ ) and we note that ( X, k · k X ) is a Banach space which is continuously embed-ded in H ( R N ) and that every h ∈ X satisfies the desired growth condition.We will apply Theorem 1.2 and to this end we let X = Y be the spaceof harmonic polynomials on R N . The space X is dense in X and it followsimmediately that part (a) of Theorem 1.2 is satisfied.For part (b), we define the map S : Y → Y by S : m X j =0 H j m X j =0 P ,k ( H j )where H j ∈ H j ( R N ). For all H ∈ Y we have that ∂/∂x k SH = H and since P ∞ n =1 ∂ n /∂x nk H is a finite sum it converges unconditionally.Next we recall that the sum P ∞ n =1 S n H converges unconditionally in X for the polynomial H ∈ Y (cf. [16, Theorem 4.2(b)]), and thus the resultfollows by Theorem 1.2.(ii) Fix 1 ≤ k ≤ N . We consider the translated harmonic function h a ( x ) := h ( x + a ) for a ∈ R N . It follows from (3.1) that h a has a unique representationof the form h a = ∞ X j =0 H a,j where H a,j ∈ H j ( R N ).For a ∈ S ( r ), it follows from (3.4) that (cid:12)(cid:12)(cid:12)(cid:12) ∂ n ∂x nk h ( a ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∂ n ∂x nk h a (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n ! p d n R − n M ( H a,n , R )for any R > r . It then follows from Lemma 3.2 that ∞ X n =0 R n n ! d n (cid:12)(cid:12)(cid:12)(cid:12) ∂ n ∂x nk h ( a ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X n =0 M ( H a,n , R ) = M ( h a , R ) ≤ C M ( h, r + R )where C is a constant that depends only on N .We recall that the sequence ( d n ) n given by (3.2) is increasing and satisfies d n = O ( n N − ) as n → ∞ . So applying our assumption we get that ∞ X n =0 R n n ! d n (cid:12)(cid:12)(cid:12)(cid:12) ∂ n ∂x nk h ( a ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C e r + R ) ( r + R ) N − / . That is, ∞ X n =0 R n + N − / n ! n N − e R (cid:12)(cid:12)(cid:12)(cid:12) ∂ n ∂x nk h ( a ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C for some constant C that only depends on r and N . By applying Lemma 1.1for α = 2 and β = N − /
2, we obtain l l X n =0 (cid:12)(cid:12)(cid:12)(cid:12) ∂ n ∂x nk h ( a ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C for every l ∈ N and a ∈ B ( r ), and for some constant C that only depends on r and N . In particular, if we select a finite family { a i ∈ S ( r ) : i = 0 , . . . , m } uniformly distributed on S ( r ) we get1 m m X i =0 l l X n =0 (cid:12)(cid:12)(cid:12)(cid:12) ∂ n ∂x nk h ( a i ) (cid:12)(cid:12)(cid:12)(cid:12) ! = 1 l l X n =0 m m X i =0 (cid:12)(cid:12)(cid:12)(cid:12) ∂ n ∂x nk h ( a i ) (cid:12)(cid:12)(cid:12)(cid:12) ! ≤ C. Taking limits as m → ∞ , we conclude that1 l l X n =0 M (cid:18) ∂ n ∂x nk h, r (cid:19) ≤ C for each l ∈ N , and it follows that h cannot be a distributionally unboundedharmonic function with respect to the operator ∂/∂x k . (cid:3) Growth of D α -Distributionally Irregular HarmonicFunctions In this section we study permissible growth rates, in terms of the sup-norm on spheres, of harmonic functions that are distributionally irregularwith respect to the partial differentiation operators D α . The hypercycliccase was studied by Aldred and Armitage [3] and the frequently hypercycliccase by Blasco et al. [16].We begin by recalling some notation and results from [2, 3] which arerequired in the sequel. Set c = 1 and for N ≥ c N = N N − Y j =1 (2 j ) j (2 j + 1) j +1 / N . It was shown in [3, Section 3.2] that for N ≥ c N > r N c N = r N o (1) , as N → ∞ . We also require a suitable antiderivative for the partial differentiation op-erators D α . Aldred and Armitage [3] used an inductive construction withthe linear maps P n,k introduced in (3.5) to define suitable antiderivatives forthe differentiation operators D α . The main properties of these antiderivat-ives are contained in the following lemma, which combines Lemmas 2 and 3from [3]. Lemma 4.1.
Let H ∈ H m ( R N ) for m ∈ N and N ≥ . Given α ∈ N N ,there exists H α ∈ H m + | α | ( R N ) such that D α H α = H with (4.2) M ( H α , ≤ ( | α | !) − M ( H, for N = 2 C | α | A c | α | N ( | α | !) − M ( H, for N ≥ where c N is as defined in (4.1) and A, C > are constants depending onlyon N . ATE OF GROWTH OF DISTRIBUTIONALLY CHAOTIC FUNCTIONS 15
We say h ∈ H ( R N ) is of exponential type τ , for 0 ≤ τ < ∞ , iflim sup r →∞ r − log M ∞ ( h, r ) ≤ τ. We require the following estimate from [18, Proposition 4] (cf. [3, Lemma4]) which gives that(4.3) M ∞ ( H, r ) ≤ p d m r m M ( H, r > H ∈ H m ( R N ) and d m is the dimension of H m ( R N ) as definedin (3.2).We also need the following lemma which is taken from [16, Lemma 2.2]. Lemma 4.2.
Let < α ≤ and β ∈ R . Then there exists a constant C > such that, for all r > ∞ X n =0 r αn ( n + 1) β n ! α ≤ C e αr r ( α +2 β − / . In the following theorem, we modify the approach taken by Bernal andBonilla [11] for entire functions to construct a class of D α -distributionallyirregular harmonic functions which satisfy the required growth estimates. Theorem 4.3.
Let N ≥ and α ∈ N N with α = 0 . Then there exists aharmonic function of exponential type c N in H ( R N ) that is distributionallyirregular for D α .Proof. Let H ∈ H | α | ( R N ) be such that D α H = 1.Let A, C > N . We begin by choosing a ∈ N such that C ∞ X n =2 a n A (cid:0) ( n − a ) | α | (cid:1) ! < . Next we choose b ∈ N with b > a .We subsequently choose a ∈ N with a > b such that C ∞ X n =2 a n A n | α | (cid:0) ( n − a ) | α | (cid:1) ! < b ∈ N with b > a .Proceeding in this way we obtain sequences ( a n ) and ( b n ) with(4.4) C ∞ X n =2 a K n A K n | α | (cid:0) ( n − a K ) | α | (cid:1) ! < K for all K ∈ N .Next we define the sets A := ∞ [ n =1 (cid:8) a n , a n + 1 , . . . , a n (cid:9) and B := ∞ [ n =1 (cid:8) b n , b n + 1 , . . . , b n (cid:9) . Notice thatdens( A ) = lim sup n →∞ |A ∩ { , . . . , n }| n ≥ lim sup n →∞ |A ∩ { , . . . , n }| a n ≥ lim sup n →∞ (cid:12)(cid:12) { a n , a n + 1 , . . . , a n } (cid:12)(cid:12) a n = lim n →∞ a n − a n + 1 a n = 1 . Hence we have that dens( A ) = 1 and similarly dens( B ) = 1.In order to define the required harmonic function h ∈ H ( R N ), we considera sequence ( ω n ) ⊂ [0 , + ∞ ), with lim n →∞ ω n = + ∞ . We set(4.5) e ω n := min { ω n , n } and we define for n ∈ N (4.6) ω ∗ n = (e ω n if n ∈ B h = ∞ X n =0 ω ∗ n +1 H nα where H nα ∈ H ( n +1) | α | ( R N )with D nα H nα = ( D α ) n H nα = H .Next we show that h is distributionally irregular for D α . Fix r >
0. If n ∈ B , then for k x k = rM ∞ ( D nα h, r ) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j =0 ω ∗ j +1 D nα H jα ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j =0 ω ∗ j +1 D nα H jα (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e ω n → ∞ . Next we prove for every r > n ∈A M ∞ ( D nα h, r ) = 0 . We fix r > ε >
0. Choose K ∈ N with K ≥ c N r and 1 /K < ε ,where c N is as given in (4.1). Let j := a K . For each j ∈ A with j ≥ j there exists a unique K ≥ K such that a K ≤ j ≤ a K . It follows that D jα h = ∞ X n = j ω ∗ n +1 D jα H nα = ∞ X n =2 a K ω ∗ n ( n − j ) α since ω ∗ n = 0 for n
6∈ B and where we set 1 α = H .It follows from the triangle inequality that for r > M ∞ (cid:0) D jα h, r (cid:1) ≤ ∞ X n =2 a K ω ∗ n M ∞ (1 ( n − j ) α , r ) . It further follows from (4.3) that(4.8) M ∞ (cid:0) ( n − j ) α , r (cid:1) ≤ (cid:0) d ( n − j ) | α | (cid:1) / r ( n − j ) | α | M (cid:0) ( n − j ) α , (cid:1) where d ( n − j ) | α | is the dimension of H ( n − j ) | α | ( R N ) as defined in (3.2). ATE OF GROWTH OF DISTRIBUTIONALLY CHAOTIC FUNCTIONS 17
Next we consider the case N ≥
3. It follows from (4.2) that(4.9) M (cid:0) ( n − j ) α , (cid:1) ≤ C ′ (( n − j ) | α | ) A ′ c ( n − j ) | α | N (( n − j ) | α | )!where A ′ , C ′ are constants depending only on N .We recall that the sequence ( d m ) given by (3.2) is increasing and satisfies d m = O ( m N − ) as m → ∞ . Combining this fact with (4.8), (4.9), (4.6),that K > c N r and (4.4) it follows that M ∞ (cid:0) D jα h, r (cid:1) ≤ ∞ X n =2 a K ω ∗ n M ∞ (1 ( n − j ) α , r ) ≤ C ′ ∞ X n =2 a K ( n | α | ) A ( c N r ) ( n − j ) | α | (( n − j ) | α | )! ≤ C ′ | α | A ∞ X n =2 a K n A K n | α | (cid:0) ( n − a K ) | α | (cid:1) ! < K ≤ K < ε (4.10)where A > N and we have taken theconstant in (4.4) to be C = C ′ | α | A .For N = 2 we have that d k = 2 for k ≥
1, so it follows from (4.8) and(4.2) that(4.11) M ∞ (cid:0) ( n − j ) α , r (cid:1) ≤ √ r ( n − j ) | α | M (cid:0) ( n − j ) α , (cid:1) (( n − j ) | α | )!and a similar calculation to above gives that M ∞ (cid:0) D jα h, r (cid:1) < ε .Thus for every given r > n ∈A M ∞ ( D nα h, r ) = 0and hence h is a distributionally irregular harmonic function for D α .Next we consider the growth of h . For N ≥
3, it follows from the triangleinequality, (4.3), (4.2) and (4.6) that M ∞ ( h, r ) ≤ ∞ X n =0 ω ∗ n +1 M ∞ ( H nα , r ) ≤ C ′ | α | A M ( H, r | α | ∞ X n =0 ( n + 1) A ( c N r ) n | α | ( n | α | )!where A, C ′ > N . By Lemma 4.2 itfollows that C ′ | α | A M ( H, r | α | ∞ X n =0 ( n + 1) A ( c N r ) n | α | ( n | α | )! ≤ Cr A e c N r for some constant C >
For N = 2 it follows from the triangle inequality, (4.3) and (4.2) that M ∞ ( h, r ) ≤ ∞ X n =0 ω ∗ n +1 M ∞ ( H nα , r ) ≤ √ r | α | M ( H, ∞ X n =0 ( n + 1) r n | α | ( n | α | )! . By another application of Lemma 4.2 we also get the required growth forthe case N = 2.Finally we verify that h is a harmonic function defined on the whole of R N . Fix r > k x k = r and K (depending on r ) largeenough, that by taking j = 0 in a calculations similar to (4.10) and (4.11)yield that the tail (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n = K ω ∗ n +1 M ∞ ( H nα , r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n = K ω ∗ n +1 M ∞ (1 ( n +1) α , r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. Hence by completeness the partial sums of h converge to a harmonic functiondefined on the whole of R N for N ≥ (cid:3) Acknowledgements
This project was initiated during a visit by C. Gilmore to the UniversitatPolit`ecnica de Val`encia and he wishes to thank the members of IUMPA fortheir hospitality and mathematical stimulation during his visit.We wish to thank Tom Carroll and Stephen Gardiner for helpful sugges-tions regarding Lemma 3.2.
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School of Mathematical Sciences, University College Cork, Ireland
E-mail address : [email protected] Institut Universitari de Matem`atica Pura i Aplicada, Universitat Polit`ecnicade Val`encia, Edifici 8E, Acces F, 4a planta, 46022 Val`encia, Spain
E-mail address : [email protected] Institut Universitari de Matem`atica Pura i Aplicada, Universitat Polit`ecnicade Val`encia, Edifici 8E, Acces F, 4a planta, 46022 Val`encia, Spain
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