Slow Growth for Universal Harmonic Functions

Abstract

Given any continuous increasing function such that , we show that there are harmonic functions on satisfying the inequality for every , which are universal with respect to translations. This answers positively a problem of D. H. Armitage (2005). The proof combines techniques of Dynamical Systems and Operator Theory, and it does not need any result from Harmonic Analysis.

1. Introduction

A classic result of Birkhoff from 1929 [1] says that there are entire functions whose translates approximate any other entire function as accurately as we want on an arbitrary compact set of the complex plane. More precisely, given any entire function , there exists an increasing sequence of integers such that , uniformly on compact sets of . Then is called a universal entire function (with respect to translation). In terms of Dynamical Systems, the (continuous and linear) operator , admits elements whose orbit is dense in . Within the framework of Operator Theory, is called a hypercyclic operator, and is a hypercyclic vector for . The same phenomenon happens for any translation operator with respect to a nonzero .

A harmonic function on is said to be universal with respect to translations if the set of translates is dense in the space of all harmonic functions on with the topology of local uniform convergence, that is, the topology of uniform convergence on compact subsets of , also called compact-open topology. Dzagnidze [2] showed that there are universal harmonic functions on (see also the recent paper [3] where it is shown that there are harmonic functions which are universal in a stronger sense, which depends on how "frequently" certain orbits visit any neighbourhood). One may think that the growth of a universal function cannot be arbitrarily "controlled" from above by certain prescribed growth. This is certainly the case for polynomial growth since polynomial growth (either in the entire or in the harmonic case) implies that the function is a polynomial and, obviously, the translation of a polynomial is another polynomial of the same degree. But it came as a surprise that if we fix any transcendental growth, one can find universal entire functions that grow more slowly [4, 5]. In the harmonic case, Armitage [6] showed that universal harmonic functions can also have slow growth. More precisely, given any , a continuous increasing function such that

(1.1)

then there is a universal harmonic function on that satisfies for all . Armitage [6] asked whether the condition (1.1) can be relaxed: is it true that, for every , one can find universal harmonic functions on with arbitrarily slow transcendental growth? In other words, can the exponent of be reduced to ? We answer positively this question. It is easy to notice that is equivalent to saying that for all , and no universal harmonic function can satisfy that for any , since this would force to be a polynomial. This is why we can only consider transcendental growths for universal harmonic functions. Let us recall that the case can be obtained as a consequence of the result in [4, 5] (as was noticed in [6]) since, for example, the real part of a universal entire function is a universal harmonic function on .

We will recall some notions. For the basic theory of harmonic functions we refer the reader to the books [7, 8]. For a good source on the theory of hypercyclic operators and related properties, we refer the reader to [9–12].

A continuous function on a metric space is topologically transitive (resp., mixing) if, for any pair of nonempty open sets, there is (resp., ) such that (resp., for every ). We will work with (continuous and linear) operators on separable, metric, and complete topological vector spaces . In our framework it is well known that topological transitivity is equivalent to hypercyclicity.

2. Universal Harmonic Functions of Slow Growth and Strong Approximation of Polynomials

Given a transcendental growth determined by , to find universal harmonic functions whose growth is bounded by (i.e., for all ), it suffices to find a Banach space consisting of harmonic functions, whose topology is finer than the topology of uniform convergence on compact sets of , containing the harmonic polynomials, and such that, on the one hand, the translation operator , , is (well-defined, continuous and) hypercyclic for every , and on the other hand every harmonic function in the unit ball of has a growth bounded by . Indeed, in this case, there are functions in the unit ball of whose orbit under is dense in , therefore every harmonic polynomial can be approximated in the topology of by a (multiple of a fixed ) translation of , and we obtain that the harmonic functions can be approximated, uniformly on compact sets of , by translations of .

Theorem 2.1.

Let be a continuous increasing function such that

(2.1)

Then there is a Banach space of harmonic functions, whose topology is finer than the topology of uniform convergence on compact sets of , containing the harmonic polynomials, such that the translation operator is a (well-defined and bounded) mixing operator, for any , and every with satisfies for all .

In particular, there are universal harmonic functions of arbitrarily slow transcendental growth.

Proof.

Let us first consider the case . We fix , and by we denote the corresponding directional derivative operator (, ). Since commutes with the Laplacian, one can easily construct a basis of the harmonic polynomials such that each is -homogeneous with , , and for every . To illustrate a particular case, for instance, let and . A construction of a basis of harmonic polynomials with the above conditions is

(2.2)

Note that the polynomials constructed above are all homogeneous, and we wrote the part of the basis until degree .

We consider a sequence of positive weights such that and , . Let us define the space of harmonic functions of growth controlled by as

(2.3)

The space is continuously included in the space of all harmonic functions on under the compact-open topology. Indeed, if , , , and , then

(2.4)

It is clear that if we select the sequence increasing fast enough, we have that every with satisfies that for all . To prove it, let . Our hypothesis on implies that

(2.5)

because each is a polynomial. Let so that if , . We suppose that

(2.6)

Given any and with , we have

(2.7)

We have that is a bounded operator which acts as

(2.8)

since , .

The space is naturally isomorphic to , where and , . Via the natural isomorphism, the operator is then conjugated to the following operator on which is the -sum of the backward shift:

(2.9)

We fix the notation . Since the translation operator equals , we thus have that is conjugated to . Then we are done if we prove that is a mixing operator on .

Given arbitrary nonempty open sets , we fix and pairs of nonempty open sets in the corresponding spaces, , such that

(2.10)

The operator is mixing on any weighted -space, as was shown in the proof of Theorem of [13]; therefore we find such that , , for every . We finally conclude that

(2.11)

and , being conjugated to , is a mixing operator.

The case is even simpler since the basis of harmonic polynomials can be taken as such that and for every , . The corresponding space is the direct sum of two weighted -spaces, and a simplification of the above argument does the job.

Remark 2.2.

What we actually showed in the proof of Theorem 2.1 is that there are with for all such that, for any harmonic polynomial , there is an increasing sequence of integers such that in the topology of , which is finer than the topology of uniform convergence on compact sets of . Moreover, the function is universal with respect to translations in the direction of , a result also stronger than Armitage's in the sense that in [6] all translations were allowed for a universal function.

3. Final Comments

It is possible to obtain the main result of the paper by using tensor product techniques developed in [14]. However the proofs seem to be technically much more complicated since one has to represent the space of harmonic functions on as a quotient of a direct sum of the complete tensor product of spaces of harmonic functions with fewer variables, while we produced here a, somehow, direct proof. We want to thank Antonio Bonilla for pointing out to us the possibility of a tensor product approach.

When we fix a nonzero , one can consider the parametric family of translation operators on our space . This is a -semigroup of operators, which we showed to be hypercyclic on . We refer to [12, 13] for the basic theory of hypercyclic -semigroups. In [15] we proved that every hypercyclic vector of a -semigroup is shared by all operators of the semigroup (except, obviously, ). In particular, if is a universal harmonic function for , then it is also universal (or a hypercyclic vector) for for all . Let us notice that the inheritance of hypercyclicity by discrete subsemigroups depends strongly on the fact that the index set is or : there are hypercyclic -semigroups of operators whose index set is a sector of the complex plane and such that no discrete subsemigroup is hypercyclic [16].

The operator can be considered as a (infinite type) differential operator since . In [17] we develop techniques based on a kind of generalized backward shifts which allow us to extend the main results in this paper and [5] to other differential operators, and then we show the existence of harmonic and entire functions of arbitrarily slow transcendental growth which are universal with respect to some differential operators.