C O L L O Q U I U M M A T H E M A T I C U M VOL. 83 2000 NO. 2

C O L L O Q U I U M M A T H E M A T I C U M

VOL. 83 2000 NO. 2

“COUNTEREXAMPLES” TO THE HARMONIC LIOUVILLE THEOREM AND HARMONIC FUNCTIONS

WITH ZERO NONTANGENTIAL LIMITS

BY

A. B O N I L L A (LA LAGUNA)

Abstract. We prove that, if µ > 0, then there exists a linear manifold M of harmonic functions in R ^N which is dense in the space of all harmonic functions in R ^N and

lim

kxk→∞

x∈S

kxk ^µ D ^α v(x) = 0

for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if µ ≥ N + 1, then every function v ∈ M satisfies

T

H v dλ = 0 for every (N − 1)-dimensional hyperplane H, where λ denotes the (N − 1)-dimensional Lebesgue measure. On the other hand, we prove that there exists a linear manifold M of harmonic functions in the unit ball B of R ^N , which is dense in the space of all harmonic functions and each function in M has zero nontangential limit at every point of the boundary of B.

1. Introduction. Liouville’s theorem states that a bounded holomor- phic function on the complex plane C is constant. A similar result holds for harmonic functions on R ^N , that is, a bounded harmonic function on R ^N is constant. Nevertheless, if the boundedness condition is slightly weakened, then nonconstant entire functions in C or harmonic functions in R ^N can be obtained.

For example, Armitage and Goldstein [3] proved that, for each µ > 0, there exists a nonconstant entire function f such that

exp(r ^µ )f (re ^iθ ) → 0 (r → ∞)

for all θ ∈ [0, 2π). And, for each µ > 0, there exists a nonconstant harmonic function v in R ⁿ such that

kxk ^µ v(x) → 0 (kxk → ∞, x ∈ L) for every semi-infinite line L in R ^N .

2000 Mathematics Subject Classification: Primary 31B05.

Key words and phrases: harmonic functions, approximation, Liouville harmonic the- orem, universal function, Radon transform, nontangential limits.

This work is supported in part by DGICYT, ref PB95-0749.

[155]

Moreover, Armitage and Goldstein [5] showed that there exists a non- constant harmonic function v on R ^N such that

(1)

\

H

v dλ = 0

for every (N − 1)-dimensional hyperplane H, where λ denotes the (N − 1)- dimensional Lebesgue measure. Also, Armitage and Gauthier [2] have proved that the space of harmonic functions on R ^N for which (1) is true is dense in h(R ^N ), the space of all harmonic functions in R ^N .

A harmonic function f is universal if to each harmonic function g in R ⁿ corresponds a sequence {a n } n≥1 depending on g and satisfying

n→∞ lim f (x + a _n ) = g(x)

uniformly on compact sets. In [2] it is proved that the set of universal har- monic functions is residual in h(R ^N ).

On the other hand, Bernal [9] proved that, given α ∈ (0, 1/2), there exists a linear manifold M of entire functions which is dense in the space of all entire functions and, in addition,

z→∞ lim

z∈S

exp(|z| ^α )f ^(j) (z) = 0

for every f ∈ M and j ∈ N, where S denotes any plane strip. See also [10]

for a strengthening of this result.

In this paper, we prove that, if µ > 0, then there exists a linear manifold M of harmonic functions in R ^N which is dense in h(R ^N ) and

kxk→∞ lim

x∈S

kxk ^µ D ^α v(x) = 0

for every v ∈ M and multi-index α, where S denotes any hyperplane strip.

Moreover, every nonnull function in M is universal.

In particular, we see that if µ ≥ N +1, then all functions in M satisfy (1).

In [7] Ash and Brown prove that there is a harmonic function, which is not identically zero, in the unit disc of the complex plane which has zero nontangential limit at every point of the boundary of the disc.

We prove that there exists a linear manifold M of harmonic functions in the unit ball of R ^N , which is dense in the space of all harmonic functions in the unit ball (with the compact-open topology) and each function in M has zero nontangential limit at every point of the boundary.

2. “Counterexamples” to the harmonic Liouville theorem. We need some notation. A hyperplane strip is the region lying between two par- allel hyperplanes in R ^N . h(R ^N ) is endowed with the compact-open topology.

(R ^N ) ^∗ is the one-point compactification of R ^N . If F is a closed subset of R ^N ,

then h(F ) is the space of all harmonic functions on a neighbourhood of F .

Theorem 1. If µ > 0, then there exists a linear manifold M of harmonic functions in R ^N with the following properties :

(a) M is dense in h(R ^N ),

(b) lim _{kxk→∞, x∈S} kxk ^µ D ^α v(x) = 0 on any hyperplane strip S, for every v ∈ M and for every multi-index α,

(c) D ^α v is bounded on any hyperplane strip, (d) Every nonnull function in M is universal.

Hence, if µ ≥ N + 1, then

(e) D ^α v is integrable with respect to N -dimensional Lebesgue measure on any hyperplane strip,

(f) D ^α v is integrable with respect to (N − 1)-dimensional Lebesgue mea- sure on any hyperplane H and

T

H D ^α v = 0.

P r o o f. Let µ > 0 and consider a denumerable family {p n } ^∞ _n=1 of har- monic polynomials dense in h(R ^N ). (Moreover, given u ∈ h(R ^N ) and a compact set K ⊂ R ^N , there exists a sequence n 1 < n 2 < . . . such that p n k → u uniformly on K.)

Let L denote the curve

L := {(x 1 , . . . , x N ) : x 1 = t, x 2 = t ² , . . . , x N = t ^N ; t ≥ 0}

and T the tract

{x ∈ R ^N : dist(x, L) < (1 + kxk) ^1/2 }.

Then there exists a disjoint sequence D ^∗ _j = D(a _j , 1 + 2 ^j ) of open discs such that D ^∗ _j ⊂ T \ L and ka j+1 k > ka j k + 2 ^j+2 for j = 1, 2, . . .

Consider now B n := {x : kxk ≤ n} and

E n := {x ∈ R ^N : kxk ≥ n + 1 and dist(x, L) ≥ (1 + kxk) ^1/2 } and define

F n := B n ∪ E n ∪ [

j>j 0

D(a j , 2 ^j )

where j ₀ is the last index such that D _j ^∗ ₀ ∩ B n+1 6= ∅. Then F n is a closed subset of R ^N , and (R ^N ) ^∗ \ F n is connected and locally connected at {∗}.

Now, let Ω B n and Ω E n be disjoint open subsets of R ^N containing B n and E _n respectively, which intersect no D _j ^∗ for j > j ₀ . Dividing the sequence {a j } into infinitely many disjoint subsequences {a i(m,j) } by setting i(m, j) = (j + m)(j + m + 1)/2 + m, we define the function u n : F n → R by

u n (z) =



 



 



p n (x), x ∈ Ω B n ,

0, x ∈ Ω E n ,

p j (x − a i(n,j) ), x ∈ D _i(n,j) ^∗ ,

0, x ∈ D _i(m,j) ^∗ , m 6= n.

Then u n ∈ h(F n ), and by Theorem 1.1 of [3], there exists v n ∈ h(R ^N ) such that

(2) |(v n − u n )(x)| < 1

n (1 + kxk) ^−µ−1 , x ∈ F n . Hence

|(v n − u n )(x)| < 1/n, x ∈ B n .

Thus the sequence {v n } ^∞ _n=1 is dense in h(R ^N ). Now, we define M as the linear span of {v n }. Evidently, M is a linear dense submanifold of h(R ^N ).

In order to verify (b), it suffices to check it for v _n . Define E _n ^∗ as

E _n ^∗ := {x ∈ R ^N : kxk ≥ n + 2 and dist(x, L) ≥ (1 + kxk) ^1/2 + 1}.

Then by using the Cauchy estimates for harmonic functions [8, p. 33], from (2), since u n = 0 in E n , we infer that

|D ^α v n (x)| ≤ C α max{|v n (y)| : ky − xk ≤ 1} ≤ C α

1 kxk ^µ+1 for all x ∈ E _n ^∗ and all multi-indices α.

Note that every fixed hyperplane strip S is wholly contained in E _n ^∗ except for a bounded set. Therefore,

kxk→∞ lim

x∈S

kxk ^µ D ^α v(x) = 0

for every v ∈ M and multi-index α on any hyperplane strip S. Moreover, every function in M satisfies (c), and (e) if µ ≥ N + 1.

Also every nonnull function of M is universal. If v ∈ M , v = P

j∈I α j v j , I finite, since every nonzero scalar multiple of a universal function is again universal, we may suppose that α j 1 = 1 with j 1 ∈ I. In order to prove that v = P

j∈I α j v j is universal, it is enough to check that

(3) lim

n→∞

 X

j∈I

α j v j (x + a i(j 1 ,n) ) − p n (x) 

= 0

uniformly on compact subsets. This is readily seen by computing

(4) max

D(0,2 ⁿ )

X

j∈I

α j v j (x + a _i(j 1 ,n) ) − p n (x)   . By the triangle inequality, (4) is less than

max

D(0,2 ⁿ ) |v j 1 (x + a _{i(j 1 ,n)} ) − p _n (x)| + max

D(0,2 ⁿ )

X

j∈I j6=j 1

|α j | · |v j (x + a _{i(j 1 ,n)} )|.

Now by (2), we conclude that v is universal.

For details of (f), see [5].

Remark 1. As in the holomorphic case (see [9]), it is possible to prove that any set M ⊂ h(R ^N ) satisfying (b) is of the first category in h(R ^N ).

3. Harmonic functions with zero nontangential limits. By h(B) we denote the space of all harmonic functions in the unit ball of R ^N . h(B) is endowed with the compact-open topology. B ^∗ is the one-point compacti- fication of B. If F is a relatively closed subset of B, then h(F ) is the space of all harmonic functions on a neighbourhood of F .

Theorem 2. There exists a dense linear manifold M of harmonic func- tions in B which have zero nontangential limit at every point of ∂B.

P r o o f. Consider a denumerable family {p _n } ^∞ _n=1 of harmonic polynomi- als dense in h(B) with the compact-open topology.

Let F n = B n ∪ E n , where B n = {x : kxk ≤ 1 − 1/n} and E n = B \ (B n+1 ∪ {x = (x 1 , . . . , x N ) ∈ B :

x _N < 0, x ∈ B((1/2, 0, . . . , 0), 1/2) \ B((3/4, 0, . . . , 0), 1/4)}).

Then F n is a relatively closed subset of B, and B ^∗ \ F n is connected and locally connected at {∗}.

Let Ω B n and Ω E n be disjoint open subsets of B containing B n and E n

respectively, and define the function u _n : F _n → R by u n (z) =  p n (x), x ∈ Ω B n ,

0, x ∈ Ω E n .

Then u n ∈ h(F n ), and by Theorem 1.1 and Remark 9.1 of [6], there exists v n ∈ h(B) such that

(5) |(v n − u n )(x)| < 1

n dist(x, ∂B), x ∈ F n . Hence

|(v n − u n )(x)| < 1/n, x ∈ B n .

Thus, the sequence {v n } ^∞ _n=1 is dense in h(B). Now, we define M as the linear span of {v n }. Evidently, M is a linear dense manifold in h(R ⁿ ).

In order to verify that every v ∈ M has zero nontangential limit at every point of ∂B, it suffices to check it for every v _n . But this is a consequence of

|v n (x)| < dist(x, ∂B), ∀x ∈ E n .

Remark 2. The set of harmonic functions in B with zero nontangential limit at all points is of the first category. Indeed, given x ∈ ∂B and n, m ∈ N, let

E n,m (x) := {u ∈ h(B) : |u(rx)| ≤ 1/n, 1 − 1/m < r < 1}

and

A := {u ∈ h(B) : lim

r→1 u(rx) = 0, ∀x ∈ ∂B}.

Then

A = \

x∈∂B

\

n

[

m

E n,m (x) and E n,m (x) is closed and nowhere dense.

Remark 3. If N = 2, then the nonzero functions u in M do not have the property that there exists a positive number α so that m(r) = sup _|z|≤r |u(z)| = o((1 − r) ^−α ) as r → 1 (see [7, Corollary 1]).

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Departamento de An´ alisis Matem´ atico Universidad de La Laguna

38271 La Laguna (Tenerife) Canary Islands, Spain E-mail: abonilla@ull.es

Received 15 March 1999 (3711)