On the Integrability of Eigenfunctions of the Laplace-Beltrami Operator ...

Potential Analysis 16: 205–220, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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On the Integrability of Eigenfunctions of the Laplace–Beltrami Operator in the Unit Ball of Cn MANFRED STOLL Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA (e-mail: [email protected]) (Received: 13 July 1999; accepted: 4 September 2000) and ∆ denote the volume measure, Abstract. Let B denote the unit ball in Cn , n 1, and let τ , ∇, gradient, and Laplacian respectively, with respect to the Bergman metric on B. For γ ∈ R and p 0 < p < ∞, we denote by Lγ the set of real, or complex-valued measurable functions f on B for which (1 − |z|2 )γ |f (z)|p dτ (z) < ∞, B

p | ∈ Lp and by Dγ the Dirichlet space of C 1 functions f on B for which |∇f γ . Also, for λ ∈ C, we 2 = λf . The main denote by Xλ the set of C real, or complex-valued functions f on B for which ∆f result of the paper is as follows: Let 0 < p < ∞ and suppose λ ∈ R with λ −n2 . Then p

Lγ ∩ Xλ = {0},

and for λ = 0,

p

Dγ ∩ Xλ = {0}

(a) for all γ n + p2 ( n2 + λ − n) when p 1, and (b) for all γ p2 (n + n2 + λ) when 0 < p < 1.

By example it is shown that the result is best possible for all values of p with p n/(n + n2 + λ). Mathematics Subject Classifications (2000): 31C25, 32A35, 46E15. Key words: Dirichlet spaces, eigenfunctions, Laplace–Beltrami operator, M-harmonic functions.

1. Introduction and ∇, Let B denote the unit ball in Cn , n 1, with boundary S. Also, let ∆, τ denote the Laplace–Beltrami operator, the gradient, and the volume measure, respectively, with respect to the Bergman metric on B. For a real, or complexvalued C 2 function f we have (z) = (f ◦ ϕz )(0) = 4(1 − |z|2 ) ∆f

n i,j =1

(δi,j − zj zi )

∂ 2 f (z) , ∂zj ∂zi

(1.1)

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MANFRED STOLL

where is the usual Laplacian. A real, or complex-valued C 2 function f on B = 0 is said to be M-harmonic or invariant harmonic on B. Also, for satisfying ∆f 1 a C function f we have n ∂f ∂f ∂f ∂f 2 2 f (z)| = 2(1 − |z| ) (δi,j − zj zi ) + . (1.2) |∇ ∂ z¯ i ∂ z¯ j ∂zj ∂zi i,j =1 are invariant under the group M of biholomorphic automorphisms and ∇ Both ∆ ◦ ψ)| = |(∇f ) ◦ ψ| for all ψ ∈ M. ◦ ψ) = (∆f ) ◦ ψ and |∇(f of B, that is ∆(f When n = 1, (z) = (1 − |z|2 )2 f (z) ∆f

and

(z)| = (1 − |z|2 )|∇f (z)|, |∇f

where ∇f = (fx , fy ) is the usual gradient of f . The M-invariant volume measure τ on B is given by dτ (w) = (1 − |w|2 )−(n+1) dm(w),

(1.3)

where m is normalized Lebesgue measure on B. The results of this paper were partially motivated by the following theorem of Yamashita. THEOREM A ([12, Theorem 1]). Let f be a solution of f = fxx + fyy = λf , λ 0, in the unit disc U ⊂ C satisfying 1 (1 − |z|2 )γ |∇f (z)|2 dx dy < ∞ (1.4) π U for some γ , 0 < γ 1, then |f |2/γ admits a harmonic majorant in U . In [11] we considered the extension of Theorem A to eigenfunctions of the Laplace– satisfying an integrability condition analogous Beltrami operator ∆ Specifically, for γ ∈ R we denote to (1.4) defined in terms of the gradient ∇. 2 by Dγ the set of real, or complex-valued C 1 functions f on B satisfying (z)|2 dτ (z) < ∞. (1.5) Dγ2 (f ) = (1 − |z|2 )γ |∇f B

The quantity Dγ2 (f ) When n = 1, Dγ2 (f )

is called the γ -weighted invariant Dirichlet integral of f . is simply the ordinary γ -weighted Dirichlet integral of f as

defined in (1.4). = λf for The initial goal in [11] was to prove that if f is a solution of ∆f 2 some λ 0 with Dγ (f ) < ∞ for some γ , 0 < γ n, then |f |n/γ admits an M-harmonic majorant on B. Although this turns out to be true when λ = 0, for = λf satisfying Dγ2 (f ) < ∞ for λ > 0 we proved that the only solution of ∆f some γ , 0 < γ n, is the zero function [11, Theorem 3.1]. In the unit disc U it is easy to find eigenfunctions of satisfying the hypothesis of Theorem A. However,

EIGENFUNCTIONS OF THE LAPLACE–BELTRAMI OPERATOR

207

and annihilate the same functions in U , the eigenspaces of ∆ and even though ∆ corresponding to a given λ are significantly different. In analogy with (1.5), for γ ∈ R and 0 < p < ∞, we denote by Dγp (B) or Dγp the set of real, or complex-valued C 1 functions f on B for which p (z)|p dτ (z) (1.6) Dγ (f ) = (1 − |z|2 )γ |∇f B

is finite, with the Dirichlet norm 1/p . f Dγp = |f (0)| + Dγp (f ) The quantity Dγp of course is only a norm for p 1. Also, for γ ∈ R, we denote by Lpγ the set of measurable functions f on B for which f p,γ < ∞, where for 0 < p < ∞, p (1.7) f p,γ = (1 − |w|2 )γ |f (w)|p dτ (w). B

Finally, for λ ∈ C, we denote by Xλ the set of real, or complex-valued C 2 func with eigenvalue λ, i.e., tions f on B that are eigenfunctions of ∆

= λf . Xλ = f ∈ C 2 (B) : ∆f The space X0 is simply the space of M-harmonic functions on B. Although we will not do so, as in Theorem 3.1 of [11], it is possible to prove that if f ∈ Dγp ∩ Xλ (λ > 0) or f ∈ Lpγ ∩ Xλ (λ 0) for some γ , 0 < γ min{n, pn}, then f (z) = 0 for all z ∈ B. Thus a natural question to ask is for what values of γ ∈ R are the spaces Lpγ ∩ Xλ and Dγp ∩ Xλ nontrivial. One of the main results of the paper, Theorem 2.4, is as follows: Let 0 < p < ∞ and suppose λ is real with λ −n2 . Then Lpγ ∩ Xλ = {0},

and for λ = 0, Dγp ∩ Xλ = {0}, √ (a) for all γ n + p2 ( n2 + λ − n) when p 1, and √ (b) for all γ p2 (n + n2 + λ) when 0 < p < 1. By example √ it is shown that the result is best possible for all values of p with p n/(n + n2 + λ). Since Dγp ∩ X0 always contains the constant functions, we consider this case separately in Section 3. When n = 1, the spaces Dγp (U ), 0 1−p. However, in Theorem 3.3 we prove that when n 2, the spaces Dγp , 0 n − 12 p. For M-harmonic functions, the best that we have been able to prove is that for n 2, if γ min{n − p, p(n − 1)}, then Dγp ∩ X0 contains only constant functions. In view of Theorem 3.3 and the result for n = 1, it is unlikely that this result is best possible.

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MANFRED STOLL

Dirichlet spaces of M-harmonic or holomorphic functions have been considp ered by several authors. The space D0 of holomorphic functions on B was considered by Arazy, Fisher, Janson and Peetre in [1]. In that paper the authors proved p that when n 2, D0 contains nonconstant holomorphic functions if and only if p > 2n. Dirichlet type spaces of holomorphic and M-harmonic functions have also been considered by Hahn and Youssfi in [2–4]. In the papers of Hahn and Youssfi these spaces are referred to as Besov Spaces. More general types of Besov spaces of holomorphic functions have also been studied by Peloso in [7]. Throughout the paper we will use C to denote a positive constant, not necessarily the same on any two occurrences. Also, if f and g are functions defined on some set S, we use the notation f (z) ≈ g(z) to mean that there exist positive constants C1 and C2 such that C1 f (z) g(z) C2 f (z) for all z ∈ S. 2. Integrability of Eigenfunctions of ∆ In this section we consider the problem of determining the values of γ for which the spaces Xλ ∩ Dγp (λ = 0) and Xλ ∩ Lpγ are nontrivial. First however, we include some preliminary results about the weighted Dirichlet space Dγp . For a ∈ B, let ϕa denote the Möbius transformation of B satisfying ϕa (0) = a and ϕa−1 = ϕa . By [9, p. 26] ϕa satisfies 1 − |ϕa (z)|2 =

(1 − |z|2 )(1 − |a|2 ) . |1 − z, a|2

(2.1)

and τ , f ◦ ϕa ∈ Dγp for all a ∈ B with If f ∈ Dγp , then by the M-invariance of ∇ Dγp (f

◦ ϕa )

1 + |a| 1 − |a|

γ Dγp (f ).

(2.2)

Similarly, if f ∈ Lpγ , then f ◦ ϕa ∈ Lpγ with f ◦ ϕa p,γ

1 + |a| 1 − |a|

γ f p,γ .

(2.3)

Also, from the definitions it follows that if f = u + iv, where u and v are realvalued, then f ∈ Dγp if and only if both u and v are in Dγp . For z ∈ B, set E(z) = {w ∈ B : |ϕz (w)| < 13 }.

(2.4)


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By [6, Theorem 2.1], if f ∈ Xλ with λ = 0, and 0 < p < ∞, then there exists a constant C = C(p, |λ|) such that (w)|p dτ (w). |f (z)|p C |∇f E(z)

From identity (2.1) it follows immediately that (1 − |w|2 ) ≈ (1 − |z|2 ) for all w ∈ E(z). Thus 2 γ p f (w)|p dτ (w). (1 − |w|2 )γ |∇ (1 − |z| ) |f (z)| C E(z)

Finally, since τ (E(z)) = τ ({w : |w| < 13 }), which is constant, by integrating the above and using Fubini’s theorem we obtain f pp,γ CDγp (f )

(2.5)

for all f ∈ Dγp ∩ Xλ , λ = 0. Although we will not require it, one also has Dγp (f ) Cf pp,γ for all f ∈ Lpγ ∩ Xλ , 0 < p < ∞, and all λ ∈ C [6]. can be obtained from the Poisson kernel P Examples of eigenfunctions of ∆ for ∆ given by (1 − |z|2 )n , |1 − z, t|2n

P (z, t) =

z ∈ B, t ∈ S.

For α ∈ C, let Pα be the function on B defined by Pα (z) = P α (z, e1 ) = exp{α log P (z, e1 )}, where e1 = (1, 0, . . . , 0). By Theorem 4.2.2 of [9], if α and λ are related by λ = −4n2 α(1 − α),

(2.6)

then Pα ∈ Xλ . Furthermore, Xλ contains every function f of the form f (z) = P α (z, ζ ) dν(ζ ), S

where ν is a complex measure on S. In particular, Xλ contains the radial function gα defined by gα (z) = P α (z, ζ ) dσ (ζ ). S

The function gα satisfies gα (z) = g1−α (z). Furthermore, if α = β + is, β, s ∈ R, then |gα (z)| gβ (z) for all z ∈ B.

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MANFRED STOLL

α |. Let λ and α be related by (2.6), with α = β + is, We next compute |∇P β and s real. Then λ = −4n2 [β(1 − β) + s 2 + is(1 − 2β)].

(2.7)

and ∇ it immediately follows that From the definition of ∆ f¯ + f¯∆f + 2|∇ f |2 . |2 = f ∆ ∆|f Thus α |2 − Pα ∆P α |2 − 2 Re λ|Pα |2 . α − P α ∆P α = ∆|P Pα |2 = ∆|P 2|∇ Since |Pα |2 = Pβ2 = P2β , we have Pα |2 = 4n2 {2 Re[α(1 − α)] − 2β(1 − 2β)}|Pα |2 = 8n2 |α|2 |Pα |2 . 2|∇ Therefore, α | = 2n|α||Pα |. |∇P

(2.8)

By Proposition 1.4.10 of [9] we have the following asymptotic estimates for gβ (z), β ∈ R. LEMMA 2.1. For β ∈ R and z ∈ B,  (1 − |z|2 )n(1−β) ,    1 , gβ (z) = g1−β (z) ≈ (1 − |z|2 )n/2 log  (1 − |z|2 )   (1 − |z|2 )nβ ,

if β > 12 , if β = 12 , if β < 12 .

THEOREM 2.2. Suppose 0 n − pn min{β, 1 − β}. p Proof. Although we could use the fact that Dγp (gα ) Cgα p,γ for all p, 0 < p < ∞, we prove this inequality directly. Since gα is a radial function, with r = |z| α (z)| = (1 − r 2 )|gα (r)|, and we have |∇g d α |gα (r)| P (re1 , ζ )dσ (ζ ). S dr But

2 nβ−1 2 d α (1 − r (1 − r ) ) d P (re1 , ζ ) = n|α| dr |1 − rζ |2 . dr |1 − rζ1 |2 1

A straightforward computation and crude estimates give d (1 − r 2 ) 6 dr |1 − rζ |2 |1 − rζ |2 . 1 1


Therefore gα (z)| 6n|α| |∇

211

P β (re1 , ζ ) dσ (ζ ) = 6n|α|gβ (r), S

and by integration in polar coordinates, 1 p (1 − r 2 )γ −n−1 r 2n−1 gβ (r) dr. Dγp (gα ) C 0

Also, since gα p,γ gβ p,γ , the conclusion follows by the estimates for gβ (r) given in Lemma 2.1. ✷ Suppose α = β + is, β, s ∈ R, and λ is defined by (2.6). Then by (2.7), except when β = 12 , λ is real if and only if s = 0, and when β = 12 , λ = −n2 − 4n2 s 2 . Hence if λ is real with λ −n2 , then α is also real. In this case, upon solving (2.6) for α we obtain 1 1 λ + n2 . (2.9) α= ± 2 2n Thus by Theorem 2.2, Dγp (gα ) and also gα p,γ are finite for all p γ > n + ( λ + n2 − n). 2 The following theorem shows that for radial functions in Xλ , where λ is real with λ −n2 , this is not only sufficient, but also necessary. THEOREM 2.3. Let 0 n + p2 ( n2 + λ − n). or

(λ = 0)

Proof. If f ∈ Dγp ∩ Xλ with λ = 0, then by inequality (2.5) f ∈ Lpγ ∩ Xλ . Thus it suffices to prove the result for f ∈ Lpγ ∩ Xλ . If f ∈ Xλ is radial, then by Theorem 4.2.3 of [9], f (z) = f (0)gα (|z|), where 2 λ and α are related by (2.6). But since λ is real with λ √ −n , α is given by either 1 1 of the two solutions in (2.9). Hence with α = 2 + 2n λ + n2 , by Lemma 2.1 it follows that f ∈ Lpγ if and only if p ✷ γ > n − pn(1 − α) = n + ( n2 + λ − n). 2 NOTE. For the case Dγp ∩ Xλ we exclude λ = 0 since if f is a radial function on B = 0, then f is constant on B and thus f ∈ Dγp for all γ . satisfying ∆f We are now ready to state and prove the main result of the paper.

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MANFRED STOLL

THEOREM 2.4. Let 0 < p < ∞ and suppose λ is real with λ −n2 . Then Lpγ ∩ Xλ = {0},

and for λ = 0, Dγp ∩ Xλ = {0} √ (a) for all γ n + p2 ( n2 + λ − n) when p 1, and √ (b) for all γ p2 (n + n2 + λ) when 0 < p < 1. Proof. As in Theorem 2.3 it suffices to prove the result for f ∈ Lpγ ∩ Xλ . (a) Suppose p 1 and f ∈ Lpγ ∩ Xλ . Then by (2.3), f ◦ ϕa ∈ Lpγ for each a ∈ B. Now by integration in polar coordinates, p (1 − |z|2 )γ |f ◦ ϕa (z)|p dτ (z) f ◦ ϕa p,γ = B 1 2 γ −n−1 2n−1 (1 − r ) r |f (ϕa (rζ ))|p dσ (ζ ) dr, = 2n 0

S

which by Hölder’s inequality p 1 2 γ −n−1 2n−1 dr. (1 − r ) r f (ϕ (rζ )) dσ (ζ ) 2n a 0

S

But by Theorem 4.2.4 of [9], since f ∈ Xλ , f (ϕa (rζ )) dσ (ζ ) = f (a)gβ (r), S

where β is √ defined to be either of the two values given in (2.9). Therefore with 1 n2 + λ (λ > −n2 ), we have β = 12 + 2n 1 p (1 − r 2 )γ −n−1 r 2n−1 gβ (r) dr f ◦ ϕa pp,γ 2n|f (a)|p 0 1 (1 − r 2 )γ −n+pn(1−β)−1 r 2n−1 dr. C|f (a)|p 0

The above integral is finite however if and only if p γ > n − pn(1 − β) = n + ( n2 + λ − n). 2 √ Thus if f ◦ ϕa p,γ < ∞ for γ n + p2 ( n2 + λ − n) we must have f (a) = 0, √ 1 n2 + λ gives the same result since (1 − hence the result. Taking β = 12 − 2n β ) = β. If β = 12 (λ = −n2 ), then p 1 1 1 r 2n−1 (1 − r 2 )γ −n+ 2 pn−1 log dr, f ◦ ϕa pp,γ C|f (a)|p (1 − r 2 ) 0 and the above integral is again finite if and only γ > n − 12 pn.


213

(b) Suppose now that 0 < p < 1 and f ∈ Lpγ ∩ Xλ . For z ∈ B, let E(z) be defined as in (2.4). For f ∈ Xλ , by [6, Theorem 2.1] there is a constant C = C(p, |λ|) such that p |f (w)|p dτ (w). |f (z)| C E(z)

Since (1 − |w|2 ) ≈ (1 − |z|2 ) for all w ∈ E(z), we have (1 − |w|2 )γ |f (w)|p dτ (w) C(1 − |z|2 )−γ , |f (z)|p C(1 − |z|2 )−γ E(z)

or |f (z)| C(1 − |z|2 )−γ /p for some positive constant C. Thus for all z with f (z) = 0, γ

|f (z)|p−1 C(1 − |z|2 ) p (1−p) . Therefore, p |f (rζ )| dσ (ζ ) = |f (rζ )||f (rζ )|p−1 dσ (ζ ) S S γ C(1 − r 2 ) p (1−p) |f (rζ )| dσ (ζ ) S γ p −γ

|f (0)|gβ (r). √ 1

0) and β = 12 + 2n n2 + λ, then by Lemma 2.1 and Therefore, if λ > −n2 (λ = the above, 1 γ (1 − r 2 ) p −n+n(1−β)−1 r 2n−1 dr. f pp,γ C|f (0)| C(1 − r ) 2

0

The above integral is finite however if and only if √ γ > pnβ = p2 (n + n2 + λ), from which the result follows. If λ = −n2 (β = 12 ), the result follows as in (a). ✷ EXAMPLE 2.5. this example we show that the result of Theorem 2.4 is sharp In√ for all p n (n + n2 + λ). √ 1 n2 + λ, by Theorem 2.2 the (a) Suppose first that p 1. With β = 12 + 2n p p function gβ is in both Dγ ∩ Xλ and Lγ ∩ Xλ for all √ γ > n − pn(1 − β) = n + p2 ( n2 + λ − n).

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MANFRED STOLL

√ 1 (b) Suppose 0 < p < 1. Consider the function Pβ with β = 12 + 2n n2 + λ. Then by identity (2.8), 1 p p 2 γ −n−1 2n−1 Dγ (Pβ ) = CPβ p,γ = C (1 − r ) r Pβ (rζ, e1 )p dσ (ζ ) dr S 0 1 (1 − r 2 )γ −n−1 r 2n−1 gpβ (r) dr. = C 0

then gpβ (r) ≈ (1 − r ) p γ > npβ = (n + n2 + λ). 2

If pβ >

1 , 2

2 n(1−pβ)

, and the above integral is finite provided

1 p then gpβ (r) ≈ (1 − |z|2 )n/2 log (1−|z| 2 ) , and Dγ (Pβ ) is again finite √ provided γ > 12 n = npβ = p2 (n + n2 + λ). But pβ 12 if and only if

If pβ =

p

1 , 2

n . √ n + n2 + λ

✷

Whether the conclusion √ of Theorem 2.4 is best possible for the values of p satisfy ing 0 −n2 . If f is a nonnegative function in Xλ ∩ Lpγ or in Xλ ∩ Dγp (λ = 0) for some γ satisfying √ √ γ max{ 12 p(n + n2 + λ), n − 12 p(n + n2 + λ)}, then f ≡ 0 on B. The result is best possible for all p, 0 < p < 1.

√ Proof. If f ∈ Xλ ∩Lpγ or f ∈ Xλ ∩Dγp (λ = 0) for some γ 12 p(n+ n2 + λ), then by Theorem 2.4 f ≡ 0 on B. On the other hand, if f ∈ Xλ is nonnegative, then by Theorem 17.11.1 of [5], f (z) = P β (z, ζ ) dν(ζ ) S

√ 1 n2 + λ. Since β > 0 we for some nonnegative measure ν on S and β = 12 + 2n β 2 nβ have P (z, ζ ) C(1 − |z| ) and thus 1 (1 − |z|2 )γ |f (z)|p dτ (z) C[ν(S)]p (1 − r 2 )γ +npβ−n−1 r 2n−1 dr. B

0

Since the integral on the right is finite if and only if √ γ > n − npβ = n − 12 p(n + n2 + λ), we must have f ≡ 0 if f ∈ Lpγ for some γ n − 12 p(n + Pβ shows that the result is best possible.

√

n2 + λ). The function ✷


215

3. Integrability of the Gradient of M-harmonic Functions f | where f is an M-harmonic In this final section we consider the integrability of |∇ function on B. Suppose f is a real-valued C 1 function on B. Then by (1.2) (z)|2 = 4(1 − |z|2 )[|∂f (z)|2 − |Rf (z)|2 ], |∇f

(3.1)

∂f ∂f , . . . , ∂z ) is the complex gradient of f and Rf is the radial where ∂f = ( ∂z n 1 derivative of f given by

Rf (z) =

n j =1

zj

∂f . ∂zj

Thus the following inequality holds: (z)|2 4(1 − |z|2 )|∂f (z)|2 . 4(1 − |z|2 )2 |∂f (z)|2 |∇f

(3.2)

We will consider the cases n = 1 and n 2 separately. The case n = 1 is special in that every M-harmonic function u on U is also euclidean harmonic and hence the real part of an analytic function. 3.1. THE CASE n = 1 Suppose u is a nonconstant real-valued function on the unit disc U with u ∈ Dγp . Since u is nonconstant we have |∂u(a)| = 0 for some a ∈ U . Since u ◦ ϕa ∈ Dγp for all a ∈ U , we can without loss of generality assume that |∂u(0)| = 0. Suppose u = Re f , where f is analytic on U . Then 2 2 2 2 u(z)| = 4(1 − |z| ) ∂u |∇ ∂z 2 2 ∂u 2 2 ∂u = (1 − |z| ) + ∂x ∂y = (1 − |z|2 )2 |f (z)|2 . Thus u ∈ Dγp (U ) if and only if f ∈ Dγp (U ), and p (1 − |z|2 )γ |∇u(z)| dτ (z) Dγp (u) = U 1 (1 − |z|2 )γ +p−2 |f (z)|p dx dy = π U 2π 1 1 (1 − r 2 )γ +p−2 |f (reiθ )|p dθr dr. = π 0 0 Since f is analytic, |f (z)|p is subharmonic on U for all p, 0 1 − p. This is best possible. For if f (z) = z and u = Re f , then u is nonconstant and Dγp (u) < ∞ for all γ > 1 − p. This then proves the following. THEOREM 3.1. For 0 1 − p. 3.2. THE CASE n 2 For M-harmonic functions on Bn , n 2, we have the following theorem. THEOREM 3.2. Let f be M-harmonic on B and 0 < p < ∞. If f ∈ Dγp for some γ , 0 < γ min{n − p, p(n − 1)}, then f is constant on B. Proof. Let f be a nonconstant real-valued M-harmonic function in Dγp . Then (z0 )| = 0 for some z0 ∈ B. But then f ◦ ϕz0 is a nonconstant M-harmonic |∇f function in Dγp with |∂f (0)| = 0. Hence without loss of generality we may assume ∂f and that ∂1 f (0) = 0. Here, as well as below, we use the notation ∂j f = ∂z j ∂f ∂ j f = ∂z . j Since f is M-harmonic on B, f (z) = f (ϕz (rζ )) dσ (ζ )

(3.3)

S

for all z ∈ B and 0 < r < 1. It is easily shown using the definition of ϕz as given in [9, p. 25] that the Möbius transformation ϕz satisfies w, z (3.4) z − sz w, (1 − w, z)ϕz (w) = 1 − 1 + sz where sz = 1 − |z|2 . From this it follows that the mapping z → ϕz (w) is C ∞ on B. Thus from (3.3) we have ∂ (f ◦ ϕz )(rζ )z=0 dσ (ζ ). ∂1 f (0) = S ∂z1 A tedious, but routine computation using the chain rule and identity (3.4) gives ∂ (f ◦ ϕz )(rζ )z=0 = X1 f (−rζ ), ∂z1 where X1 f (w) = ∂1 f (w) − w 1

n k=1

w k ∂ k f (w).


Therefore,

217

|∂1 f (0)|

|X1 f (−rζ )| dσ (ζ ) =

|X1 f (rζ )| dσ (ζ ).

S

S

(a) Suppose 1 p < ∞. Then by Hölder’s inequality, p |∂1 f (0)| |X1 f (rζ )|p dσ (ζ ).

(3.5)

S

Since f is real-valued, |∂ k f | = |∂k f |. Therefore, |X1 f (w)| |∂1 f (w)| +

n

|wk ∂k f (w)|

k=1

(1 + |w|)|∂f (w)| 2|∂f (w)|. Thus by (3.2) we obtain f (w)|. (1 − |w|2 )|X1 f (w)| |∇ Therefore f (z)|p dτ (z) (1 − |z|2 )γ |∇ B 1 2n−1 2 γ +p−n−1 r (1 − r ) |X1 (rζ )|p dσ (ζ ) dr, 2n 0

S

which by (3.5)

1

2n|∂1 f (0)|

p

r 2n−1 (1 − r 2 )γ +p−n−1 dr.

0

The above integral however is finite if and only if γ > n − p. Hence if f ∈ Dγp for some γ n − p, then f must be constant. (b) Suppose 0 < p < 1. As in Section 2, let E(z) = {w : |ϕz (w)| < 13 }. Then by [6, Theorem 2.1] (w)|p dτ (w). (z)|p C |∇f |∇f E(z)

Since (1 − |w|2 ) ≈ (1 − |z|2 ) for all w ∈ E(z), we have f (z)| CDγp (f )(1 − |z|2 )−γ /p |∇ (z)| = 0, for all z ∈ B. Thus for all z with |∇f f (z)|p−1 C(1 − |z|2 )−γ +γ /p . |∇

218

MANFRED STOLL

Hence

Dγp (f )

=

(w)|p dτ (w) (1 − |w|2 )γ |∇f

B

f (w)||∇ f (w)|p−1 dτ (w) (1 − |w|2 )γ |∇ (w)| dτ (w) = CDγ1 /p (f ). C (1 − |w|2 )γ /p |∇f =

B

B

Therefore if Dγp (f ) < ∞ for some γ p(n − 1), then Dγ1 /p (f ) < ∞ for some γ n − 1. But then by part (a), f must be constant on B. ✷ p Whether the conclusion of Theorem 3.2 is best possible for M-harmonic functions when n 2 is not known to the author. However, as we prove in the following theorem, the conclusion is certainly not best possible for holomorphic functions on Bn , n 2. THEOREM 3.3. For n 2, 0 n − 12 p. Proof. Suppose f is a nonconstant holomorphic function in Dγp (Bn ) with n 2. Without loss of generality we may assume that ∂1 f (0) = 0. For λ ∈ U , set sλ = 1 − |λ|2 . If z = (z2 , . . . , zn ) ∈ Bn−1 , then (λ, sλ z ) ∈ Bn . that For a holomorphic function f on Bn , it follows from the definition of ∇f (λ, sλ z )|2 |∇f

2 n 2(1 − |λ|2 )(1 − |z |2 )−sλ ∂1 f (λ, sλ z ) + λ zj ∂j f (λ, sλ z ) . (3.6) j =2

Now by integration in polar coordinates, 1 f (rζ )|p dσ (ζ ) dr. r 2n−1 (1 − r 2 )γ −n−1 |∇ Dγp (f ) = 2n 0

Sn

By [8, Lemma 3.2], for n 2, f (rζ )|p dσ (ζ ) |∇ Sn n−1 2 n−2 (rλ, sλ rζ )|p dσn−1 (ζ ) dx dy, (1 − |λ| ) |∇f = π U Sn−1 which by (3.6)

(1 − |λ| )

C(1 − r )

2 p/2

2 n−2+p/2

U

Sn−1

|hλ (rζ )|p dσ (ζ ) dx dy,


219

where hλ (z ) = −sλ ∂1 f (rλ, sλ z ) + λ

n

zj ∂j f (rλ, sλ z ).

j =2

But for fixed λ ∈ U , hλ as a function of z is holomorphic on Bn−1 . Therefore, for all p, 0 < p < ∞, |hλ (rζ )|p dσn−1 (ζ ) |hλ (0 )|p = (1 − |λ|2 )p/2 |∂1 f (rλ, 0 )|p . Sn−1

Thus

f (rζ )|p dσn (ζ ) C(1 − r 2 )p/2 |∇

Sn

(1 − |λ|2 )n+p−2 |∂1 f (rλ, 0 )|p dx dy

U

C(1 − r )

2 p/2

|∂1 f (0)|p .

The last inequality follows from the fact that for fixed r, λ → ∂1 f (rλ, 0 ) is holomorphic in U . Hence 1 p p r 2n−1 (1 − r 2 )γ +p/2−n−1 dr, Dγ (f ) C|∂1 f (0)| 0

and the above integral is finite if and only if γ > n − 12 p. Hence if f ∈ Dγp for some γ n − 12 p, f must be constant. On the other hand, if f (z) = z1 , then Dγp (f ) < ∞ for all γ > n − 12 p. Hence the result is best possible. ✷ An alternate method of proving the previous theorem is to use the techniques of p Arazy, Fisher, Janson and Peetre in proving that D0 contains nonconstant holomorphic functions when n 2 if and only if p > 2n. For further details the reader is referred to [1, Lemma 4.1]. References 1. 2. 3. 4. 5.

6.

Arazy, J., Fisher, S., Janson, S. and Peetre, J.: ‘Membership of Hankel operators on the ball in unitary ideals’, J. London Math. Soc. (2) 43 (1991), 485–508. Hahn, K. T. and Youssfi, E. H.: ‘Möbius invariant Besov p-spaces and Hankel operators in the Bergman space on the ball in Cn ’, Complex Variables 17 (1991), 89–104. Hahn, K. T. and Youssfi, E. H.: ‘M-harmonic Besov p-spaces and Hankel operators in the Bergman space on the ball in Cn ’, Manuscripta Math. 71 (1991), 67–81. Hahn, K. T. and Youssfi, E. H.: ‘Tangential boundary behavior of M-harmonic Besov functions in the unit ball’, J. Math. Anal. Appl. 175 (1993), 206–221. Karpelevic, F. I.: The Geometry of Geodesics and the Eigenfunctions of the Beltrami–Laplace Operator on Symmetric Spaces, Trans. Moscow Math. Soc. for the Year 1965, American Mathematical Society, Providence, RI, 1967, pp. 51–199. Pavlovic, M.: ‘Inequalities for the gradient of eigenfunctions of the invariant Laplacian in the unit ball’, Indag. Math., N.S. 2 (1991), 89–98.

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Peloso, M. M.: ‘Möbius invariant spaces on the unit ball’, Michigan Math. J. 39 (1992), 509– 536. Ramey, W. and Ullrich, D.: ‘The pointwise Fatou theorem and its converse for positive pluriharmonic functions’, Duke Math. J. 49 (1982), 655–675. Rudin, W.: Function Theory in the Unit Ball of Cn , Springer-Verlag, New York, 1980. Stoll, M.: Invariant Potential Theory in the Unit Ball of Cn , London Math. Soc. Lect. Notes Series 199, 1994. Stoll, M.: ‘Holomorphic and M-harmonic functions with finite Dirichlet integral on the unit ball of Cn ’, Illinois J. Math. (to appear). Yamashita, S.: ‘Dirichlet-finite functions and harmonic majorants’, Illinois J. Math. 25 (1981), 626–631.