Bloch-type space of temperature functions on a finite cylinder

c 2007, Scientific Horizon 

JOURNAL OF FUNCTION SPACES AND APPLICATIONS Volume 5, Number 3 (2007), 287–297

http://www.jfsa.net

Bloch-type space of temperature functions on a finite cylinder Marcos L´ opez-Garc´ıa (Communicated by Hans Triebel ) 2000 Mathematics Subject Classification. Primary: 46E22, 47B38; Secondary: 35K05. Keywords and phrases. Bloch space, duality, temperature functions.

Abstract. We define the Bloch-type space BT as the linear space of temperature functions on the cylinder ST = S1 × (0, T ) such that (x, t)| < ∞, ΩT = [0, 2] × (0, T ); we prove that (b1 (ST ))∗ = BT , sup t| ∂u ∂t (x,t)∈ΩT

where b1 (ST ) is the Bergman space of temperature functions on ST belonging to L1 (ΩT , dxdt).

1.

Introduction

For D the open unit disk in the complex plane C, the classic Bergman space Ap is the subspace of analytic functions f : D → C such that f ∈ Lp (D), 0 < p < ∞. If 1 ≤ p < ∞ then Ap is a Banach space, and for p > 1 it is well known that (Ap )∗ = Aq under the integral pairing (see [4, Theorem 1.16])  f, g = f (z) g (z)dA (z) , f ∈ Ap , g ∈ Aq , D

288

Bloch-type space of temperature functions

where dA(z) = π −1 dxdy, and q is the conjugate exponent of p : p−1 +q −1 = 1. An analytic function f in D is in the Bloch space B if sup(1 − z∈D

2

|z| ) |f  (z)| < ∞. When normed with 2

f B = |f (0)| + sup(1 − |z| ) |f  (z)| , z∈D

the Bloch space B is a Banach space. In addition, we have (A1 )∗ = B under the integral pairing (see [4, Theorem 1.21])  f ∈ A1 , g ∈ B. f, g = lim− f (rz) g (z)dA (z) , r→1

For Ω ⊂

R2+

D

an open set, let   ∂2u ∂u H(Ω) = u ∈ C 2 (Ω) : on Ω . = ∂x2 ∂t

We shall call the elements of H (Ω) temperature functions on Ω. For T > 0 finite, we define the linear space of temperature functions on the cylinder ST = S1 × (0, T ) as H (ST ) = {u ∈ H (R × (0, T )) : u(x, t) = u(x + 2, t)}. Roughly speaking, a function in H (ST ) determines a temperature distribution on a circular ring of fine wire with radius 1/π , up to a certain time T. In [7] we introduced the weighted Bergman-type spaces bpα (ST ) consisting of temperature functions on the cylinder ST that belong to Lpα (ΩT ) = Lp (ΩT , tα dxdt) , where ΩT = [0, 2] × (0, T ) , 1 ≤ p < ∞, and α > −1. Notice that if u ∈ bpα (ST ) then u|ΩT ∈ Lpα (ΩT ) , hence bpα (ST ) ⊂ Lpα (ΩT ) . As in the analytic case, we proved that (bpα (ST ),  · Lp (ΩT ) ) is a Banach ∗ space, and (bpα (ST )) = bqβ− α q (ST ) under the integral pairing (see [7, ( p) Theorem 1.2])  u (z) v (z)tβ dxdt, z = (x, t) , u, vβ = ΩT

  provided that α, β > −1, p > max 1, 1+α 1+β . In particular, for α > −1 we

have that b2α (ST ) is a Hilbert space with respect the inner product inherited from L2α (ΩT ) ,  (u, v)b2α (ST ) =

u (z) v (z)tα dxdt,

ΩT

and the corresponding orthogonal projection Pα : L2α (ΩT ) → b2α (ST ) is called the weighted Bergman projection on b2α (ST ) .

M. L´ opez-Garc´ıa

289

In this paper we define the Bloch-type space  linear space of  BT as the  temperature functions v on ST such that sup t  ∂v ∂t (x, t) : (x, t) ∈ ΩT < ∞. When BT is endowed with the norm 

      ∂v   T     + sup t  vBT = max v x, (x, t) : (x, t) ∈ ΩT , 2  ∂t x∈[0,2]

we obtain that BT , ·BT is a Banach space. Furthermore, we show that

∗ 1 b (ST ) = BT under the integral pairing  u, v0 = u (z) (L10 v) (z)dxdt, u ∈ b1 (ST ) , v ∈ BT , ΩT

where L10 : BT → L∞ (ΩT ) is a bounded operator such that P0 L10 v = v for every v ∈ BT . In fact, we have the following results, Theorem 1. For α > −1, there is a bounded linear differential operator L1α : BT → L∞ (ΩT ) such that Pα L1α v = v for all v ∈ BT .

∗ Theorem 2. If α > −1 then b1α (ST ) = BT under the integral pairing  u (z) (L1α v) (z)tα dxdt, u ∈ b1α (ST ) , v ∈ BT . u, vα = ΩT

The proof of this theorem is based on the methods and techniques in [1], [3], [5], [6], which involve the construction of the so-called Bell operator L1 satisfying P L1 v = v for every v ∈ B, where P is the Bergman projection and B the Bloch space in the corresponding setting (harmonic functions on the unit ball in Rn , holomorphic functions on strictly pseudoconvex domains with C 4 boundary in Cn ).

2.

Notation and preliminary results

Throughout this paper we will write z = (x, t), w = (y, τ ), dz = dxdt, dw = dydτ, S1 = {eπiθ : θ ∈ [0, 2]} , Z∗ = {n ∈ Z : n = 0}, and α will be a real number greater than −1. Also K(x, t) will denote the GaussWeierstrass kernel. For t > 0 , let  K(x + 2n, t) θ (x, t) = n∈Z

=

1  −π2 n2 t+πnix e , 2 n∈Z

ϕ (x, t)

= −2

∂θ (x, t) , ∂x

[9, Chapter V, Theorem 4.1]

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Bloch-type space of temperature functions

and for t ≤ 0, let K = θ = ϕ = 0 . It follows that  2  ∞ θ (x, t) dx = K (x, t) dx = 1, for all t > 0. 0

−∞

Let Q = (0, 1) × (0, 1), Γ = Γ1 ∪ Γ2 ∪ Γ3 (the parabolic boundary of Q ), where Γ1 = {0} × [0, 1) , Γ2 = {1} × [0, 1) , Γ3 = (0, 1) × {0} , and let λ be the one-dimensional Lebesgue measure on Γ.  (x, t; ξ, τ ) on Q × Γ defined as follows Consider the heat kernel K ⎧ ξ = 0, 0 ≤ τ < 1, ⎨ ϕ (x, t − τ ) ,  (x, t; ξ, τ ) = ϕ (1 − x, t − τ ) , ξ = 1, 0 ≤ τ < 1, K ⎩ θ (x − ξ, t) − θ (x + ξ, t) , τ = 0, 0 < ξ < 1. It is well known that if u ∈ H (Q) ∩ C Q then (see [2, Theorem 6.3.1])   (x, t; ξ, τ ) u (ξ, τ ) dλ (ξ, τ ) , for all (x, t) ∈ Q. K (1) u(x, t) = Γ

Conversely, if v ∈ C (Γ) then   (x, t; ξ, τ ) v (ξ, τ ) dλ (ξ, τ ) (2) u(x, t) = K Γ

is a temperature function on Q. Remark 3. Suppose f is a 2 -periodic continuous function on R. Then u is a temperature function on R2+ , 2 -periodic in the variable x, and u (x, t) → f (x) as t → 0 uniformly on [0, 2] if and only if (see [9, Chapter V, Theorem 8])  2 θ(x − y, t)f (y)dy. u(x, t) = 0

Next we give a basic but useful result.

  Proposition 4. If u ∈ H (ST ) then |u(x, t)| ≤ max u(x, T2 ) for all T 2

x∈[0,2]

< t < T.

Proof. Let u ∈ H (ST ) , by the remark above and the uniqueness of the solution of the heat equation on a finite cylinder (see [9, p. 100]), we can write 

  2

T T T < t < T, u(x, t) = θ x − y, t − u y, dy, for all 2 2 2 0 therefore |u(x, t)| ≤ =



    T  ∞ T  max u x, K x − y, t − dy x∈[0,2]  2  −∞ 2 

  T  T max u x, , for all t > . 2  2 x∈[0,2] 



M. L´ opez-Garc´ıa

291

We define the linear space of continuous functions on the cylinder ST as C ST = {u ∈ C (R × [0, T ]) : u(x, t) = u(x + 2, t)} . For 1 ≤ p < ∞, α > −1 the subspace H (ST ) ∩ C(ST ) is dense in bpα (ST ) ([7, Corollary 3.8]). We define the incomplete gamma function γ as  σ tβ−1 e−t dt if β, σ > 0. γ (β, σ) = 0 ∗

For any n ∈ Z , let



un (x, t) =

2α π 2(1+α) n2(1+α) −π2 n2 t+πnix e , γ (1 + α, 2π 2 n2 T )

 2 and u0 (x, t) = 2T1+α 1+α , then {un }n∈Z is an orthonormal basis of bα (ST ) ([7, Lemma 3.9]). The weighted Bergman projection Pα : L2α (ΩT ) → b2α (ST ) can be written as the integral operator  Nα (z, w) u (w) τ α dw, Pα u (z) = ΩT

where Nα (z, w) is a symmetric, real-valued function called the Bergman reproducing kernel of b2α (ST ), it is given by   Nα (z, w) = un (z)un (w) = un (x, t)un (y, τ ), n∈Z

n∈Z

b2α (ST )

and satisfies Nα (·, w) ∈ for every w ∈ ST (see [7, Section 3]).



2 2 Since γ 1 + α, 2π n T ≥ γ 1 + α, 2π 2 T , and e−x ≤ Cλ x−λ for x > 0 , λ ≥ 0 we have that the series defining Nα converges absolutely and uniformly on Ω × R2+ provided Ω ⊂ R2+ is compact, furthermore,  the function R2+ , therefore 2Nα and

its derivatives are bounded on Ω × ∞ 2 2 Nα ∈ C R+ × R+ and Nα (·, w) ∈ H(ST ) for all w ∈ R+ . On the other hand, the function  2 2 1 n2(1+α) e−π n (t+τ )+πni(x−y) , Θα (z, w) = θα (x − y, t + τ ) = π 2(1+α) 2 n∈Z

satisfies ([8, Lema 2.14])  |Θα (z, w)| τ β dw ≤ Cα,β tβ−α if α > β > −1. ΩT

Finally, during the proof of Theorem (1.3) in [8] we have constructed an isomorphism Mα : b1β (ST ) → b1β (ST ) , applying the theory of Fourier

292

Bloch-type space of temperature functions

multipliers on L1 S 1 , such that 

n ∂ Nα n Mα (z, w) = in Θα+ n2 (z, w) , if α + > β > −1 ∂xn 2 

n ∂ Nα n Mα (z, w) = (−1) Θα+n (z, w) , if α + n > β > −1 ∂tn where Mα is acting on the variable w. In particular, we have        ∂Nα  α   α −1   τ dw ≤ Cα 1 (z, w) τ dw ≤ Cα t 2 , (z, w) (3) Θ α+ 2  ∂x  Ω Ω   T  T  α  ∂Nα   |Θα+1 (z, w)| τ α dw ≤ Cα t−1 .  ∂t (z, w) τ dw ≤ Cα ΩT ΩT

3.

Basic properties of the Bloch-type space BT

We recall the definition of the normed linear space BT . Definition 5. For T > 0, let         ∂v BT = v ∈ H (ST ) : sup t  (x, t) : (x, t) ∈ ΩT < ∞ , ∂t and  

     ∂v   T   : (x, t) ∈ ΩT .  (x, t) + sup t vBT = max v x,  ∂t  x∈[0,2] 2  The following result is about the growth of functions in BT . Lemma 6. If v ∈ BT then

 T (4) |v (x, t)| ≤ vBT 1 + ln , for all (x, t) ∈ ΩT . t Proof. For t < T2 we have 

 1 

       T d v x, T − v (x, t) =  v x, σ + (1 − σ) t dσ     2 2 0 dσ  1

  

  ∂v T T =  −t x, σ + (1 − σ) t dσ  2 ∂t 2 0       ∂v (x, t) : (x, t) ∈ ΩT ≤ sup t  ∂t T

 1 2 −t × dσ T 0 σ 2 + (1 − σ) t T ≤ vBT ln . 2t When t > T2 the inequality holds by Proposition 4. 

M. L´ opez-Garc´ıa

293

Remark 7. From the 2 -periodicity in the variable x of the functions in BT , together with (4), we note that norm convergence in BT implies the uniform convergence on compact subsets in R × (0, T ). Next we show that the normed linear space (BT ,  · BT ) is complete. Proposition 8. For every T > 0, BT is a Banach space. Proof. Let (uj ) a Cauchy sequence in BT , by the remark above ∂ there is a function u such that u = lim uj . Likewise, the sequence ∂t uj ∂uj converges uniformly on compact subsets in R × (0, T ) , therefore ∂t → ∂u ∂t on R × (0, T ) . 2 Pick (x0 , t0 ) ∈ ΩT . Let R = (a, b) × (c, d) , with d − c = (b − a) , such that (x0 , t0 ) ∈ R and R ⊂ ΩT . Consider the mapping Ψ : Q → R given by Ψ (ξ, τ ) = ((b − a) ξ + a, (d − c) τ + c) . Since uj ∈ H (R) ∩ C R , (1) implies that 

 Ψ−1 (x, t) ; Ψ−1 (ξ, τ ) uj (ξ, τ ) dλR (ξ, τ ) , for (x, t) ∈ R, K uj (x, t) = ΓR

where ΓR = Ψ (Γ) and λR is the one-dimensional Lebesgue measure normalized on each segment of ΓR . By the dominated convergence theorem we have 

 Ψ−1 (x, t) ; Ψ−1 (ξ, τ ) u (ξ, τ ) dλR (ξ, τ ) . K u(x, t) = ΓR

Since the function u is continuous on ΓR then u is a temperature function on R. Since (x0 , t0 ) ∈ ΩT was arbitrary we conclude that u ∈ H (ST ) , and  u − uj BT → 0 as j → ∞. Proposition 9. Let 1 ≤ p < ∞ and α > −1. If v ∈ BT then v ∈ bpα (ST ) . Therefore Pα v = v for all v ∈ BT . Proof. Since tα dxdt is a finite measure on ΩT , by the previous lemma it is T p sufficient to show that 0 ln Tt  tα dt < ∞. Let ε > 0 such that ε < α + 1, p then |ln u| /uα+1−ε → 0 as u → ∞, so   ∞  ∞  T p  T p α |ln u| 1 ln  t dt = T α+1 du ≤ C du < ∞.  α,T  t α+2 1+ε u u 0 1 1 Proposition 10. For α > −1 the weighted Bergman projection Pα maps   ∂ L∞ (ΩT ) continuously into BT . Additionally, t1/2 ∂x (Pα ϕ)L∞ (ΩT ) < ∞ for every ϕ ∈ L∞ (ΩT ) . Proof. Let ϕ ∈ L∞ (ΩT ) , differentiating under the integral sign yields  ∂ (Pα ϕ) ∂Nα (z) = (z, w) ϕ (w) τ α dw, z ∈ ΩT ∂t ΩT ∂t

294

Bloch-type space of temperature functions

by (3) we get

   ∂ (Pα ϕ)  −1   ≤ Cα ϕ ∞ (z) , L (ΩT ) t  ∂t 

z ∈ ΩT . 

The proof of the other conclusion is similar. Finally, we prove the main results.

Proof of Theorem 1. Let ψα ∈ C ∞ (R) a nonnegative function satisfying that ψα ≡ 1/ (α + 1) for t ≤ T2 and ψα ≡ 0 for t > T − δ, with δ > 0 small enough. Thus, for v ∈ BT we define 

  1 ∂2 ∂ + (ψα (t) tv (x, t)) Lα v (x, t) = v (x, t) − αψα (t) v (x, t) + ∂t ∂x2 ∂v = {1 − (1 + α) ψα (t) − tψα (t)} v (x, t) − 2tψα (t) (x, t) . ∂t Clearly L1α is a linear differential operator on BT . By Proposition 4, it follows that   1  L v (x, t) ≤ Cψα v α BT for all (x, t) ∈ ΩT .   ∂ ∂2 Letting g (x, t) = αψα (t) v (x, t) + ∂t + ∂x (ψα (t) tv (x, t)) , we claim 2 2 that g ∈ Lα (ΩT ) , and (g, u)L2 (ΩT ) = 0 for all u ∈ H (ST ) ∩ C ST . So, α Pα L1α v = Pα v − Pα g = Pα v = v since v ∈ b2α (ST ) . To prove the claim, we rewrite g = ((1 + α) ψα + tψα ) v + 2tψα ∂v ∂t then g ∈ L2α (ΩT ) . By the other hand, if u ∈ H (ST ) ∩ C ST then integrating by parts we get for ε > 0 small enough,   T 2

∂ ∂2 u + 2 (ψα tv) tα dz ∂t ∂x ε 0 x=2  T ∂u  ∂v −v = u ψα tα+1 dt ∂x ∂x x=0 ε  T 2  2 t=T ψα tα+1 uv t=ε dx − αψα uvtα dz + 0



= −

0

2 α+1

ε

0

ε u (x, ε)v (x, ε) dx − α+1

 T ε

0

2

αψα uvtα dz.

Lemma 6 and the dominated convergence theorem imply (g, u)L2α (ΩT ) = 0.  Corollary 11. The mapping Pα : L∞ (ΩT ) → BT is onto. From Proposition 10 we have the following   ∂v  (x, t) : (x, t) ∈ ΩT < ∞. Corollary 12. If v ∈ BT then sup t1/2  ∂x

M. L´ opez-Garc´ıa

295

Proposition 13. For α > −1, the bilinear form ·, ·α : b1α (ST ) × BT → C given by  u, vα =

ΩT

u (z) (L1α v) (z)tα dz

is continuous, and u, vα = (u, v)b2 (ST ) for u ∈ b2α (ST ) , v ∈ BT . α

Proof. For u ∈

b1α

(ST ) , v ∈ BT , we have   |u, vα | ≤ ub1 (ST ) L1α v L∞ (ΩT ) ≤ C ub1 (ST ) vBT . α

α

b2α

(ST ) , v ∈ BT

u, vα = u, L1α v b2 (S ) T α

= Pα u, L1α v b2 (ST )

α = u, Pα L1α v b2 (ST )

On the other hand, if u ∈

α



∗ Proof of Theorem 2. Consider the linear mapping Φα : BT → b1α (ST ) given by Φα v = ·, vα . By the previous proposition Φα is continuous, and injective: If Φα v = 0 then v, vα = (v, v)b2 (ST ) = 0, therefore v ≡ 0. α

∗ Let F ∈ b1α (ST ) , by the Hahn-Banach theorem the operator F can be extended continuously to L1 (ΩT , tα dz) , and the Riesz representation theorem implies the existence of ϕ ∈ L∞ (ΩT ) such that  Fu = u (z) ϕ (z)tα dz, for all u ∈ b1α (ST ) . =

(u, v)b2 (ST ) . α

ΩT

In particular, for u ∈ b2α (ST ) we have Fu = =

(u, ϕ)L2α (ΩT ) = (Pα u, ϕ)L2α (ΩT ) (u, Pα ϕ)b2α (ST ) = u, Pα ϕα ,

where the last equality follows from the previous proposition. Since b2α (ST ) is dense in b1α (ST ) , then F = Φα (Pα ϕ) . By the open mapping theorem  we have that Φα is an isomorphism. We finish with a simple example of Bloch-type temperature functions, Example 14. Suppose (an )n is a bounded sequence of complex numbers √ and suppose (λn )n is a sequence of positive integers with λn /λn−1 ≥ λ , where 1 < λ < ∞. Let ∞  2 2 an e−π λn t+πλn ix . v(x, t) = n=1

Since e

−x

−σ

≤ Cσ x

for x > 0 , σ ≥ 0 then v ∈ H(ST ), and   ∞   ∂v  2 2  (x, t) ≤ M λ2n (e−π t )λn ,  ∂t  n=1

296

Bloch-type space of temperature functions

where M > 0 is a constant. 2

Setting z = e−π t , and K = λ/(λ−1) we have that 0 < z < 1, 1 < K < ∞. Then, 2

2

2

λ21 |z|λ1 ≤ |z| + · · · + |z|λ1 ≤ K(|z| + · · · + |z|λ1 ). Since λ2n ≤ K(λ2n − λ2n−1 ) for every n ≥ 2, we have 2

2

2

λ2n |z|λn ≤ K(|z|λn−1 +1 + · · · + |z|λn ). 2

Let δ > 0 small enough and Cδ > 0 a constant such that t ≤ Cδ (1 − e−π t ) for 0 < t < δ, therefore   ∞   ∂v  MK Cδ M K  (x, t) ≤ M K . |z|n ≤ ≤  ∂t  −π 2 t t 1 − e n=1 Since

∂v ∂t

∈ C(R × [δ, ∞)) we obtain that v ∈ BT .

Acknowledgement. This article was written while visiting the Universitat de Val`encia, Spain, on posdoctoral stay; the author has been supported by Programa de Apoyos para la Superaci´ on del Personal Acad´emico de la DGAPA of the Universidad Nacional Aut´ onoma de M´exico. The author thanks the Departament de An` alisi Matem`atica of the Universitat de Val`encia for its hospitality and support.

References [1] S. Bell, A duality theorem for harmonic functions, Michigan Math. J., 29 (1) (1982), 123–128. [2] J.R. Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications, 23, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. [3] B. Coupet, Le dual des functions holomorphes int´egrables sur un domaine strictement pseudo-convexe, In : Proceedings of the American Mathematical Society, 102 (3), (1988), 493–501. [4] H. Hedenmalm, B. Korenblaum and K. Zhu, Theory of Bergman Spaces, Graduate Texts in Mathematics, Vol. 199, Springer Verlag, New York, 2000. [5] E. Ligocka, On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball in Rn , Studia Math., 87 (1) (1987), 23–32. [6] J. Motos and S. P´erez-Esteva, On Bell’s duality theorem for harmonic functions, Studia Math., 137 (1) (1999), 49–60.

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[7] M. L´ opez-Garc´ıa, Bergman spaces of temperature functions on a cylinder, Int. J. Math. Math. Sci., 2003 (19) (2003), 1193–1213. [8] M. L´ opez-Garc´ıa, An atomic decomposition for the Bergman space of temperature functions on a cylinder, Bol. Soc. Mat. Mexicana, 11 (3) (2005), 101-119. [9] D.V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67, Academic Press, New York, 1975. Instituto de Matem´ aticas Universidad Nacional Auton´ oma de M´exico Circuito Exterior Ciudad Universitaria M´exico D.F., C.P. 04510 M´exico (E-mail : fl[email protected] ) (Received : August 2006 )

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