Another proof of boundedness of the Cèsaro operator on Hp

A N N A L E S

U N I V E R S I T A T I S M A R I A E C U R I E { S K Š O D O W S K A L U B L I N { P O L O N I A

VOL. L IV, 7 SECTIO A 2000

MARIA NOWAK

Another proof of boundedness of the Ces` aro operator on

Dedicated to Professor Zdzis law Lewandowski on his 70th birthday

Abstract. We give a new short proof of the boundedness of the Ces´aro operator on H^p, 0 < p < ∞.

1. Introduction. Let H^p, 0 < p < ∞, be the standard Hardy space on the unit disc D. For f (z) =P∞

n=0anzⁿ in H^p, the Ces`aro operator is given by the formula

C(f )(z) =

∞

X

n=0

1 n + 1

n

X

k=0

ak

! zⁿ.

It follows from [H], [S1], [S2] and [M] that the Ces`aro operator is a bounded operator on H^p for 0 < p < ∞. In [S2] the author proved the case p = 1 and he remarked that the proof cannot be adapted for the other values of p. Here we present a modification of his proof which works for all positive p.

1991 Mathematics Subject Classification. 47B38, 30D55.

Key words and phrases. Ces`aro operator, Hardy spaces.

76 M. Nowak

H^∞, the space of bounded analytic functions on D, is not mapped into it- self by the Ces`aro operator. However, C is a bounded operator from H^∞into the space BMOA ([DS], see also [EX, p. 191]). Recently J. Shi and G. Ren [SR] have proved that C is a bounded operator on a mixed norm space Hp,q(ϕ), and as a special case, it is bounded on the weighted Bergman space.

For an analytic function f on D and for 0 < p, q, γ < ∞, we define

Mp(r, f ) = 1 2π

Z 2π 0

|f (re^iθ)|^pdθ

^1/p

, kf k_p= sup

0≤r<1

Mp(r, f )

and

Mp,q,γ(f ) =

Z 1 0

(1 − r)^qγ−1M_p^q(r, f )dr

^1/q

Our proof is based on the Hardy-Littlewood inequality and its dual inequal- ity due to T. M. Flett. We state these results as the following lemmas.

Lemma HL [HL, p. 411]. Let f be an analytic function on D and let

0 < p < q, α = 1 p −1

q, l ≥ p.

Then

M_q,l,α^l (f ) = Z 1

0

M_q^l(r, f )(1 − r)^lα−1dr ≤ CM_p^l(r, f ).

The next two lemmas are special cases of Theorem 2 in [F, p.750].

Lemma F1. Let 0 < p ≤ 2, and let f be an analytic function on D such that f (0) = 0. If Mp,p,1(f⁰) < ∞, then f ∈ H^p and

kf k_p≤ CM_p,p,1(f⁰).

Lemma F2. Let 0 < s < p < ∞, γ = 1 −

1 s− ¹_p

> 0 and let f be an analytic function on D such that f (0) = 0. If Ms,p,γ(f⁰) < ∞, then f ∈ H^p and

kf k_p ≤ CM_s,p,γ(f⁰).

Another proof of boundedness of the Ces`aro operator on H^p 77

2. The Proof. Put F (z) = zCf (z). A computation shows that F⁰(z) =

1

1−zf (z), z ∈ D. Assume first that 0 < p ≤ 2. If β > 1, _β¹ + _β¹0 = 1 and pβ⁰ > 1, then the H¨older inequality and the lemma in [D, p.65] give

M_p,p,1^p (F⁰) = Z 1

0

(1 − r)^p−1M_p^p(r, F⁰)dr

= Z 1

0

(1 − r)^p−1 Z 2π

0

|f (re^iθ)|^p

|1 − re^iθ|^pdθdr

≤ Z 1

0

(1 − r)^p−1M_βp^p (r, f )

Z 2π 0

dθ

|1 − re^iθ|^pβ0

^1/β

0

dr

≤ Z 1

0

(1 − r)^β01−1M_pβ^p (r, f )dr = Z 1

0

(1 − r)^−1βM_pβ^p (r, f )dr

≤ Ckf k^p_p,

where the last inequality follows from Lemma HL. Thus, by Lemma F1, kF k^p_p≤ Ckf k^p_p.

Now assume that 2 < p < ∞. If _p+1¹ < α < 1, then γ = 1−

1 αp −¹_p

> 0 and

M_pα,p,γ^p (F⁰) = Z 1

0

(1 − r)^p−α1

Z 2π 0

|f (re^iθ)|^αp

|1 − re^iθ|^αpdθ

^1/α dr.

Take β > 1, α as above and such that αβ > 1 and pαβ⁰ > 1 (e.g. α =

2

p+1, β = p + 1). Then, in much the same way as in the first case, one can get

M_pα,p,γ^p (F⁰) ≤ Z 1

0

(1 − r)^−αβ1 M_pαβ^p (r, f )dr ≤ Ckf k^p_p. Thus Lemma F2 implies the desired result.

References

[NS] Danikas, N., A.G. Siskasis, The Ces`aro operator on bounded analytic functions, Analysis 13 (1993), 295-299.

[D] Duren, P.L., Theory of H^p Spaces, Academic Press, New York 1970.

[F] Flett, T.M., The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746-765.

[EX] Ess´en, M., J. Xiao, Some results on Qp spaces,0 < p < 1, J. reine angew. Math.

485 (1997), 173-195.

[H] Hardy, G.H., Notes on some points in the integral calculus LXVI, Messenger of Math. 58 (1929), 50-52.

[HL] Hardy, G.H., J.E. Littlewood, Some properties of fractional integrals. II, Math. Z.

34 (1932), 403-439.

78 M. Nowak

[M] Miao, J., The Ces`aro operator is bounded on H^pfor 0 < p < 1, Proc. Amer. Math.

Soc. 116 (1992), 1077-1079.

[S1] Siskasis, A.G., Composition semigroup and the Ces`aro operator on H^p, J. London Math. Soc.(2) 36 (1987), 153-164.

[S2] Siskasis, A.G., The Ces`aro operator is bounded on H¹, Proc. Amer. Math. Soc.

110 (1990), 461-462.

[SR] Shi, J., G. Ren, Boundedness of the Ces`aro operator on mixed norm spaces, Proc.

Amer. Math. Soc. 126 (1998), 3553-3560.

Instytut Matematyki UMCS received December 20, 1999 pl. Marii Curie-Sk lodowskiej 1

20-031 Lublin, Poland

e-mail: nowakm@golem.umcs.lublin.pl