On the Taylor Coefficients of Functions in the Hardy Space Over the Bidisc
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Complex Analysis and Operator Theory
On the Taylor Coefficients of Functions in the Hardy Space Over the Bidisc O. Blasco1
· I. García-Bayona1
Received: 28 January 2020 / Accepted: 11 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we analyze the Taylor coefficients of functions in Hardy spaces on the bidisc. We present the two-dimensional versions of some classical Hardy and Paley inequalities and we find certain conditions on the Taylor coefficients for the converse of these inequalities to hold. Keywords Hardy spaces · Bidisc · Hardy inequalities · Double sequences · Bi-monotonically increasing sequences Mathematics Subject Classification 42A32 · 42B30 · 30H10 · 32A35 · 40B05
1 Introduction and Preliminaries Consider D and T to be the unit disc and the torus in C, and let D2 := {(z, w) : |z| < 1, |w| < 1} and T2 := {(ξ1 , ξ2 ) ∈ C2 , |ξ j | = 1, j = 1, 2} be the bidisc and the bitorus in C2 respectively. We write dm 1 and dm 2 for the normalized Lebesgue measures over T and T2 respectively. Recall that for 1 ≤ p < ∞, an analytic function f defined in D (respectively D2 ) is said to belong to H p (D) (respectively H p (D2 )) whenever f p = supr M p (r , f ) < ∞ where
Communicated by Joseph Ball. O. Blasco and I. García-Bayona were partially supported by MTM2014-53009-P (MINECO Spain) and I. García-Bayona was also supported by FPU14/01032 (MCIU Spain).
B
I. García-Bayona [email protected] O. Blasco [email protected]
1
Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain 0123456789().: V,-vol
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O. Blasco, I. García-Bayona
p
M p (r , f ) :=
T
| f (r ξ )| p dm 1 (ξ ) =
1 2π
π −π
| f (r ei x )| p d x
(respectively p M p (r ,
1 f ) := (2π )2
π
−π
π
−π
| f (r ei x , r ei y )| p d xd y).
k k l n We shall write f (z) = n≥0 an z and f (z, w) = k,l≥0 cl z w for analytic p p 2 functions f belonging to H (D) and H (D ), respectively. It is well known (see [2,11]) that Hardy spaces can be viewed as closed vector subspaces of complex L p spaces on the torus or bitorus respectively. In particular, we can consider H p (that we denote H p (T) or H p (T2 )) as functions in L p (T) or L p (T2 ) whose Fourier coefficients satisfy fˆ(n) = 0 for n < 0 and fˆ(k, l) = 0 for (k, l) such that k < 0 or l < 0, respectively. One of the basic properties of coefficients of functions in Hardy spaces can be expressed in terms of the following inequalities: Case 1 ≤ p ≤ 2: There exists C p > 0 such that ⎛ ⎝
n≥0
⎞1/ p
|p
|an ⎠ (n + 1)2− p
≤ C p f p , ∀ f (z) =
an z n ∈ H p (D).
(1)
n≥0
p p−2 < ∞ then f (z) = Case p ≥ 2: If (an ) satisfies that n≥0 |an | (n + 1) n p n≥0 an z ∈ H (D) and there exists C p > 0 such that ⎛ f p ≤ C p ⎝
n≥0
|p
⎞1/ p
|an ⎠ (n + 1)2− p
.
(2)
Let us point out that inequalities (2) and (1) for 1 < p ≤ 2 were shown by Paley (see [12, Theorem 5.1 , Chapter XII]) while the case p = 1 in inequality (1) seems that appeared for the first time in the work of Hardy
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