VOLTERRA TYPE OPERATORS ON QK SPACES. Songxiao Li and Hasi Wulan. Abstract. The boundness of Volterra type operators on QK space is inves-.
TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 1, pp. 195-211, February 2010 This paper is available online at http://www.tjm.nsysu.edu.tw/
VOLTERRA TYPE OPERATORS ON QK SPACES Songxiao Li and Hasi Wulan
Abstract. The boundness of Volterra type operators on QK space is investigated in this paper. A new generalized Carleson measure and Logarithmic QK spaces has been introduced and studied. In addition, we give a new characterization of QK space and QK,0 space.
1. INTRODUCTION Let D = {z : |z| < 1} be the unit disk of complex plane C and ∂D be the boundary of D. Denote by H(D) the class of functions analytic in D. For a ∈ D, g(z, a) = log |ϕa1(z)| is the Green function in D, where ϕ a (z) = (a − z)/(1 − az) is the Mo¨ bius map of D. An f ∈ H(D) is said to belong to the Bloch space B if B(f ) = supz∈D (1 − |z|2 )|f (z)| < ∞. The expression B(f ) defines a seminorm while the natural norm is given by f B = |f (0)| + B(f ). The norm makes B into a conformally invariant Banach space. For any nonnegative, nondecreasing and Lebesgue measurable function K : (0, ∞) → [0, ∞), we say that f belongs to the space QK if 2 (1) f QK = sup |f (z)|2 K(g(z, a))dA(z) < ∞, a∈D D
where dA is an area measure on D normalized so that A(D) = 1. It is easy to see that QK is Mo¨ bius invariant, that is, f ◦ ϕaQK = f QK , whenever f ∈ QK and a ∈ D. Received May 29, 2007, accepted April 11, 2008. Communicated by Der-Chen Chang. 2000 Mathematics Subject Classification: Primary 47B38, Secondary 30H05. Key words and phrases: Volterra type operator, QK space, Carleson measure.
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The space QK,0 consists of analytic functions f on D for which (2) |f (z)|2 K(g(z, a))dA(z) = 0. lim |a|→1 D
It is easy to see that QK,0 is a closed subspace in Q K . We know that the Green function g(a, z) in (1) and (2) can be replaced by the expression 1 − |ϕa(z)|2 (see [8]). For 0 < p < ∞, K(t) = tp gives the space Qp . K(t) = 1 gives the Dirichlet space D. For more results on Qp spaces and QK spaces, see [5, 6, 8, 9, 19-22]. If the function K is only defined on (0, 1], then we extend it to (0, ∞) by setting K(t) = K(1) for t > 1. We define an auxiliary function (see [9] or [21]) as follow: (3)
K(st) , 0 < s < ∞. 0 1/2, taking fa = log 1−¯ az , then fa ∈ QK . Hence |g(z)||fa (z)|(1 − |z|2 ) = |(Ig fa ) (z)|(1 − |z|2 ) ≤ Ig fa B Ig fa QK
(14)
Ig faQK . Letting z = a, we get (15)
|¯ a||g(a)| Ig fa QK Ig log
1 Q . 1−z K
Taking supremum in the last inequality over the set 1/2 ≤ |a| < 1 and noticing that by the maximum modulus principle there is a positive constant C independent of g ∈ H(D) such that (16)
sup |g(a)| ≤ C a∈D
sup 1/2≤|a| 3/4, let I be the subarc centered at a/|a| of length (1−|a|) 2π . Consider Sn = {z ∈ D : |z −
a | ≤ 2n (1 − |a|)}, n = 1, 2, · · · . |a|
We have that 1 1 − |a|2 2n , z ∈ Sn \ Sn−1 , n = 2, · · · . |1 − a ¯z|2 2 |I| Thus
p 2 K(1 − |ϕa(z)|2 )dµ(z) log |1 − az| D p (1 − |z|2 )(1 − |a|2 ) 2 K log dµ(z) = |1 − az| |1 − a ¯z|2 D p (1 − |z|2 )(1 − |a|2 ) 2 K log dµ(z) |1 − az| |1 − ¯az|2 S1 p ∞ (1 − |z|2 )(1 − |a|2 ) 2 K log dµ(z) + |1 − az| |1 − a ¯z|2 n=2 Sn \Sn−1 p ∞ 1 − |z| 2 dµ(z) K log C+ |1 − az| 22n |I| Sn \Sn−1 n=2
∞ n=2
Putting
1−|z| 2n |I|
2 log n 2 |I|
p
1 − |z| dµ(z). sup K 1−|z| 2n |I| z∈Sn K( n ) Sn 2 |I| K( 21−|z| 2n |I| )
= t, we have sup
z∈Sn
K( 21−|z| 2n |I| ) K( 1−|z| 2n|I| )
K(2−n t) = ϕK (2−n ). K(t) 0≤t≤1
≤ sup
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Songxiao Li and Hasi Wulan
Since µ is a p−logarithmic K−Carleson measure, p 1 − |z| 2 dµ(z) 1, K log n 2 |I| 2n |I| Sn for all n = 1, 2, · · · . Thus ∞
n=2 ∞
2 log n 2 |I|
ϕK (2−n )
n=2
Therefore
p
1 − |z| dµ(z) sup K 1−|z| 2n |I| z∈Sn K( n ) Sn 2 |I| K( 21−|z| 2n |I| )
0
1
ϕK (s) ds. s
2 )pK(1 − |ϕa(z)|2 )dµ(z) sup (log |1 − az| a∈D D
0
1
ϕK (s) ds < ∞. s
The proof is completed. Carefully check the proof of the above theorem, we have the following result. We omit the details. Theorem 3.2. Let 0 ≤ p < ∞ and µ be a positive Borel measure on D. Let K satisfy (4). Then µ is a vanishing p−logarithmic K−Carleson measure if and only if p 2 (21) K(1 − |ϕa(z)|2 )dµ(z) = 0. log lim |1 − az| |a|→1 D 4. THE LOGARITHMIC QK SPACES log
From the above section, it is natural to consider the following spaces QK and Qlog K,0 defined as follows. For any nonnegative, nondecreasing and Lebesgue measurable function K : (0, ∞) → [0, ∞), we say that f belongs to the logarithmic QK space, denoted by Qlog K , if 2 2 2 |f (z)|2K(g(z, a))dA(z) < ∞, log f Qlog = sup |1 − az| K a∈D D and f belongs to the space Qlog K,0 if 2 2 |f (z)|2 K(g(z, a))dA(z) = 0. log lim |1 − az| |a|→1 D
Volterra Type Operators on QK Spaces
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log To study the spaces Qlog K and QK,0 , we consider the logarithmic Bloch space log
Blog and the little logarithmic Bloch space B 0 . We say f ∈ Blog if f Blog = sup |f (z)|(1 − |z|2 ) log z∈D
log
f belongs to the little logarithmic Bloch space B0 lim |f (z)|(1 − |z|2 ) log
|z|→1
2 < ∞. 1 − |z|2 if
2 = 0. 1 − |z|2
In [4], Attete proved that if f ∈ L 1a then the Hankel operator Hf¯ is bounded on L1a if and only if f ∈ Blog . log The first result concering the relationship between Q log K and B , is follows. log log log Theorem 4.1. Qlog K ⊂ B ; QK,0 ⊂ B0 .
Proof. For 0 < r < 1, let D(a, r) = {a ∈ D : |ϕa(z)| < r} be the pseudohyperbolic disk with center a ∈ D and radius r. By [27] we see that 1 1 1 1 , z ∈ D(a, r). 2 2 2 2 2 |1 − az| (1 − |z| ) (1 − |a| ) |D(a, r)| Choose an r0 ∈ (0, 1) such that g(z, a) ≥ log r10 for z ∈ D(a, r). By the subharmonicity, we obtain 2 2 |f (z)|2 K(g(z, a))dA(z) log |1 − az| D 2 2 1 |f (z)|2 dA(z) log K(log ) r0 D(a,r0) |1 − az| 2 2 1 |f (z)|2 dA(z) K(log ) log r0 1 − |a|2 D(a,r0 ) 2 2 1 (1 − |a|2 )2 |f (a)|2 , K(log ) log r0 1 − |a|2 log log log which means that Q log K ⊂ B . The proof of the inclusion QK,0 ⊂ B0 is similar to the former.
Theorem 4.2. If (22)
0
then (i)
Qlog K
Proof.
1
K(log(1/r))(1 − r 2 )−2 rdr < ∞,
log = Blog ; (ii) Qlog K,0 = B0 .
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Songxiao Li and Hasi Wulan
log log (i) From Theorem 4.1, we know that Qlog K ⊂ B . Now we assume that f ∈ B and observe that 2 2 |f (z)|2 K(g(z, a))dA(z) log |1 − az| D 2 2 |f (z)|2 K(g(z, a))dA(z) log 1 − |z| D 2 f Blog (1 − |z|2 )−2 K(g(z, a))dA(z) D 1 K(log(1/r))(1 − r 2 )−2 rdr < ∞. f 2Blog 0
Hence f ∈ Qlog K . (ii) From Theorem 4.1, it suffices to prove that B0log ⊂ Qlog K,0 . Suppose that
f ∈ B0log . From the assumption, for given ε > 0 there exists an r, 0 < r < 1, such that 1 K(log(1/r))(1 − r 2 )−2 rdr < ε. r
Thus,
(23)
2 2 |f (z)|2 K(g(z, a))dA(z) |1 − az| D\D(a,r) 2 2 |f (z)|2 K(g(z, a))dA(z) log 1 − |z| D\D(a,r) (1 − |z|2 )−2 K(g(z, a))dA(z) f 2Blog f 2Blog
log
D\D(a,r) 1 r
K(log(1/r))(1 − r 2 )−2 rdr
f 2Blog ε. Since f ∈ B0log , we see that for given ε > 0, there existing δ > 0, such that for δ < |z| < 1 2 (1 − |a|2 )|f (a)| < ε. log 1 − |a| For z ∈ D(a, r), we can choose ρ, 0 < ρ < 1, such that ρ < |a| < 1 implies δ < |z| < 1. Then for ρ < |a| < 1
Volterra Type Operators on QK Spaces
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2 2 |f (z)|2 K(g(z, a))dA(z) |1 − az| D(a,r) 2 2 |f (z)|2 K(g(z, a))dA(z) log 2 1 − |z| D(a,r) 2 (1 − |z|2 )−2 K(g(z, a))dA(z) ε D(a,r) r K(log(1/r))(1 − r 2 )−2 rdr. ε2
(24)
log
0
Combining (23) and (24), we get lim
|a|→1 D
log
2 2 |f (z)|2K(g(z, a))dA(z) = 0, |1 − az|
which shows that f ∈ Q log K,0 . We complete the proof. Theorem 4.3. Let K satisfy (4) and f ∈ H(D). Then the following statements are equivalent. log (a) f ∈ QK . (b) |f (z)|2 dA(z) is a 2-logarithmic K-Carleson measure. log
Proof. (a) ⇒ (b). Suppose that f ∈ Q K , by 1 − |ϕa (z)|2 ≤ g(z, a), we obtain (25)
sup
a∈D D
log
2 2 |f (z)|2 K(1 − |ϕa(z)|2 )dA(z) < ∞. |1 − az|
From Theorem 3.1, we see that (b) holds. Assume that (b) holds, i.e (25) holds. From the proof of Theorem 4.1 we know that (25) implies f ∈ B log . Therefore
(26)
2 2 |f (z)|2 K(g(z, a))dA(z) |1 − az| |g(z,a)|>1 2 2 |f (z)|2 K(g(z, a))dA(z) log 1 − |z| |g(z,a)|>1 2 (1 − |z|2 )−2 K(g(z, a))dA(z) f Blog |g(z,a)|>1 1 2 )dA(w). (1 − |w|2 )−2 K(log f Blog |w| |w|
