Carleson measures, Riemann-Stieltjes and multiplication operators on a general family of function spaces

Let $\mu$ be a nonnegative Borel measure on the unit disk of the complex plane. We characterize those measures $\mu$ such that the general family of spaces of analytic functions, $F(p,q,s)$, which contain many classical function spaces, including the Bloch space, $BMOA$ and the $Q_s$ spaces, are embedded boundedly or compactly into the tent-type spaces $T^{\infty}_{p,s}(\mu)$. The results are applied to characterize boundedness and compactness of Riemann-Stieltjes operators and multiplication operators on $F(p,q,s)$.


INTRODUCTION
The Carleson measure was introduced by Carleson [9] for studying the problem of interpolation by bounded analytic functions and for solving the famous corona problem. Later on variations of the Carleson measure have been also introduced and studied. It was found that Carleson type measures are usually closely related to certain function spaces. For example, the original Carleson measure is closely related to Hardy spaces H p . This feature makes Carleson type measures important tools for the modern function theory and operator theory.
In this paper we study Carleson measures related to the general analytic function space F (p, q, s) on the unit disk of the complex plane, introduced by the second author in [27]. Basically, we are going to characterize those nonnegative Borel measures µ such that the embedding from F (p, q, s) to a certain tent-type space is bounded or compact. The results will be applied to get characterizations for the bounded and compact Riemann-Stieltjes operators and pointwise multiplication operators on F (p, q, s).
Our work involves several spaces of analytic functions. Let us first review these spaces. Let D be the unit disk on the complex plane. Let H(D) be the space of all analytic functions on D. For α > −1 and p > 0, the weighted Bergman space A p α consists of all functions f ∈ H(D) such that Here dA α (z) = (α + 1)(1 − |z| 2 ) α dA(z), and dA is the normalized area measure on D.
The weighted Dirichlet space D p α consists of all functions f ∈ H(D) such that Key words and phrases. Carleson measures, F (p, q, s) spaces, tent-type spaces, Riemann-Stieltjes operators, multiplication operators.
It is well-known that D 2 1 = H 2 , the Hardy space, and for α > p − 1, D p α = A p α−p , with equivalence of norms.
Next, we recall the Bloch type spaces (also called α-Bloch spaces) B α . Let α > 0. The α-Bloch space B α consists of all analytic functions f on D such that The little α-Bloch space B α 0 consists of all analytic functions f on D for which lim |z|→1 |f ′ (z)|(1 − |z| 2 ) α = 0.
It is known that B α is a Banach space under the norm and B α 0 is a closed subspace of B α . As α = 1, B 1 = B, the Bloch space. Here are some related facts about B α . When 0 < α < 1, B α = Lip 1−α , the Lipschitz type space, which contains analytic functions f on D for which there is a constant C > 0 such that |f (z) − f (w)| ≤ C|z − w| 1−α for all z, w ∈ D. As a simple consequence, we know that, when 0 < α < 1, B α ⊂ H ∞ , the space of all bounded analytic functions f on D with When α > 1, an analytic function f ∈ B α if and only if sup z∈D |f (z)|(1 − |z| 2 ) α−1 < ∞, and the above supremum is comparable to | f | B α . For all these results, see [31]. Next, we introduce the spaces F (p, q, s). For a point a ∈ D, let ϕ a (z) = (a − z)/(1 − az) denote the Möbius transformation of D that interchanges 0 and a. An easy calculation shows ϕ ′ a (z) = − 1 − |a| 2 (1 −āz) 2 . For 0 < p < ∞, −2 < q < ∞, 0 ≤ s < ∞. The space F (p, q, s) is defined as the space of all functions f ∈ H(D) such that It is known that, for p ≥ 1, F (p, q, s) is a Banach space under the norm | f | F (p,q,s) = |f (0)| + f F (p,q,s) .
For 0 < p < 1, the space F (p, q, s) is a complete metric space with the metric given by d(f, g) = | f − g | p F (p,q,s) . In other words, it is an F -space, in the terminology introduced by Banach [6]. The family of spaces F (p, q, s) was introduced in [27]. It contains, as special cases, many classical function spaces, such as the analytic Besov spaces, weighted Bergman spaces, weighted Dirichlet spaces, the α-Bloch spaces, BMOA and the recently introduced Q s spaces.
Among these F (p, pα − 2, s) spaces, the case α = 1 is particularly interesting, since the spaces F (p, p − 2, s) are Möbius invariant, in the sense that for any function f ∈ F (p, p − 2, s) and any a ∈ D, one has f • ϕ a F (p,p−2,s) = f F (p,p−2,s) .
As mentioned above, when s > 1 we have that F (p, p − 2, s) = B, the Bloch space. As p = 2, F (p, p − 2, s) = Q s , the Q s spaces, introduced in [4]. As p = 2 and s = 1, F (p, p − 2, s) = BM OA, the space of analytic functions of bounded mean oscillation. See [27] for details of all of the above facts about F (p, q, s) spaces.
The tent-type spaces used in this paper are defined as follows. Let I be an arc on the unit circle ∂D. Denote by |I| the normalized arc length of I so that |∂D| = 1. Let S(I) be the Carleson box defined by Let 0 ≤ s < ∞ and 0 < p < ∞. For a nonnegative Borel measure µ on the unit disk D, we define T ∞ p,s (µ) as the space of all µ-measurable functions f on D satisfying By a standard argument we can show that for p ≥ 1, T ∞ p,s (µ) is a Banach space. Next, we introduce the Carleson measures needed in this paper. We say that a nonnegative Borel measure µ on D is a (p, s)-logarithmic Carleson measure if there is a constant C > 0 such that We will denote the class of all (p, s)-logarithmic Carleson measures on D by LCM p,s , and denote by When p = 0, the (0, s)-logarithmic Carleson measures are called s-Carleson measures, and we will denote by CM s = LCM 0,s and µ CMs = µ LCM0,s . We say µ is a vanishing (p, s)-logarithmic Carleson measure if A vanishing (0, s)-logarithmic Carleson measure is also called a vanishing s-Carleson measure.
Logarithmic Carleson measures were first introduced by the second author in [29]. In this paper, we are going to characterize the measures µ such that the identity operator I : F (p, q, s) → T ∞ p,s (µ) is bounded or compact. The results will be applied to give characterizations for the bounded and compact Riemann-Stieltjes operators and pointwise multiplication operators on the F (p, q, s) spaces. Our results generalize some recent results by Xiao in [26] and by Pau and Peláez in [18].
The paper is organized as follows. Section 2 is devoted to some preliminary results. In section 3 we characterize boundedness and compactness of I : F (p, p−2, s) → T ∞ p,s (µ). In section 4 we characterize boundedness and compactness of I : F (p, pα − 2, s) → T ∞ p,s (µ) for α = 1. In section 5 we use these results to characterize bounded and compact Riemann-Stieltjes operators on F (p, q, s), and in section 6 we characterize bounded and compact pointwise multiplication operators on F (p, q, s).

PRELIMINARY RESULTS
In this Section we state and prove some preliminary results needed for the rest of the paper. Some of them may have independent interest. We begin with the following lemma.
Let µ be a nonnegative Borel measure on D such that the point evaluation is a bounded functional on T ∞ p,s (µ). Then I : F (p, q, s) → T ∞ p,s (µ) is a compact operator if and only if f n T ∞ p,s (µ) → 0 whenever {f n } is a bounded sequence in F (p, q, s) that converges to 0 uniformly on every compact subset of D.
Using the fact that the point evaluations on F (p, q, s) and T ∞ p,s (µ) are bounded functionals (see Proposition 2.17 of [27] for the case of F (p, q, s)) , the proof of Lemma 2.1 is standard. See, for example, the proof of Proposition 3.11 in [10]. We omit the details here.
We also need the following equivalent description of (p, s)-logarithmic   We also need the following equivalent definition for T ∞ p,s (µ).
The following lemma is from [17].
Hence we can use the second inequality of Lemma 2.4 to get The proof is complete.
Using this corollary, we are able to prove the following result. Lemma 2.6. Let 0 < p < ∞, 0 < α < ∞ and 0 < s < ∞ satisfy s + pα > 1. Let Then Proof. A simple computation using the well-known identity Since s + pα − 2 > −1, pα > 0, 2s > 0, and we can use Corollary 2.5 to get The proof is complete. Lemma 2.7. Let 0 < p < ∞, and γ > −1 with γ > p − 2. For any f ∈ D p γ , and any a ∈ D, Proof. If p > 1, the result follows from Lemma 2.1 of [8]. If 0 < p ≤ 1 we use the atomic decomposition for D p γ (see [30,Theorem 32]): there is a sequence {a k } in D and a sequence of numbers {λ k } ∈ ℓ p such that Note that D p γ = A p γ−p in the notation of [30]. Now, since 0 < p ≤ 1, we obtain The last inequality holds by Corollary 2.5. The proof is complete.
. For any f ∈ F (p, pα − 2, s) and any a ∈ D, Proof. Changing the variable z = ϕ a (w), we get by Lemma 2.7, The proof is complete.
for any arc I with |I| < 1 − r. Now fix the above r. Consider any arc I on the unit circle ∂D. Let α = 1 − r, and n = [|I|/α]. Then nα ≤ |I| < (n + 1)α. Clearly, we can cover I by n + 1 arcs I 1 , I 2 , ..., I n+1 with |I k | = 1 − r = α for k = 1, 2, ..., n + 1. Let µ r = µ| D\Dr . Then In this section we describe the boundedness and compactness of the embedding from the Möbius invariant space F (p, p − 2, s) to the tent-type space T ∞ p,s (µ). The main result in this section is the following. Theorem 3.1. Let s > 0 and 0 < p < ∞ satisfy s + p > 1, and let µ be a nonnegative Borel measure on D. Then The following two conditions are equivalent: (a) For any bounded sequence {f n } in F (p, q, s) satisfying f n (z) → 0 uniformly on every compact subset of D, Remark. When p = 2, the result is obtained by Xiao in [26] using techniques from [18].
In order to prove Theorem 3.1, we need some results on Carleson measures for weighted Dirichlet spaces D p α . A nonnegative Borel measure µ on D is said to be a Carleson measure and is called a compact Carleson measure for D p α if lim n f n L p (µ) = 0 whenever {f n } is a bounded sequence in D p α converging to zero uniformly on compact subsets of D. Carleson measures for the weighted Bergman spaces A p α = D p α+p , with α > −1 and p > 0 are described as those positive Borel measures on D such that This result was proved by several authors, including Oleinik and Pavlov [16] (for p > 1), Stegenga [24] (for α = 0), and Hastings [12]. One can also find a proof in [14]. The next result will be essential in order to prove Theorem 3.1.
(a) If µ ∈ LCM p,s then µ is a Carleson measure for D p s+p−2 . Moreover Proof. The case p = 2 with 0 < s < 1 of part (a) was proved in [18] using the description of Carleson measures for D p s+p−2 given in [5]. Here we will prove the result directly from the definition of Carleson measures for D p s+p−2 . We first consider the case p > 1. Let p ′ denote the conjugate exponent of p. Since Using the reproducing formula for the Bergman space A p α (see p.80 of [32]), and then integrating along the segment [0, z] it follows that It is easy to check that This inequality together with Fubini's theorem and Lemma 2.2 yields Hence µ is a Carleson measure for D p s+p−2 . Now we consider the case 0 < p ≤ 1. For r > 0, fix an r-lattice {a n } in the Bergman metric. This means that the hyperbolic disks D(a n , r) = {z : β(z, a n ) < r} cover the unit disk D and β(a i , a j ) ≥ r/2 for all i and j with i = j. Here β(z, w) denotes the Bergman or hyperbolic metric. If {a k } is an r-lattice in D, then it also has the following property: for any R > 0 there exists a positive integer N (depending on r and R) such that every point in D belongs to at most N sets in {D(a k , R)}. There are elementary constructions of r-lattices in D. See [32,Chapter 4] for example. Note that by subharmonicity we have for all j = 1, 2, . . . Let β be a sufficiently large number so that β ≥ s + p − 2 and βp > s − p. It follows that where N is a positive integer such that each point of D belongs to at most N of the sets D(a j , 2r). This finishes the proof of (a).
To prove (b), we must show that if {f n } is a bounded sequence in D p s+p−2 converging to zero uniformly on compact subsets of D, then f n L p (µ) → 0. Suppose first that p > 1. Let α = s + p − 2. Since µ is a vanishing (p, s)-logarithmic Carleson measure on D, for any ε > 0, there is an r, 0 < r < 1, such that Since {f n } converges to zero uniformly on compact sets, the same is true for the sequence of its derivatives {f ′ n }, and hence there is a positive integer n 0 such that for all n ≥ n 0 sup For the first term, we have D |w|≤r For the second term, proceeding as in the proof of part (a), we have D |w|>r This finishes the proof of (b) for p > 1. For 0 < p ≤ 1, we observe first that the condition on µ implies that µ is a vanishing s-Carleson measure. Therefore, given ε > 0, there is an r 0 with 0 < r 0 < 1 such that where β is a sufficiently large number so that β ≥ s + p − 2 and βp > s − p.
Let {f n } be a bounded sequence in D p s+p−2 converging to zero uniformly on compact subsets of D, and for r > 0 fix an r-lattice {a k } in the Bergman metric. Since |a k | → 1, there are only k 0 points a k with |a k | ≤ r 0 . Now, the same argument used in part (a) gives Since {f n } converges to zero uniformly on compact sets, the same is true for the sequence of its derivatives {f ′ n }, and hence there is a positive integer n 0 such that for all n ≥ n 0 This gives Also, using (1), the same proof given in part (a) yields where N is a positive integer such that each point of the unit disk belongs to at most N of the sets D(a k , 2r). This finishes the proof of the lemma. Now we are ready to prove our main result in this section.
3.1. Proof of Theorem 3.1. We first prove (i). Suppose µ is a (p, s)-logarithmic Carleson measure. Let s > 0 and p > 0 satisfy s + p > 1. By Lemma 3.2, µ is a Carleson measure for D p s+p−2 . Given any subarc I of ∂D, let w = (1 − |I|)ζ and ζ be the center on I. An easy computation shows that, for any z ∈ S(I), Take any function f ∈ F (p, p − 2, s). By Corollary 2.8 in [27], F (p, p − 2, s) ⊂ B. Thus, from a well-known estimate of Bloch functions (see, for example [32]), This, together with the fact that µ ∈ LCM p,s , gives On the other hand, if we let then, applying Lemma 3.2, we get , with C being a positive constant independent of w. Indeed, notice first that and this, together with Proposition 2.8, yields This shows that the identity operator I : F (p, p−2, s) → T ∞ p,s (µ) is a bounded operator.
Conversely, suppose that the identity operator I : F (p, p − 2, s) → T ∞ p,s (µ) is bounded. Given any arc I ⊂ ∂D, let a = (1 − |I|)ζ, where ζ is the center of I, and consider the function f a (z) = log 2 1 −āz .
Then by Lemma 2.6, there is a constant It is easy to see that f a (z) ≈ log 2 |I| for any z ∈ S(I). Combining the above two inequalities we get Thus µ is a (p, s)-logarithmic Carleson measure, completing the proof of (i).
To prove (ii), first let µ be a vanishing (p, s)-logarithmic Carleson measure and {f n } be a bounded sequence in F (p, p − 2, s) with f n (z) → 0 uniformly on every compact subset of D. We need to prove that f n T ∞ p,s (µ) → 0. The case s ≥ 1 is easier. Let D r = {z ∈ D : |z| ≤ r}, and µ r = µ| D\Dr be the restriction of µ on D \ D r . By Proposition 2.9, if r → 1 then For 0 < s < 1 we don't have Proposition 2.9 at our disposal, so that the proof must follow a different route that also works for s ≥ 1. For any arc I, let w = w I = (1 − |I|)ζ, with ζ being the center of I. Then we have Since µ is a vanishing (p, s)-logarithmic Carleson measure on D, for any ε > 0, there is an r, 0 < r < 1, such that This, together with the facts that {f n } is a bounded sequence in F (p, p − 2, s) and the inequality |f n (w)| ≤ C| f n | F (p,p−2,s) log 2 |I| gives sup |I|<1−r |f n (w)| p µ S(I) |I| s < Cε.
If |I| ≥ 1 − r then |w| ≤ r, and since {f n } converges to zero uniformly on compact subsets of D, there is a positive integer n 0 such that sup |I|≥1−r |f n (w)| p µ S(I) |I| s < ε for n ≥ n 0 .
For the other term, we have Fix a sufficiently large number β so that β ≥ s + p − 2 and p(1 + β) > 2s. The proof of part (b) in Lemma 3.2 gives Since {f n } is a bounded sequence in F (p, p − 2, s), it follows from the proof of the boundedness part that f n,w D p s+p−2 ≤ C f n F (p,p−2,s) ≤ C, with C independent of n and w.
It is clear that f n,w (0) = (1 − |w| 2 ) s p (f n (0) − f n (w)). It is well-known F (p, p − 2, s) ⊂ B, the Bloch space. Hence {f n } is a bounded sequence in B and so there is a constant K > 0 such that f n B ≤ K. Thus and so for any ε > 0 and any n ∈ N there is an r ∈ (0, 1) such that |f n,w (0)| p < ε whenever r < |w| < 1. On the other hand, since f n (w) → 0 uniformly on compact subsets of D, we know that there is an N > 0 such that if n ≥ N then |f n,w (0)| p = (1 − |w| 2 ) s |f n (0) − f n (w)| p < ε for all |w| ≤ r. Combining the above arguments we know that sup w∈D |f n,w (0)| p < ε if n is sufficiently large.
It remains only to deal with the term We have that A(n, w) ≤ max(1, 2 p−1 ) A 1 (n, w) + A 2 (n, w) with Since {f ′ n } converges to zero uniformly on compact subsets of D, it is clear that sup w∈D A 1 (n, w) < ε for n big enough. Similarly, if 0 < r 0 < 1 is fixed, it follows from the fact that {f n } converges uniformly to zero on compact subsets that lim n→∞ sup |w|≤r0 A 2 (n, w) = 0.
Since µ is a vanishing (p, s)-logarithmic Carleson measure, we can choose r 0 so that

This together with Lemma 2.4 gives
Now it is easy to deduce that lim n→∞ sup |w|>r0 A 2 (n, w) = 0 completing this part of the proof.
Conversely, suppose that for any bounded sequence {f n } in F (p, p−2, s) with f n (z) → 0 uniformly on every compact subset of D we have f n T ∞ p,s (µ) → 0. Let {I n } be a sequence of subarcs of ∂D such that |I n | → 0. Let ζ n be the center of I n , w n = (1 − |I n |)ζ n , and Arguing as in Lemma 2.6, we easily see that {f n } is a bounded sequence on F (p, p − 2, s), and f n (z) → 0 uniformly on every compact subset of D. Thus proving that µ is a vanishing (p, s)-logarithmic Carleson measure. The proof is complete.
Using Lemma 2.1 we immediately get the following corollary of Theorem 3.1.
In the previous section we studied the embedding I : F (p, p − 2, s) → T ∞ p,s (µ). In this section we consider the embedding I : F (p, pα − 2, s) → T ∞ p,s (µ) when α = 1. First, we look at the case 0 < α < 1. We need first the following lemma from [15]. Remark. It is clear from the proof given in [15] that the above lemma also holds for Y = T ∞ p,s (µ) in the case 0 < p < 1. Theorem 4.2. Let 0 < α < 1, s > 0 and p > 0 satisfy s + pα > 1, and let µ be a nonnegative Borel measure on D. Then the following conditions are equivalent: Thus µ is an s-Carleson measure. (iii)=⇒(i). Suppose that µ is an s-Carleson measure. By Corollary 2.8 in [27], F (p, pα− 2, s) ⊂ B α . If we can prove that I : B α → T ∞ p,s (µ) is bounded then we are done. Since B α ⊂ H ∞ for 0 < α < 1, a standard application of the Closed Graph Theorem shows (iii)=⇒(ii). We further prove that (iii) implies (ii). Since F (p, pα − 2, s) ⊂ B α (see Corollary 2.8 in [27]), it is enough to prove that I : B α → T ∞ p,s (µ) is compact. Since we just proved that I : B α → T ∞ p,s (µ) is bounded, by Lemma 4.1, we need only prove that f n T ∞ p,s (µ) → 0, whenever {f n } is a bounded sequence in B α that converges to 0 uniformly on D. Since {f n } converges to 0 uniformly on D. for any given ε > 0, there exists a positive integer N such that |f n (z)| < ε for any n > N . Hence, for any n > N we have Therefore lim n→∞ f n p T ∞ p,s (µ) = 0, and so I : . This is obvious. The proof is complete.
Next, we consider the case α > 1.
We prove first that µ must be a β-Carleson measure. For each a ∈ D, consider the test functions Then sup a∈D f a D p β+p−2 ≤ C, and since µ is a Carleson measure for D p β+p−2 , one obtains Hence µ is a β-Carleson measure. Now, by Proposition 2.3, we have that f p for all t > 0. Fix a ∈ D, and let t = s + 2p(α − 1), and f ∈ F (p, pα − 2, s). Then Since F (p, pα − 2, s) ⊂ B α , and α > 1, one has This, together with the fact that µ is an s + p(α − 1) -Carleson measure and Lemma 2.2, gives . It remains to deal with the term I 2 (a). Set Since µ is a Carleson measure for D p s+pα−2 , we obtain By (2), one has On the other hand, by Lemma 2.2 and Proposition 2.8,  Proof. (i) Suppose first that µ is an [s + p(α − 1)]-Carleson measure. In all the cases considered, this is equivalent to µ being a Carleson measure for D p s+αp−2 (see [25]. Note that for 0 < p ≤ 1, the implication needed here is proved in our proof of Lemma 3.2). Thus, by Theorem 4.3 it follows that I : By Lemma 2.6, there is a constant C > 0 such that | f a | F (p,pα−2,s) ≤ C. Since I : It is easy to see that f a (z) ≈ |I| 1−α for any z ∈ S(I). Combining the above two inequalities we get 1 |I| s+p(α−1) µ(S(I)) ≤ C.
Next, we prove (ii). Let µ be a vanishing [s + p(α − 1)]-Carleson measure and {f n } be a bounded sequence in F (p, pα − 2, s) with f n (z) → 0 uniformly on every compact subset of D. We first deal with the cases considered with s + p(α − 1) ≥ 1. Let D r = {z ∈ D : |z| ≤ r}, and µ r = µ| D\Dr be the restriction of µ on D \ D r . By Proposition 2.9, if r → 1 then Recall that, in the introduction we have seen that, when α > 1, if g ∈ B α then there is a constant C > 0, independent of g, such that By Corollary 2.8 in [27], Hence, given any arc I ⊂ ∂D with w = (1−|I|)ζ and ζ is the center of I, for our sequence {f n } we can find a uniform constant C > 0 such that The fact that the limit f n T ∞ p,s (µ) → 0 now follows from part (i) and The proof for 0 < p ≤ 1 follows a similar argument as the corresponding one in Theorem 3.1. We left the details to the interested reader.
Conversely, suppose that for any bounded sequence {f n } in F (p, pα−2, s) with f n (z) → 0 uniformly on every compact subset of D we have f n T ∞ p,s (µ) → 0. Let {I n } be a sequence of subarcs of ∂D such that |I n | → 0. Let ζ n be the center of I n , w n = (1−|I n |)ζ n , and f n (z) = 1 − |w n | 2 (1 −w n z) α . By a proof similar to that of Lemma 2.6, we can easily see that {f n } is a bounded sequence on F (p, pα − 2, s), and f n (z) → 0 uniformly on every compact subset of D. Clearly, for any z ∈ S(I), |f n (z)| ≈ |I n | 1−α . Thus Hence µ is a vanishing [s + p(α − 1)]-Carleson measure. The proof is complete.
Using Lemma 2.1 we immediately get the following corollary of Theorem 4.4.

Corollary 4.5.
Let α > 1, s > 0 and p > 0 satisfy s + αp > 1, and let µ be a nonnegative Borel measure on D. Suppose that the point evaluation is a bounded functional on T ∞ p,s (µ). If 0 < p ≤ 1; or p > 1 with s + p(α − 1) > 1; or 1 < p ≤ 2 with s + p(α − 1) = 1, then In this section we look at applications of our main theorems to Riemann-Stieltjes integral operators. Recall that, for g ∈ H(D), the Riemann-Stieltjes integral operator J g is defined by for f ∈ H(D). The operators J g were first used by Ch. Pommerenke in [20] to characterize BMOA functions. They were first systematically studied by A. Aleman and A. G. Siskakis in [2]. They proved that J g is bounded on the Hardy space H p if and only if g ∈ BM OA. Thereafter there have been many works on these operators. See, [1], [3], [13], [19], and [22] for a few examples. Here we are considering boundedness and compactness of these operators on F (p, q, s).
For 0 < p < ∞, −2 < q < ∞, 0 ≤ s < ∞ such that q + s > −1, we define the space F L (p, q, s), called logarithmic F (p, q, s) space, as the space of analytic functions f on D satisfying We also say f ∈ F L,0 (p, q, s), if The following result establishes the boundedness of J g on F (p, q, s). Theorem 5.1. Let α > 0, s > 0 and p > 0 satisfy s + pα > 1. Then we have the following results.
(ii) J g is bounded on F (p, p − 2, s) if and only if g ∈ F L (p, p − 2, s).
(i) As 0 < α < 1, by Theorem 4.2, (3) is equivalent to that µ g is an s-Carleson measure. By Lemma 2.2, this means that which is the same as Thus g ∈ F (p, pα − 2, s).
Remark. Parts of the above results have been proved before. When p = 2, α = 1 and s = 1, it is known that F (2, 0, 1) = BM OA, the space of analytic functions of bounded mean oscillation. In this case, the above result was proved by Siskakis and the second author in [22]. When p = 2, α = 1, 0 < s < 1, we know that F (p, pα − 2, s) = Q s . In this case, the result was proved by Xiao in [26]. Note also that in the case α > 1 and γ = s + p(α − 1) > 1, by Theorem 1 in [28] (also see Theorem 1.3 in [27]), we know that F (p, p − 2, γ) = B, the Bloch space.
We can also use our previous results to characterize compactness of J g on F (p, q, s).
(ii) J g is compact on F (p, p − 2, s) if and only if g ∈ F L,0 (p, p − 2, s).
Proof. As in the proof of Theorem 5.1, we know that the operator J g is compact on Conversely, suppose I g is compact on F (p, pα−2, s). Then I g is bounded on F (p, pα− 2, s), and so by (i), g ∈ H ∞ . Since F (p, pα − 2, s) ⊂ B α , we know that I g is compact from F (p, pα − 2, s) to B α . Let {a n } be any sequence of points in D such that |a n | → 1, and let h n (z) = 1 − |a n | 2 α(1 −ā n z) α . As in the proof of (i), we know that sup n≥1 h n F (p,pα−2,s) < ∞.
Since g ∈ H ∞ , we must have g = 0. The proof is complete. Proof. (i) If 0 < α < 1, then F (p, αp − 2, s) ⊂ H ∞ , and it is easy to see that the space F (p, αp − 2, s) is an algebra. In order to prove (ii) and (iii), notice that By a computation similar to the proof of Lemma 2.6, we get that sup a∈D ψ a F (p,pα−2,s) ≤ C < ∞.
It is also easy to check that ψ a (a) = 0, ψ ′ a (a) =ā(1 − |a| 2 ) −α . Since M g is bounded on F (p, pα − 2, s) and F (p, pα − 2, s) ⊂ B α , we get that M g is bounded from F (p, pα − s, s) to B α . Hence, there is a constant C > 0 such that Hence g ∈ H ∞ . By Proposition 6.1, I g is bounded on F (p, pα − 2, s). Hence, from (5) we know that J g is also bounded on F (p, pα − 2, s). The results now follow from Theorem 2.3 and the fact that g ∈ H ∞ . When α > 1, in all the cases considered, we obtain M (F (p, αp − 2, s)) = H ∞ ∩ F (p, p − 2, γ), and the statements of (iii) are consequences of F (p, p − 2, γ) ⊂ H ∞ for 0 < p ≤ 1, and the fact that H ∞ ⊂ B and F (p, p − 2, γ) = B for γ > 1. The proof is complete.
Since g ∈ H ∞ , we must have g = 0. The proof is complete.

AN OPEN QUESTION
Finally, we want to mention a natural question that remains open. The question concerns the embedding I : F (p, pα − 2, s) → T ∞ p,s (µ) in the case α > 1. From Theorem 4.3 we know that µ being a Carleson measure for D p s+αp−2 is a sufficient condition for the boundedness. On the other hand, it is easy to see that if the embedding is bounded, then µ must be an [s + p(α − 1)]-Carleson measure. In the cases 0 < p ≤ 1; or s + p(α − 1) > 1; or 1 < p ≤ 2 and s + p(α − 1) = 1 it is well known that the two conditions are equivalent, allowing to obtain a complete description in that case (see Theorem 4.4). However, it is known that the two conditions are no longer equivalent in the remaining cases. So, what is the criterion for the boundedness and compactness of the embedding I : F (p, pα−2, s) → T ∞ p,s (µ) in these cases? Is the converse of Theorem 4.3 true? How about the boundedness of the Riemann-Stieltjes operator J g and the multiplication operator M g on F (p, pα−2, s) for this case?