Some questions regarding verification of Carleson measures

Abstract In this paper, we give a survey of some recent progress in terms of verifying Carleson measures; this includes the difference between two definitions of a Carleson measure,the Bergman tree condition, the T1 condition for Besov-Sobolev spaces on a complex ball,vector-valued Carleson measures, Carleson measures in strongly pseudoconvex domains and reverse Carleson measures.

Key words Carleson measures; Besov-Sobolev spaces; Hardy spaces; Bergman spaces

1 Introduction

The Carleson measure has its root in the famous works of Carleson(cf. [7,8]),motivated by the corona problem for the Banach algebra H∞(D) of bounded holomorphic functions defined in the unit disc D; namely, if we let M be the maximal ideal space of H∞(D), is the “corona”M¯D empty? In the language of function theory,it can be proven that this problem is equivalent to the Bezout formulation, which is as follows:

If we let f1,f2,··· ,fk∈H∞(D) satisfy

In [7], Carleson found a connection between the corona problem and the interpolating sequences in the unit disc for the Banach algebra H∞(D). He first proved the following important result on the interpolating sequences:

Theorem 1.1 ([7], Theorem 3) Let {zj}j∈Nbe an infinite sequence of distinct points of D. Then the following conditions are equivalent:

(i)any bounded sequence{wj}j∈Ncan be interpolated by a bounded holomorphic function f, i.e., there exists f ∈H∞(D) satisfying f(zj)=wj,j ≥1;

(ii) the sequence {zj}j∈Nis uniformly seperated in the sense that

where d is the pseudohyperboloc distance in D.

In the proof of this theorem, Carleson introduced an important notion of measure (now named after himself): a finite positive Borel measure μ on the unit disc D is said to be a Carleson measure for Hardy space Hp(D) if the inclusion map Hp(D) →Lp(μ) is continuous.In [8], Carleson further gave a geometric characterization of Carleson measure which requires a preliminary definition: let μ be a finite positive Borel measure on D, we say that μ is a geometric Carleson measure if

for all arcs I ⊂∂D, where S(I)={reiθ:θ ∈I,0 ＜1-r ＜|I|}, is called the Carleson window.

He then obtained the following result:

Theorem 1.2 ([8], Theorem 1) Let μ be a finite positive Borel measure on D. Then the following properties are equivalent:

(i) μ is a geometric Carleson measure;

(ii) μ is a Carleson measure.

This means that a measure μ is a Carleson measure for Hp(D)if and only if the μ-measure of any Carleson window is controlled by its side length. This is the starting point for various research programmes about the conditions for verifying Carleson measures. While doing research on the interpolating sequences of H∞(D), Carleson[7] observed that for the case of two generators, if one of the two functions is a Blaschke product whose zero set is an interpolating sequence, the corona theorem then holds. In modern language, his interpolating sequence condition can be stated as follows:

The first inequality of the above two conditions means that {zj}i∈Nis uniformly discrete(see Section 4 below). This idea can be easily generalized to the case of n functions, of which one function is a Blaschke product. Then, by Theorem 1.2 and a hard lemma concerning the Blaschke product, Carleson [8] finally proved that the corona is indeed empty.

Nowadays, the Carleson measure has become a fundamental tool in analysis for studying the structure of function spaces,interpolating sequences,and relevant operators. Such measures have also become interesting objects of research in their own right. In this survey, we are only concerned with questions related to our own work and the other progress with which we are most familiar. The paper is organized as follows: in Section 2, for Besov-Sobolev spaces on the complex ball, we show the difference between the two Carleson measures defined by the embedding of function spaces and by the Carleson box,respectively,and we state the Bergman tree condition and the T1-condition that are used to verify the Carleson measures. In Sections 3-5, we successively present the results regarding Carleson measures with values in Banach spaces, Carleson measures for Bergman spaces in strongly pseudoconvex domains and reverse Carleson measures.

2 Carleson Measures for Besov-Sobolev Spaces on Complex Ball

2.1 Difference Between Two Carleson Measures

Let B ={z ∈Cn:|z |＜1}be the unit ball of Cn(n ＞1),and let S ={z ∈Cn:|z |=1}be its boundary. Let dυ denote the normalized Lebesgue measure of B, i.e., let υ(B)=1, and dσ denote the normalized rotation invariant Lebesgue measure of S satisfying σ(S) = 1. Let dλ(z)=(1-|z |2)-n-1dυ(z) be the invariant measure on the ball.

In Theorem 2.1, we consider not only the case σ = 0 but also the case 0 ＜σ ＜∞, and these results hold for all the ranges 1 ＜p ＜∞. Although we give only the necessary condition and the sufficient condition, we find that the sufficient condition can be given by increasing the arbitrarily small ε ＞0 in the necessary condition. This result cannot be improved, since even in the one-dimensional situation, a unified necessary and sufficient condition holds only for p ＜q. When p = q, in the unit disc, the Carleson measures for Besov-Sobolev spaces are characterized by using an appropriate capacity condition μ(T(E)) ≤Ccapp(E), where for all compact subsets E in the unit disc, T(E) denotes a Carleson tent and

2.2 Bergman Tree Condition

and if t ＞0 is such that Bβ(0,r)=B(0,t)(note from(2.4)that Bergman metric balls centered at the origin are Euclidean balls), then the β-balls are ellipsoids.

Note that the set

is a Euclidean sphere (with different radius) centered at the origin for each r ＞0.

Applying an abstract construction to the spheres Srfor r ＞0, one has

2.3 T1 Condition

Volberg and Wick settle the important open question about characterizing the Carleson measures for the Besov-Sobolev space of analytic functions Bσ2on the complex ball of Cn. In particular, they demonstrate that for σ ＞0, the Carleson measures for the space are characterized by a T1 Condition.

In Cn, let B denote the open unit ball and consider the kernels given by

is a Bergman-type Calder´on-Zygmund operator of order m on the closed unit ball which satisfies the hypotheses of Theorem 1 of [21] on the kernel Kmand measure μ, but with respect to the quasi-metric Δ(z,w). If μ is compactly supported in the ball, the integral above converges absolutely. Otherwise we will consider the approximations of μ as having supports compactly contained in the unit ball, and we will be interested in the uniform eatimates of such operators.

The above mentioned (quasi)-metric on the spherical layer around ∂B is

3 Carleson Measures with Values in Banach Spaces

From the viewpoint of functional analysis, the range of complex functions is essentially in a Hilbert space. If the range of functions are in general Banach spaces, we can ask: what are the new properties of spaces of such functions? what is the relationship between the function spaces and the construction of a Banach space as a range? In[17],we answered thess questions.

This paper concerns the vector-valued version of (3.1). More precisely,we are interested in characterizing Banach spaces B for which one of the two inequalities in(3.1)holds for B-valued functions f. Given a Banach space B, let Lp(T;B) denote the usual Lp-space of Bochner pintegrable functions on T with values in B. The space BMO(T;B) of B-valued functions on T is defined in the same way as in the scalar case just by replacing the absolute value of C by the norm of B. We have

Theorem 3.1 ([17], Theorem 1.1) Let B be a Banach space. Then,

(i) There exists a positive constant c such that

holds for all trigonometric polynomials f with coefficients in B if and only if B admits an equivalent norm that is 2-uniformly smooth.

This theorem means that the characterizations of Carleson measures for some spaces of functions with values in Banach spaces depend on the geometrical properties of Banach spaces as a range. We can see that this result is different from that for complex function spaces. The uniformly convex and uniformly smooth items mentioned in the theorem are the concepts of the geometry of Banach spaces; they characterize the geometric belongingness of spaces in terms of a space norm as follows:

We have the modulus of convexity and modulus of smoothness of B by

The above result is a continuation of studying the vector-valued Littlewood-Paley theory; the argument is engaged in more general Euclidean space Rn, and depends on Calder´on-Zygmund singular integral theory. Here we point out that the two one-sided inequalities in Theorem 3.1 simultaneously hold if and only if the space B is isomorphic to a Hilbert space,the special case of which is just as f with complex values.

4 Carleson Measures in Strongly Pseudoconvex Domains

In several complex variables, it is more natural to characterize the object by the invariant metric, not the Euclidean, and to understand the Levi geometry of the boundary of a domain

Here σ is the Lebesgue measure on Un.

This theorem has a direct corollary, which is as follows:

Theorem 4.2 ([12], Corollary) Let μ be a finite positive Borel measure on the unit disc,and let Ap(D) be the Bergman space of D. Then μ is a Carleson measure if and only if, for any Carleson boxes

one has μ(Sθ0,h)≤Ch2for all h and all θ0, where C is a constant.

In other words, for the Bergman space Ap(D), μ is a Carleson measure if and only if the μ-measure of any Carleson box is controlled by its Lebesgue area. Cima and Wogen [9] gave a corresponding result for the Bergman space of the unit ball, again by the generalized Carleson boxes in the unit ball, but these characterizations are not intrinsic; i.e., the Carleson boxes are not a biholomorphic invariant object. In [15], Luecking showed that, at least for the Bergman spaces in the unit disk, Carleson measures can be characterized by the pseudohyperbolic distance. Duren and Weir [10] genaralized this to the setting of the unit ball. They also gave a characterization by the Berezin transform. Recall that for any finite positive Borel measure μ in the unit ball B, its Berezin transform is given by

where K(ζ,z) is the Bergman kernel of B. Duren and Weir’s result is as follows:

Theorem 4.3 ([10], Theorem 2, Theorem 3) Let μ be a finite positive Borel measure on the unit ball B in Cn, and let Ap(B) be the Bergman space of B. Let ν be the Lebesgue measure in R2n,normalized so that ν(B)=1. Then,for each fixed p(0 ＜p ＜∞),the following four statements are equivalent:

(i) μ is a Carleson measure for Ap(B);

(ii) μ(β(α,r))≤Cν(β(α,r)) holds for some r (0 ＜r ＜1), for some constant C, and for all pseudohyperbolic balls β(α,r),α ∈B;

(iii) μ(β(α,r)) ≤Cν(β(α,r)) holds for every r (0 ＜r ＜1), for some constant C, and for all pseudohyperbolic balls β(α,r),α ∈B.

(iv) The Berezin transform Bμis bounded.

One would like to generalize this theorem to strongly pseudoconvex domains. This was done by Abate and Saracco [3] in 2011. Their proofs depend on a detailed understanding of the Kobayashi geometry of strongly psendoconvex domains. Let us first recall some devices.

If X is a complex manifold, the Lempert function δX:X ×X →R+is defined by

Given θ ＞0, we call μ a θ-Carleson measure if, for all r ∈(0,1), there exists Cr＞0 such that

Obviously when θ = 1, we get the notion of a Carleson measure. Abate et al. [2] gave a characterization for the θ-Carleson measure:

Theorem 4.5 ([2], Theorem 1.2) Let D be a bounded strongly pseudoconvex domain in Cn, and let μ be a finite positive Borel measure on D, and 1 ＜p ＜r ＜∞. Then the following are equivalent:

(i) Tμ:Ap(D)→Ar(D) continuously;

(ii) μ is a (1+1/p-1/r)-Carleson measure.

In 2020, Abate et al. [1] further considered weighted Bergman spaces and obtained a finer theorem with p,r ∈(0,+∞), which generalized the relevant result of Pau and Zhao [19]on the unit ball.

In [3], Abate and Saracco provided a class of examples for Carleson measures through uniformly discrete sequences. A sequence Γ = {xj}j∈N⊂X is called uniformly discrete if there exists δ ＞0 such that kX(xj,xk) ≥δ for all j /= k. Given z0∈X,r ＞0, let N(z0,r,Γ)be the number of points of Γ contained in the ball of center z0and radius r with distance ρX=tanh kX. Then we have

Theorem 4.6 ([3], Theorem 4.2) Let D ⊂Cnbe a bounded strongly pseudoconvex domain with distance ρD. Let Γ={xj}j∈Nbe a sequence in D. Then the following statements are equivalent:

5 Reverse Carleson Measures

where SI= {z ∈D : |z| ≥1-h,arg z ∈I}, and h is the length of I. Recently, Hartmann et al. [11] pointed out that the Carleson condition can be dropped in the result of [14], and gave a complete characterization of reverse Carleson measure in the unit disc. Their result is as follows:

Theorem 5.1 ([11], Theorem 2.1) Let 1 ＜p ＜∞and let μ be a finite positive Borel measure on D. Then the following assertions are equivalent:

where E is a Borel subset of ¯D and |·| denotes the normalized Lebesgue measure on T. For Hardy space Hp(D), Lef`evre et al. [6] gave an explicit characterization of such operators in terms of the Carleson measure mφ.

Theorem 5.2 ([14], Theorem 5.1) Let φ : D →D be a nonconstant analytic self map.Then the composition operator Cφ: Hp(D) →Hp(D),1 ≤p ＜∞has a closed range if and only if there is a constant c ＞0 such that, for 0 ＜h ＜1,

Here dA is the Lebesgue measure on D, Nφis the Nevanlinna counting function, and S(ξ,h)={z ∈D : |z-ξ| ≤h}, which is equivalent in size with the Carleson window. Since Lef`evre et al. [13] proved the equivalence between Carleson measures and Nevanlinna counting functions,the above theorem implies that Cφhas a closed range if and only if mφis a reverse Carleson measure.

It should be noted that these results on reverse Carleson measures only concern the unit disc in the complex plane. It is interesting to consider the analogue in the case of several complex variables. In particular, for composition operators in the unit ball, it is important to think about how to characterize the ones with a closed range. For example,for the Bloch space in the unit ball, we know the necessary and sufficient conditions for characterizing composition operators with closed range, but the two conditions are not unified in general. Examining the above theorem, we think that the reverse Carleson measure might provide the device for cracking this problem.