Compact differences of weighted composition operators

Compact differences of two weighted composition operators acting from the weighted Bergman space $A^p_\omega$ to another weighted Bergman space $A^q_\nu$, where $0<p\le q<\infty$ and $\omega,\nu$ belong to the class $\mathcal{D}$ of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proof a new description of $q$-Carleson measures for $A^p_\omega$, with $\omega\in\mathcal{D}$, in terms of pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of $q$-Carleson measures for the classical weighted Bergman space $A^p_\alpha$ with $-1<\alpha<\infty$ to the setting of doubling weights.


Introduction and main results
Let HpDq denote the space of analytic functions in the unit disc D " tz P C : |z| ă 1u. For a nonnegative function ω P L 1 pr0, 1qq, the extension to D, defined by ωpzq " ωp|z|q for all z P D, is called a radial weight. For 0 ă p ă 8 and a radial weight ω, the weighted Bergman space A p ω consists of f P HpDq such that where dApzq " dx dy π is the normalized Lebesgue area measure on D. As usual, A p α stands for the classical weighted Bergman space induced by the standard radial weight ωpzq " p1´|z| 2 q α , where´1 ă α ă 8.
For a radial weight ω, write p ωpzq " ş 1 |z| ωpsq ds for all z P D. In this paper we always assume p ωpzq ą 0, for otherwise A p ω " HpDq for each 0 ă p ă 8. A weight ω belongs to the class p D if there exists a constant C " Cpωq ě 1 such that p ωprq ď C p ωp 1`r 2 q for all 0 ď r ă 1. Moreover, if there exist K " Kpωq ą 1 and C " Cpωq ą 1 such that p ωprq ě C p ω`1´1´r K˘f or all 0 ď r ă 1, then we write ω P q D. In other words, ω P q D if there exists K " Kpωq ą 1 and C 1 " C 1 pωq ą 0 such that p ωprq ď C 1 ż 1´1´r K r ωptq dt, 0 ď r ă 1.
The intersection p D X q D is denoted by D, and this is the class of weights that we mainly work with.
Each analytic self-map ϕ of D induces the composition operator C ϕ on HpDq defined by C ϕ f " f˝ϕ. The weighted composition operator induced by u P HpDq and ϕ is uC ϕ and sends f P HpDq to u¨f˝ϕ P HpDq. These operators have been extensively studied in a variety of function spaces. See for example [4,5,6,7,8,22,23,25,26]. XXX If now ψ is another analytic self-map of D, the pair pϕ, ψq induces the operator C ϕ´Cψ . One of the most important problem considering these operators is to characterize compact differences in Hardy spaces. Shapiro and Sundberg [24] studied this problem in 1990. Very recently, Choe, Choi, Koo and Yang [3]have solved this problem. For more about difference operators, see [2,8,10,11,20]. Moorhouse [11,12] obtain some important results on this operator in weighted Bergman spaces. He showed [11], among other things, that C ϕ´Cψ is Saukko [20,21] generalized this result by showing that if either 1 ă p ď q, or p ą q ě 1, then C ϕ´Cψ : A p α Ñ A q β is compact if and only if the operators δ 1 C ϕ and δ 1 C ψ are both compact from A p α to L q β . Very recently, Acharyya and Wu [1] characterized the compact differences of two weighted composition operators uC ϕ´v C ψ between different weighted Bergman spaces A p α and A q β , where 0 ă p ď q ă 8 and´1 ă α, β ă 8. Their result states that, if α`2 p ď β`2 q and u, v P HpDq satisfy sup zPD p|upzq|`|vpzq|q p1´|z| 2 q β`2 q´α`2 p ă 8, In this paper we characterize compact differences of two weighted composition operators from the weighted Bergman space A p ω to another weighted Bergman space A q ν with 0 ă p ď q ă 8 and ω, ν P D. To state the result, write δ 2 pzq " δ 2,ϕ,ψ pzq " ψpzq´ϕpzq 1´ψpzqϕpzq , z P D, and observe that |δ 1 | " |δ 2 | on D. Our main result reads as follows.
We need two specific tools for the proof of Theorem 1. The first one concerns continuous embeddings A p ω Ă L q µ . Recall that a positive Borel measure µ on D is a q-Carleson measure for A p ω if the identity operator I d : A p ω Ñ L q µ is bounded. A complete characterization of such measures in the case ω P p D can be found in [15], see also [13,17]. In particular, it is known that if q ě p and ω P p D, then µ is a q-Carleson measure for A p ω if and only if sup aPD µpSpaqq ωpSpaqq q p ă 8.
Here and from now on Spaq " tz : 1´|a| ă |z| ă 1, | arg z´arg a| ă p1´|a|q{2u is the Carleson square induced by the point a P Dzt0u, Sp0q " D and ωpEq " ş E ωdA for each measurable set E Ă D. We will need a variant of this result and its "compact" counterpart for ω P D where the Carleson squares are replaced by pseudohyperbolic discs. To this end, denote ϕ a pzq " a´z 1´az for a, z P D. The pseudohyperbolic distance between two points a and b in D is ρpa, bq " |ϕ a pbq|. For a P D and 0 ă r ă 1, the pseudohyperbolic disc of center a and of radius r is ∆pa, rq " tz P D : ρpa, zq ă ru. It is well known that ∆pa, rq is an Euclidean disk centered at p1´r 2 qa{p1´r 2 |a| 2 q and of radius p1´|a| 2 qr{p1´r 2 |a| 2 q. Moreover, if µ is a q-Carleson measure for A p ω , then the identity operator satisfies Another result needed is a lemma that allows us to estimate the distance between images of two points, say z and a, under f sufficiently accurately whenever z is close to a in the sense that z P ∆pa, rq, and f P A p ω with ω P D. For the statement, denote r ωpzq " p ωpzq{p1´|z|q for all z P D.
This lemma plays an important role in the proof of Theorem 1 when we show that (1.5) and (1.6) are sufficient conditions for the compactness. By [18,Proposition 5] we know that provided ω P D. This explains the appearance of the weight r ω on the right hand side of (1.9). It is worth observing that, despite of (1.10), the strictly positive weight r ω cannot be replaced by ω in the statement because ω P D may vanish in pseudohyperbolic discs of fixed radius that tend to the boundary.
The rest of the paper contains the proofs of the results stated above. We first prove Lemma 3 in the next section. The proof of the result on Carleson measures, Theorem 2, is given in Section 3, and finally, Theorem 1 is proved in Section 4.
To this end, couple of words about the notation used in the sequel. The letter C " Cp¨q will denote an absolute constant whose value depends on the parameters indicated in the parenthesis, and may change from one occurrence to another. We will use the notation a b if there exists a constant C " Cp¨q ą 0 such that a ď Cb, and a b is understood in an analogous manner. In particular, if a b and a b, then we write ab and say that a and b are comparable.

Proof of Lemma 3
It is known that if ω P D, then there exist constants 0 ă α " αpωq ď β " βpωq ă 8 and C " Cpωq ě 1 such that In fact, this pair of inequalities characterizes the class D because the right hand inequality is satisfied if and only if ω P p D by [17, Lemma 2.1] while the left hand inequality describes the class q D in an analogous way, see [14, (2.27)]. The chain of inequalities (2.1) will be frequently used in the sequel.

(2.2)
Let R P pr, 1q and set R 1 " r`R 2 . Further, let 0 ă s ă 1. Then the Cauchy integral formula for the derivative and the subharmonicity of |f | p yield Fix now s " spr, Rq P p0, 1q sufficiently small such that ∆pϕ a pwq, sq Ă ∆pa, Rq for all w such that |w| " R 1 . Further, an application of the right hand inequality in (2.1) shows that p ωpζqp ωpaq for all ζ P ∆pa, Rq. Therefore, by combining (2.2)  This proves the case p " q because |ϕ a pzq| " ρpz, aq for all a, z P D. This part of the proof is valid for all f P HpDq if ω P p D. Let now q ą p, and observe that trivially |f pzq´f paq| q " p|f pzq´f paq| p q But (1.10) guarantees }f } A p r ω -}f } A p ω ď 1, and thus the assertion in the case q ą p follows from the above estimate.

Proof of Theorem 2
To prove (i), assume first (1.7) and let 0 ă r ă 1. The fact that |f | p is subharmonic in D together with Minkowski's inequality in continuous form (Fubini's theorem in the case q " p) and (1.7) imply Since ω P D by the hypothesis, we may apply the right hand inequality in (2.1) to deduce ωp∆pζ, rqq p ωpζqp1´|ζ|q, ζ P D. , and hence }f } L q µ }f } A p ω for all f P HpDq by (1.10). Thus µ is a q-Carleson measure A p ω . Conversely, assume that µ is a q-Carleson measure A p ω . For each a P D, consider the function induced by ω and 0 ă γ, p ă 8. Then [17, Lemma 2.1] implies that for all γ " γpω, pq ą 0 sufficiently large we have }f a } A p ω -1 for all a P D. Therefore the assumption yields that is, µp∆pa, rqq pp ωpaqp1´|a|qq q p for all a P D. Since ω P D Ă q D by the hypothesis, there exists K " Kpωq ą 1 and C " Cpωq ą 1 such that p ωprq ě C p ω`1´1´r K˘f or all 0 ď r ă 1 by the definition. Fix now r " rpKq P p0, 1q sufficiently large such that Then, as ω P D Ă p D, the right hand inequality in ( The claim (1.7) now follows from these estimates for all r " rpωq P p0, 1q sufficiently large.
To prove (ii), assume first that I d : A p ω Ñ L q µ is compact. An application of [17, Lemma 2.1] and the right hand inequality in (2.1) ensure that we may choose γ " γpp, ωq ą 0 sufficiently large such that }f a } A p ω -1 for all a P D, and f a Ñ 0 uniformly on compact subsets of D, as |a| Ñ 1´. Therefore the closure of the set tf a : a P Du is compact in L q µ . Since for each ε ą 0 the open balls Bpf a , εq " tf P L q µ : }f a´f } L q µ ă εu cover tf a : a P Du, there exists a finite subcover tBpf an , εq : n " 1, . . . , N " N pεqu. Let now a P D be arbitrary, and let j " jpaq P t1, . . . , N u such that f a P Bpf an , εq. Then, for each R P p0, 1q, we have By fixing R P p0, 1q sufficiently large, and taking into account that ε ą 0 was arbitrary, we deduce for all r P p0, 1q. Now fix r " rpωq as in the case (i) to have p ωpaqp1´|a|q ωp∆pa, rqq for all a P D. Then we obtain (1.8).
Conversely, assume (1.8). Let tf k u kPN be a sequence in A p ω such that sup kPN }f k } A p ω " M ă 8. Then it is easy to see that tf k u kPN is uniformly bounded on compact subsets of Dthis follows, for example, from (4.3) below. Therefore tf k u kPN constitutes a normal family by Montel's theorem, and hence we may extract a subsequence tf k j u jPN that converges uniformly on compact subsets of D to a function f which belongs to HpDq by Weierstrass' theorem.
Since ε ą 0 was arbitrary, we have lim sup jÑ8 ż D |f pzq´f k j pzq| q dµpzq " 0, and hence I d : A p ω Ñ L q µ is compact. This completes the proof of the theorem.

Proof of Theorem 1
With the auxiliary results proved in the previous sections we are ready for the proof of the main result. We will follow the arguments used in [1] with appropriate modifications. The following two propositions will prove Theorem 1. The first one gives necessary conditions for uC ϕ´v C ψ : A p ω Ñ A q ν to be compact. Proof. Consider the test functions f a defined in (3.2), and set F a pzq " ϕ a pzqf a pzq for all a, z P D. Obviously, }F a } A p ω ď }f a } A p ω for all a P D. Further, by the proof of Theorem 2, both f a and F a tend to zero uniformly on compact subsets of D as |a| Ñ 1´, and }f a } A p ω -1 for all a P D if γ " γpω, pq ą 0 is sufficiently large. Since uC ϕ´v C ψ : A p ω Ñ A q ν is compact by the hypothesis, we therefore have lim |a|Ñ1´} uC ϕ pf a q´vC ψ pf a q} A q ν " 0 (4.1) and lim |a|Ñ1´} uC ϕ pF a q´vC ψ pF a q} A q ν " 0.
Sufficient conditions for the compactness of uC ϕ´v C ψ : A p ω Ñ A q ν are given in the next result. Proof. It suffices to show that for any norm bounded sequence tf n u in A p ω which tends to zero uniformly on compact subsets of D as n Ñ 8, we have }puC ϕ´v C ψ qpf n q} A q ν Ñ 0 as n Ñ 8. For simplicity, assume }f n } A p ω ď 1 for all n. Fix 0 ă r ă R ă 1, and denote E " tz P D : |δ 1 pzq| ă ru and E 1 " DzE. Write puC ϕ´v C ψ qpf n q " puC ϕ´v C ψ qpf n qχ E 1`pu´vqC ψ pf n qχ E`u pC ϕ´Cψ qpf n qχ E , and observe that it is enough to prove that each of the three quantities tends to zero as n Ñ 8.
We begin with considering the first two quantities in (4.8). By the definition of the set E we have the estimates |puC ϕ´v C ψ qpf n qχ E 1 | ď 1 r p|δ 1 uC ϕ pf n q|`|δ 1 vC ψ pf n q|q and |pu´vqC ψ pf n qχ E | ďˆ1 1´r˙γ |1´ψδ 2 | γ |u´v||C ψ pf n q| on D. Therefore it suffices to prove that δ 1 uC ϕ , δ 1 vC ψ and p1´ψδ 2 q γ pu´vqC ψ are compact operators from A p ω to L q ν . We show in detail that δ 1 uC ϕ is compact -the same argument shows the compactness of the other two operators.
Let µ be a finite nonnegative Borel measure on D and h a measureable function on D. For an analytic self-map ϕ of D, the weighted pushforward measure is defined by for each measurable set M Ă D. If µ is the Lebesgue measure, we omit the measure in the notation and write ϕ˚phqpM q for the left hand side of (4.9). By the measure theoretic change of variable [9, Section 39], we have }δ 1 uC ϕ pf q} L q ν " }f } L q ϕ˚p|δ 1 u| q νq for each f P A p ω . Therefore Theorem 2 shows that δ 1 uC ϕ : A p ω Ñ L q ν is compact if and only if ϕ˚p|δ 1 u| q νqp∆pa, rqq ωp∆pa, rqq q p " ş ϕ´1p∆pa,rqq |δ 1 pzqupzq| q νpzq dApzq ωp∆pa, rqq This is what we prove next. Define W a,r " sup zPϕ´1p∆pa,rqqˇp Then W a,r Ñ 0, as |a| Ñ 1´, by the hypothesis (1.5). Moreover, for a P D and z P ϕ´1p∆pa, rqq, (2.1) yields |δ 1 pzqupzq| q W a,r pp ωpaqp1´|a|qq q p p νpzqp1´|z|q , and therefore, for each ε P p0, 1q we have ϕ˚p|δ 1 u| q νqp∆pa, rqq "  Before proceeding further, we indicate how to get to this point with the operator p1ψ δ 2 q γ pu´vqC ψ . After the measure theoretic change of variable and an application of Theorem 2, consider It remains to deal with the third term in (4.8). By Lemma 3, Fubini's theorem and (2.1), we have }upC ϕ´Cψ qpf n qχ E } q A q ν " ż E |upzq| q |f n pϕpzqq´f n pψpzqq| q νpzq dApzq ż Since the identity operator from A p ω to L q ϕ˚p|δ 1 u| q νq is compact, it is also bounded. This and Theorem 2 yield }upC ϕ´Cψ qpf n qχ E } q , 0 ă r ă 1.
By choosing 0 ă r ă 1 sufficiently large, the last term can be made smaller than a pregiven ε ą 0. For such fixed r, the first term tends to zero as n Ñ 8 by the uniform convergence. Therefore }upC ϕ´Cψ qpf n qχ E } q A q ν Ñ 0, n Ñ 8, and hence also the last term in (4.8) tends to zero. This finishes the proof of the proposition.