Compact differences of composition operators on Bergman spaces induced by doubling weights

Bounded and compact differences of two composition operators acting from the weighted Bergman space $A^p_\omega$ to the Lebesgue space $L^q_\nu$, where $0<q<p<\infty$ and $\omega$ belongs to the class $\mathcal{D}$ of radial weights satisfying a two-sided doubling condition, are characterized. On the way to the proofs a new description of $q$-Carleson measures for $A^p_\omega$, with $p>q$ and $\omega\in\mathcal{D}$, involving 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. The case $\omega\in\widehat{\mathcal{D}}$ is also briefly discussed and an open problem concerning this case is posed.


Introduction and main results
Each analytic self-map ϕ of the unit disc D " tz P C : |z| ă 1u induces the composition operator C ϕ , defined by C ϕ f " f˝ϕ, acting on the space HpDq of analytic functions in D. These operators have been extensively studied in a variety of function spaces, see for example [4,24,25,29]. Some studies on topological properties of the space of composition operators attracted attention towards differences of composition operators. The question of when the difference C ϕ´Cψ of two composition operators is compact on the Hardy space H p was posed by Shapiro and Sundberg [26]. Differences of composition operators have been studied ever since by many authors on several function spaces, see, for example, [1,3,6,10]. In 2004, Nieminen and Saksman [14] showed that the compactness of C ϕ´Cψ on the Hardy space H p , with 1 ď p ă 8, is independent of p. By using this, Choe, Choi, Koo and Yang [2] then characterized compact operators C ϕ´Cψ on Hardy spaces by using Carleson measures in 2020. Further, Moorhouse [12,13] characterized the compactness of C ϕ´Cψ on the standard weighted Bergman space A 2 α . Saukko [22,23] generalized Moorhouse's results by characterizing the compactness from A p α to A q β if either 1 ă p ď q, or p ą q ě 1. Very recently, Shi, Li and Du [28] extended Saukko's results for the complete range 0 ă p, q ă 8 and to higher dimensions.
In this paper we are interested in the compactness of C ϕ´Cψ on weighted Bergman spaces induced by doubling weights. We proceed with necessary definitions. For 0 ă p ă 8 and a positive Borel measure ν on D, the Lebesgue space L p ν consists of complex valued ν-measurable functions f on D such that If ν is continuous with respect to the Lebesgue measure, that is, dνpzq " ωpzqdApzq for some non-negative function ω, then we adopt the notation L p ν " L p ω without arising any confusion. For a nonnegative function ω P L 1 pr0, 1qq, its 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 analytic functions in L p ω . 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 the theory of these spaces consult [5,8].
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 radial 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 The intersection p D X q D is denoted by D, and this is the class of weights that we mainly work with. For recent developents on Bergman spaces induced by weights in p D, see [17] and the reference therein.
In this paper we consider compact differences of two composition operators from the weighted Bergman space A p ω to the Lebesgue space L q ν when 0 ă p, q ă 8 and ω P p D. To state the first main result, write The next result generalizes [23, Theorem 1.2] to the setting of doubling weights.
Theorem 1. Let 0 ă q ă p ă 8 and ω P D, and let ν be a positive Borel measure on D. Let ϕ and ψ be analytic self-maps of D. Then the following statements are equivalent: (iii) δC ϕ and δC ψ are compact (or equivalently bounded) from A p ω to L q ν . The proof of Theorem 1 is organized as follows. We first show that C ϕ´Cψ is compact if δC ϕ and δC ψ are bounded. The proof of this implication is straightforward and relies on the fact that the norm of f P HpDq in A p ω is comparable to the L p ω -norm of the non-tangential maximal function N pf qpzq " sup ζPΓpzq |f pζq|, where Γpzq " is a non-tangential approach region with vertex at z. Then, as each compact operator is bounded, the proof boils down to showing that δC ϕ and δC ψ are compact whenever C ϕ´Cψ is bounded. This part of the proof is more laborious. We first observe that for each ρ-lattice tz k u the function belongs to A p ω for all b " tb k u P ℓ p and its A p ω -norm is dominated by a universal constant times }b} ℓ p . Then we use this function for testing and apply Khinchine's inequality. After duality arguments we finally arrive to a situation where we must understand well the 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 : 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 [18], see also [15,16,19]. In particular, it is known that if p ą q and ω P p D, then µ is a q-Carleson measure for A p ω if and only if the function belongs to L p p´q ω . Here and from now on T pzq " tζ P D : z P Γpζqu is the tent induced by z P Dzt0u. Further, ωpEq " ş E ωdA for each measurable set E Ă D. Observe that in (1.1) we may replace the tent T pzq by the Carleson square Spzq " tζ : 1´|z| ă |ζ| ă 1, | arg ζ´arg z| ă p1´|z|q{2u because ωpT pzqq -ωpSpzqq for all z P Dzt0u if ω P p D. To complete the proof of Theorem 1 we will need a variant of the above characterization of Carleson measures in the case ω P D. Our statement involves pseudohyperbolic discs and an auxiliary weight associated to ω. The pseudohyperbolic distance between two points a and b in D is ρpa, bq " |pa´bq{p1´abq|. 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. We denote r ωpzq " p ωpzq{p1´|z|q for all z P D and note that provided ω P D, by [17]. Our embedding theorem generalizes the case n " 0 of [11,Theorem 1] to doubling weights and reads as follows.
Theorem 2. Let 0 ă q ă p ă 8 and ω P D, and let µ be a positive Borel measure on D.
Then the following statements are equivalent: We may not replace L p p´q r ω by L p p´q ω in part (iii) of Theorem 2. A counter example can be constructed as follows. Write Dpz, rq for the Euclidean disc tζ : |ζ´z| ă ru. Let r n " 1´2´n and A n " Dp0, r n`1 qzDp0, r n q for all n P N. Pick up an ω P D such that it vanishes on A 2n for all n P N. A simple example of a such weight is ř nPN χ A 2n`1 . Then choose µ such that for some ε ą 0 its support is contained in the union of the discs ∆pa n , εq which have the property that for some fixed r P p0, 1q we have ∆pz, rq Ă A 2n for all z P ∆pa n , εq and for all n P N. The choice a n " pr 2n`r2n`1 q{2 works if 0 ă r ă 1 and ε " εprq ą 0 are sufficiently small. Then, for such an r, the norm }Θ ω µ } L p p´q ω vanishes and thus it cannot be comparable to The second main result of this study concerns the case when p ă q and ω P p D. It completes in part the main result in [10] which concerns the class D only. An analogue of this result was proved for the Hardy spaces in [27]. Therefore Theorem 3 takes care of the gap consisting of small Bergman spaces that exists between the Hardy and the standard weighted Bergman spaces.
Theorem 3. Let 0 ă p ă q ă 8 and ω P p D, and let ν be a positive Borel measure on D. Let ϕ and ψ be analytic self-maps of D. Then C ϕ´Cψ : A p ω Ñ L q ν is bounded (resp. compact) if and only if δC ϕ and δC ψ are bounded (resp. compact) from A p ω to L q ν .
If q " p then the boundedness (resp. compactness) of δC ϕ and δC ψ implies the same property for C ϕ´Cψ by Proposition 4 below. Further, Proposition 5 below shows that C ϕ´Cψ is compact if δC ϕ and δC ψ are bounded when p ą q. But we do not know if the boundedness of C ϕ´Cψ implies that of δC ϕ and δC ψ if ω P p DzD if p ě q. The methods used in this paper do not seem to give this implication and hence this case remains unsettled.
The rest of the paper is organized as follows. In the next section we consider Carleson embeddings and prove Theorem 2. The sufficiency of the conditions on δC ϕ and δC ψ are established in Section 3, while Section 4 is devoted to their necessity. Finally, in Section 5 we indicate how our main findings on C ϕ´Cψ follow from these results.
To this end, couple of words about the notation used in this paper. 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.

Carleson measures
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, these inequalities characterize the class D because the right hand inequality is satisfied if and only if ω P p D by [15, Lemma 2.1], while the left hand inequality describes the class q D in an analogous manner [17, (2.27)]. The inequalities (2.1) will be frequently used throughout the paper.
Proof of Theorem 2. If µ is a q-Carleson measure for A p ω , then I : A p ω Ñ L q µ is automatically compact by [19,Theorem 3(iii)]. Therefore it suffices to show that (i) and (iii) are equivalent and establish (1.3).
Assume first Θ ω µ P L p p´q r ω for some r P p0, 1q. The subharmonicity of |f | q , Fubini's theorem, Hölder's inequality and (1.2) imply Therefore µ is a q-Carleson measure for A p ω and }I} q Conversely, assume that µ is a q-Carleson measure for A p ω . Then (1.2) shows that µ is also a q-Carleson measure for A p r ω and the corresponding operator norms are comparable. Further, since ω P D by the hypothesis, an application of (2.1) shows that r ω P D. Therefore [18, is a non-tangential approach region with vertex at z. Further, by [19,Theorem 3 Let now r P p0, 1q be given. For K ą 1 and z P DzDp0, 1´1 K q write z K " p1´Kp1| z|qqe i arg z . Pick up K " Kprq ą 1 and R " Rprq P p1´1 K , 1q sufficiently large such that ∆pz K , rq Ă Γpzq for all z P DzDp0, Rq. Straightforward applications of the left hand inequality in (2.1) show that r ωpT pζqq ωpSpζqq, as |ζ| Ñ 1´, and r ωpzq r ωpz K q for all z P DzDp0, Rq. In an analogous way we deduce ωpSpζqq ωpSpz K qq for all ζ P ∆pz K , rq and z P DzDp0, Rq by using the right hand inequality. Therefore

Sufficient conditions
In this section we establish sufficient conditions for C ϕ´Cψ : A p ω Ñ L q ν to be bounded or compact. All these results are valid under the hypothesis ω P p D despite the main results stated in the introduction concern only the class D. We begin with the case p ď q.
Proposition 4. Let 0 ă p ď q ă 8 and ω P p D, and let ν be a positive Borel measure on D. Let ϕ and ψ be analytic self-maps of D. If δC ϕ and δC ψ are bounded (resp. compact) from A p ω to L q ν , then C ϕ´Cψ : A p ω Ñ L q ν is bounded (resp. compact).
Proof. We begin with the statement on the boundedness. Let first q ą p. Let f P A p ω with }f } A p ω ď 1. Fix 0 ă r ă R ă 1, and denote E " tz P D : |δpzq| ă ru and E 1 " DzE. Write pC ϕ´Cψ qpf q " pC ϕ´Cψ qpf qχ E 1`pC ϕ´Cψ qpf qχ E , and observe that it is enough to prove that the quantities are bounded. We begin with considering the first quantity in (3.1). By the definition of the set E we have the estimate on D. Since the operators δC ϕ and δC ψ both are bounded from A p ω to L q ν by the hypothesis, the first term in (3.1) is bounded by (3.2).
We next show that also the second term in (3.1) is bounded. 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 ϕ˚ph, µqpM q " 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 (3.3). By the measure theoretic change of variable [7, Section 39], we have }δC ϕ pf q} L q ν " }f } L q ϕ˚p|δ| q νq for each f P A p ω . Therefore the identity operator from A p ω to L q ϕ˚p|δ| q νq is bounded by the hypothesis. Hence ϕ˚p|δ| q νqp∆pζ, Rqq ωpSpζqq q p for all ζ P Dzt0u by [18,Theorem 1(c)]. Further, by [10, Lemma 3], with ω " 1 and q " p, there exists a constant C " Cpp, r, Rq ą 0 such that |f pzq´f paq| q ď C ρpz, aq p p1´|a|q 2 ż ∆pa,Rq |f pζq| p dApζq, a P D, z P ∆pa, rq, for all f P A p ω with }f } A p ω ď 1. This and Fubini's theorem yield Therefore also the second term in (3.1) is bounded. This finishes the proof of the case q ą p.
This estimate implies (3.5), and thus the case q " p is proved.
To obtain the compactness statement, it suffices to show that the quantities }pC ϕ´Cψ qpf n qχ E 1 } L q ν , and }pC ϕ´Cψ qpf n qχ E } L q ν (3.7) tend to zero as n Ñ 8 for each sequence tf n u nPN in A p ω which tends to zero uniformly on compact subsets of D as n Ñ 8 and satisfies }f n } A p ω ď 1 for all n P N. Since δC ϕ and δC ψ are compact from A p ω to L q ν by the hypothesis, an application of (3.2) to f " f n shows that the first quantity in (3.7) tends to zero as n Ñ 8. As for the second quantity, observe that (3.4) implies }pC ϕ´Cψ qpf n qχ E } q L q ν ż D |f n pζq| q ϕ˚p|δ| q νqp∆pζ, Rqq p1´|ζ|q 2 dApζq, n P N. (3.8) Let first q ą p. Since the identity operator from A p ω to L q ϕ˚p|δ| q νq is compact by the hypothesis, we have ϕ˚p|δ| q νqpSpζqq{ωpSpζqq q p Ñ 0 as |ζ| Ñ 1´by [19,Theorem 3(ii)]. Now, for each ζ P Dzt0u pick up ζ 1 " ζ 1 pζ, Rq P D such that arg ζ 1 " arg ζ, ∆pζ 1 , Rq Ă Spζq and 1´|ζ 1 | -1´|ζ| for all ζ P Dzt0u. Then and hence sup ζPDzt0u ϕ˚p|δ| q νqp∆pζ, Rqq{ωpSpζqq q p ă 8 and, for a given ε ą 0, there exists η " ηpεq P p0, 1q such that ϕ˚p|δ| q νqp∆pζ, Rqq{ωpSpζqq q p ă ε for all ζ P DzDp0, ηq. Further, by the uniform convergence, there exists N " N pε, η, qq P N such that |f n | q ď ε on Dp0, ηq for all n ě N . These observations together with the proof of the boundedness case and (3.8) yield Thus also the second quantity in (3.7) tends to zero as n Ñ 8 in the case q ą p.
Finally, let q " p. The compactness of the identity operator from A p ω to L p ϕ˚p|δ| p νq implies ϕ˚p|δ| p νqpSpζqq{ωpSpζqq Ñ 0 as |ζ| Ñ 1´by [19,Theorem 3(ii)]. By following the proof above the only different step consists of making the quantity Standard arguments can now be used to make the right hand side smaller than a pregiven ε ą 0 for η sufficiently large by using ϕ˚p|δ| p νqpSpζqq{ωpSpζqq Ñ 0 as |ζ| Ñ 1´, see, for example, [16, pp. 26-27] for details. This completes the proof of the proposition.
The next result is a counter part of Proposition 4 in the case p ą q.
Proposition 5. Let 0 ă q ă p ă 8 and ω P p D, and let ν be a positive Borel measure on D. Let ϕ and ψ be analytic self-maps of D. If δC ϕ and δC ψ are bounded from A p ω to L q ν , then C ϕ´Cψ : A p ω Ñ L q ν is compact.
Proof. Let tf n u be a bounded sequence in A p ω such that f n Ñ 0 uniformly on compact subsets of D. Since δC ϕ and δC ψ are bounded from A p ω to L q ν by the hypothesis, they are also compact by [19,Theorem 3(iii)], and therefore lim nÑ8`} δf n pϕq} L q ν`} δf n pψq} L q ν˘" 0. (3.9) Let 0 ă r ă R ă 1, and denote E " tz P D : |δpzq| ă ru and E 1 " DzE. To prove the compactness of C ϕ´Cψ : A p ω Ñ L q ν , it suffices to show that lim nÑ8 p}pC ϕ´Cψ qpf n qχ E } L q ν`} pC ϕ´Cψ qpf n qχ E 1 } L q ν q " 0 since }pC ϕ´Cψ qpf n q} q By using (3.2) and (3.9), it is easy to show that lim nÑ8 }pC ϕ´Cψ qpf n qχ E 1 } L q ν " 0.

Necessary conditions
In this section we establish necessary conditions for C ϕ´Cψ : A p ω Ñ L q ν to be bounded or compact. In the case p ă q we work with the whole class p D, but the arguments employed in the case p ě q rely strongly on the hypothesis ω P D. We begin with the case p ď q. Proposition 6. Let either 0 ă p ă q ă 8 and ω P p D or p " q and ω P D, and let ν be a positive Borel measure on D. Let ϕ and ψ analytic self-maps of D. If C ϕ´Cψ : A p ω Ñ L q ν is bounded (resp. compact), then δC ϕ and δC ψ are bounded (resp. compact) from A p ω to L q ν .
Proof. Let first p ă q and ω P p D. We begin with the boundedness and show in detail that δC ϕ is bounded from A p ω to L q ν . For each a P D, consider the function f a pzq "ˆ1´| a| 1´az˙γ ωpSpaqq´1 p , z P D, induced by ω and 0 ă γ, p ă 8. Then [15,Lemma 2.1] implies that for each γ " γpω, pq ą 0 sufficiently large we have }f a } A p ω -1 for all a P D. Fix such a γ. Since C ϕ´Cψ is bounded, According to [22, p. 795], for each 0 ă γ ă 8 and 0 ă r ă 1 there exist a constant C " Cpγ, rq ą 0 such thaťˇˇˇ1´ˆ1´a C|a|ρpz, wq, z P ∆pa, rq, a, w P D. It follows that ϕ˚p|δ| q νq is a bounded q-Carleson measure for A p ω by [18, Theorem 1(c)], and hence δC ϕ : A p ω Ñ L q ν is bounded. The same argument shows that also δC ψ is bounded. For the compactness statement, first observe that f a tends to zero uniformly on compact subsets of D as |a| Ñ 1´. Then, if C ϕ´Cψ is compact, we have lim |a|Ñ1´} pC ϕ´Cψ qpf a q} L q ν " 0. By arguing as above we deduce lim |a|Ñ1´ϕ˚p |δ| q νqp∆pa, rqq pωpSpaqqq q p " 0. Therefore δC ϕ : A p ω Ñ L q ν is compact by [19,Theorem 3]. The same argument shows that also δC ψ is compact.
Let now p " q and ω P D. The statement follows from the proof above with the modification that [10, Theorem 2], valid for ω P D, is used instead of [18, Theorem 1(c)] and [19,Theorem 3]. The only extra step is to observe that for each ω P D there exists r " rpωq P p0, 1q such that ωpSpaqq -ωp∆pa, rqq for all a P Dzt0u. This follows from (2.1). With this guidance we consider the proposition proved.
The next result establishes a counter part of Proposition 6 when p ą q. Proposition 7. Let 0 ă q ă p ă 8 and ω P D, and let ν be a positive Borel measure on D. Let ϕ and ψ be analytic self-maps of D. If C ϕ´Cψ : A p ω Ñ L q ν is bounded, then δC ϕ and δC ψ are both bounded from A p ω to L q ν .
Proof. Let tz k u kPN be a ρ-lattice such that it is ordered by increasing modulii and z k ‰ 0 for all k. Then by [21, Theorem 1] there exist constants M " M pp, ωq ą 1 and C " Cpp, ωq ą 0 such that the function belongs to A p ω and satisfies }F } A p ω ď C}b} ℓ p for all b " tb k u P ℓ p . Since C ϕ´Cψ : A p ω Ñ L q ν is bounded by the hypothesis, we deduce We now replace b k by b k r k ptq, integrate with respect to t from 0 to 1, and then apply Fubini's theorem and Khinchine's inequality to get By applying (4.1) and the estimate |1´z k z| -1´|z k |, valid for all z P ∆pz k , ρq and k P N, we obtaiňˇˇˇˇˆ1´| and hence If q ě 2 then the inequality ř j c x j ď´ř j c j¯x , valid for all c j ě 0 and x ě 1, imply pωpT pz k qqq q p χ ϕ´1p∆pz k ,ρqq pzq¸dνpzq. To get the same estimate for 0 ă q ă 2 we apply Hölder's inequality. It together with the fact that the number of discs ∆pz k , rq to which each ϕpzq may belong to is uniformly bounded yields ż Pick up an r " rpρq P p0, 1q such that ∆pz, ρq Ă ∆pz k , rq for all z P ∆pz k , ρq and k P N.
The right hand inequality in (2.1) shows that p ωpzqp ωpz k q and ωpSpzqq -ωpSpz k qq for all z P ∆pz k , ρq and k P N. Then, as tz k u kPN is a ρ-lattice, we deduce ż Therefore ϕ˚p|δ| q νq is a q-Carleson measure for A p ω by Theorem 2. For the same reason, ψ˚p|δ| q νq is a q-Carleson measure for A p ω . The proof is complete.

Proofs of main theorems
Here we shortly indicate how the main results stated in the introduction easily follow from the propositions proved in the previous two sections.
Proof of Theorem 1. The theorem follows by Propositions 5 and 7. Namely, if δC ϕ and δC ϕ are bounded from A p ω to L q ν , then C ϕ´Cψ : A p ω Ñ L q ν is compact, and thus bounded as well, by Proposition 5. Conversely, if C ϕ´Cψ : A p ω Ñ L q ν is bounded, then δC ϕ and δC ϕ are bounded by Proposition 7. l Proof of Theorem 3. The theorem is an immediate consequence of Propositions 4 and 6. l