Bergman spaces with exponential weights

Difference of Composition operators over Bergman spaces with exponential weights

“…In fact, its proof is the same with the case of a doubling measure ∆η which is found in lemma 19 of [16]. For the references in our setting, you can refer to [10,18,21] for the proofs of Lemma 3.3. derivsubmean Lemma 3.3. Let ω = e −η , where η is a subharmonic function and 0 < p < ∞.…”

“…One may refer to the proof of Lemma 2.1 of [21] for the inequality above. In [10], when τ satisfies the conditions τ (r) ց 0 and τ ′ (r) → 0 as r → 1 − , the authors gave the following useful inequality to estimate the reproducing kernel function of A 2 (ω): for each positive number M > 0, there exists a constant C(M ) > 1 such that , we consider an arbitrary curve γ connecting z and w with γ(0) = z and γ(1) = w. We can get the minimum value 0 < t 0 ≤ 1 such that |z − γ(t 0 )| = δτ (z).…”

“…In [2,10], it was proved that P f = f for f ∈ A p (ω) for 0 < p < ∞ which is used in the proof of Proposition 4.1. The Bergman kernel estimates of Bergman spaces with exponential weights ω = e −η have been established by [1,2,10,14] for example: for ω ∈ W and τ (z) = (∆η(z)) −1/2 , there are positive constants σ, C > 0 such that…”

“…We remark that W is a subclass of the so-called fast weights considered by Kriete and MacCluer in [6,12]. We can check η(z) = 1 1−|z| is contained in W as a typical example and the readers can refer to [10,21] to study other examples belonging to W and why these conditions are needed. On the other hand, we note that the standard weights η(r) = −α log(1 − r), α > −1, are not contained in the class W, since we can not find τ such that τ ′ (r) converge to 0 when r → 1 − .…”

Toeplitz operators on Bergman spaces with exponential weights

“…Another class of weight was introduced by Borichev, Dhuez and Kellay [5] in 2007, denoted by BDK, which has later been considered in [1,3,4,21,22]. As stated in [17], the weight W 0 covers BDK, but there is no inclusion relation between the weight W 0 and OP. It deserves to be mentioned that the weight W 0 contains non-radial weights.…”