# Geometric Qp Functions (Frontiers in Mathematics) - download pdf or read online

By Jie Xiao

This booklet records the wealthy constitution of the holomorphic Q functionality areas that are geometric within the feel that they remodel clearly less than conformal mappings, with specific emphasis on fresh improvement in accordance with interplay among geometric functionality and degree conception and different branches of mathematical research, together with strength concept, harmonic research, sensible research, and operator conception. mostly self-contained, the ebook features as a tutorial and reference paintings for complex classes and learn in conformal research, geometry, and serve as areas. Self-contained, the booklet features as an educational and reference paintings for complicated classes and examine in conformal research, geometry, and serve as areas.

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1. Carleson Measures under an Integral Operator Proof. Assuming and 49 dµf,p (z) = |f (z)|2 (1 − |z|2 )p dm(z) dµTa,b f,p (z) = |Ta,b f (z)|2 (1 − |z|2 )p+2a−2 dm(z), we verify that µf,p CMp < ∞ yields µTa,b f,p CMp < ∞. To that end, we use 2k I as the subarc of T with the same center as I and the length 2k |I| for k ∈ N with k ≤ − log2 |I|. Given any Carleson box S(I). Under µf,p CMp < ∞ we make the following estimates: µTa,b f,p S(I) |Ta,b f (z)|2 (1 − |z|2 )p+2a−2 dm(z) = S(I) ≤ + S(I) S(I) S(2I) S(2I) |f (w)|(1 − |w|2 )b−1 dm(w) |1 − wz| ¯ a+b + S(I) = D\S(2I) |f (w)|(1 − |w|2 )b−1 dm(w) |1 − wz| ¯ a+b D\S(2I) 2 |f (w)|(1 − |w|2 )b−1 dm(w) |1 − wz| ¯ a+b 2 dm(z) (1 − |z|2 )2−p−2a dm(z) (1 − |z|2 )2−p−2a 2 dm(z) (1 − |z|2 )2−p−2a Int1 + Int2 .

4]. 46 Chapter 2. 4. 4 is taken from [Xi5] whose Berezin-type characterization is a by-product of [Kw] that describes the holomorphic diagonal Besov space Bp = {f ∈ H : f ∈ Ap,p−2 }, p ∈ (1, ∞) respectively the Bloch space B by means of D D |f (z) − f (w)|p Ξ(f ; z, w) |1 − wz| ¯ 4 α dm(z)dm(w) < ∞, where α ∈ [0, 2 + p), respectively sup a∈D D D |f (z) − f (w)|p Ξ(f ; z, w) |1 − wz| ¯ 4 α (1 − |σa (w)|2 )2 dm(z)dm(w) < ∞, where p ∈ (0, ∞), α ∈ [0, 2 + p). 1 (i) due to [KimKw] and [Lue]. 5. 5 comprises [RocWu, Theorem 1] and its proof and conformally invariant version (cf.

It suﬃces to consider the case β = γ since β ≥ γ implies 2γ−β (1 − |z|2 )β (1 − |w|2 )β |1 − z¯w|3+2α+2β ≤ (1 − |z|2 )β (1 − |w|2 )γ |1 − z¯w|3+2α+β+γ ≤ 2β−γ (1 − |z|2 )γ (1 − |w|2 )γ . |1 − z¯w|3+2α+2γ If α = 1/2, then a direct computation yields D |f (z) − f (0)|2 (1 − |z|2 )β dm(z) D |f (z)|2 (1 − |z|2 )β+2 dm(z). Applying this inequality to f ◦ σw , we derive I(f ; w) = D = |f (z) − f (w)|2 |1 − z¯w|−4−2β (1 − |z|2 )β dm(z) (1 − |w|2 )−(2+β) (1 − |w|2 )−(2+β) D D D |f ◦ σw (z) − f ◦ σw (0)|2 (1 − |z|2 )β dm(z) |(f ◦ σw ) (z)|2 (1 − |z|2 )β+2 dm(z) |f (z)|2 |1 − z¯w|−2β−4 (1 − |z|2 )β+2 dm(z).