Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion

The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

“…Remark. We have that an inequality r µ (ν) c ≤ C( µ ∞ ) ν − µ c holds, which was observed in [26,Remark 5.1] by examining the proof of [8, Lemma 10] (see also [31]). This implies that the map r µ : M 0 (D) → M 0 (D) is locally bounded.…”

“…It was noticed in our recent paper [31] that the pull-back operator P h does not necessarily maps VMO(R) into itself if h is strongly quasisymmetric defined on R. However, under an extra assumption that h is uniformly continuous, the operator P h preserves VMO(R).…”

Two Equivalent Conditions for Muckenhoupt Weights

“…−y/2,u) du can be estimated similarly (24). together with (22) and (23) implies|B 3 | ≤ 1 t t 0 1 y I |G(x + y) − G(x − y)|dx dy ≤ 2ε.…”

“…This class was first studied in [19] when Shen discussed the characterizations of VMO-Teichmüller space on the real line and its complex Banach manifold structure. Later, it was investigated further in [23,24]. In particular, it was proved that h is strongly symmetric if and only if it can be extended to a quasiconformal homeomorphism of R 2 + onto itself whose Beltrami coefficient µ induces a vanishing Carleson measure |µ(z)| 2 y −1 dxdy on R 2 + .…”

“…Since the logarithmic derivative does not have conformal invariance, the notion of strongly symmetric homeomorphism on the real line is different from the one on the unit circle. In fact, the space of strongly symmetric homeomorphisms on S is smaller than that on R in the sense of conformal automorphism (see [24]).…”