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All these observations deduce the proof.

end{aligned}Applying Schauder estimates in Proposition we immediately deduce the proof of Theorem 8.1.

and follow the argument of (1) term by term to deduce the proof of (2) results.

Taking into account the relation (vert frac{1-overline {a_{k}}}{1-a_{k}} cdotfrac{a_{k+1}}{1- vert a_{k}vert ^{2}}vert leq1) and using (6) in Theorem 1, we can readily deduce the following corollary (the proof is omitted here).

If we do not assume that ∂γ Ω ⊂ Ω, then we deduce from the proof of the Theorem that is equivalent to (4).

end{aligned} By this and Theorem 1.1, we deduce that the proof of Theorem 1.2 can be reduced to proving that begin{aligned} bigl| {M_{Phi}(Tb }bigr| _{L^{1}(mu }leq C|{b}|_{H^{1,infty}_{atb}(mu)}.

For example the ascending HNN extensions of free groups are known to be residually finite and even virtually residually nilpotent (proved by Borisov and the third author [6, 7]) but the only upper bound one can deduce from the proof is exponential.

We then deduce the desired result and this completes the proof.

Part of the following result can be deduced from the proof of [[3], Theorem 3.1].

This result can be deduced by the proof of Lemma 3.2, with slight modifications.

Firstly, we give the following equivalence which can be deduced in the proof of Theorem 4 in [6].