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Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to \mathbb …

The question I would be asking is roughly : do the smooth Sobolev functions defined on an open bounded domain extend to smooth Sobolev functions on the Euclidean space ? For detail : Fix $p \geq 1. …

It is true, you have to use Kellog-Warschowski's theorem (wikipidea, will give link soon),for $C^{0,\alpha}$ -maps on $S^1$, and note that $|t-\zeta|^{\alpha}$ is a $C^{0,\alpha}(S^1)$ map, with the H …

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question: Consider an orientation-preserving quasi …

I asked the question before, but didn't get any reply, so I took the liberty to ask again. Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, is th …

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following Theorem: …

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data $ …