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Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings. (English) Zbl 1217.46019

Let \((X,d,\mu)\) be a metric space \(X\) with metric \(d\) and Radon measure \(\mu\). The case of preference is \(\mathbb R^n\) with the Euclidean distance and the Lebesgue measure. Let \(s,p \in (0,\infty)\) and \(q \in (0,\infty]\). Then \(\dot{M}^s_{p,q} (X)\) collects all \(u\) with
\[ |u(x) - u(y)| \leq 2^{-ks} \big( g_k (x) + g_k (y) \big), \quad d(x,y) \sim 2^{-k}, \]
and
\[ \| u \, | \dot{M}^s_{p,q} (X) \| = \inf \| \{ g_k \} \, | L_p (X, \ell_q ) \| < \infty, \]
where the infimum is taken over all admissible \(\{ g_k \}\); similarly, \(\dot{N}^s_{p,q} (X)\) with \(\ell_q \big( L_p (X) \big)\) in place of \(L_p (X, \ell_q)\). It is one aim of the paper to prove that
\[ \dot{M}^s_{p,q} (\mathbb R^n) = \dot{F}^s_{p,q} (\mathbb R^n), \quad \dot{N}^s_{p,q} (\mathbb R^n) = \dot{B}^s_{p,q} (\mathbb R^n), \]
if \(s \in (0,1)\), \(p \in (\frac{n}{n+s}, \infty)\), \(0<q \leq \infty\) (modification for \(F\)-spaces). Here, \(\dot{B}^s_{p,q} (\mathbb R^n)\), \(\dot{F}^s_{p,q} (\mathbb R^n)\) are the well-known scales of homogeneous Besov-Triebel-Lizorkin spaces on \(\mathbb R^n\). This modifies the (fractional) Hajłasz-Sobolev spaces, where
\[ |u(x) - u(y)| \leq d(x,y)^s [g(x) + g(y)], \quad g \in L_p (X), \]
\(s \in (0,1]\). It is the second major aim of this paper to prove that \(\dot{M}^s_{n/s,q} (\mathbb R^n)\) (and, as a consequence, \(\dot{F}^s_{n/s,q} (\mathbb R^n)\)) is invariant under quasi-conformal mappings if \(s \in (0,1)\), \(q\in [0,\infty]\). There are counterparts in some metric spaces \((X,d,\mu)\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30L10 Quasiconformal mappings in metric spaces
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