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Characterizations of Besov spaces via variable differences. (English) Zbl 0562.46022
The paper deals with equivalent quasi-norms for Besov spaces $$B^ s_{p,q}(R_ n)$$, where $$0<p,q\leq \infty$$ and $$s>n(\frac{1}{\min (p,1)}-1).$$ It is well-known that for the above restrictions of parameters p,q,s Besov spaces can be characterized in terms of modulus of continuity, i.e. difference norms. The study of Besov spaces on Lie groups, for example, provokes the question, if this concept may be generalized to ”variable” differences. We give an affirmative answer for two types of differences $$\Delta^ M_{h+\epsilon (x,h)}f(x)=\Delta^ M_ yf(x)|_{y=h+\epsilon (x,h)}$$, $${\underline \Delta}{}^ M_{h+\epsilon (x,h)}f(x)=\Delta_{h+\epsilon (\cdot,h)}...\Delta_{h+\epsilon (\cdot,\epsilon)}f(x)$$, provided the smooth perturbation $$\epsilon$$ (x,h) is ”small”. This means $$\| f| L_ p\| +(\int^{\delta}_{0}| h|^{-sq}\| {\underline \Delta}^ M_{h+\epsilon (x,h)}f(x)| L_ p\|^ q(dt/t))^{1/q}$$ is an equivalent quasi-norm for $$B^ s_{p,q}(R_ n).$$ Here $${\underline \Delta}$$ can be replaced by $$\Delta$$. In the proofs maximal functions are used instead of the usual multipliers.
##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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