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Hypersurfaces in $${\mathbb{R}}^ n$$ whose unit normal has small BMO norm. (English) Zbl 0733.42014
Let M be an unbounded connected hypersurface in $${\mathbb{R}}^{d+1}$$ which is assumed a priori to be smooth. Denote by n a continuous unit normal vector field on M, and by $$n_{x,r}$$ the average of n over the surface ball $$M\cap \{y\in {\mathbb{R}}^{d+1}:\;| y-x| <r\}$$ with respect to the surface measure. The BMO norm $$\| n\|_*$$ of n is the maximal average of the function $$| n-n_{x,r}|$$ over all such surface balls. Define also $$\theta (R)=\sup | <x-y,n_{x,r}>| r^{-1},$$ where the sup is over all $$x,y\in M$$ and $$r\in (0,R)$$ with $$| x- y| \leq r.$$
The author first proves that if $$\| n\|_*$$ is small if $$\theta$$ (R) is small for some R, then a Poincaré inequality holds on M, with constant depending only on the dimension d. He then proves his main result, which asserts that if $$\| n\|_*\leq \epsilon (d),$$ then in fact $$\theta (\infty)\leq C(d)\| n\|_*.$$ Thus, sufficient smallness of $$\| n\|_*$$ is by itself sufficient to render M a “chord-arc surface with small constant”. These surfaces have been studied by the author in two concurrent papers (see below). Chord-arc surfaces are higher dimensional generalizations of chord-arc curves $$\Gamma$$ in $${\mathbb{R}}^ 2$$, which are defined by the property $$arclength \Gamma (x,y)\leq C_ M| x-y|$$ for all $$x,y\in \Gamma.$$ Such curves have been extensively studied. For example, a theorem of D. S. Jerison and C. E. Kenig [Math. Scand. 50, 221-247 (1982; Zbl 0509.30025)] asserts they are bi-Lipschitz images of $${\mathbb{R}}$$. The author raises the question of whether the analogue of this result holds in $${\mathbb{R}}^ d$$ when the surface has small constant.

##### MSC:
 42B99 Harmonic analysis in several variables 53A05 Surfaces in Euclidean and related spaces 30G35 Functions of hypercomplex variables and generalized variables
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