Hypersurfaces in \({\mathbb{R}}^ n\) whose unit normal has small BMO norm.

*(English)*Zbl 0733.42014Let 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.

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.

Reviewer: A.Baernstein II (St.Louis)

##### MSC:

42B99 | Harmonic analysis in several variables |

53A05 | Surfaces in Euclidean and related spaces |

30G35 | Functions of hypercomplex variables and generalized variables |