Locally polynomially integrable surfaces and finite stationary phase expansions
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MARK AGRANOVSKY To Larry Zalcman, with friendship and appreciation Abstract. Let M be a strictly convex smooth connected hypersurface in Rn its convex hull. We say that M is locally polynomially integrable if for every and M by a parallel point a ∈ M the (n − 1)-dimensional volume of the cross-section of M translation of the tangent hyperplane at a to a small distance t depends polynomially on t. It is conjectured that only quadrics in odd-dimensional spaces possess such a property. The main result of this article partially confirms the conjecture. The study of integrable domains and surfaces is motivated by a conjecture of V. I. Arnold about algebraically integrable domains. The result and the proof are related to studying oscillating integrals for which the asymptotic stationary phase expansions consist of a finite number of terms.
1
Introduction
In [1], the following definition was introduced for bounded domains in Rn . The body K ⊂ Rn is called polynomially integrable if the Radon transform of its characteristic function χK , χK (x)dVn−1 (x), AK (ξ, p) = voln−1 ({x ∈ K : x, ξ = p}) = x,ξ=p
is a polynomial in p. The same term will be used for the boundary ∂K which in our case will be assumed to be a smooth hypersurface. The function A(ξ, p), which we call the (sectional) volume function, evaluates the (n − 1)-dimensional volume of the cross-section of K by the hyperplane {x ∈ Rn : x, ξ = p}. It was proved that there are no polynomially integrable bodies with C∞ boundary in R2k , while in odd dimensions only solid ellipsoids are polynomially integrable [9], [1], and even rationally or real-analytically integrable [2], domains with C∞ boundary. 23 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0124-5
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M. AGRANOVSKY
In this article, we extend the notion of polynomial integrability to the case of open hypersurfaces. In this case, the volume function AM (ξ, p) is defined for the cross-section of M with hyperplanes close to their tangent. To guarantee the finiteness of the volumes, we assume that M is strictly convex. The main question with which we are concerned is: what are polynomially integrable hypersurfaces? The quadrics in odd-dimensional spaces deliver examples of such hypersurfaces and it is expected that there are no other examples. The main result of this article is towards this conjecture. Namely, we prove that, under certain additional conditions, the polynomiality of the volume function A(ξ, t) implies that the hypersurface M is a quadric, namely, an elliptic paraboloid, in R2m+1 . The study of polynomially integrable bodies and surfaces has been motivated by the works [3] and [10], devoted to algebraically integrable bodies. They are bodies K for which the two-valued volume function VK± (ξ, t), where V + (ξ, t) =
t
−∞
A(ξ, u)du, V −(ξ, t) =
t
+∞
A(ξ, u)du,
is algebraic. The celebrated Newton Lemma about ovals from [8] states that there is no such domain (with infinitely smooth boundary) in the plane. V. I. Arnold ([3], problems 1987-14, 1988-13
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