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Published online by Cambridge University Press:  01 April 1999

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Given a smooth, compactly supported hypersurface S in ℝn that does not pass through the origin, and denoting by tS the surface dilated by a factor t>0, we can consider the averaging operator defined for functions f∈[Sscr ], the Schwartz class of functions, by

formula here

where dσ is Lebesgue measure on S. We can now define a maximal average operator,

formula here

In the case where S is Sn−1, the sphere of unit radius in ℝn, we are looking at Stein's spherical maximal function, as treated in his paper [10]. Stein proved that M is a bounded operator on the Lp spaces if and only if p>n/(n−1) when n>3. Subsequently, Bourgain [2] showed that if S is any compactly supported smooth curve with non-vanishing Gaussian curvature, then M will be bounded on Lp, if and only if p>2, thus dealing with the case of the circular maximal function in the plane. (For related results, see also [6] and [7]). In the case where S is a curve whose curvature vanishes to order at most m−2 at a single point, Iosevich [4] showed that M is bounded on Lp for p>m, and unbounded if p = m. If we study curves given by Γ(s) = (s, γ(s)+1), s∈[0, 1], for some suitably smooth γ, where γ(0) = γ′(0) = … = γ(m−1)(0) ≠ γ(m)(0)>0, then we can reinterpret his results as follows. Define

formula here

for Schwartz functions f. Iosevich proved that

formula here

for p>2. If we note that κ(s), the curvature of the curve Γ(s) is approximately 2k(m−2) whenever s∈[2k, 21−k], then we have that the operator

formula here

is bounded on Lp for some p>2, if σ is sufficiently large, since

formula here

which is finite so long as σ>(m/p−1) (m−2)−1. If we want to choose σ independent of m>2, the type of the curve, such that Mσ is bounded on Lp for some fixed p>2, then clearly we can take σ = 1/p. In this paper we show that Mσ will be bounded on Lp for p>max{σ−1, s} for a class of infinitely flat, convex curves in the plane. Counterexamples will show that this is the best possible result, in that there exist flat curves for which Mσ is unbounded for 2<p[les ]σ−1.

Notes and Papers
The London Mathematical Society 1999

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