© 1999 by London Mathematical Society
© The London Mathematical Society
Maximal Functions with Mitigating Factors in the Plane
34 Springfield Road, Brighton BN1 6DA gianfranco{at}marletta.freeserve.co.uk
Received 1 December 1995. Revision received 21 October 1996.
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
S, the Schwartz class of functions, by
 |
where d
is Lebesgue measure on S. We can now define a maximal average operator,
 |
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 
(s) = (s, 
(s)+1), s[0, 1], for some suitably smooth , where (0) = '(0) = ... = (m–1)(0) 
(m)(0)>0, then we can reinterpret his results as follows. Define
 |
for Schwartz functions f. Iosevich proved that
||Mk||Lp–Lp
cp2–k(1–m/p)
for p>2. If we note that 
(s), the curvature of the curve (s) is approximately 2–k(m–2) whenever s[2–k, 21–k], then we have that the operator
 |
is bounded on Lp for some p>2, if is sufficiently large, since
 |
 |
which is finite so long as >(m/p–1) (m–2)–1. If we want to choose independent of m>2, the type of the curve, such that M is bounded on Lp for some fixed p>2, then clearly we can take = 1/p. In this paper we show that M will be bounded on Lp for p>max{–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 M is unbounded for 2<p–1.