MAXIMAL FUNCTIONS WITH MITIGATING FACTORS IN THE PLANE

Abstract

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 L^p 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 L^p, 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 L^p 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 2^−k(m−2) whenever s∈[2^−k, 2^1−k], then we have that the operator

formula here

is bounded on L^p 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 M_σ is bounded on L^p for some fixed p>2, then clearly we can take σ = 1/p. In this paper we show that M_σ will be bounded on L^p 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 M_σ is unbounded for 2<p[les ]σ⁻¹.