Maximal Functions with Mitigating Factors in the Plane

Given a smooth, compactly supported hypersurface S in R n 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, byA t f ( x ) = ∫ t sf ( x ‐ y ) d σ ( y )where d σ is Lebesgue measure on S . We can now define a maximal average operator, M f ( x ) = sup t > 0| A t f ( x ) |In the case where S is S n −1 , the sphere of unit radius in R n , 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. DefineM k f ( x ) = sup t > 0| ∫ 2 ‐ k2 1 ‐ kf ( x ‐ t Γ ( s ) ) d s |for Schwartz functions f . Iosevich proved that‖ M k ‖L p − L p ⩽ c p 2 − k ( 1 − m / p )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 operatorM σ f ( x ) = sup t > 0| ∫ 0 1 f ( x ‐ t Γ ( s ) )( κ ( s ) ) σ d s |is bounded on L p for some p >2, if σ is sufficiently large, since‖ M σ ‖L p ‐ L p⩽ C ∑ k ⩾ 02 ‐ k ( m ‐ 2 ) σ‖ M k ‖L p ‐ L p⩽ C ∑ k ⩾ 02 ‐ k (( m ‐ 2 ) σ + 1 ‐ m / p )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 ⩽σ −1 .