ar X iv : m at h / 02 05 15 6 v 1 [ m at h . C A ] 1 4 M ay 2 00 2 SINGULAR MAXIMAL FUNCTIONS AND RADON TRANSFORMS NEAR L

We show that some singular maximal functions and singular Radon transforms satisfy a weak type L log logL inequality. Examples include the maximal function and Hilbert transform associated to averages along a parabola. The weak type inequality yields pointwise convergence results for functions which are locally in L log logL. 1.Introduction Let Σ be a compact smooth hypersurface of R, and let μ be a compactly supported smooth density on Σ, i.e. μ = χdσ where χ ∈ C 0 (R ) and dσ is the surface carried measure on Σ. Unless stated otherwise we shall always make the following Curvature Assumption. The Gaussian curvature does not vanish to infinite order on Σ. We consider a group of dilations on R, given by t = exp(P log t), t > 0, and we assume that P is a d× d matrix whose eigenvalues have positive real part. For k ∈ Z we set δk = 2 and define the measure μk by (1.1) 〈μk, f〉 = 〈μ, f(δk·)〉. We shall consider the convolutions μk ∗ f and study the behavior of the maximal function (1.2) Mf(x) = sup k∈Z |μk ∗ f(x)| and some related singular integrals. By a rescaling we may assume that the measure μ is supported in the unit ball {x : |x| ≤ 1}. The first complete L bounds (1 < p < ∞) for a class of such operators (Hilbert transforms on curves) seems to be due to Nagel, Rivière and Wainger [9]. A classical reference is the article by Stein and Wainger [17] containing many related results; see also the paper by Duoandikoetxea and Rubio de Francia [6] which contains general results for maximal functions and singular integrals generated by singular measures, with decay assumptions on the Fourier transform. Concerning the behavior on L it is presently not known even for the special classes considered here whether the maximal operator M is of weak type (1, 1), i.e. 1991 Mathematics Subject Classification. 42B20. The first author is supported in part by a grant from the National Science Foundation. The second author is a Clay Prize fellow and is supported by the Sloan and Packard foundations. Typeset by AMS-TEX 1 whether it maps L to the Lorentz space L. This question had been raised in [17]. For some ’flat’ cases counterexamples are in [3], but these do not seem to apply in the case of our curvature assumption. We shall examine the behavior of the maximal function on spaces “near” L. Two results in this direction are known: Christ and Stein [4] showed by an extrapolation argument that if f is supported in a cube Q and f ∈ L logL(Q) then the maximal function Mf belongs to L (again under substantially weaker finite type assumptions). Moreover Christ [2] showed that the lacunary spherical maximal function maps the standard Hardy space H(R) to L, and that maximal functions and Hilbert transforms associated to a parabola in R map the appropriate Hardy space with respect to nonisotropic dilations to L. Weak L (see also Grafakos [8] and our recent paper [12] for related results). For the two operators associated to the parabola (t, t) it is also known ([11]) that they map the smaller product-type Hardy space H prd(R× R) to the smaller Lorentz space L . We recall that for f to belong to a Hardy space H a rather substantial cancellation condition has to be satisfied. If locally the cancellation is missing one has a restriction on the size of f ; more precisely if a function f ∈ H is single signed in an open ball then f belongs to L logL(K) for all compact subsets K of this ball. This can be deduced from the maximal function characterization of H and the fact that f0 ∈ L logL(q0) if f0 is supported on the cube q0 and the appropriate variant of the Hardy-Littlewood maximal function of f0 belongs to L (q0), see [15, §I.5.2 (c)]. Here we are interested in the behavior in Orlicz spaces near L without assuming additional cancellation conditions. Our main result is that the maximal operator acts well on L log logL and the global version satisfies weak type L log logL inequalities. We first give a Definition. Let Φ : R → R be a convex function and let T be an operator mapping simple functions on R to measurable functions. T is of weak type Φ(L) if there is a constant C so that the inequality (1.3) ∣{x ∈ R : |Tf(x)| > α} ∣∣ ≤ ∫ Φ (C|f(x)| α ) dx holds for all α > 0. Abusing the notation slightly we shall say that T is of weak type L log logL if there is a constant C so that the inequality (1.3) holds with Φ(t) = t log log(e + t). Theorem 1.1. The maximal operator M is of weak type L log logL. We also prove a related theorem on singular convolution operators with kernels supported on hypersurfaces (assuming our finite type curvature assumption). Let μk be as in (1.1) and assume that in addition