The Kakeya Maximal Function and the Spherical Summation Multipliers

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. I. Introduction. The purpose of this paper is to start the program of getting a real variable understanding of the Bochner-Riesz spherical summation operators. These operators are defined on functions on R' by the formula TAf ()=mA (t) f ) where mx ( I) = (1-j2)x if < < 1 and mx (t) = 0 otherwise. They were first studied by Bochner [1] and Stein [11], [12], [13] in connection with summation of multiple Fourier Series. If X is bigger than a critical exponent depending of the dimension (X > (n -1)/2), then the kernel of TX is integrable and therefore TX is bounded on every LP(Rn). Stein [11] and Calderon and Zygmund [2], showed that TA is bounded on LP, 1 < p < oo whenX=critical index=(n-1)/2. The problem then, arises when we consider X smaller than that critical index. Herz [9] (See also Fefferman [4]) pointed out that TX is unbounded outside the range p (X)2 n +1+ < P < n -2S-n (X).n+I+2/X n -I-2X- Fefferman [5] showed that T is never bounded on L P except for the obvious cases: n = 1 or p =2; and also Fefferman [4] proved that TX is bounded on L P (Rn) provided that p (X) < p < p'(X) and X > (n -1)/4. This result has been sharpened by Tomas [15] to X > (n -1)/2(n + 1). Finally Carleson and Sjolin [3], Fefferman [6] and Hormander [10] proved that, in R2, TA is bounded on LP whenever X >0 and p(X) <p p'(X).So for n >2 we have the natural question: is TX (X >0) bounded on L P(R'), p(X) < p < p'(X)?Our approach to the problem is inspired by the work of Fefferman and it is as follows: The multiplier theorem for TX can be easily reduced to this problem:

A bilinear approach to the restriction and Kakeya conjectures

“…By Theorem 3.1 the two estimates are equivalent in the triangular region below the dashed line. The point 1 is the trivial L 1 → L ∞ estimate, while the point 2 represents the higher-dimensional analogue of Cordoba's argument ( [8], [2]), and 3 is the "bush" argument as given by Bourgain [2] (see also [9], [8]). Proposition 3.2, the bilinear improvement to Cordoba's argument, is the point 4.…”

Untitled

“…In the present context, (2.11) is a simple consequence of a variable coefficient version of a maximal theorem of Córdoba [3] (see also [17]) involving averages of functions of two variables. We postpone the straightforward argument until the end of this section.…”

“…Here |Ω ∩ γ α x | denotes one-dimensional Lebesgue measure. Results of Córdoba [3] imply that such a set must have full Hausdorff dimension. For analogous sets in R 3 it is conjectured that the same should be true.…”

“…For analogous sets in R 3 it is conjectured that the same should be true. By taking projections, the results of [3] immediately imply that such sets must have dimension at least 2.…”

Covering lemmas revisited

“…(c) The family of rectangles in lacunary directions in U 2 . To see that the maximal operator here is strong type 2, order the rectangles by longest sidelength; Cordoba and Fefferman [4] observed that if R a £R, then |/?…”