On Almost Everywhere Convergence of Bochner-riesz Means in Higher Dimensions

In Rn define (TXirf)~(£) = /(£)(! k_1í2l)+If n > 3, A > ¿(n-l)/(n+l)and2 and the associated maximal operators are r;/(x) = suP|(/-(i-Ki2)i)-|(x). r>0 It is conjectured that, when A > 0, T\ is bounded on Lp if and only if pó(A) < p < Po(A), where po(A) = 2n/(n — 1 — 2A). That the restrictions on p and A are necessary was shown by Herz [7]. Carleson and Sjölin [3] proved the conjecture in dimension two. Moreover, in R2 Carbery [1] has established boundedness of Tx on Lp, and hence almost everywhere convergence of Bochner-Riesz means, for the same range of p and A except for the added restriction p > 2. In dimensions greater than two it is known by work of C. Fefferman, Stein and Tomas [5, 6, 10) that T\ is bounded on Lp, provided p'0(X) < p < po(A) and A > ^(n — l)/(n + 1), but the remaining cases have not been resolved. Our principal result is THEOREM 1. TJ is bounded on Lp(Rn) whenever 2 < p < p0(A) and A > \ (n l)/(n + 1) for alln>3. Interest in Lp bounds for T£ is due to the consequence COROLLARY. If f G Lp(Rn), n > 3, A > |(n l)/(n + 1) and 2 < p < p0(A), then lim(/-(l-|e/r|2)i)-(x) = /(x) a.e. r—>oo The proof is based on the L2 restriction theorem of Tomas and Stein [10]. Our second result is a small extension of that theorem, related in spirit to Theorem 1. THEOREM 2. Suppose p. is a nonnegative radial measure on RTM, satisfying p({0}) =0, andn > 2. Let 7 = n(nl)/(n+l). Suppose 1< p < 2(n+l)/(n + 3) and q = ((n — l)/(n + l))p'. Then a necessary and sufficient condition that the weighted norm inequality ||/||l9(R",<í^) ^ C||/IIlp(R") hold is that there exist A < 00 such that, for each 0 < r < 00, p{r < |£| < 2r} < Ar1. Received by the editors August 9, 1984. 1980 Mathematics Subject Classification. Primary 42B15, 42B25. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 16 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use BOCHNER-RIESZ MEANS 17 To begin the proof of Theorem 1 let us recall (Stein and Weiss [9] and Carbery [2]) that, for any e > 0, oo HTaVIIp < C\\Mf\\v + C(e) 52 2-fc(A-1/2-£)||S2-*/||p, fc=i where M is the maximal operator of Hardy and Littlewood and each Ss is a square function defined as follows: Consider any a G C7q°(R) supported in \-\, \] and satisfying ||Z)ao|[oo < 1 for all 0 < a < 10. Let «¿(¿f) = 4>(6)(í) = a(è~l(\ |£|)), 4>t(0 = Wt) and (roo \ 1/2 j \f*j>t\2(x)t-ldt\ . (To be more precise, each 52-* appearing in the above expression controlling II^a/IIp is defined by means of an auxiliary function a = ak which depends on k.) Define p0 = 2(n + l)/(n 1) and r = (±p0)' = (n + l)/2. LEMMA 1. If Ss is defined as above, then \\Ssf\\p < C(p)61-n/2r\\f\\p for all 2 \(n l)/(n + 1), then T{ is bounded on Lp. The full conclusion of Theorem 1 may then be obtained by interpolating with the more elementary result: T£ is bounded on Lp, for all 1 < p < oo, once A is greater than the critical index j(n — 1). (For in that case Tx is dominated pointwise by the Hardy-Littlewood maximal operator, since \((i-\t:\2)x+y(x)\ Mrh should have such a simple form is unexpected; in dimension two the weighted inequality established by Carbery for the Bochner-Riesz multipliers involves averaging over rectangles with eccentricity ¿>_1//2 and arbitrary orientations. Our proof relies on an argument given by Stein (see [6]), who reduces the problem of estimating the Bochner-Riesz operators to a local one on a fixed cube, and treats the local problem by applying the L2 restriction theorem. For the local problem one has not only Lp° boundedness, but boundedness of the operator in question from L2 to Lpo. Our sole innovation is the observation that this stronger local information automatically carries with it a weighted inequality. To formulate this principle abstractly we suppose that R" has been partitioned into a regular lattice of cubes of sidelengths 23, and that T is a sublinear operator with the property that, whenever / is supported in a cube Q of the lattice, Tf is supported in a fixed dilate Q* of Q. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use