A convolution estimate for a measure on a curve in ℝ

This paper contains almost-sharp LP Lq convolution estimates for measures on the curve (t, t2, t3, t4) in JR4. Let -y be the curve in ]R4 defined by -y(t) = (tt2,t3,t4), t e IR. For an interval I C R1 we consider the convolution operator T on R4 defined by r~~~~~~~~~ Tf (x)f (x -7-(t)) dt:5 x E R4 Our interest is in determining the type-set T of T the set of all ( , 5 ) E [0, 1] x [0, 1] such that T(LP(R4)) C Lq(JR4). Let A = (0,0), B = 3 1) C = 5 4 )5 D= (6, 15),E= (E70 56) and F = (1, 1). Since T is a convolution operator, T lies on or below the closed segment [A, F]. Estimating Tf when f is the characteristic function of a small ball shows that T lies on or above the line through A and B and so, by duality, on or below the line through F and E. Estimating Tf when f is the characteristic function of [0, 6] X [0,562] X [0,E3] X [0, E4] shows that T lies on or above the line through B and E. Thus T lies inside the closed quadrilateral ABEF (the closed convex hull of A, B, E, and F). If the interval of integration I is unbounded, then homogeneity considerations show that T C [B, E]. It is conjectured that these necessary conditions for the boundedness of T are also sufficient. By duality and interpolation, it would suffice to show that B E T if I = (0, oc). The best previous result [01] is that [C, D] C T for any I. Here we establish an almost sharp result when I is bounded. Theorem. If I is bounded, then, excepting possibly the segments (A, B], [B, C), (D, E], and [E, F), the closed quadrilateral ABEF lies in 77. Proof. Let G = ( 2 v 2). By duality, interpolation, and the result of [01], it is enough to show that [G, B) C 77. And a homogeneity argument shows that it is enough to do this for I = [0, 1]. To this end we will study an analytic family T, of convolution operators given formally by 1 j1 J1 Tz f (x = roy(Z ) //f (x y(t) u-// (t)) Jl'b dub dt. Received by the editors May 12, 1997. 1991 Mathematics Subject Classification. Primary 42B15. The author was partially supported by a grant from the National Science Foundation. (@)1999 American Mathematical Society 217 This content downloaded from 157.55.39.243 on Thu, 06 Oct 2016 04:53:25 UTC All use subject to http://about.jstor.org/terms 218 DANIEL M. OBERLIN We will establish, with acceptable estimates on the growth of the operator norms, that 2 ~~~~~3 (1) T; is bounded on L2(R14) when Re(z) =-2 and, if H= (3 that (2) Tz maps LP(]R4) into L 4) if( E [G,H) and Rez= -2. p q ~~~~~~3 Analytic interpolation and the fact that T-1 is a multiple of T then show that [G, B) C T as claimed. To establish (1) we need to estimate the Fourier transform T_3 +is given formally by -2+iS(Y F((1 +is)/2) j eiY(7(t)?u , s, v)-=24(s _ t)2 1 ( _ t)2 6(uz + v)j|. Let m denote Lebesgue measure on R4. Rom (8) it follows that if U C R4 is open and if the measure Au is defined on Ob(U) by /(U) hdAu = J ho q(t, U, s, v)J(t, u, s, v) d(t, u, s, v), then (10) m(E) 0, JXe x Qe~fl{ g~x~v) I J(t, u, s, v)d(t, u, s, v) aj(t,u,s,v)j } a4 Re This content downloaded from 157.55.39.243 on Thu, 06 Oct 2016 04:53:25 UTC All use subject to http://about.jstor.org/terms 220 DANIEL M. OBERLIN This will follow from the inequality, uniform in A > 0 and (s, v) E Qe, Ijefl{J~t~ ,s~v) 0 and 6 < Jkl < 12 we have (11) tt-kj dt < CA4. {t2 It2 _k<A} But this inequality is easily checked by considering separately the cases of large and small A. The first case follows from the fact that if Itl is large, then t2 t2 -k t4' leading to (11). In the case of small A, t2jt2 kI < A implies either t2 t2 k, Ct2 or t2t2 -k Clt ? vkj, leading to estimates of CA3 or CA2 respectively for the left-hand side of (11). These complete the proof of (4). Now define operators Vk by r1 Vkf(x)= j jk<ju<2k+ f(x -y(t) u-"/(t)) dudt. Let IlVkllpq denote the norm of Vk from LP(]R4) to Lq(]R4). Then, since