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• W2065706799 abstract "Let y: [-1, 11 -J RX be an odd curve. Set H,f(x) = PV ff(x y(t)) (dt/t) and M.J(x) sup h-' if(x y(t))l dt. We introduce a class of highly monotone curves in RI, n > 2, for which we prove that H. and Mr are bounded operators on L2(Rn). These results are known if has nonzero curvature at the origin, but there are highly monotone curves which have no curvature at the origin. Related to this problem, we prove a generalization of van der Corput's estimate of trigonometric integrals. Introduction. Let y: [-1, 1] -->RW be an odd continuous curve. For a test function f on Rn we define the Hilbert transform along y of f by rx T~ I X dt (Hyf)(x) = PVJ f(x -y(t)) t and the function along y of f by (Myf)(X) = sup f(xJ y(t))l dt. O 0, y([-e, e]) lies in the linear span of {y(k)(O): k = 1, 2, 3, . .. }. Stein and Wainger [8], [9] have shown that (1) and (2) hold if is well-curved and 1 0 and 4 is increasing on [0, 1]. Note that ?(t) = sgn(t)exp(-jtl-') satisfies (4), but that is not well-curved. Nagel and Wainger [4] have shown that (1) holds for 3 X > Ofor a 1, where Dn is a suitable differential operator of order n. In ?3 we use this estimate to prove Theorem 1 for highly monotone curves. To prove Theorem 2 we use the method of introduced by E. M. Stein in [5] and [6]. This technique has been used to prove many maximal theorems; for examples see Stein and Wainger [8], Nagel, Stein and Wainger [3] and Wainger [10]. This content downloaded from 207.46.13.148 on Sun, 11 Sep 2016 04:22:54 UTC All use subject to http://about.jstor.org/terms SINGULAR INTEGRALS AND MAXIMAL FUNCTIONS 437 The use of g-functions allows one to use the Fourier transform and reduce the maximal theorem to estimating trigonometric integrals. In ?4 we use a variant of the g-function in Stein and Wainger [9, p. 1292] and our generalization of van der Corput's lemma to prove Theorem 2 for highly monotone curves. It should be noted that the first application of the Fourier transform to the study of My was made by Nagel, Riviere and Wainger [2] in the special case -y(t) = (t, t2). I take this opportunity to thank my teacher and advisor, Professor Stephen Wainger, for the suggestions and encouragement he has given me in the course of this work. (I would also like to thank Professor Alexander Nagel and my fellow graduate students, Jim Vance and Dave Weinberg for many useful discussions.) 1. An estimate for trigonometric integrals. In this section, we prove a lemma which will be of use in ??3 and 4. Before stating the lemma we must introduce some notation. Given a smooth function a: [a, b] -> (0, oo) we define a differential operator Da by DJ(t) = (f/a)'(t). If a1, . .. , a , are n such functions, then we inductively define operators D', . . , D n D' = Dal; Dk+1 = D Dk for I S k X > O for each t in [a, b], then exp[if(t)] dt A. Now bef(t) dt b | t {ef(t) it so integration by parts yields the estimate ell (t)dt f'(b) + f'(a) f f' ) f'(b) A Now assume that the lemma is true for a given n > 1. Assume that f' is monotone and that D n+ 'f(t) > A for a < t < b. Set h = Dnf. Then D 'n+f = (h/la+1)'. Choose c in [a, b] so that h/lan+l is positive on (c, b) and h/a,n+, is negative on (a, c). Such a value of c exists, and is unique, since h/ /an+ 1 is increasing. This content downloaded from 207.46.13.148 on Sun, 11 Sep 2016 04:22:54 UTC All use subject to http://about.jstor.org/terms 438 W. C. NESTLERODE Write fb e 0() dt = fJ + fb = P + Q and estimate P and Q separately. To estimate P suppose that a < u < c. Then [P | | JUefj(') dt + c u. If a S t < u, then _ _ h h _ _h a+ (t) < (u) = (c) '(s) ds < -X(c u). an+1 an+ I an+1Ian Hence we have Dnf(t) = h(t) < -Xan+1(t)(c u) < -Xa,,+(a)(c u) for a < t < u. By the induction hypothesis IP| < Cn(Xal(a) an(a)an+1(a)(c u))'1l + c u for a < u < c. This estimate actually holds for each u < c since IPI S c a. Set c u = (Xa,(a) .* an+ l(a))-'/(n+ 1). Then we get IPI S (Cn + l)(ALa1(a) . an , (a))l The estimate of Q is made in a similar manner. Q.E.D. 2. Highly monotone curves. Let y: [0, N] Rn be a curve of class Cn with y(O) = 0. We inductively define functions a1, ... , an as follows. a 1; ak+I = D kyk for I < k < n. Here D 1, .n. , D' are the differential operators associated with a1, . .. , a,n as in ? 1. At each stage of this definition we must assume that ak is positive on (0, N) so that the operator Dk is well defined. We now consider the matrix W, = [DkyjIl<kj<n. It is easy to see that WY is upper triangular: D 'Y1 D 172 * * D y,n D 2Y2 ... D" @default.
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• W2065706799 date "1981-02-01" @default.
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• W2065706799 title "Singular integrals and maximal functions associated with highly monotone curves" @default.
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