A remark on singular Calderón-Zygmund theory

Abstract

ABSTRACT. It is shown that in Rn the operator /+oo f(X! -t,...Xn-tn)t-ldt ■oo maps L(logL) to weak L1 locally. A slight variant of the Calderón-Zygmund procedure provides a new approach to the previously known Lp boundedness of H, 1 < p < oo. Relatively sharp bounds are obtained as p -> 1+, and extrapolation produces the result for L(logL).Let a = al < ■ ■ ■ < an G Z, and for all t G R let -y(i) = (tai,..., ta") G Rn. The Lp boundedness of operators such as Hf(x) -pv J f(x -7(í))í_1 dt has been studied by a number of authors [SW,JRdF,G,8,Cl,C2,PS]. In all this work there are two main steps: First one proves L2 bounds, and then further arguments are used to pass to Lp. A prototypical technique for deducing Lp bounds from L2 bounds is the Calderón-Zygmund theory of singular integrals. As is well known [Si], if K agrees with a function away from the origin, K G L°° and K satisfies the Hörmander condition (1) below, then convolution with K is bounded on all IP and is of weak type onL1. H is a limiting case just outside the scope of that theory, for it is given by convolution with a distribution K which is homogeneous under the family of dilations 8rx = (raix\,..., ranxn), but which is equal to a difference of two Dirac measures when restricted to the unit sphere. The Hörmander condition fails, but several substitute arguments have been found [SW, JRdF, Cl]. Our purpose here is to indicate a variant of the Calderón-Zygmund procedure which does apply to H ; the main point will be that H satisfies a certain generalization of the Hörmander condition. Our method applies to a more general class of convolution operators as well as to related maximal functions. This technique appears to be slightly more precise than the reasoning used in previous studies of H and related operators. Fix a bounded subset B of Rn. L1'00 denotes the usual space weak L1, equipped with the natural quasi-norm. THEOREM 1. H is a bounded operator from L(logL)(B) to L1'co(B). The same holds for the maximal function M1f(x) = supr>0 r~l f0<t<r \f(x -l(t))\ dt.COROLLARY. limr_>0?'~1/0<i<r./'(x _ l{t))dt = f(x) a.e. for all f locally in L(logL).Define \x\ to be the reciprocal of that value of r for which 8rx G S"-1 if x ^ 0, and 101 = 0. Then a triangle inequality \x + y\ < Co(\x\ + \y\) holds. By a ball we

MAXIMAL OPERATORS AND HILBERT TRANSFORMS ALONG FLAT CURVES NEAR L1

“…The nonvanishing curvature hypothesis in Theorem 1.1 guarantees an estimate of the form (1.2) via van der Corput's lemma. This L log L result essentially reproved earlier work of Christ and Stein [9] who, for example, considered the class of homogeneous curves in R d given by (t) = (t α 1 , . .…”

Imaginary powers of Laplace operators

“…We will follow the idea of C. Fefferman [11] of using more information of the L 2 operator norm (in our case, L iα L 2 →L 2 = 1) by smoothing out the bad functions b i at a scale smaller than the size of its support and considering this part of the good function where L 2 estimates can be used (see also [4]). In our case for each…”

Strongly singular integrals along curves

“…We now turn our attention to the proof of Theorem 1.2, which relies on the result obtained by Christ and Stein [1987]. Indeed, we shall show that the general statement they proved applies to the operator (2).…”

“…He also pointed out that H 1 → L 1 boundedness cannot hold. A previous result of Christ and Stein [1987] had MSC2000: 44A12, 42B20. Keywords: strongly singular integrals, Radon transforms.…”