𝐿^{𝑝} estimates for maximal functions and Hilbert transforms along flat convex curves in 𝐑²

Carlsson, Hasse; Christ, Michael; Córdoba, Antonio; Duoandikoetxea, Javier; Francia, José Luis Rubio de; Vance, James; Wainger, Stephen; Weinberg, David A.

doi:10.1090/s0273-0979-1986-15433-9

Cited by 51 publications

(72 citation statements)

References 10 publications

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“…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.…”

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confidence: 99%

A remark on singular Calderón-Zygmund theory

Christ¹,

Stein²

1987

Proc. Amer. Math. Soc.

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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 f00?'~1/0

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confidence: 99%

A remark on singular Calderón-Zygmund theory

Christ¹,

Stein²

1987

Proc. Amer. Math. Soc.

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“…An interesting problem is to establish the L p theory for flat manifolds, which are in lack of the finite type condition. We have fairly good understanding of the flat curves of the form (t, γ(t)) in [1,4,6,8], even though appropriate extensions for general flat surfaces are not well known. In [4,6], the L p boundedness of the maximal operators and singular integrals associated with the convex curves of the form (t, γ(t)) where γ(0) = γ (0) = 0 has been obtained under the doubling type condition of γ , that is, γ (Ct) ≥ 2γ (t) for all t > 0 with some C > 0.…”

Section: Introductionmentioning

confidence: 99%

Multiparameter singular integrals and maximal operators along flat surfaces

Cho¹,

Hong²,

Kim³

et al. 2008

Rev. Mat. Iberoam.

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We study double Hilbert transforms and maximal functions along surfaces of the form (t 1 , t 2 , γ 1 (t 1 )γ 2 (t 2 )). The L p (R 3 ) boundedness of the maximal operator is obtained if each γ i is a convex increasing and γ i (0) = 0. The double Hilbert transform is bounded in L p (R 3 ) if both γ i 's above are extended as even functions. If γ 1 is odd, then we need an additional comparability condition on γ 2 . This result is extended to higher dimensions and the general hyper-surfaces of the form (t 1 , . . . , t n , Γ(t 1 , . . . , t n )) on R n+1 .

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“…For instance, there are smooth Ψ such that the operators H Ψ and M Ψ are unbounded on L p (R 2 ), 1 ≤ p < ∞, see [12]; even in the case when Ψ is convex, [8] and [13] (see also Remarks 1.4 below). On the other hand, there are a number of results giving sufficient conditions on a convex Ψ so that the corresponding operators are bounded on L p (R 2 ), 1 < p < ∞; see for example, [2], [4] and [8].…”

Section: Introductionmentioning

confidence: 99%

“…• When N = 1 and G(t) = t 0 , the singular integral operator H Ψ is the Hilbert transform along an even convex curve in the plane and a necessary and sufficient condition for H Ψ to be bounded on L 2 in this case is known, see [8] (this was later extended to all L p , 1 < p < ∞ in [4]). The curve we construct will not satisfy this condition and so the associated H Ψ will not be bounded on any L p (R 2 ) in the case N = 1.…”

Section: Introductionmentioning

confidence: 99%