Maximal functions and Hilbert Transforms along variable flat curves

Abstract

Abstract. In this work we establish L p boundedness for maximal functions and Hilbert transforms along variable curves in the plane, via L 2 estimates for certain singular integral operators with oscillatory terms. §1. IntroductionIn this paper, we study the L p (R 2 ) boundedness for the maximal function M and the Hilbert transform H along variable curves. In our discussions, these are defined a priori on functions inandwhere S(x, y) is a suitable real-valued function vanishing on the diagonal. We shall also consider the singular integral operators T λ (acting on functions on the real line) which are of the formLocal versions of the operators T λ have been studied by Phong and Stein [PS], and by Pan [P] who proved the L p boundedness of T λ with bounds independent of λ when the mixed derivative S xy does not vanish to infinite order at any diagonal point (x 0 , x 0 ). In [S] Seeger showed that for a certain class of phases S without this finite type property, the associated operators T λ are also uniformly bounded on L 2 . Here we extend the Seeger-type result, for a different, closely related (but not always directly comparable) class of phases, to all other p, 1 < p < ∞.

Singular Radon transforms and maximal functions under convexity assumptions

“…L p theorems in somewhat different variable coefficient settings are in [3], [4] and in [2]. More closely related to the setting here is the recent paper by Carbery and Pérez [1] who proved L p bounds for the semi-translation-invariant case under more restrictive third order assumptions. Optimal results on the Heisenberg group related to Theorem A were obtained by J. Kim [15], [16].…”

“…To fix our notation let Ω 0 , Ω 1 , Ω be open sets in R 2 with compact closure, so that Ω ⊂⊂ Ω 1 ⊂⊂ Ω 0 . We assume that for each x ∈ Ω 0 we are given a curve Here ε ≤ c 0 .…”

Hilbert transforms and maximal functions along variable flat curves

“…Seeger [10] and Carbery and Pérez [3] have obtained results for translation-invariant families of flat curves that do, however, take one out of the realm of translation-invariant operators. Another approach to the study of non-translation-invariant operators …”

Oscillatory hyper Hilbert transforms along curves

“…For the nature and motivation of studying such kind of convolution operators, we refer the reader to see our previous paper [1], as well as [2][3][4][5][6][7][8][9][10][11], and references therein. We also encourage the reader to see the most recent papers [12][13][14][15] on Hilbert transforms along variable curves.…”