The Hilbert Transform and Maximal Function Along Flat Curves, Dilations, and Differential Equations

MAXIMAL OPERATORS AND HILBERT TRANSFORMS ALONG FLAT CURVES NEAR L1

“…Carbery et al [6] showed that M and H are bounded on L p for all p ∈ (1, ∞) if (t) = (t, γ 2 (t), . .…”

“…Further key ingredients to the proof are an appropriate Calderón-Zygmund theory and suitable control on the dilations in hand. In our flat setting determined by the curvature condition (1.3), we rely on [4] and [6] to provide these ingredients.…”

Singular integral and maximal operators associated to hypersurfaces:L p theory

“…The first and third inequalities use the corresponding Littlewood-Paley theory (see [2] and [3]) and the second inequality uses the hypothesis that the maximal operator σ is bounded on all L p , p > 1, exactly as in [5].…”

Lp-boundedness of the Hilbert transform and maximal function along flat curves in ℝn

“…The convexity hypothesis for plane curves we replace by the "convexity" hypothesis used in the n-dimensional results of [6] and [4].…”

“…In [4] the authors find conditions on Γ under which (12c) holds, (12a) and (12b) being easily satisfied with an appropriate choice of the dilation matrices.…”