Operators associated to flat plane curves: Lp estimates via dilation methods

Multiparameter singular integrals and maximal operators along flat surfaces

“…Interestingly it turns out that we do not need any additional condition such as the doubling type condition of γ and h in any direction t 1 or t 2 . However as we extend this to general Γ, we come up with a curvature condition which is close to infinitesimal doubling condition of curve theory in [1].…”

“…Singular integral for odd γ was proved to be bounded in L 2 if and only if for some C > 0, h(Ct) ≥ 2h(t) for all t > 0 where h(t) = tγ (t) − γ(t) in [8]. The L p boundedness with p = 2 was also obtained in [1] if there is > 0 such that h (t) > h(t)/t for all t > 0. Singular integrals associated with higher dimensional flat submanifold of the form (t, γ(|t|)) : t ∈ R n have been considered in [7,11,12,14].…”

“…By inductive application of Theorem 4 and the fact that γ infinitesimal doubling implies h (where h(t) = γ (t)t − γ(t)) infinitesimal doubling in [1], we obtain that…”

“…It is interesting to weaken the hypothesis of (5.2) by using the appropriate h function of Γ and making a general theorem corresponding to curve theory in [1,8]. It is known in [2] (2 j 1 t 1 , .…”

“…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.…”