Singular Integrals Associated to Hypersurfaces: L2 Theory

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

“…The space E 0 , 0 = r, corresponding to the convex polynomial H in R r consists of only the zero vector 0 and hence Theorem 4 in [14] shows that M is bounded on all L p , 1 < p ≤ ∞. This will complete the proof of Theorem 1.1 for M Ψ once (9) has been established.…”

“…As a step in this direction, a study was initiated in [14] to understand the L 2 phenomenon for the singular integral operator H Ψ where the level sets of Ψ are parameterized by a fixed convex function G of finite type. More precisely, instead of Euclidean spheres parameterized by G(y) = |y| 2 , one considers a general convex function G of finite type at the origin (that is, the graph defined by G has no lines tangent to infinite order at 0) such that G(0) = ∇G(0) = 0.…”

“…In [14] it was shown that for general φ ∈ C 1 , the L 2 boundedness of H Ψ depends on the codimension of E 0 where 0 is the smallest positive integer such that…”

“…• The case when E 0 = {0} has already been treated; see Theorem 4 in [14]. In fact, the strategy of the proof of Theorem 1.1 is to reduce ourselves to this case.…”

Multiparameter singular integrals and maximal operators along flat surfaces

“…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].…”