Singular Integrals and Maximal Functions Associated to Surfaces of Revolution

Abstract

We obtain V estimates for singular integrals and maximal functions associated to hypersurfaces F in R n+1 , n ^ 2, which are obtained by rotating a curve around one of the coordinate axes.

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

“…Applying this result in the radial hypersurface setting where Ψ(x) = φ(|x| 2 ), G(x) = |x| 2 and so 0 = 2 and the codimension of E 2 = {0} is N , we recover the result in [7] regarding H Ψ when N ≥ 2. In the radial case as we mentioned earlier, these L 2 bounds do not in general extend to L p bounds, although when γ(…”

“…However there is a further interesting phenomenon related to the singular integral operator H Ψ in the radial hypersurface case; namely, that H Ψ is bounded on L 2 for any measurable φ as long as N ≥ 2; see [7]. This does not extend to L p , p = 2, see [10].…”

“…However there are better results for certain classes of hypersurfaces; for instance when Ψ(y) = φ(|y|) defines a radial hypersurface (in this case, Ψ is convex precisely when the function φ defined on R + is convex). Here one can exploit the nonvanishing curvature of the level sets of Ψ (which are Euclidean spheres -when N = 1 these spheres become two point sets and the underlying curvature is lost) to show that the corresponding operators H Ψ and M Ψ are bounded on L p , 1 < p < ∞; see [7]. We stress that this is valid for any convex radial hypersurface as long as N ≥ 2.…”

“…When the convexity assumption on φ is dropped, H Ψ and M Ψ can be unbounded on L p for nontrivial ranges of p (for all finite p in the case of the maximal operator M Ψ ); see [7] and [10]. However there is a further interesting phenomenon related to the singular integral operator H Ψ in the radial hypersurface case; namely, that H Ψ is bounded on L 2 for any measurable φ as long as N ≥ 2; see [7].…”

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].…”