Singular convolution operators on the Heisenberg group

Singular Convolution Operators on the Heisenberg Group

1. Statement of results and outline of method. The purpose of this note is to announce results dealing with convolution operators on the Heisenberg group. As opposed to the well-known situation where the kernels are homogeneous and C°° away from the origin, the kernels we study are homogeneous but have singularities on a hyperplane. Convolution operators with such kernels arise in the study of ...

Strongly Singular Convolution Operators on the Heisenberg Group

We consider the L mapping properties of a model class of strongly singular integral operators on the Heisenberg group H; these are convolution operators on H whose kernels are too singular at the origin to be of Calderón-Zygmund type. This strong singularity is compensated for by introducing a suitably large oscillation. Our results are obtained by utilizing the group Fourier transform and unif...

Weak Type Estimates for a Singular Convolution Operator on the Heisenberg Group

On the Heisenberg group Hn with coordinates (z, t) ∈ Cn × R, define the distribution K(z, t) = L(z)δ(t), where L(z) is a homogeneous distribution on Cn of degree −2n, smooth away from the origin and δ(t) is the Dirac mass in the t variable. We prove that the operator given by convolution with K maps H1(Hn) to weak L1(Hn).

p-ESTIMATES FOR SINGULAR INTEGRALS AND MAXIMAL OPERATORS ASSOCIATED WITH FLAT CURVES ON THE HEISENBERG GROUP

The maximal function along a curve (t, γ (t), tγ (t)) on the Heisenberg group is discussed. The L p-boundedness of this operator is shown under the doubling condition of γ ′ for convex γ in R. This condition also applies to the singular integrals when γ is extended as an even or odd function. The proof is based on angular LittlewoodPaley decompositions in the Heisenberg group.

Inequalities for Strongly Singular Convolution Operators