Smoothing theorems for Radon transforms over hypersurfaces and related operators

We extend the theorems of [M. Greenblatt, L^{p} Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron, J. Funct. Anal. 276 2019, 5, 1510–1527] on {L^{p}} to {L^{p}_{s}} Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving {L^{p}} to {L^{q}_{s}} boundedness results for such operators. Here {q\geq p} but s can be positive, negative, or zero. For many such operators we will have a triangle {Z\subset(0,1)\times(0,1)\times{\mathbb{R}}} such that one has {L^{p}} to {L^{q}_{s}} boundedness for {({1\over p},{1\over q},s)} beneath Z, and in the case of Radon transforms one does not have {L^{p}} to {L^{q}_{s}} boundedness for {({1\over p},{1\over q},s)} above the plane containing Z, thereby providing a Sobolev space improvement result which is sharp up to endpoints for {({1\over p},{1\over q})} below Z. This triangle Z intersects the plane {\{(x_{1},x_{2},x_{3}):x_{3}=0\}}, and therefore we also have an {L^{p}} to {L^{q}} improvement result that is also sharp up to endpoints for certain ranges of p and q.

Averages along Polynomial Sequences in Discrete Nilpotent Lie Groups: Singular Radon Transforms

This chapter concentrates on the averages of functions along polynomial sequences in discrete nilpotent groups, illustrating the problems that arise from studying these averages. Though special polynomial sequences can still use the Fourier transform in the central variables to analyze the operators, it appears that one needs to proceed in an entirely different way in the case of general polynomial maps, when the Fourier transform method is not available. This chapter is the first attempt to treat discrete Radon transforms along general polynomial sequences in the non-commutative nilpotent settings. It does so by analyzing the problem of L² boundedness of singular Radon transforms.