Singular spherical maximal operators on a class of degenerate two-step nilpotent Lie groups

Singular Spherical Maximal Operators on a Class of Step Two Nilpotent Lie Groups

Let H ∼= R ⋉ R be the Heisenberg group and let μt be the normalized surface measure for the sphere of radius t in R. Consider the maximal function defined by Mf = supt>0 |f ∗ μt|. We prove for n ≥ 2 that M defines an operator bounded on L(H) provided that p > 2n/(2n− 1). This improves an earlier result by Nevo and Thangavelu, and the range for L boundedness is optimal. We also extend the result...

Singular Spherical Maximal Operators on a Class of Two Step Nilpotent Lie Groups

Let H be the Heisenberg group and let μt be the normalized surface measure for the sphere of radius t in R. Consider the maximal function defined by Mf = supt>0 |f ∗μt|. We prove for n ≥ 2 that M defines an operator bounded on L(H) provided that p > 2n/(2n − 1). This improves an earlier result by Nevo and Thangavelu, and the range for L boundedness is optimal. We also extend the result to a mor...

SINGULAR SPHERICAL MAXIMAL OPERATORS ONA CLASS OF TWO STEP NILPOTENT LIE GROUPSDetlef M uller

Geometry of 2-step Nilpotent Lie Groups

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Some Two–Step and Three–Step Nilpotent Lie Groups with Small Automorphism Groups

We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are “small” in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we also get new examples of compact manifolds covered by two-step simply connected nilpotent Lie groups which do not admit Anosov automorphisms...