Circular maximal functions on the Heisenberg group

Abstract

We prove the $L^p$ boundedness of the circular maximal function on the Heisenberg group ${\mathbb{H}}^1$ for $2<p\le \infty$. The proof is based on the square sum estimate associated with the $2\times 2$ cone $|({\xi }_1^{\prime },{\xi }_2^{\prime })|=|({\xi }_3^{\prime },{\xi }_4^{\prime })|$ of the phase space arising from the vector fields $X_1,X_2,t{X}_3,\partial /\partial t$ on the Heisenberg group, rather than the $2\times 1$ cone $|({\xi }_1,{\xi }_2)|=|{\xi }_3|$ of the frequency space arising from $\partial /\partial x_1,\partial /\partial x_2,\partial /\partial t$ on the Euclidean space.

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References in corpus (1)

• Annulus Maximal Averages on Variable Hyperplanes