A maximal function on harmonic extensions of $H$-type groups

Vallarino, Maria

Annales mathématiques Blaise Pascal, Tome 13 (2006) no. 1, pp. 87-101.

Let $N$ be an $H$-type group and $S\simeq N\times {ℝ}^{+}$ be its harmonic extension. We study a left invariant Hardy–Littlewood maximal operator $M_{\rho }^{ℛ}$ on $S$, obtained by taking maximal averages with respect to the right Haar measure over left-translates of a family $ℛ$ of neighbourhoods of the identity. We prove that the maximal operator $M_{\rho }^{ℛ}$ is of weak type $(1,1)$.

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