Boundedness of the dyadic maximal function on graded Lie groups

Michael Ruzhansky, Julio Delgado, Duván Cardona

Abstract: Let

1 < p l e q i n f t y

and let

n g e q 2 .

It was proved independently by C. Calder'on, R. Coifman and G. Weiss that the dyadic maximal function �egin{equation*} mathcal{M}^{dsigma}_Df(x)=sup_{jinmathbb{Z}}left|smallintlimits_{mathbb{S}^{n-1}}f(x-2^jy)dsigma(y) ight| end{equation*} is a bounded operator on

L^{p} (m a t h b b R^{n})

where

d s i g m a (y)

is the surface measure on

m a t h b b S^{n - 1} .

In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure

d s i g m a

with compact support on a graded Lie group

G,

we associate the corresponding dyadic maximal function

m a t h c a l M_{D}^{d s i g m a}

using the homogeneous structure of the group. Then, we prove a criterion in terms of the order (at zero and at infinity) of the group Fourier transform

w i d e h a t d s i g m a

of

d s i g m a

with respect to a fixed Rockland operator

m a t h c a l R

on

G

that assures the boundedness of

m a t h c a l M_{D}^{d s i g m a}

on

L^{p} (G)

for all

1 < p l e q i n f t y .

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