On the Lp boundedness of the non-centered Gaussian Hardy-Littlewood Maximal Function
نویسندگان
چکیده
The purpose of this paper is to prove the L p (R n ; dd) boundedness, for p > 1, of the non-centered Hardy-Littlewood maximal operator associated with the Gaussian measure dd = e ?jxj 2 dx. Let dd = e ?jxj 2 dx be a Gaussian measure in Euclidean space R n. We consider the non-centered maximal function deened by Mf(x) = sup x2B 1 (B) Z B jfj dd; where the supremum is taken over all balls B in R n containing x. P. Sjj ogren 2] proved that M is not of weak type (1,1) with respect to dd. A more general result was obtained by A. Vargas 3], who characterized those radial and strictly positive measures for which the corresponding maximal operator is of weak type (1,1). However, these papers leave open the question of the L p (dd) boundedness of M for p > 1 and n > 1: The main result in this paper is Theorem 1 M is a bounded operator on L p (dd) for p > 1, that is, there exists a constant C = C(n; p) such that for f 2 L p (dd); kMfk L p (dd) Ckfk L p (dd) : We denote S n?1
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