Journal für die reine und angewandte Mathematik

On quasiradial Fourier multipliers and their maximal functions

By Andreas Seeger at Darmstadt

In this paper we consider quasiradial Fourier multipliers m о ^, where m:(0, oc)—> С and q e С'^(Щ) is positive in Щ and homogeneous with respect to a dilation group (А^),

( 1 ) QiAtX)^tQ{x); xeM\ r^O.

The dilation matrix A^ is defined by Л^ = t^; P is a real n x n-matrix whose eigenvalues have positive real parts, q is called an v4j-homogeneous distance function.

We use standard notations; с is a general constant which need not be the same in different estimates. The domain of integration is omitted if we integrate over M".

Multiplier criteria and results for maximal functions can be deduced from mapping properties of the following square function:

which was first introduced by E. M. Stein [19].

The following LMnequalities are known (cf. [13], [6], [2], [14], [17]):

UÀf ) \\çuc\\f\\ „

( 3 )

Я>п ( ' - - | ) + | , \<VU2 and ^>«({-;^) + ;^' 2^P<oo-

This is best possible for \<vuX also the assumption eeC"(№g) is not really necessary; however in two dimensions, Carbery [1], [2] gave an essential improvement

for2<p<oo , e(<^) = |^|:

( 4 ) \\g,{f)\\Ln«^uc\\f\\LH«^), A>2(^|--j^2-

The purpose of this paper is to prove a partial extension of (4) in higher dimensions.