Multipliers in Hp(Rn), 0< p<∞

Abstract

In this paper we develop some recent results of Calderdn and Torchinsky [2] concerning H p multipliers in order to present sharp conditions in terms of directional derivatives of the multiplier function m(~) which will assure that the associated translation invariant operator T defined by means of its Fourier transform by0~ suppf, preserve the Hardy spaces HP(R"), 0<p< ~o.In our context a tempered distribution f is in HP(R ") if M(u,x) = sup lu(y, t)l Q(x--y)~=t is in LP(R~), ItfLlup=llM(u)IIp, 0<p<~, where O(x) denotes the parabolic metric associated to the group {tP}t>0 with (Px, x)~(x,x), trace P=~, and u(y,t)= (f,q~t)(Y) is an extension of f to R+ +1 by means of convolution with a function q~t(y)=t-r~o(t-Py) in the Schwartz class S with non-vanishing integral, see [1].When P=I, ~=n and ~(x)=[x I these spaces coincide with the H p spaces of several real variables considered in [5].A bounded function re(C) is an H p multiplier with norm ~K if [IZfl]np<=gIIfll~t,.Since HP=L p for p>l and m is a multiplier in L p if and only if it is an L p' multiplier with lip § lip'= 1 we will assume throughout that p<=2.Bounded functions m(r are the L2(R ") multipliers.We study here conditions on the smoothness of rn(~) and on its decay, together with its derivatives, at infinity that will imply that m(~) is also a multiplier for some p<2.

Locations

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