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REMARK ON COMMUTATIVE APPROXIMATE IDENTITIES ON HOMOGENEOUS GROUPS

By Jacek Dziubañski and Communicated J. Marshall Ash

Abstract

Abstract. We give, using the functional calculus of Hulanicki [4], a construction of a commutative approximate identity on every homogeneous group. In [2] Folland and Stein asked, whether on every homogeneous group yT there is a function <p in the Schwartz class S"(yf) with the following properties: JJ/.tp(x)dx = 1, tpt * (ps = 4>s * <Pt where <pt(x) = t~°-cf>(ôt-ix), and Q is the homogeneous dimension of JV. The family {</>,} is then called a commutative approximate identity and is used for characterizing Hardy spaces Hp (Jf), cf. [2]. Folland and Stein [2] produced a commutative approximate identity in the case when Jf is graded. Glowacki showed in [3] that for every homogeneous group the densities of the stable semigroup of symmetric measures generated by the functional (where £2 is a nonzero nonnegative smooth away from the origin homogeneous of degree 0 symmetric function and | • | is a smooth homogeneous norm on yf) belong to 3?(Jf), cf. [2, p. 253]. The construction of a commutative approximate identity from such a semigroup of measures was presented in [2, pp. 258-260]. The purpose of this note is to give (using the functional calculus of Hulanicki [4]) an alternative construction of a commutative approximate identity from the distribution P. We prove that on every homogeneous group yT the following theorem holds. Theorem 1. If F £ Cc(-l, 1), F = 1 in a neighborhood ofO, then there is a function tp £ S^iJV) such that /0° ° FiX) dEPiX)f = f * tp, where EP is the spectral resolution of the operator f>-, Pf — f * P. Corollary 1. Since ¡J/.(pix)dx = F(0) and /0°°F(tX)dEP(X)f = f *</>,, the family {4>t} forms a commutative approximate identity. Let U = {x £ JV: \x \ < 1} and t(x) = inf{«: x G U"). For every a> 0 the function wa(x) = (1 + r(x))a is submultiplicative. Moreover, there are Received by the editors August 28, 1990

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.4889
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