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generalization of the Fourier transform of a radial integrable function onEuclidean «-space) has [(« - l)/2] derivatives; in [6] Schwartz has improvedSchoenberg's result, actually obtaining a stronger version of Theorem A incase p = 1. Our main result, Theorem 1, represents a combination of theresults of Schwartz and Tomas in case of ordinary derivatives; the extensionto fractional ones is new. Furthermore, Theorem 1 allows us to deducenecessary conditions for radial Fourier Mpq multipliers (in an elementary way).A comparison of these conditions with sufficient ones enlightens to someextent the structure of radial Fourier multipliers.The author is obliged to the referee for pointing out the papers ofSchoenberg [5] and Schwartz [6].The following notation will be used: y, z E R", s, ..., x E R; S is the set

Some necessary conditions for radial Fourier multipliers