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The antilocal property (to all directions)of an operator A states that, if f=Af=0 in a domain U, then / is identically zero on the whole space. (For the precise definition see §2). This property is known to hold for the class of fractional powers of the Laplacian: A―Ax, where i is a non-integral real number, by Goodman-Segal [6] when the space dimension is odd and by Murata [14] in general. Liess [13] showed this property for the fractional powers of the ellipticdifferentialoperator with analytic coefficients using the theory of pseudodifferential operators with analytic symbol. As is well known, the fractional power Aa/2is the generator of the isotropic stable process in case 0 0, then /=0 on (0, + oo) but not necessarily f=Q in (-co, 0). The result has been shown for aG(0, 1) and will be extended to fl£(0,1)W(1, 2) in §4, b), 2). [It is stillopen if the asymmetric Cauchy generators (the generalized case for a=l) have the similar property.] The proof is carried out on the similar line as in Liess [13], and so Sato-Kashiwara-Kawai theory is essentialin the proof. The firstaim of this note is to show what type of antilocal property holds for a finitesum of one-dimensional stable generators, for example, the sum o1 a one-sided a-stable generator to the right and a one-sided /3-stablegeneratoi to the left. The solution will be given in §4, b), 3). In fact the example citec

Remarks on transmission, antitransmission and antilocal properties for sums of stable generators