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The characterization of the Hardy space, H of the plane, as those integrable functions whose first order Riesz transforms are (or whose maximal function is) integrable is well known. J.-A. Chao and M. Taibleson have shown that there is a conjugate system characterization of H of a local 1 2 field that parallels the Riesz system characterization of H (R ). C. Fefferman has conjectured that "nice" conjugate systems, such as the second order Riesz 1 2 transforms would also give a characterization of H (R ). In the present paper a counter example of A. Gandulfo and M. Taibleson is described that shows that any conjugate system generated by an even kernel will fail to characterize H of a local field. A counter example of J. Garcia-Cuerva is described that shows that the second order Riesz system for the Euclidean plane (which is generated 1 2 by an even kernel) will fail to characterize H (R ) in the above sense. Let ƒ G L(R) and let ƒ*(*) = supy>0\f(x, y)\, where ƒ(*, y) is the Poisson integral off. We say that ƒ G H(R) iff ƒ* G L(R). Let (r, 0) be the polar representation of (xv x2) G R , and let ( • )" and ( • )" represents the Fourier transform and its inverse. The following characterization of H(R) is in [5, §8]: THEOREM A Iff is real-valued andfEL^R), then ƒ G H^R) iff (efy GL'iR). Similarly, if AT is a local field, e.g., a p-adic field, we may define f*(x) = Pfcez ' ƒ(*> *)l > where f(x, k) is the regularization of/. (See [6, Chapter IV].) We say that ƒ G H(K) iff ƒ* EL^K). The following characterization of H(K) follows from results of Chao and Taibleson [3] and Chao [1], [2]. THEOREM B. Suppose n is a multiplicative character on K that is unitary, ramified of degree 1, homogeneous of degree 0 and odd. If fEL(K) then fEH\K)iff{Tify EL\K). AMS (MOS) subject classifications (1970). Primary 42A18, 42A40; Secondary 12B99, 46J15.

Conjugate system characterizations of 𝐻¹: Counter examples for the Euclidean plane and local fields