Real Analytic Functions on Product Spaces and Separate Analyticity

Let f be a function on the product space V × W, where V and W are analytic manifolds, both either real or complex. The function f is said to be analytic (or bi-analytic) on V × W if it is analytic in the analytic structure induced on V × W by the corresponding structures on V and W. The function f is said to be separately analytic on V × W if, for each x in V, the function f(x, .) is analytic on W while, for each y in W, the function f (. ,y) is analytic on V. In the case of complex analytic manifolds, the classical theorem of Hartogs (3, chapter VII) states that the two notions of analyticity and separate analyticity are equivalent. For real analytic manifolds, it is known that such an equivalence does not hold, even if one adds the additional hypothesis that f is infinitely differentiate on V × W.