PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 3, Pages 841–845 S 0002-9939(03)07092-8 Article electronically published on August 19, 2003 THE CAUCHY PROBLEM FOR A CLASS OF KOVALEVSKIAN PSEUDO-DIFFERENTIAL OPERATORS ROSSELLA AGLIARDI AND MASSIMO CICOGNANI (Communicated by David S. Tartakoff) Abstract. We prove the H ∞ well-posedness of the forward Cauchy problem for a pseudo-differential operator P of order m ≥ 2 with the Log-Lipschitz continuous symbol in the time variable. The characteristic roots λ k of P are distinct and satisfy the necessary Lax-Mizohata condition Imλ k ≥ 0. The Log- Lipschitz regularity has been tested as the optimal one for H ∞ well-posedness in the case of second-order hyperbolic operators. Our main aim is to present a simple proof which needs only a little of the basic calculus of standard pseudo- differential operators. Introduction Let us consider the Cauchy problem  Pu(t, x)= f (t, x), 0 ≤ t ≤ T, x ∈ R n , ∂ j t u(0,x)= g j (x), 0 ≤ j ≤ m − 1 (1.1) for a pseudo-differential operator of Kovalevskian type P = D m t + m−1  j=0 A j (t, x, D x )D j t , A j (t) ∈ OPS m−j , (1.2) of order m ≥ 2 in [0,T ] × R n . One says that problem (1.1) is well posed in the Sobolev space H ∞ = H ∞ (R n )=  µ H µ (R n ) if for every f ∈C ([0,T ]; H ∞ ) and g j ∈ H ∞ , 0 ≤ j ≤ m, there is a unique solution u ∈C m ([0,T ]; H ∞ ). In this paper we are concerned with the question of what kind of regularity in the time variable t one has to assume for the A j ’s in (1.2) in order to obtain such a well-posedness. From [1] and [2] we know that for second-order strictly hyperbolic differential operators the sharp regularity is the Log-Lipschitz continuity: a function a(t) is said to be Log-Lipschitz continuous, in short a ∈ LL([0,T ]), if it satisfies |a(t) − a(s)|≤ C|t − s|   log |t − s|   , 0 < |t − s| < 1 2 . Received by the editors September 30, 2002 and, in revised form, November 5, 2002. 2000 Mathematics Subject Classification. Primary 35G10, 35L30. Key words and phrases. Strictly hyperbolic operators, energy estimates, Log-Lipschitz continuity. c 2003 American Mathematical Society 841 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use