Fourier Analysis 2013
DOI: 10.1007/978-3-319-02550-6_3
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A Note on Complete Hyperbolic Operators with log-Zygmund Coefficients

Abstract: Abstract. The present paper is the continuation of the recent work [7], and it is devoted to strictly hyperbolic operators with non-regular coefficients. We focus here on the case of complete operators whose second order coefficients are log-Zygmund continuous in time, and we investigate the H ∞ well-posedness of the associate Cauchy problem. Mathematics Subject Classification (2000). Primary 35L15; Secondary 35B65, 35S50, 35B45.

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Cited by 8 publications
(30 citation statements)
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“…Recently, Colombini and Del Santo, in [7] (for a rst approach to the problem see also [10], where smoothness in space were required), came back to the Cauchy problem for the operator (1), mixing up a Tarama-like hypothesis (concerning the dependence on the time variable) with the one of Colombini and Lerner (with respect to x). More precisely, they assumed a punctual log-Zygmund condition in time and a punctual log-Lipschitz condition in space, uniformly with respect to the other variable (see conditions (9) and (10) below).…”
Section: Lu(t ·) H S Dtmentioning
confidence: 99%
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“…Recently, Colombini and Del Santo, in [7] (for a rst approach to the problem see also [10], where smoothness in space were required), came back to the Cauchy problem for the operator (1), mixing up a Tarama-like hypothesis (concerning the dependence on the time variable) with the one of Colombini and Lerner (with respect to x). More precisely, they assumed a punctual log-Zygmund condition in time and a punctual log-Lipschitz condition in space, uniformly with respect to the other variable (see conditions (9) and (10) below).…”
Section: Lu(t ·) H S Dtmentioning
confidence: 99%
“…For rst and third inequalities, the proof is the same as in [7]. We have to pay attention only to (23).…”
Section: 3mentioning
confidence: 99%
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“…By analogy with what done in [5] (see also [8], [6] and [7] for the case of localized energy), we immediately link the approximation parameter ε with the dual variable ξ, setting eq:param eq:param (38) …”
Section: Energymentioning
confidence: 99%