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ak+1 assumptions Banach space belongs bounded coefficients bounded variation Cauchy»s problem characteristic polynomial Consider continuous functions converges defined denoted derivatives of order double differential operator double order dual dual space duality energy integral equivalence fact finishes the proof finite following lemma formula function f Hahn-Banach theorem Hence hermitian Hilbert space homogeneous hyperbolic equations hyperbolic operators implies inequalities for partial infinitely differentiable integrable functions left side Leray let f linear functional linear homeomorphism Lip0 Lip1 Lipp*q locally integrable locally integrable function LP*q norm normal notation operator of order Pa(x Pak+1 prove the lemma remains to prove Remark replaced respectively result right side satisfying scalar product second kind sense shows strongly dense suffices to prove tends to zero term theorem 7.1 theory total order unique solution vanishing in neighborhoods variable vector space weakly dense write