Multiplier operators connected with the cauchy problem for the wave equation. Difference regularization

For the operator M_t^at > 0, a + n/2¹0, -1, -2, …, defined on Fourier transforms of Schwartz functions to w Î(Rⁿ) by the relation (Here formula is presented) the question of extension to a bounded linear operator m₁^a L_p^r® L_q^sis considered, where L_p^rand L_q^sare Lebesgue spaces of Bessel potentials, 1 £ p, q £ ¥, and -¥<r, s < ¥. Sharp conditions are obtained under which such an extension is possible. An explicit representation of m_t^af is given for a < 0 and f Î L_p^r1 £ p < ¥, r ³ 0, in the form of a difference hypersingular integral converging in the L_p^r-norm and almost everywhere. For the operator generated by the Fourier multiplier (Here formula is presented) an assertion is obtained regarding the convergence of m₁^abj, jÎ L_pas t ® 0 in the L_q^snorm and almost everywhere which generalizes a familiar result of Stein corresponding to the case b = 0. The results are applied to the investigation of the Cauchy problem for the wave equation in the scale of spaces L_p.