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Multiplier operators associated with the Cauchy problem for the wave equation. Difference regularization. (Russian) Zbl 0703.35099

The solution of the wave equation \(\square u=v\), \(u(0,x)=u_ 1\), \(\partial_ tu(0,x)=u_ 2\), can be represented by the aid of the Fourier operators \[ M^{\alpha}_ t: f\to M^{\alpha}_ t(f)=(2\pi)^{-n} \int e^{-ix\xi} m_{\alpha}(t| \xi |)\hat f(\xi)d\xi, \] where \(m_{\alpha}(\rho)=(n/2+\alpha)(\rho /2)^{1-n/2- \alpha} J_{n/2+\alpha -1}(\rho)\), \(J_ s(\rho)\) is the Bessel function of order s and \(\hat f(\xi)\) is the Fourier transform of \(f\in S({\mathbb{R}}^ n)\). The first goal of the work is to give precise description of the parameters \(\alpha\in {\mathbb{R}}\), \(r\in {\mathbb{R}}\), \(s\in {\mathbb{R}}\), \(p\in [1,\infty]\), \(q\in [1,\infty]\), for which \(M^{\alpha}_ t\) can be extended as a linear bounded operator \(M^{\alpha}_ t: L^ r_ p\to L^ s_ q\), where \(L^ r_ p\) is the Sobolev space with norm \[ \| \int e^{ix\xi} <\xi>^ r f(\xi)d\xi \|_{L^ p},\quad <\xi>=(1+| \xi |^ 2)^{1/2}. \] The second goal of the work is to approximate \(M^{\alpha}_ tf\), \(\alpha <0\), by a difference hypersingular integral. Namely, for \(f\in S({\mathbb{R}}^ n)\) set \[ g_ f(\eta)= (1/2)\Gamma(n/2+\alpha) \pi^{- n/2} \eta^{n/2-1}+ \int_{{\mathbb{S}}^{n-1}} f(x-\sigma \sqrt{\eta})d\sigma. \] Then the integral \[ M^{\alpha}_{t,\epsilon}f=t^{2-n-3} C_{\alpha,1}\int^{\infty}_{\epsilon}u^{\alpha - 1}(\sum^{\ell}_{j=0} \binom{\ell}{j}(-1)^ j g_ f(t^ 2- (1+j)u))du \] represents an approximation of \(M^{\alpha}_ t\) provided \(C_{\alpha,1}\) are suitable constants.
Reviewer: V.Georgiev

MSC:

35L05 Wave equation
35G05 Linear higher-order PDEs
42A45 Multipliers in one variable harmonic analysis

Keywords:

Sobolev space
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