## 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

Sobolev space
Full Text: