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Multiplier operators connected with the Cauchy problem for the wave equation. Difference regularization
B. S. Rubin
Abstract:
For the operator
$M_{t^\alpha}$,
$t>0$,
$\alpha+n/2\ne0,-1,-2,\dots$, defined on Fourier transforms of Schwartz functions
$\omega\in S(\mathbf R^n)$ by the relation
$$
F[M_{t^\alpha}\omega](\xi)=m_\alpha(t|\xi|)F[\omega](\xi),\quad m_\alpha(\rho)=\Gamma\biggl(\frac n2+\alpha\biggr)\biggl(\frac\rho2\biggr)^{1-n/2-\alpha}J_{n/2+\alpha-1}(\rho),
$$
the question of extension to a bounded linear operator
$\mathscr M_{t^\alpha}\colon L_p^r\to L_q^s$ is considered, where
$L_p^r$ and
$L_q^s$ are Lebesgue spaces of Bessel potentials,
$1\leqslant p\leqslant\infty$,
$1\leqslant q\leqslant\infty$, and
$-\infty<r<\infty$,
$-\infty<s<\infty$. Sharp conditions are obtained under which such an extension is possible. An explicit representation of
$\mathscr M_{t^\alpha}f$ is given for
$\alpha<0$ and
$f\in L_p^r$,
$1\leqslant p<\infty$,
$r\geqslant0$, in the form of a difference hypersingular integral converging in the
$L_q^s$-norm and almost everywhere. For the operator
$M_{t^{\alpha,\beta}}$ generated by the Fourier multiplier
$$
\mu_{t,\alpha,\beta}(\xi)=(1+|\xi|^2)^{-\beta/2}m_\alpha(t|\xi|),
$$
an assertion is obtained regarding the convergence of
$M_{t^{\alpha,\beta}}\varphi$,
$\varphi\in L_p$, as
$t\to0$ in the
$L_q^s$-norm and almost everywhere which generalizes a familiar result of Stein corresponding to the case
$\beta=0$. The results are applied to the investigation of the Cauchy problem for the wave equation in the scale of spaces
$L_p^r$.
Figures: 4.
Bibliography: 43 titles.
UDC:
517.983
MSC: 35L05,
35L15,
42B15 Received: 27.04.1987
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