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This article is cited in 3 scientific papers (total in 3 papers)
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.
Received: 27.04.1987
Citation:
B. S. Rubin, “Multiplier operators connected with the Cauchy problem for the wave equation. Difference regularization”, Math. USSR-Sb., 68:2 (1991), 391–416
Linking options:
https://www.mathnet.ru/eng/sm1674https://doi.org/10.1070/SM1991v068n02ABEH002109 https://www.mathnet.ru/eng/sm/v180/i11/p1524
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