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

B. S. Rubin

Sbornik. Mathematics, Tome 68 (1991) no. 2, pp. 391-416

Résumé

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 $-\infty$, $-\infty$. Sharp conditions are obtained under which such an extension is possible. An explicit representation of $\mathscr M_{t^\alpha}f$ is given for $\alpha0$ 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.