On the boundedness of pseudo-differential operators
Alberto P. Calderón, Rémi Vaillancourt · Journal of the Mathematical Society of Japan · 1971
In this note we show that a bounded symbol $p(x, \xi)$ with bounded deriva- tives $\partial_{x}^{\beta}\partial_{\xi}^{a}p(x, \xi)$ defines a bounded pseudo-differential operatorormander [1] by the inequality $|\partial_{x}^{\beta}\partial_{\xi^{a}}p(x, \xi)|\leqq C_{\alpha\beta}(1+|\xi|)^{-\rho|\alpha|}$ are of this form.The result is new for $\rho=0$.Our proof makes use of a modification of a lemma of Cotlar (see [2]) for almost orthogonal operators in a Hilbert space.This problem was proposed to us by Hitoshi Kumano-go who will present shortly applications to parabolic and semi-elliptic1) operators.THEOREM.Let the symbol $p(x, \xi)$ be a matrix of functions $p_{ij}(x, \xi)$ defined on $R_{x}^{n}\times R_{\xi^{n}}$ such that $|\partial_{x_{n}}^{\theta n}\cdots\partial_{x_{1}^{1}}^{\beta}\partial_{\epsilon_{n}^{c_{n}}}^{t}\cdots\partial_{\xi^{1}}^{a_{1}}p_{if}(x, \xi)|\leqq C_{a,\beta}$ for $\alpha_{k},$ $\beta_{\iota}=0,1,2,3$ and all $x$ and $\xi$ .Then the pseudo-differential operator $(Pf)(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}p(x, \xi)f(\xi)d\xi$ , $f\in S$ , can be extended to a bounded operator from $L^{2}$ into $L^{2}$ .We state immediately the auxiliary lemma which will be proved later.LEMMA.Let $A_{z}$ be a z-weakly measurable and uniformly bounded family of operators in $L^{2},$ $\Vert A_{t}\Vert\leqq M_{0}$ for all $z$ in a measure space $Z$ with element of measure $dz$ .If the inequalities $\Vert A_{z}A_{l}^{*}\Vert\leqq h^{g}(z, z^{J})$ and $\Vert A_{l}^{*}A_{\iota^{\prime}}\Vert\leqq h^{2}(z, z^{J})$ hold with a nonnegative function $h(z, z^{J})$ which is the kernel of a bounded integral operator $H$ in $L^{2}$ with norm $M$, then the operator ' Research supported by the Office of Naval Research.1) For the larger class of $\lambda$ -elliptic operators, see Nagase and Shinkai [3].$P_{Seudo\cdot di_{JT^{erential}}}$ operators