Tachizawa, Kazuya

Kyoto University Research Information Repository

TitleA generalization of Calderon-Vaillancourt's Theorem(Harmonic Analysis and Nonlinear Partial DifferentialEquations)Author(s)Tachizawa, KazuyaCitation数理解析研究所講究録 (1999), 1102: 64-75Issue Date1999-06URL http://hdl.handle.net/2433/63184RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityA generalization of $\circ \mathrm{a}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}\acute{\mathrm{O}}\mathrm{n}- \mathrm{v}_{\mathrm{a}}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{t}’ \mathrm{s}$ Theorem- (Kazuya Tachizawa)1 IntroductionIn this paper we study $L^{2}$ boundedness of pseudodifferential operators with Weylsymbols. We give a generalization of Calder\’on-Vaillancourt’s theorem.For a symbol $\sigma\in S’(\mathrm{R}^{2}d)$ we associate a pseudodifferential operator $L_{\sigma}$ . If a $\in$$L^{2}(\mathrm{R}^{2d})$ , then we can easily prove that $L_{\sigma}$ extends to a bounded operator on $L^{2}(\mathrm{R}^{d})$ .On the other hand Calder\’on and Vaillancourt gave another condition for the $L^{2}$ bound-edness of pseudodifferential $\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}_{0}\mathrm{r}\mathrm{s}([1], [2]).\dot{\mathrm{T}}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{r}$ condition is about pseudodifferen-tial operators with Kohn-Nirenberg symbols. Similar results hold for the Weyl symbolcase$([3], [10])$ . A generalization of their results is known$([11])$ . This generalization doesnot contain the $L^{2}$ symbol case. In this paper we give a generalization of both results.First we give the definition of pseudodifferential operators with Weyl symbols.Let $W(f, g)$ be the Wigner transform of $f,$ $g\in S(\mathrm{R}^{d})$ , that is,$W(f, g)(x, \xi)=\int_{\mathrm{R}^{d}}e^{-2}f\pi i\xi\cdot p(x+\frac{p}{2})g(x-\frac{\overline p}{2})dp$for $x,$ $\xi\in \mathrm{R}^{d}$ .For $\sigma\in S’(\mathrm{R}^{2}d)$ and $f\in S(\mathrm{R}^{d})$ , we define $L_{\sigma}f\in S’(\mathrm{R}^{d})$ as$(L_{\sigma}f, g)=(\sigma, W(g, f))$1102 1999 64-75 64for all $g\in S$ , where we use the notation $(F, f)=\langle F, \overline{f}\rangle$ for $F\in S’,$ $f\in S$ . It turns outthat $L_{\sigma}$ is a continuous linear operator from $S$ to $S’$ . We call $L_{\sigma}$ a pseudodifferentialoperator with Weyl symbol $\sigma(\mathrm{c}\mathrm{f}.[10])$ .In Folland [10] it is proved that if $\sigma\in C^{2d+1}(\mathrm{R}^{2}d)$ and$\sum_{|\alpha|+|\beta|\leq 2d+1}||\partial_{x\xi}^{\alpha_{\partial^{\beta}\sigma||_{\infty}}}<\infty$,then $L_{\sigma}$ extends to a bounded operator on $L^{2}(\mathrm{R}^{d})$ . This is the Calder\’on-Vaillancourttheorem for pseudodifferential operators with Weyl symbols.In [11] Gr\"ochenig and Heil proved a generalization of Calder\’on-Vaillancourt’s the-orem. If $\sigma\in M_{\infty,1}(\mathrm{R}^{2d})$ , then $L_{\sigma}$ extends to a bounded operator on $L^{2}(\mathrm{R}^{d})$ , where$M_{\infty,1}(\mathrm{R}^{2d})$ denotes the set of $F\in S’(\mathrm{R}^{2d})$ satisfying$\int_{\mathrm{R}^{2d}}\sup_{2\mathrm{R}d}|(Fx\in’)e-)^{2}|i\xi\cdot-(\cdot xd\xi<\infty$.In this paper we give a generalization of these results.Let(1) $\varphi(x)=2d/4e-\pi|x|^{2}$ , $x\in \mathrm{R}^{d}$and$\phi(x, y)=W(\varphi, \varphi)(_{X}, y)=2^{d}e-2\pi(|x|^{2}+|y|^{2})$, $(x, y)\in \mathrm{R}^{2d}$ .For $\sigma\in S’(\mathrm{R}^{2d})$ and $\alpha,$ $\beta\in \mathrm{R}^{2d}$ we define$S_{\phi}(\sigma)(\alpha, \beta)=(\sigma, e\phi 2\pi i\beta\cdot(\cdot-\alpha))$ .For $\alpha=(\alpha_{1}, \alpha_{2}),$ $\beta=(\beta_{1}, \beta_{2}),$ $\alpha_{1},$ $\alpha_{2},$ $\beta 1,$ $\beta_{2}\in \mathrm{R}^{d}$ , we set$N( \alpha, \beta)=(-\frac{\alpha_{1}+\beta_{1}}{2}, \frac{\alpha_{2}+\beta_{2}}{2}, \alpha_{2}-\beta_{2}, \alpha_{1}-\beta 1)$.For $\xi,$ $\eta\in \mathrm{R}^{2d}$ we set$F(\xi, \eta)=|S_{\phi}(\sigma)(N(\xi, \eta))|$65and.(2) $k(\xi)=(1+|\xi|)-2d-1$ .For each positive integer $n$ and $a,$ $b\in \mathrm{R}^{2d}$ we set$G_{n}(a, b)$ $=$ $[ \int_{\mathrm{R}^{2d}}\cdots\int_{\mathrm{R}^{2d}}k(a-\eta 1)F(\xi_{1}, \eta 1)k(\xi 1^{-}\xi_{1}’)F(\xi_{1}/, \eta’1)k(\eta 1^{-\eta}2)/$$\cross F(\xi_{2}, \eta_{2})k(\xi_{2^{-}}\xi_{2}’)F(\xi’2’\eta 2)\prime k(\eta’2-\eta 3)\cdots$$\cross F(\xi_{n}, \eta_{n})k(\xi_{n}-\xi_{n}’)F(\xi;n’\eta n)\prime k(\eta n-\prime b)d\xi_{1}d\eta_{1}\cdots d\xi’nd\eta’n]1/n$Theorem 1.1 We assume$G_{n}(a, b)<\infty$for every $n\in \mathrm{N},$ $a,$ $b\in \mathrm{R}^{2d}$ and$\sup_{n\in \mathrm{N}}\sup_{a\in \mathrm{R}}G_{n}2d(a, a)<\infty$.Then $L_{\sigma}$ extends to a bounded operator on $L^{2}(\mathrm{R}^{d})$ .Corollary 1.1 Let $F(\xi, \eta)$ be a function as above. For $1\leq p\leq\infty$ and $p^{-1}+p^{\prime-1}=1$we assume$\{\int_{\mathrm{R}^{2d}}(\int_{\mathrm{R}}2d\eta F(\xi,)^{p}d\xi)^{p’}/pd\eta\}^{1/p}’<\infty$and$\{\int_{\mathrm{R}^{2d}}(\int_{\mathrm{R}^{2d}}F(\xi, \eta)\mathrm{P}d\eta)^{p/p}d’\xi\}^{1/p’}<\infty$ .Then $L_{\sigma}$ extends to a bounded operator on $L^{2}(\mathrm{R}^{d})$ .Remark 1.1 The case $p=1$ or $p=\infty$ is mentioned in Gr\"ochenig and Heil [11]. Theyused this fact to prove their result on $L^{2}$ boundedness of pseudodifferential operators.When $p=2$ , the conditions in the corollary is equivalent to saying $\sigma\in L^{2}(\mathrm{R}^{2d})$ . When$1<p<2$ or $2<p<\infty$ , our corollary gives a new resul.$t$ .66Remark 1.2 We can prove similar results about the boundedness of pseudodifferential$operator\mathit{8}$ on $So\dot{b}olev$ spaces.2 Proof of Theorem 1.1First we recall the definition of the Weyl-Heisenberg frame. For $a=(x, y)\in \mathrm{R}^{d}\cross \mathrm{R}^{d}$and $f\in L^{2}(\mathrm{R}^{d})$ we set$\rho(a)f(t)=\rho(X, y)f(t)=e^{\pi}eix\cdot y2\pi iy\cdot tf(t+x)$ ,where $t\in \mathrm{R}^{d}$ .Let $\varphi(x)$ be the function defined by (1). Let $I= \mathrm{Z}^{d}\cross\frac{1}{2\pi}\mathrm{Z}^{d}$ . Then $\{\rho(a)\varphi\}_{a}\in I$ is aframe of $L^{2}(\mathrm{R}^{d})$ , that is, there exist positive constants $C_{1}$ and $C_{2}$ such that$C_{1}||f||_{2}2 \leq a\in\sum I|(f, \rho(a)\varphi)|^{2}\leq C_{2}||f||_{2}^{2}$for all $f\in L^{2}(\mathrm{R}^{d})$ . The dual frame of $\{\rho(a)\varphi\}_{a}.\in I$ is given by $\{\rho(a)\tilde{\varphi}\}_{a}\in I$ , where $\tilde{\varphi}$ isthe function in $S(\mathrm{R}^{d})$ which is constructed by $\varphi$ . Furthermore we have$C_{2}^{-1}||f||^{2}2 \leq\sum_{Ia\in}|(f, \rho(a)\tilde{\varphi})|2\leq c_{1}-1||f||22$for all $f\in L^{2}(\mathrm{R}^{d})(\mathrm{c}\mathrm{f}.[5])$ .By the frame theory we have the following proposition.Proposition 2.1 (i) For every $f\in L^{2}(\mathrm{R}^{d})$ we have(3) $f$ $=$$\sum_{a\in I}(f, \rho(a)\varphi)\rho(a)\tilde{\varphi}$(4) $=$$\sum_{a\in I}(f, \rho(a)\tilde{\varphi}\rangle\rho(a)\varphi$which converge in $L^{2}(\mathrm{R}^{d})$ .67(ii) There exists a $K>0$ such that$|| \sum_{a\in I}C_{a}\beta(a)\varphi||_{2}\leq K(\sum_{a\in I}|c_{a}|^{2)^{1}}/2$for all $\{c_{a}\}\in\ell^{2}(I)$ .(iii) For every $f\in S(\mathrm{R}^{d})$ we have the $expansion\mathit{8}(\mathit{3})$ and (4) in $S$ .(iv) For every $f\in S’(\mathrm{R}^{d})$ we have the expansions (3) and (4) in $S’$ .The proofs of (i) and (ii) is in [5]. The properties (iii) and (iv) are consequences ofFeichtinger and Gr\"ochenig’s result $([7], [8], [9], [14])$ .Let $f\in S(\mathrm{R}^{d})$ . By (iii) and (iv) of Proposition 2.1 we have$L_{\sigma}f= \sum_{a\in I}(L\sigma f, \rho(a)\tilde{\varphi})\rho(a)\varphi$in $S’$ .If we show$\sum_{a\in I}|(L_{\sigma}f, \rho(a)\tilde{\varphi})|2<\infty$,then we conclude $L_{\sigma}f\in L^{2}(\mathrm{R}^{d})$ and$||L_{\sigma}f||^{2}2 \leq C\sum_{a\in I}|(L_{\sigma}f, \beta(a)\tilde{\varphi})|^{2}$,where we used (ii) of Proposition 2.1.Here we have$\sum_{a\in I}|(L_{\sigma}f, \rho(a)\tilde{\varphi})|2\sum_{I}=|a\in b\in\sum I(f, \rho(b)\tilde{\varphi})(L_{\sigma}\rho(b)\varphi, \rho(a)\tilde{\varphi})|^{2}$.If the infinite matrix $\{(L_{\sigma}\rho(b)\varphi, \rho(a)\tilde{\varphi})\}a,b\in I$ is bounded on $\ell^{2}(I)$ , then we concludethat$||L_{\sigma}f||2C \sum_{a}2\leq\in I|(f, \rho(a)\tilde{\varphi})|2\leq C’||f||_{2}^{2}<\infty$ .68Therefore $L_{\sigma}$ extends to a bounded operator on $L^{2}(\mathrm{R}^{d})$ .Now we use the following lemma to show the boundedness of an infinite matrix on$\ell^{2}$ .Lemma 2.1 ([4]) Let $A=(a_{ij})$ be an infinite matrix which acts on the sequence space$\ell^{2}(\mathrm{N})$ . Then the boundedness of $A$ from $\ell^{2}(\mathrm{N})$ to $\ell^{2}(\mathrm{N})$ is equivalent to the followingtwo conditions.(a) For every $n\in \mathrm{N},$ $(A^{*}A)^{n}$ is well defined.(b)$\sup_{n\in \mathrm{N}i}\sup_{\in \mathrm{N}}|[(A*A)^{n}]ii|^{1}/n<\infty$ ,where $[(A^{*}A)^{n}]_{ii}$ is the $(i, i)$ component of $(A^{*}A)^{n}$ .Remark 2.1 In [$\mathit{4}f$ Crone gave an additional condition. In [13] Maddox and Wicksteadshowed that the condition is unnecessary $(cf.[\mathit{1}\mathit{2}])$ .We shall show that the infinite matrix $\{(L_{\sigma}\rho(b)\varphi, \rho(a)\tilde{\varphi})\}a,b\in I$ satisfies the conditionsof Lemma 2.1. In order to prove this we use the following lemma by Gr\"ochenig andHeil [11].Lemma 2.2 For $\sigma\in S’(\mathrm{R}^{2d})$ and $f,$ $g\in S(\mathrm{R}^{d})$ , we have$(L_{\sigma}f, g)= \int_{\mathrm{R}^{2d}}\int_{\mathrm{R}^{2d}}s_{\phi()}\sigma(N(\alpha, \beta))e^{-}(2\pi i[\alpha,\beta]f, \rho(\beta)\varphi)(\rho(\alpha)\varphi, g)d\alpha d\beta$ ,where $[ \alpha, \beta]=\frac{1}{2}(\alpha_{2}\cdot\beta_{1^{-\alpha\cdot\beta_{2})}}1$ for $\alpha=(\alpha_{1}, \alpha_{2}),$ $\beta=(\beta_{1}, \beta_{2})\in \mathrm{R}^{d}\mathrm{x}\mathrm{R}^{d}$ .First we show the condition (a) of the Lemma 2.1.Let $A=\{(L_{\sigma}\rho(b)\varphi, \rho(a)\tilde{\varphi})\}a,b\in I,$ $n\in \mathrm{N}$ and $c,$ $d\in I$ . The $(c, d)$ component of theinfinite matrix $(A^{*}A)^{n}$ is given by$a_{1}, \ldots,a_{n’ 1}b\sum_{n-},\ldots,b1\in I\overline{(L\rho\sigma(C)\varphi,\rho(a1)\tilde{\varphi})}(L\rho\sigma(b1)\varphi, \rho(a_{1})\tilde{\varphi})\overline{(L\sigma\rho(b_{1})\varphi,\rho(a_{2})\tilde{\varphi})}$$\mathrm{x}(L_{\sigma}\rho(b2)\varphi, \rho(a_{2})\tilde{\varphi})\cdot*\cdot(L_{\sigma}\rho(bn-1)\varphi, \rho(a_{n})\tilde{\varphi})(L_{\sigma}\rho(d)\varphi, \rho(an)\tilde{\varphi})$ ,69where we will show that the series in the above sum absolutely converges.By Lemma 2.2 we have$\overline{(L_{\sigma}\rho(C)\varphi,\rho(a1)\tilde{\varphi})}(L\sigma\rho(b_{1})\varphi, \rho(a_{1})\tilde{\varphi})\overline{(L_{\sigma}\rho(b_{1})\varphi,\rho(a_{2})\tilde{\varphi})}$$\cross(L_{\sigma^{\beta}}(b_{2})\varphi, \rho(a2)\tilde{\varphi})\cdots\overline{(L\sigma\rho(b_{n}-1)\varphi,\rho(a_{n})\tilde{\varphi})}(L\sigma\rho(d)\varphi, \rho(an)\tilde{\varphi})$$=$ $\int\int\overline{S_{\emptyset}(\sigma)(N(\xi 1,\eta 1))}e(\rho(C)\varphi, \beta(\eta 1)\varphi)(\rho(\xi 1)2\pi i[\xi 1,\eta_{1}]\overline{(\varphi,pa1)\tilde{\varphi})}d\xi 1d\eta 1$$\cross\int\int s_{\phi}(\sigma)(N(\xi\prime 1’\eta_{1})’)e-2\pi i[\xi’1’\eta’1](\rho(b_{1})\varphi, \beta(\eta_{1})\varphi)(_{\beta(\xi)}\prime\prime 1\varphi, \rho(a_{1})\tilde{\varphi})d\xi_{1}\prime d\eta_{1}’\cdots$$\cross\int\int\overline{S_{\emptyset}(\sigma)(N(\xi_{n},\eta n))}e^{2}’\overline{(\rho(b_{n}-1)\varphi,\rho(\eta_{n})\varphi)(\beta(\xi n)}\pi i[\xi n\eta n](\varphi,$$\rho a_{ })\tilde{\varphi})d\xi_{n}d\eta n$$\cross\int\int S_{\emptyset}(\sigma)(N(\xi_{n}’, \eta’n))e^{-}n’\eta_{n}]’)2\pi i[\xi’(\rho(d)\varphi, \rho(\eta n)\varphi)(\rho(\xi_{n}’)\varphi, \rho(an\tilde{\varphi})\prime d\xi’nd\eta n’$.Let $k(\xi)$ be the function defined by (2). Since$|(\rho(b)\varphi, \rho(\eta)\varphi)|\leq|(\rho(b-\eta)\varphi, \varphi)|\leq ck(b-\eta)$and$|(\rho(\xi)\varphi, \rho(a)\tilde{\varphi})|\leq|(\rho(\xi-a)\varphi,\tilde{\varphi})|\leq ck(\xi-a)$for all $a,$ $b\in I$ and $\xi,$ $\eta\in \mathrm{R}^{2d}$ , we have$\sum_{a_{1},\ldots,a_{n},b1bn-1\in I},\ldots,\int\cdots I|S_{\phi}(\sigma)(N(\xi 1, \eta_{1}))||(\rho(c)\varphi, \rho(\eta_{1})\varphi)||(\beta(\xi 1)\varphi, \beta(a1)\tilde{\varphi})|$$\cross|S_{\emptyset}(\sigma)(N(\xi’1’\eta’1))||(\rho(b1)\varphi, \rho(\eta_{1})’\varphi)||(p(\xi_{1}’)\varphi, \beta(a1)\tilde{\varphi})|\cdots$$\cross|S_{\emptyset}(\sigma)(N(\xi n’\eta n))||(\rho(b_{n-}1)\varphi, \rho(\eta n)\varphi)||(\rho(\xi n)\varphi, \rho(an)\tilde{\varphi})||s\phi(\sigma)(N(\xi_{n}’, \eta’n))|$$\cross|(\rho(d)\varphi, \rho(\eta_{n}’)\varphi)||(\rho(\xi_{n}’)\varphi, \rho(a_{n})\tilde{\varphi})|d\xi_{1}d\eta_{1}d\xi^{\prime_{d\eta_{1}\cdot\cdot d}}1\xi’\cdot d\eta nnd\xi\prime nd\eta_{n}’$$\leq \mathit{0}^{n}\sum_{a_{1},\ldots,a_{n,1}b,..b_{n-}1\in I}.,\int\cdots\int F(\xi 1, \eta_{1})k(c-\eta 1)k(\xi_{1^{-}}a1)F(\xi’1’\eta_{1})k(b_{1}-\eta_{1}’)’$$\cross k(\xi_{1^{-}}’a1)F(\xi 2, \eta_{2})\cdots k(\xi n-an)F(\xi/n’\eta n’)k(d-\eta n’)k(\xi\prime n-an)d\xi_{1}\cdots d\eta_{n}’$.Since$\sum_{b\in I}k(\xi-b)k(\xi’-b)\leq Ck(\xi-\xi’)$70for all $\xi,$ $\xi’\in \mathrm{R}^{d}$ , the above quantity is bounded by$C^{n} \int\cdots\int k(c-\eta_{1})F(\xi_{1,\eta_{1})k(}\xi_{1^{-}}\xi_{1}’)F(\xi_{1}’,/)\eta_{1}k(\eta_{1}’-\eta_{2})F(\xi_{2}, \eta_{2})\cdots$$\mathrm{x}k(\xi_{n}-\xi_{n}’)F(\xi_{n}’, \eta’n)k(d-\eta’n)d\xi 1\ldots d\eta n$’$=$ $C^{n}G_{n}(c, d)^{n}<\infty$ .Hence we conclude that the $(c, d)$ component of the infinite matrix $(A^{*}A)^{n}$ is welldefined.Next we check the condition (b). By similar calculations we have$|_{a_{1},\ldots,a_{n}},b_{1},..b_{n}-1 \in I\sigma\sum.,\overline{(Lp(C)\varphi,\rho(a1)\tilde{\varphi})}(L_{\sigma}p(b_{1})\varphi, p(a_{1})\tilde{\varphi})\overline{(L\beta(\sigma b_{1})\varphi,\beta(a2)\tilde{\varphi})}$$\cross(L_{\sigma}\rho(b2)\varphi, \rho(a_{2})\tilde{\varphi})\cdots\overline{(L\rho\sigma(b_{n}-1)\varphi,\rho(a_{n})\tilde{\varphi})}(L_{\sigma}\rho(d)\varphi, \rho(a_{n})\tilde{\varphi})|$$\leq$ $C^{n}G_{n}(a, a)^{n}$ .Hence n-th root of the quantity of the left hand side is bounded by $CG_{n}(a, a)$ . By theassumption we get the condition (b).3 Proof of Corollary 1.1We shall prove Corollary 1.1 for $1<p<\infty$ . The proof for $p=1$ or $p=\infty$ is similar.We set$\{\int_{\mathrm{R}^{2d}}(\int \mathrm{R}2d)^{p’}F(\xi, \eta)^{p}d\xi/pd\eta\}^{1/p}’=K_{1}$ ,$\{\int_{\mathrm{R}^{2d}}(\int_{\mathrm{R}^{2d}}F(\xi, \eta)pd\eta)^{p/p}d\xi\}\prime 1/p’=K_{2}$and $K= \max\{K_{1}, K_{2}\}$ . We shall estimate(5) $\int\cdots I^{k(a-}\eta_{1})F(\xi_{1}, \eta 1)k(\xi 1^{-}\xi_{1}’)F(\xi/1’\eta 1)k(\eta_{1^{-}\eta_{2}})F(’/\xi_{2}, \eta 2)$$\cross k(\xi_{2^{-\xi_{2}’)\cdots F}}(\xi_{n}, \eta n)k(\xi_{n}-\xi_{n}’)F(.\xi_{n}’,’\eta_{n})k(\eta_{n}’-b)d\xi_{1\eta}d1\ldots d\xi\prime nd\eta_{n}’$71By H\"older’s inequality we have$\int k(a-\eta 1)F(\xi_{1,\eta}1)d\eta 1$$\leq$$( \int k(a-\eta 1)^{p}d\eta_{1})^{1/p}\prime\prime(\int F(\xi_{1}, \eta_{1})pd\eta_{1})^{1/}p$$\leq$$C_{\mathrm{i}}( \int F(\xi_{1}, \eta_{1})pd\eta 1)^{1/}p$ ,where we set$C_{1}=( \int k(a-\eta_{1})^{p}d\eta 1)^{1/}\prime p’$Hence (5) is bounded by$C_{1} \int\cdots\int(\int F(\xi_{1}, \eta_{1})pd\eta 1)1/p(k\xi_{1}-\xi\prime 1)F(\xi_{1}’, \eta_{1}^{;})k(\eta’1-\eta_{2})F(\xi 2, \eta_{2})k(\xi 2-\xi_{2}’)$. . . $F(\xi_{n}, \eta_{n})k(\xi_{n}-\xi’n)F(\xi’n’\eta’n)k(\eta_{n}’-b)d\xi 1d\xi/1\ldots d\xi^{;}nd\eta_{n}’$ .Now we have$\int(\int F(\xi_{1,\eta_{1}})^{p}d\eta_{1})^{1/p}k(\xi_{1}-\xi\prime 1)F(\xi_{1}’, \eta’1)d\xi 1d\xi_{1}’$$\leq\{\int\int(\int F(\xi_{1}, \eta 1)pd\eta_{1})^{p’/p}k(\xi 1^{-}\xi’)d\xi 1d\xi_{1}/\}11/p’$$\cross(\int\int k(\xi_{1}-\xi’1)F(\xi_{1}’’, \eta 1)pd\xi 1d\xi_{1}’)1/p$$\leq C_{2}K(\int F(\xi_{1}’, \eta_{1}’)^{\mathrm{P}}d\xi_{1}/)1/p$ ,where$C_{2}= \int k(\mathrm{t})dt$ .Hence (5) is bounded by$C_{1}C_{2}K \int\cdots\int(\int F(\xi\prime 1’\eta_{1}’)pd\xi_{1}’)1/p-k(\eta_{1}’-\eta 2)F(\xi 2, \eta 2)k(\xi_{2}\xi_{2}’)$. . . $F(\xi_{n}, \eta_{n})k(\xi n-\xi’n)F(\xi/n’\eta n’)k(\eta_{n}-\prime b)d\eta_{12}^{\prime_{d\xi\xi_{n}n}}\cdots d\prime d\eta’.$.By repeating this calculation, we can estimate (5) by72.$C_{1}C_{2}^{2n}-1K2n-1 \int(\int F(\xi_{n}’, \eta_{n})\prime pd\xi’n)1/p\eta’k(n-b)d\eta_{n}’$$\leq c_{1}c_{2}^{212n-}n-K1\{\int(\int F(\xi_{n}’, \eta_{n})^{p}\prime d\xi_{n)}’p’/pd\eta’n\}1/p’((\int k\eta_{n}-b’)pd\eta_{n})\prime 1/p$$\leq c_{1}c_{23}2n-1oK^{2}n<\infty$ ,where$C_{3}=( \int k(\eta_{n}-/b)pd\eta’n)1/p$Hence the condition (a) of Theorem 1.1 is satisfied. Furthermore we have$\sup_{n\in \mathrm{N}}\sup_{a\in \mathrm{R}}G_{n}2d(a, a)\leq CK^{2}$ .Therefore we proved the $L^{2}$ boundedness by Theorem 1.1.$*_{\vee}’\not\equiv\overline{\overline{\mathrm{x}}}\mathrm{f}\mathrm{f}\mathrm{l}$[1] A. Calder\’on and R. Vaillancourt, On the boundedness of pseudodifferential oper-ators, J. Math. Soc. Japan, 23, 1971, 374-378.[2] A. Calder\’on and R. Vaillancourt, A class of bounded pseudodifferential operators,Proc. Nat. Acad. Sci. USA, 69, 1972, 1185-1187.[3] $\mathrm{H}.\mathrm{O}$ . Cordes, On compactness of commutators of multiplications and convolutions,and boundedness of pseudodifferential operators, J. Funct. Anal., 18, 1975, 115-131.[4] L. Crone, A characterization of matrix operators on $\ell^{2}$ , Math. Z., 123, 1971, 315-317.[5] I. Daubechies, Ten lectures on wavelets, SIAM 1992.73[6] $\mathrm{H}.\mathrm{G}$ . Feichtinger, Atomic characterizations of modulation spaces through Gabor-type representations, Rocky Mountain J. Math., 19, 1989, 113-126.[7] $\mathrm{H}.\mathrm{G}$ . Feichtinger and K. Gr\"ochenig, Banach spaces related to integrable grouprepresentations and their atomic decompositions I, J. Funct. Anal., 86, 1989, 307-340.[8] $\mathrm{H}.\mathrm{G}$ . Feichtinger and K. Gr\"ochenig, Banach spaces related to integrable grouprepresentations and their atomic decompositions II, Monatsh. Math., 108, 1989,129-148.[9] H. G. Feichtinger and K. Gr\"ochenig, Gabor wavelets and the Heisenberg Group:Gabor expansion and short time Fourier transform from the group theoretical pointof view, $\mathrm{C}.\mathrm{K}$ . Chui ed., Wavelets-A tutorial in theory and applications, AcademicPress Inc., 1992, 359-397.[10] $\mathrm{G}.\mathrm{B}$ . Folland, Harmonic analysis in phase space, Annals of Mathematics StudiesNo. 122, Princeton University Press, 1989.[11] K. Gr\"ochenig and C. Heil, Modulation spaces and pseudodifferential operators,preprint.[12] $\mathrm{I}.\mathrm{J}$ . Maddox, Infinite matrices of operators, Lecture Notes in Mathematics, 786,Springer, Berlin, 1980.[13] $\mathrm{I}.\mathrm{J}$ . Maddox and $\mathrm{A}.\mathrm{W}$ . Wickstead, Crone’s theorem for operators, Math. Colloq.Univ. Cape Town XI, 1977, 33-45.74[14] R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and$\mathrm{i}_{\mathrm{o}\mathrm{c}\mathrm{a}1\mathrm{r}\mathrm{i}\mathrm{g}}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{o}\dot{\mathrm{m}}$etric bases, H.Feichtinger and T.Strohmer ed., Gabor analysis andalgorithms: Theory and applications, Birkh\"au,ser, 1998, 171-192.Mathematical InstituteTohoku UniversitySendai 980-8578, JapanE-m$ail$ address: tachizaw@math.tohoku.ac.jp75

A generalization of Calderon-Vaillancourt's Theorem (Harmonic Analysis and Nonlinear Partial Differential Equations)

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A generalization of Calderon-Vaillancourt's Theorem (Harmonic Analysis and Nonlinear Partial Differential Equations)

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