Content-Type: multipart/mixed; boundary="-------------9911301400176" This is a multi-part message in MIME format. ---------------9911301400176 Content-Type: text/plain; name="99-455.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-455.keywords" Schrodinger Operator, Periodic Petential, Absolute Continuous Spectrum, Weighted Uniform Sobolv Inequalities. ---------------9911301400176 Content-Type: application/x-tex; name="s11.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="s11.tex" %%%%%%%%%%%%%%%%%%%%% % % Last updated Nov. 8 % %%%%%%%%%%%%%%%%%%%%%%%%% \input amstex \loadeufm \documentstyle{amsppt} \magnification=\magstep1 \baselineskip=18pt \parskip=5pt \NoBlackBoxes \define\ba{{\bold{a}}} \define\bb{{\bold{b}}} \define\bx{{\bold{x}}} \define\bby{{\bold{y}}} \define\bd{{\bold{D}}} \define\bn{{\bold{n}}} \define\bk{{\bold{k}}} \define\oo{{\Omega}} \define\e{{\varepsilon}} \define\z{{\Bbb Z}} \define\br{{\Bbb R}} \define\cc{{\Bbb C}} \define\bh{{\Bbb H}} \define\loc{{\text{loc}}} \define\im{{\text{Im}}} \define\re{{\text{Re}}} \define\domain{{\text{Domain}}} \define\per{{\text{per}}} \define\bt{{\Bbb{T}}} \centerline{\bf The Periodic Schr\"odinger Operators with Potentials} \centerline{\bf in the C.~Fefferman-Phong Class} \medskip\medskip \centerline{\bf Zhongwei Shen\footnote{Research supported in part by the NSF grant DMS-9732894.}} \centerline{Department of Mathematics} \centerline{University of Kentucky} \centerline{Lexington, KY 40506} \centerline{Email: shenz\@ms.uky.edu} \bigskip \medskip \noindent{\bf Abstract.} We consider the periodic Schr\"odinger operator $-\Delta +V(\bx)$ in $\br^d$, $d\ge 3$ with potential $V$ in the C.~Fefferman-Phong class. Let $\oo$ be a periodic cell for $V$. We show that, for $p\in((d-1)/2, d/2]$, there exists a positive constant $\e$ depending only on the shape of $\oo$, $p$ and $d$ such that, if $$ \limsup_{r\to 0} \, \sup_{\bx\in \oo} r^2\left\{\frac{1}{|B(\bx,r)|} \int_{B(\bx,r)} |V(\bby)|^p d\bby\right\}^{1/p} < \e, $$ then the spectrum of $-\Delta +V$ is purely absolutely continuous. We obtain this result as a consequence of certain weighted $L^2$ Sobolev inequalities on the d-torus. It improves an early result by the author for potentials in $L^{d/2}$ or weak-$L^{d/2}$ space. \noindent{\bf 1991 Mathematics Subject Classification.} Primary 35J10. \noindent{\bf Key Words and Phrases.} Schr\"odinger Operator, Periodic Potential, Absolute Continuous Spectrum, Weighted uniform Sobolev Inequalities. \bigskip \centerline{\bf 1. Introduction} \medskip Consider the Schr\"odinger operator $$ -\Delta +V(\bx)\ \ \ \text{ in } \ \ \br^d \tag 1.1 $$ with a real periodic potential $V$. In a celebrated work \cite {27}, L.~Thomas showed that, if $d=3$ and $V\in L_{\loc}^2 (\br^3)$, the spectrum of $-\Delta +V$ is purely absolutely continuous. Thomas' result was extended subsequently by M.~Reed-B.~Simon \cite{20}, L.~Danilov \cite{7}, R.~Hempel -I.Herbst \cite {10,11}, M.~Birman -T.~Suslina \cite {1,2,3}, A.~Morame \cite{19}, and A.~Sobolev \cite {22}. We remark that \cite {7} studied the Dirac operator with a periodic scalar potential and \cite {1,2,3,10,11,19,22} investigated the magnetic Schr\"odinger operator $(-i\nabla -\ba (\bx))^2 +V(\bx )$ with periodic potentials $\ba$ and $V$. In particular, the results in \cite {2,3}, pertaining to the case $\ba =0$, give the absolute continuity of $-\Delta +V$ for $V\in L^r_{\loc} (\br^d)$ where $r>1$ if $d=2$, $r=d/2$ if $d=3$ or $4$, and $r=d-2$ if $d\ge 5$. Recently in \cite{21}, the author was able to establish the absolute continuity of $-\Delta +V$ for $V\in L_{\loc}^{d/2}(\br^d)$, $d\ge 3$. In the context of $L^p$ spaces, this is best possible. \cite {21} also gives the optimal condition for the absolute continuity of $-\Delta +V$ with periodic potentials in Lorentz spaces. See \cite{Theorem 1.2, 21}. The purpose of this paper is to study the spectral properties of the periodic Schr\"odinger operators with potentials in the C.~Fefferman-Phong class. Our result improves that in \cite {21}. \noindent{\bf Definition 1.2.} Let $g\in L^p_\loc (\br^d)$ for some $p\ge 1$. We say $g\in F_\loc ^p (\br^d)$ if $$ \sup \Sb 0(d-1)/2$. Then there exists a positive constant $\e$ depending only on the shape of $\oo$, $A$, $p$ and $d$ such that, if $$ \limsup_{r\to 0} \, \sup_{\bx\in \oo} \, r^2 \left\{ \frac{1}{r^d} \int_{Q(\bx,r)} |V(\bby)|^pd\bby\right\}^{1/p} < \e, \tag 1.4 $$ then the spectrum of $\bd A\bd^T +V$ is purely absolutely continuous. \endproclaim \noindent{\bf Remark 1.5.} Our main theorem contains Theorem 1.2 in \cite {21}. This follows from the estimate, $$ \aligned \limsup_{r\to 0} \, \sup_{\bx\in \oo} & \, r^2\left\{ \frac{1}{r^d} \int_{Q(\bx,r)} |V(\bby)|^pd\bby\right\}^{1/p}\\ & \le C_{p,d} \limsup_{t\to\infty} \, t|\left\{ \bx\in\oo: |V(\bx)|>t\right\} |^{2/d} \endaligned \tag 1.6 $$ where $1t\right\} |^{2/d} < \e_0. $$ Let $N=\e_0/r^2$. If $r$ is sufficiently small, $$ \aligned r^2& \left\{ \frac{1}{r^d} \int_{Q(\bx,r)} |V(\bby)|^pd\bby\right\}^{1/p}\\ &\ \ \ \ \ \ \ \le r^2 N +r^2 \left\{ \frac{1}{r^d} \int_{\{ \bby\in\oo: |V(\bby)|>N\} } |V(\bby) |^pd\bby\right\}^{1/p}\\ &\ \ \ \ \ \ \ =\e_0 + r^2\left\{ \frac{1}{r^d} \int_N^\infty p\, t^{p-1} |\{ \bby\in\oo: |V(\bby)|>t\} | dt\right\}^{1/p}\\ &\ \ \ \ \ \ \ \le \e_0 +r^2 \left\{ \frac{1}{r^d} \int_N^\infty p\, t^{p-1}\cdot \frac{(\e_0)^{d/2}}{t^{d/2}} dt\right\}^{1/p}\\ &\ \ \ \ \ \ \ =C_{p,d}\, \e_0. \endaligned $$ Hence, $$ \limsup_{r\to 0}\, \sup_{\bx\in \oo} \, r^2\left\{ \frac{1}{r^d} \int_{Q(\bx,r)} |V(\bby)|^pd\bby\right\}^{1/p} \le C_{p,d}\, \e_0. $$ This gives (1.6). On the other hand, we can construct a periodic potential $V$ which satisfies (1.4) and $$ V(\bx)\approx \frac{1}{|\bx^\prime|^\alpha}\ \ \text{ near }\bx=0, $$ where $0<\alpha<2$ and $\bx^\prime=(x_2,\dots, x_d)$. Note that $V\notin$ weak-$L^{d/2}(\oo)$ if $\alpha>2(d-1)/d$. \medskip \noindent{\bf Remark 1.7.} Under the assumption that $V$ is periodic and satisfies condition (1.4) for some $p>1$, one may define the self-adjoint operator $-\Delta +V$ by a quadratic form. See section 2. However the problem of absolute continuity of $-\Delta +V$ remains open in the case $d\ge 4$ and $1 N\}$ ($N$ is a fixed large number), and $1(d-1)/2$, $\bh_0(z)=(\bd +z\ba +\bb)A(\bd + z\ba +\bb)^T$, and $C$ depends only on $p$, $A$ and $d$. We remark that (1.11) should be considered as a weighted $L^2$ uniform Sobolev inequality on the d-torus. In the case of $\br^d$, similar estimates were established by S.~Chanillo -E.~Sawyer \cite{4} and F.~Chiarenza -A.~Ruiz \cite {6} in the study of unique continuation problems. The weighted $L^2$ estimates in \cite {4,6} for the operator $\Delta +\ba\cdot\nabla +b$ extend the uniform $L^p-L^q$ estimates obtained in \cite {14} by C.~Kenig - A.~Ruiz -C.~Sogge. To show (1.11), we will appeal to the localization argument developed in \cite {14}. It reduces (1.11) to the following estimate, $$ \aligned \|\psi\|_{L^2(\bt^d, \omega d\bx)} &\le C(p, c_0, A, d) \sup\Sb \bx\in\oo\\ 0(d-1)/2$, $\xi\in \cc$ and $\re\sqrt{\xi}\ge c_0>0$. (1.12) may be proved by using the Stein complex interpolation theorem, together with a weighted norm inequality for a fractional integral, as in case of $\br^d$ \cite{4}. See Theorem 4.3. However, in the localization process, we will need the following weighted spectral projection estimate for the elliptic operator $\bd A\bd^T $ on the d-torus, $$ \aligned &\|\sum_{|\bn B|\in [k, k+1)} \hat{\psi}(\bn) e^{i\bn\bx}\|_{L^2(\bt^d, \omega d\bx )}\\ &\le C \, k^{1/2} \left\{ \sup\Sb \bx\in\oo\\ 01$. In section 3 we use the Thomas approach to prove that our main theorem follows from the estimate (1.11). Sections 4-6 are devoted to the proof of (1.11) (Theorem 3.10). In section 4 we give the proof of (1.12) (Theorem 4.3). Section 5 contains the proof of the weighted spectral projection estimate (1.13). See Theorem 5.1. Finally in section 6 we use a localization argument to deduce (1.11) from (1.12)-(1.13). Throughout the rest of this paper, we will assume that $d\ge 3$, $\oo=[0,2\pi)^d$, and $V$ is periodic with respect to the lattice $(2\pi \z)^d$. We will use $C$ to denote positive constants which may depend on $d$, $p$ and the matrix $A$, which are not necessarily the same at each occurrence. \medskip \centerline{\bf 2. The C.~Fefferman-Phong Class} We begin with the definition of the self-adjoint operator $\bd A\bd^T +V$ on $L^2(\br^d)$. Let $\psi\in C_0^1(\br^d)$. Then $$ \big|\psi (\bby)-\frac{1}{|\oo|}\int_\oo \psi(\bx) d\bx\big| \le C\, I_1 (|\nabla\psi|\chi_\oo)(\bby) \tag 2.1 $$ where $I_\alpha$ denotes the Riesz potential $$ I_\alpha (f)(\bx) =C_{\alpha, d} \int_{\br^d} \frac{ f(\bby)\, d\bby }{ |\bx -\bby|^{d-\alpha}} \tag 2.2 $$ (see \cite{Lemma 7.16, 9}). It follows that $$ \int_\oo |\psi|^2 |V| d\bx \le C\int_{\br^d} |I_1(|\nabla\psi|\chi_\oo)|^2 |V|\chi_\oo d\bx +C\int_\oo |V|\, d\bx \cdot \int_\oo |\psi|^2 d\bx. $$ We now use the C.~Fefferman-Phong estimate \cite {Lemmas B and C, 8}, $$ \int_{\br^d} |I_1(f)|^2 |V|\, d\bx \le C \sup\Sb r>0\\ \bx\in \br^d \endSb r^2\left\{ \frac{1}{r^d} \int_{Q(\bx,r)} |V(\bby)|^p d\bby\right\}^{1/p} \int_{\br^d} |f|^2 d\bx \tag 2.3 $$ where $10\\ \bx\in \br^d\endSb r^2\left\{ \frac{1}{r^d} \int_{Q(\bx,r)} \left(|V|\chi_\oo\right)^p d\bby\right\}^{1/p} \int_\oo |\nabla\psi|^2 d\bx\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +C\int_\oo |V|\, d\bx\cdot\int_\oo |\psi|^2 d\bx\\ & \le C \sup\Sb 0 +\right\} d\bx \tag 2.5 $$ for $f$, $g \in C_0^1(\br^d)$. \proclaim{\bf Theorem 2.6} Let $V$ be a real periodic function and $V\in F^p_\loc(\br^d)$ for some $p\in (1, d/2]$. Then there exists a positive constant $\e$ depending only on $A$, $p$ and $d$ such that, if $V$ satisfies the condition (1.4), then there exists a unique self-adjoint operator, which we denote by $\bd A\bd^T +V$, such that $$ q[f,g]=\int_{\br^d} <(\bd A\bd^T +V)f, g> \, d\bx $$ for $f\in \domain (\bd A\bd^T +V)$ and $g\in H^1(\br^d)$. \endproclaim The following lemma will be needed in the proof of Theorem 2.6. \proclaim{\bf Lemma 2.7} Let $1N$ and $V_N(\bx)=0$ if $|V(\bx)|\le N$. \endproclaim \demo{Proof} By (1.4), there exists $r_0>0$ such that $$ \sup\Sb \bx\in \oo\\ 0 d\bx\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ +\left\{ C\int_\oo |V|\, d\bx +N\right\} \int_{\br^d} |\psi|^2 d\bx \\ &\le C\e \int_{\br^d} < A\nabla \psi, \nabla\psi> d\bx + C_{\e, V} \int_{\br^d} |\psi|^2 d\bx \endaligned $$ if $N$ is large. This implies that, if $\e$ is so small that $C\, \e<1/2$, the symmetric quadratic form $q$ is semi-bounded from below and closable on $H^1(\br^d)$. Hence it defines a unique self-adjoint operator. \medskip In the rest of this section we will establish two propositions concerning the periodic functions which are locally in the C.~Fefferman-Phong class. They will be used in the subsequent sections. \proclaim{\bf Proposition 2.8} Suppose $V$ is periodic with respect to the lattice $(2\pi\z)^d$ and $V\in F_\loc^p (\br^d)$ for some $p\in (1, d/2]$. Let $11$, by the well known estimates for the maximal function \cite{24}, $$ \aligned r^2_0\left\{ \frac{1}{|Q|}\int_Q \omega_1^p\, d\bby\right\}^{1/p} & \le Cr^2_0\left\{ \frac{1}{|2Q|} \int_{2Q} |V|^p d\bx\right\}^{1/p}\\ &\le C \sup\Sb 02r_0$ such that $$ \omega_2(\bx_0) \le C\,\left\{ \frac{1}{t^d} \int_{Q(\bx_0,t)} |V|^q d\bx\right\}^{1/q}. $$ In view of (2.11), we may assume that $t\le 2\pi$. We obtain $$ \aligned r_0^2\left\{\frac{1}{|Q|} \int_Q \omega_2^p d\bx\right\}^{1/p} & \le Cr_0^2 \left\{ \frac{1}{t^d} \int_{Q(\bx_0, t)} |V|^q d\bx\right\}^{1/q}\\ &\le C\, t^2 \left\{ \frac{1}{t^d} \int_{Q(\bx_0, t)} |V|^p d\bx\right\}^{1/p}\\ &\le C\sup\Sb 00} \left\{ \frac{1}{2t} \int_{x_1-t}^{x_1+t} |\omega (s,x_2,\dots, x_d)|^q ds\right\}^{1/q} \tag 2.14 $$ where $(d-1)/2(d-1)/2$ in the last inequality. Thus we have proved that $$ \aligned r^2& \left\{\frac{1}{|Q|} \int_Q \omega_*^p (\bby) d\bby\right\}^{1/p}\\ &\le C\left\{ 1+\sum_{j=1}^{N} (2^j)^{\frac{d-1}{p}-2}\right\} \sup \Sb 0 + \right\}d\bx \tag 3.2 $$ for $\phi,\psi\in H^1(\bt^d)$, where $\overline{\bk}$ denotes the conjugate of $\bk$. It follows from the proof of Theorem 2.6 that, if $V$ satisfies (1.4) for some $p>1$, $$ \int_\oo |\psi|^2 |V| d\bx \le C\, \e \int_\oo <(\nabla\psi)A,\nabla\psi> d\bx +C_{\e,V} \int_\oo |\psi|^2 d\bx \tag 3.3 $$ for any $\e>0$. This implies that, if $\e$ is so small that $C\,\e<1/2$, the form $q(\bk)$ is strictly m-sectorial. Hence there exists a unique closed operator, which we denote by $(\bd +\bk)A(\bd +\bk)^T +V$, such that $$ q(\bk)[\phi,\psi] =\int_\oo <\left\{(\bd+\bk)A(\bd+\bk)^T+V\right\}\phi, \psi> d\bx \tag 3.4 $$ for any $\phi\in \domain \left( (\bd+\bk)A(\bd+\bk)^T+V\right) $ and $\psi\in H^1(\bt^d)$ \cite{12}. Also $$ \aligned \domain & \left( (\bd+\bk)A(\bd+\bk)^T+V \right)\\ &=\left\{ \phi\in H^1(\bt^d):\ \left\{(\bd+\bk)A(\bd+\bk)^T+V \right\}\phi\in L^2(\bt^d)\right\} \\ &=\left\{ \phi\in H^1(\bt^d):\ (\bd A\bd^T+V)\phi \in L^2 (\bt^d)\right\}, \endaligned \tag 3.5 $$ is independent of $\bk$. Choose $\ba\in\br^d$ such that $$ |\ba |=1,\ \ \ba A=(s_0,0,\dots, 0) \text{ and } s_0>0. \tag 3.6 $$ Let $$ L=\left\{\bb\in \br^d: \ |\bb|<\sqrt{d}\ \text{ and } <\bb,\ba>=0\right\}. \tag 3.7 $$ For a fixed $\bb\in L$, we consider the family of operators $$ \bh_V(z) =(\bd +z\ba +\bb) A (\bd +z\ba +\bb)^T,\ \ \ z\in\cc \tag 3.8 $$ defined by the form (3.2). Since $\domain (\bh_V(z)) \subset H^1(\bt^d)$, $\bh_V(z)$ has compact resolvent. \proclaim{\bf Proposition 3.9} If, for every $\bb\in L$, the family of operators $\{ \bh_V(z): z\in \cc\}$ has no common eigenvalue, then the spectrum of the operator $\bd A\bd +V$ defined in Theorem 2.6 is purely absolutely continuous. \endproclaim \demo{Proof} See \cite {21} and \cite{20}. \enddemo To show that $\{ \bh_V(z), z\in \cc\}$ has no common eigenvalue under the assumptions of the main theorem, we consider $\bh_V(\delta_0 +i\rho)$ for $\rho\in\br$, where $$ \delta_0 =\frac{1}{a_1}\left(\frac12 -b_1\right). $$ We will show that $\{ \bh_V(\delta_0 +i\rho):\, \rho\in\br\}$ has no common eigenvalue. This will be done by establishing certain uniform estimates for $$ \bh_0(\delta_0 +i\rho) =(\bd +(\delta_0+i\rho)\ba +\bb) A(\bd +(\delta_0+i\rho)\ba +\bb)^T $$ on the weighted $L^2$ spaces. \proclaim{\bf Theorem 3.10} Let $\omega\in L^p(\bt^d)$ for some $p\in ((d-1)/2,d/2]$. Suppose $\omega\ge c_\omega>0$ and $$ \|\omega\|_{F^p(\bt^d)} \equiv \sup \Sb 0 (d-2)/2$ if $d\ge 4$. Suppose $\omega\ge c_\omega>0$. Then, if $\psi\in H^1(\bt^d)$ and $\bh_0(\delta_0+i\rho)\psi \in L^2(\bt^d,\frac{d\bx}{\omega})$, we have $$ c_\rho\cdot ( c_\omega)^{1/2} \|\psi\|_{L^2(\bt^d,\frac{d\bx}{\omega})} \le \left( \| \omega\|_{L^p(\bt^d, d\bx)}\right)^{1/2} \cdot \| \bh_0(\delta_0 +i\rho)\psi\|_{L^2(\bt^d,\frac{ d\bx}{\omega})} \tag 3.14 $$ where $c_\rho\to\infty$ as $|\rho|\to\infty$. \endproclaim We now give the \noindent{\bf Proof of the main theorem, assuming Theorems 3.10 and 3.13.} In view of Proposition 3.9, we suppose that $E$ is an eigenvalue for $\bh_V(z)$ for all $z\in \cc$. Let $z=\delta_0 +i\rho$. There exists $\psi_\rho\in \domain(\bh_V(z))$ such that $$ \| \psi\|_{L^2(\bt^d,d\bx)}=1 \ \ \ \text{ and }\ \ \bh_V(z)\psi_\rho =E\psi_\rho. $$ Suppose $V$ satisfies (1.4). By Lemma 2.7, we may choose $N$ so large that $$ \sup\Sb 00$. Note that $\omega$ is periodic with respect to $(2\pi \z)^d$. Also by (2.9) and (3.15), $$ \sup\Sb 00$, we have $\psi_\rho \in L^2(\bt^d,\frac{d\bx}{\omega})$. It follows that $$ \bh_0(\delta_0 +i\rho)\psi_\rho \in L^2(\bt^d,\frac{d\bx}{ \omega}) $$ and $$ \|\bh_0(\delta_0+i\rho)\psi_\rho\|_{L^2(\bt^d, \frac{d\bx}{\omega})} \le \{ |E| +N\}\| \psi_\rho\|_{L^2(\bt^d, \frac{d\bx}{\omega})} +\| \psi_\rho\|_{L^2(\bt^d, \omega d\bx)}. \tag 3.18 $$ Now, by Theorem 3.10 and (3.17), $$ \|\psi_\rho\|_{L^2(\bt^d, \omega d\bx)} \le C_0\, \e_0 \| \bh_0 (\delta_0 +i\rho)\psi_\rho\|_{L^2(\bt^d,\frac{d\bx}{ \omega})} $$ where $\rho\ge 1$ and $C_0$ depends only on $p$, $q$, $A$ and $d$. In view of (3.18), we see that, if $\e$ is chosen so small that $C_0\, \e \le 1/2$, we have $$ \|\bh_0(\delta_0+i\rho)\psi_\rho\|_{L^2(\bt^d, \frac{d\bx}{\omega})} \le 2 \{ |E| +N\}\| \psi_\rho\|_{L^2(\bt^d, \frac{d\bx}{\omega})}. $$ This, together with Theorem 3.13, implies that $$ c_\rho\cdot (c_\omega)^{1/2} \le \left( \| \omega\|_{L^p(\bt^d, d\bx)}\right)^{1/2} \cdot 2\left\{ |E| +N\right\}. $$ We obtain a contradiction since $c_\rho\to\infty$ as $\rho\to\infty$. This shows that $\{ \bh_V(z): z\in\cc\}$ has no common eigenvalue. By Proposition 3.9, the spectrum of $\bd A\bd^T +V$ on $L^2(\br^d)$ is purely absolutely continuous. \medskip We end this section with the proof of Theorem 3.13. The proof of Theorem 3.10 is much more involved, as we discussed in the introduction. It will be given in sections 4-6. \noindent{\bf Proof of Theorem 3.13.} Theorem 3.13 is a consequence of Theorem 2.1 in \cite{21}. Indeed, it follows from Theorem 2.1 in \cite{21} that, if $\psi\in H^1(\bt^d)$ and $\bh_0(\delta_0+i\rho)\psi \in L^{p_0}(\bt^d, d\bx)$, we have $$ c_\rho \|\psi\|_{L^2(\bt^d, d\bx)} \le \| \bh_0(\delta_0 +i\rho)\psi\|_{L^{p_0}(\bt^d, d\bx)} $$ where $c_\rho\to\infty$ as $|\rho| \to\infty$, $1\le p_0\le 2$ if $d=3$ and $2(d-2)/d(d-2)/2$ if $d\ge 4$. The proof of Theorem 3.13 is finished. \medskip \centerline{\bf 4. A Weighted Uniform Sobolev Inequality} For $$ \psi(\bx) =\sum_{\bn\in \z^d} \hat{\psi}(\bn) e^{i\bn \bx}, \tag 4.1 $$ and $\bb\in\br^d$, $|\bb|<\sqrt{d},$ we define $$ S\psi(\bx) =\sum_{\bn\in\z^d} \frac{\hat{\psi}(\bn)e^{i\bn\bx}} {(\bn +\bb)A(\bn +\bb)^T +z} \tag 4.2 $$ where $z\in \cc$, $\im\, z \ne 0$. We choose a branch of $\sqrt{z}$ so that $\re\sqrt{z}\ge 0$. This section is devoted to the proof of the following theorem. \proclaim{\bf Theorem 4.3} Let $\omega\in L^p(\bt^d)$ for some $p\in ((d-1)/2, d/2]$. Suppose that $\omega\ge c_\omega>0$ and $\| \omega\|_{F^p(\bt^d)}<\infty$. Let $z\in\cc$ such that $\im\, z\ne 0$ and $\re\sqrt{z}\ge c_0>0$. Then, for any $\psi\in L^2(\bt^d, d\bx)$, $$ \| \omega^{1/2} S(\psi \omega^{1/2}) \|_{L^2(\bt^d,d\bx)} \le C\| \omega\|_{F^p(\bt^d)} \|\psi\|_{L^2(\bt^d,d\bx)} \tag 4.4 $$ where $C$ depends only on $p$, $d$, $A$, and $c_0$. \endproclaim \demo{Proof} For $\xi\in \cc$, we consider two analytic families of operators, $$ S_\xi\psi (\bx) =\sum_{\bn\in\z^d} \frac{\hat{\psi}(\bn)e^{i\bn\bx}} {\left[ (\bn +\bb)A(\bn +\bb)^T +z\right]^\xi } \tag 4.5 $$ and $$ T_\xi \psi =\omega^{\xi/2} S_\xi (\omega^{\xi/2}\psi). \tag 4.6 $$ Clearly, if $\re\, \xi =0$, $$ \| T_\xi \psi\|_{L^2(\oo, d\bx)} \le C e^{c|\im \, \xi |} \|\psi\|_{L^2(\oo, d\bx)}. \tag 4.7 $$ If $\re \, \xi =(d-1)/2$, we use the estimate obtained in \cite{21} for the integral kernel of $S_\xi$. We have $$ S_\xi \psi (\bx) =\int_\oo G_\xi (\bx-\bby) \psi(\bby)\, d\bby \tag 4.8 $$ and $$ |G_\xi (\bx)| \le C e^{c|\im\, \xi|} \left\{ 1+\sum_{|\bx +2\pi\bn|\le C} \frac{1}{|\bx +2\pi \bn|}\right\}. \tag 4.9 $$ See (6.10) in \cite{21}. It follows that $$ \aligned |S_\xi \psi(\bx)| &\le Ce^{c|\im \, \xi|} \left\{ \int_\oo |\psi(\bby)|d\bby + \int_\oo \sum_{|\bx-\bby+2\pi \bn|\le C} \frac{|\psi(\bby)| \, d\bby} {|\bx-\bby +2\pi\bn|} \right\}\\ &\le Ce^{c|\im \, \xi|} \left\{ \int_\oo |\psi (\bby)| d\bby + I_{d-1} (|\psi|\chi_{\oo^\prime}) (\bx) \right\} \endaligned $$ where $ \oo^\prime =\cup_{|\bn|\le C} (\oo +2\pi\bn)$, and in the last inequality we used the fact that $\psi$ is periodic. Hence, if $\re\, \xi =(d-1)/2$, $$ \aligned & |T_\xi\psi(\bx)|\\ & \le Ce^{c|\im\, \xi|} \left\{ (\omega(\bx))^{\frac{d-1}{4}} \int_\oo |\psi(\bby)| (\omega(\bby))^{\frac{d-1}{4} }d\bby + (\omega(\bx))^{\frac{d-1}{4}} I_{d-1} (|\psi|\chi_{\oo^\prime} \omega^{\frac{d-1}{4}})(\bx) \right\} \endaligned $$ This, together with the weighted norm inequality for the fractional integral $I_{d-1}$ \cite{5,15}, gives $$ \aligned &\| T_\xi \psi\|_{L^2(\oo,d\bx)}\\ &\le Ce^{c|\im\, \xi|} \left\{ \int_\oo \omega^{\frac{d-1}{2}} d\bx \cdot \|\psi\|_{L^2(\oo,d\bx)} + \|\chi_{\oo^\prime} \omega^{\frac{d-1}{4}} I_{d-1} (|\psi| \chi_{\oo^\prime} \omega^{\frac{d-1}{4}}) \|_{L^2(\br^d, d\bx)}\right\}\\ &\le C_q e^{c|\im\, \xi|} \sup \Sb r>0\\ \bx\in \br^d \endSb r^{d-1} \left\{ \frac{1}{r^d} \int_{Q(\bx,r)} (\chi_{\oo^\prime} \omega^{\frac{d-1}{2}})^q d\bby\right\}^{1/q} \cdot \| \psi \chi_{\oo^\prime}\|_{L^2(\br^d, d\bx)} \endaligned $$ where $q>1$. Note that, since $\psi$ and $\omega$ are periodic, we have $$ \| \psi \chi_{\oo^\prime}\|_{L^2(\br^d,d\bx)} \le C\| \psi\|_{L^2(\oo, d\bx)} $$ and $$ \aligned \sup\Sb r>0\\ \bx\in \br^d \endSb r^{d-1} & \left\{ \frac{1}{r^d} \int_{Q(\bx, r)} (\omega^{\frac{d-1}{2}}\chi_{\oo^\prime})^q d\bby\right\}^{1/q}\\ &\le C\sup\Sb 01$. Thus we obtain that, if $\re \, \xi =(d-1)/2$, $$ \| T_\xi \psi\|_{L^2(\oo, d\bx)} \le Ce^{c|\im \, \xi|} \left( \| \omega\|_{F^p(\bt^d)}\right)^{ \frac{d-1}{2}} \|\psi\|_{L^2(\oo, d\bx)} \tag 4.10 $$ Finally, we write $$ 1= t\cdot\frac{d-1}{2} +(1-t)\cdot 0\ \ \ \ \ \text{where } t=\frac{2}{d-1}. $$ It follows from (4.7), (4.10) and the Stein interpolation theorem \cite{p.385, 25} that $$ \| T_1\psi \|_{L^2(\oo, d\bx)} \le C \|\omega\|_{F^p(\bt^d)} \| \psi\|_{L^2(\oo, d\bx)}. $$ Since $T_1\psi =\omega^{1/2} S(\omega^{1/2} \psi )$, (4.4) is proved. \enddemo \medskip \centerline{\bf 5. A Weighted $L^2$ Estimate for the Spectral Projection} The goal of this section is to establish the following theorem for the spectral projection associated with the operator $\bd A\bd^T$ on the d-torus. \proclaim{\bf Theorem 5.1} Let $\omega\in L^p(\bt^d)$ for some $p\in ((d-1)/2,d/2]$. Suppose that $\omega\ge c_\omega>0$ and $\|\omega\|_{F^p(\bt^d)} <\infty$. Then, for any $\psi\in L^2(\bt^d, d\bx)$ and $k\ge 0$, $$ \| \sum_{|\bn B|\in [k, k+1)} \hat{\psi}(\bn) e^{i\bn\bx} \|_{L^2(\bt^d,\omega d\bx)} \le C (k+1)^{1/2} \left(\| \omega\|_{F^p(\bt^d)}\right)^{1/2} \|\psi\|_{L^2(\bt^d, d\bx)} \tag 5.2 $$ where $B=B_{d\times d}=\sqrt{A}\ge 0$ and $C$ depends only on $p$, $d$ and $A$. \endproclaim It is not hard to see that (5.2) is a direct consequence of the weighted $L^2$ inequality in the next theorem. \proclaim{\bf Theorem 5.3} Let $\omega\in L^p(\bt^d)$ for some $p\in ((d-1)/2, d/2]$. Suppose that $\omega\ge c_\omega>0$ and $\|\omega\|_{F^p (\bt^d)}<\infty$. Then, for any $\psi\in C^\infty(\bt^d)$ and $k\ge 1$, $$ \|\psi\|_{L^2(\bt^d,\omega d\bx)} \le C k^{-1/2} \left(\|\omega\|_{F^p(\bt^d)}\right)^{1/2} \| (\bd A \bd^T -k^2 -2ik)\psi \|_{L^2(\bt^d, d\bx)}. \tag 5.4 $$ \endproclaim The proof of Theorem 5.3 relies on a weighted version of the Stein oscillatory integral theorem in $\br^d$ \cite{p.380, 25}. \proclaim{\bf Theorem 5.5} Let $\omega\ge 0$ and $\omega\in F^p_\loc (\br^d)$ for some $p\in ((d-1)/2, d/2)$. Suppose that $$ \|\omega\|_{F^p(\br^d)} \equiv \sup\Sb r>0\\ \bx\in \br^d \endSb r^2\left\{\frac{1}{r^d} \int_{Q(\bx,r)} \omega^p \, d\bby\right\}^{1/p} <\infty. \tag 5.6 $$ Let $a\in C^\infty(\br^d\times\br^d)$ and $$ \text{supp}\, a \subset \left\{ (\bx,\bby)\in \br^d\times\br^d: 1/2\le |\bx-\bby|\le 2\right\}. \tag 5.7 $$ Then, for any $f\in L^2(\br^d, d\bx)$, $$ \aligned \|\int_{\br^d} e^{i\lambda |\bx-\bby|} & a(\bx,\bby) f(\bby) d\bby\|_{L^2(\br^d, \omega d\bx)}\\ & \le C |\lambda|^{\frac{2-d}{2}} \left(\|\omega\|_{F^p(\br^d)}\right)^{1/2} \| f\|_{L^2(\br^d, d\bx)} \endaligned \tag 5.8 $$ where $\lambda\in\br$, $|\lambda|\ge 1$ and $C$ depends only on $p$, $d$ and the size of finite many derivatives of the function $a$. \endproclaim \demo{Proof} We adapt an argument of Stein found in \cite{pp.380-386, 25}. First, note that, using partition of unity and (5.7), we may assume that $$ \aligned \text{supp}\,a \subset \big\{ (\bx,\bby)\in \br^d\times\br^d: &\ |\bx-\bx_0|<\delta, \ |\bby-\bby_0|<\delta\\ &\ \ \ \text{ and } 1/2\le |\bx-\bby|\le 2\big\} \endaligned \tag 5.9 $$ for some $\bx_0,\ \bby_0\in \br^d$, where $\delta$ is sufficiently small. Also, by considering $$ \widetilde{\omega}(\bx) =\omega(\bx)+\frac{\e}{|\bx|^2} $$ and then letting $\e\to 0$, we may assume that $\omega>0$. Next we observe that, by duality, it suffices to show that $$ \aligned \| \int_{\br^d} e^{i\lambda |\bx-\bby|} & a(\bx,\bby) f(\bby)\, d\bby\|_{L^2(\br^d, d\bx)}\\ & \le C|\lambda|^{\frac{2-d}{2}} \left(\|\omega\|_{F^p(\br^d)}\right)^{1/2} \| f\|_{L^2(\br^d,\frac{d\bx}{\omega})}. \endaligned \tag 5.10 $$ By translation, we may assume that $\bx_0=0$. Since $a(\bx,\bby)=0$ if $|x_1|>\delta$, we need only to show that, for any fixed $x_1\in [-\delta, \delta]$, $$ \aligned \int_{\br^{d-1}} \big| \int_{\br^d} e^{i\lambda|\bx-\bby|} & a(x_1,\bx^\prime, \bby) f(\bby) d\bby\big|^2 d\bx^\prime\\ & \le C |\lambda|^{2-d} \|\omega\|_{F^p(\br^d)} \int_{\br^d} |f(\bold{z})|^2 \frac{d\bold{z}}{\omega (\bold{z})} \endaligned \tag 5.11 $$ where $\bx^\prime =(x_2,\dots, x_d)\in \br^{d-1}$. To this end, we let $$ S_\lambda f(\bx^\prime) =\int_{\br^d} e^{i\lambda |\bx-\bby|} a(x_1,\bx^\prime, \bby) f(\bby) d\bby \tag 5.12 $$ with $\bx =(x_1,\bx^\prime)$ and $x_1\in [-\delta, \delta]$ fixed. It is not hard to see that (5.11) would follow from $$ \| \omega^{1/2} S_\lambda^* S_\lambda (\omega^{1/2} f) \|_{L^2(\br^d, d\bby)} \le C |\lambda|^{2-d} \|\omega\|_{F^p(\br^d)} \| f\|_{L^2(\br^d, d\bby)}. \tag 5.13 $$ Note that $$ S_\lambda^* S_\lambda f(\bby) =\int_{\br^d} K_\lambda(\bby, \bold{z}) f(\bold{z}) d\bold{z} \tag 5.14 $$ where $$ K_\lambda (\bby, \bold{z}) =\int_{\br^d} e^{-i\lambda(|\bby-\bx|-|\bold{z}-\bx|)} \overline{a} (x_1,\bx^\prime, \bby) a(x_1,\bx^\prime, \bold{z}) d\bx^\prime. \tag 5.15 $$ By \cite{25}, there exists an analytic family of operators $$ U_\lambda^sf(\bby) =\int_{\br^d} K_\lambda^s (\bby, \bold{z}) f(\bold{z}) d\bold{z}, \tag 5.16 $$ such that $$ \align U_\lambda^0 & =S_\lambda^* S_\lambda, \tag 5.17\\ \| U_\lambda^s f\|_{L^2(\br^d, d\bby)} &\le \frac{C}{|\lambda|^d} \| f\|_{L^2(\br^d, d\bby)} \ \ \text{ if } \re\, s=1, \tag 5.18\\ |K_\lambda^s(\bby,\bold{z})| & \le \frac{C}{1+|\lambda|\, |\bby-\bold{z}|}\ \ \ \ \ \ \ \ \ \text{if } \re\, s=-\frac{d-3}{2}. \tag 5.19 \endalign $$ We remark that $K_\lambda^s(\bby,\bold{z})$ is given by $$ \int_{\br^d} e^{-i\lambda[\Phi(u,\bx^\prime, \bby) -\Phi(u,\bx^\prime,\bold{z})]} \overline{a}(x_1,\bx^\prime,\bby) a(x_1,\bx^\prime,\bold{z}) \alpha_s (u) d\bx^\prime du \tag 5.20 $$ where $$ \Phi(u,\bx^\prime,\bby) =|\bby-(x_1,\bx^\prime)| +u\, \phi_0(\bby), \ \ \ u\in\br,\ \ \bx^\prime\in \br^{d-1} \tag 5.21 $$ and $\phi_0$ is chosen so that $$ \text{det} \left(\frac{\partial^2}{\partial \widetilde{x}_j \partial y_k} \Phi(u,\bx^\prime,\bby)\right) \neq 0,\ \ \ \ \widetilde{\bx}=(u,\bx^\prime), \tag 5.22 $$ if $|(x_1,\bx^\prime)|\le\delta$ and $|\bby-\bby_0|\le \delta$. Also in (5.20), $\{ \alpha_s\}$ is a family of distributions on $\br^1$. When $\re\, s>0$, $$ \alpha_s(u) =\left\{ \aligned \frac{e^{s^2}}{\Gamma(s)} u^{s-1}\eta(u),\ \ \ \text{ if } u>0,\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \text{if } u\le 0 \endaligned\right. \tag 5.23 $$ where $\eta\in C_0^\infty(\br)$, $\eta(u)=1$ for $|u|\le 1$. For $\re\, s\le 0$, $\alpha_s$ is defined by analytic continuation. See \cite{p.381, 25}. By (5.20), $$ K_\lambda^s(\bby,\bold{z}) =K_\lambda(\bby,\bold{z}) \hat{\alpha}_s (\lambda \phi_0(\bby)-\lambda\phi_0(\bold{z})). \tag 5.24 $$ (5.19) follows from $$ |K_\lambda(\bby,\bold{z})| \le \frac{C}{\left \{ 1+|\lambda|\, |\bby-\bold{z}|\right\}^{(d-1)/2}} \tag 5.25 $$ and $$ |\hat{\alpha}_s(u)| \le \frac{C}{(1+|u|)^\sigma}, \ \ \ \ \re\, s=\sigma\ \ \text{and }\ \sigma\le 1. \tag 5.26 $$ (see \cite{p.382, 25}). Finally we consider the analytic family of operators $$ W^s_\lambda (f) =\omega^{\frac{1-s}{2}} U^s_\lambda \left( \omega^{\frac{1-s}{2}} f\right). \tag 5.27 $$ It follows from (5.18) that, if $\re\, s=1$, $$ \| W_\lambda^s f\|_{L^2(\br^d, d\bby)} \le \frac{C}{|\lambda|^d} \| f\|_{L^2(\br^d, d\bby)}. \tag 5.28 $$ If $\re \, s=-(d-3)/2$, by (5.19), we have $$ \left| W_\lambda^s (f)\right| \le \frac{C}{|\lambda|} \omega^{\frac{d-1}{4}} I_{d-1} \left( \omega^{\frac{d-1}{4}} |f|\right). \tag 5.29 $$ This, together with the weighted norm inequality for $I_{d-1}$ \cite{5,15}, gives $$ \| W_\lambda^s (f)\|_{L^2(\br^d,d\bby)} \le \frac{C_q}{|\lambda|} \sup\Sb r>0\\ \bx\in\br^d \endSb r^{d-1} \left\{ \frac{1}{r^{d}} \int_{Q(\bx,r)} \omega^{\frac{(d-1)q}{2}} d\bby\right\}^{1/q} \| f\|_{L^2(\br^d, d\bby)} $$ where $q>1$. Letting $q=2p/(d-1)>1$, we have $$ \| W^s_\lambda (f) \|_{L^2(\br^d, d\bx)} \le \frac{C}{|\lambda|} \left( \| \omega\|_{F^p(\br^d)}\right)^{\frac{d-1}{2}} \| f\|_{L^2(\br^d, d\bx)} \tag 5.30 $$ if $\re\, s=-(d-3)/2$. With (5.28) and (5.30), we apply the Stein interpolation theorem \cite{p.385, 25}. Since $$ 0=t\cdot (-\frac{d-3}{2}) +(1-t)\cdot 1\ \ \ \ \text{ where } t=\frac{2}{d-1}, $$ we obtain $$ \| W_\lambda^0 (f)\|_{L^2(\br^d, d\bby)} \le \frac{C}{|\lambda|^{d-2}} \|\omega\|_{F^p(\br^d)} \| f\|_{L^2(\br^d, d\bby)}. $$ Note that $W_\lambda^0 (f)= \omega^{1/2} U^0_\lambda (\omega^{1/2} f) =\omega^{1/2} S_\lambda^* S_\lambda (\omega^{1/2} f)$. (5.13) is proved. \enddemo To prove Theorem 5.3, let $F_z(\cdot)$ be the Fourier transform of $[|\cdot|^2 +z]^{-1}$ in $\br^d$ where $\re\, \sqrt{z}\ge 1$. We have $$ \aligned F_z(\bx) &= \frac{1}{(2\pi)^d} \int_{\br^d}\frac{e^{-i\bx\bby}}{|\bby|^2 +z}\, d\bby\\ &= c_d \left(\frac{z}{|\bx|^2}\right)^{\frac12(\frac{d}{2}-1)} K_{\frac{d}{2}-1} (\sqrt{z}|\bx|) \endaligned \tag 5.31 $$ where $K_{d/2-1}$ denotes the modified Bessel function of the third kind of order $d/2-1$ \cite{18}. By a change of variables, it is easy to see that the Fourier transform of $[|\bby B|^2 +z]^{-1}$ is given by $$ \aligned F_{z,B}(\bx) & =\frac{1}{\text{det}(B)} F_z(\bx B^{-1})\\ & =\frac{c_d}{\text{det}(B)} \left(\frac{z}{|\bx B^{-1}|}\right)^{\frac12(\frac{d}{2}-1)} K_{\frac{d}{2}-1} (\sqrt{z}|\bx B^{-1}|) \endaligned \tag 5.32 $$ where $B=\sqrt{A}$. \proclaim{\bf Theorem 5.33} Let $\eta\in C_0^\infty((-2,2))$ such that $\eta(r)=1$ if $|r|\le 1$. Let $\omega$ be a nonnegative function on $\br^d$ satisfying $\| \omega\|_{F^p(\br^d)}<\infty$ for some $p\in ((d-1)/2,d/2]$. Then $$ \aligned \|\int_{\br^d} \eta(|\bx-\bby|)\, & F_{z,B}(\bx-\bby) f(\bby) d\bby\|_{L^2(\br^d,\omega d\bx)}\\ &\le \frac{C}{|z|^{1/4}}\cdot \left(\|\omega\|_{F^p(\br^d)}\right)^{1/2} \|f\|_{L^2(\br^d,d\bx)} \endaligned \tag 5.34 $$ where $\re\, \sqrt{z}\ge 1$ and $C$ depends only on $p$, $d$, $B$ and the size of finite many derivatives of $\eta$. \endproclaim \demo{Proof} Theorem 5.33 follows from Theorem 5.5 by using partition of unity and standard rescaling argument. We omit the details. See \cite{pp.134-135, 23} for the case of $L^2\to L^q$ estimates. \enddemo We are now ready to give the \noindent{\bf Proof of Theorem 5.3.} Let $\psi\in C^\infty(\bt^d)$. Fix $\bx_0\in\oo$. We choose $\widetilde{\eta}\in C_0^\infty(Q(\bx_0,1/2))$ such that $\widetilde{\eta}\equiv 1$ on $Q(\bx_0, 1/4)$. Then, for $z=-(k+i)^2$ with $k\ge 1$, $$ \aligned &\psi(\bx)\widetilde{\eta}(\bx)^2\\ & = \widetilde{\eta}(\bx) \int_{\br^d} F_{z,B}(\bx-\bby) (\bd A\bd^T +z)(\psi \, \widetilde{\eta})\, d\bby\\ &= \widetilde{\eta}(\bx) \int_{\br^d} F_{z,B}(\bx-\bby) \left\{ (\bd A\bd^T +z)\psi\cdot \widetilde{\eta} -2\bd \psi A (\bd \widetilde{\eta})^T -\psi \bd A\bd^T \widetilde{\eta}\right\} d\bby\\ &= \widetilde{\eta}(\bx) \int_{\br^d} F_{z,B}(\bx-\bby)\eta(|\bx-\bby|)\cdot\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot\left\{ (\bd A\bd^T +z)\psi\cdot \widetilde{\eta} -2\bd \psi A (\bd \widetilde{\eta})^T -\psi \bd A\bd^T \widetilde{\eta}\right\} d\bby \endaligned $$ where $\eta\in C_0^\infty((-2,2))$, $ \eta(r)=1$ if $|r|\le 1$. It follows from Theorem 5.33 that $$ \aligned &\|\psi \widetilde{\eta}^2\|_{L^2(\br^d, \omega d\bx)}\\ &\le C k^{-1/2} \left(\| \omega \widetilde{\eta}^2\|_{F^p(\br^d)} \right)^{1/2}\cdot \\ &\ \cdot \left\{ \| [(\bd A\bd^T +z)\psi]\widetilde{\eta}\|_{L^2(\br^d,d\bx)} +\| \bd\psi A (\bd\widetilde{\eta})^T\|_{L^2(\br^d,d\bx)} +\|\psi \bd A\bd^T\widetilde{\eta}\|_{L^2(\br^d,d\bx)} \right\}\\ &\le C k^{-1/2} \left(\| \omega \widetilde{\eta}^2\| _{F^p(\br^d)} \right)^{1/2}\cdot \\ &\ \ \ \cdot \left\{ \| (\bd A\bd^T -k^2 -2ik)\psi\|_{L^2(\bt^d,d\bx)} +\|\bd \psi\|_{L^2(\bt^d, d\bx)} +\| \psi\|_{L^2(\bt^d, d\bx)}\right\} \endaligned \tag 5.35 $$ where we have used the fact that $\psi$ is periodic in the last inequality. Note that, since $\omega$ is periodic, $$ \aligned \|\omega\widetilde{\eta}^2\|_{F^p(\br^d)} &\le C\sup \Sb 00$ and $\| \omega\|_{F^p(\bt^d)} <\infty$. We need to show that, for any $\psi\in L^2 (\bt^d,\frac{d\bx}{\omega})$, $$ \| \bh_0 (\delta_0 +i\rho)^{-1} \psi \|_{L^2(\bt^d, \omega d\bx)} \le C \| \omega\|_{F^p(\bt^d)} \|\psi\|_{L^2(\bt^d,\frac{d\bx}{\omega})} \tag 6.1 $$ where $$ \bh_0(\delta_0 +i\rho)^{-1}\psi (\bx) =\sum_{\bn\in \z^d} \frac{\hat{\psi}(\bn) e^{i\bn\bx}} {(\bn +\bk )A (\bn +\bk)^T} \tag 6.2 $$ and $\bk =(\delta_0 +i\rho)\ba +\bb$. As in \cite{21}, we write $\psi =\sum_{j=-\infty}^\infty \psi_j$ where $$ \aligned \psi_j & =\sum \Sb \bn\in \z^d\\ n_1\in [2^{j-1},2^j -1] \endSb \hat{\psi}(\bn) e^{i\bn \bx}, \ \ \ \ \ \text{ for } j\ge 1,\\ \psi_j & =\sum \Sb \bn\in \z^d\\ n_1\in [-2^{-j}+1,-2^{-j-1}] \endSb \hat{\psi}(\bn) e^{i\bn \bx}, \ \ \ \ \ \text{ for } j\le -1,\\ \psi_0 & =\sum \Sb \bn\in \z^d\\ n_1=0 \endSb \hat{\psi}(\bn) e^{i\bn \bx}. \endaligned \tag 6.3 $$ \proclaim{\bf Proposition 6.4} If estimate (6.1) holds for $\psi_j$ with a constant $C$ independent of $j$, then it holds for $\psi$. \endproclaim \demo{Proof} Let $\omega_*$ be defined by (2.14). Note that $\omega_*\ge \omega$. By Proposition 2.13 and Remark 2.16, $ \| \omega_*\|_{F^p(\bt^d)} \le C\| \omega\|_{F^p(\bt^d)}$ and $\omega_*(\cdot,x_2,\dots, x_d)$, $\omega_*(\cdot, x_2,\dots, d_d)^{-1}$ are $A_1(\br)$ weights uniformly in $\bx^\prime=(x_2,\dots, x_d)\in\bt^{d-1}$. This , together with the weighted one-dimensional Littlewood-Paley theory on $\bt^1$ \cite{17}, gives $$ \aligned \| \bh_0 (\delta_0 +i\rho)^{-1}\psi\|_{L^2(\bt^d, \omega d\bx)} &\le \| \bh_0 (\delta_0 +i\rho)^{-1}\psi\|_{L^2(\bt^d, \omega_* d\bx)}\\ &\le C \| \left( \sum_{j=-\infty}^\infty |\bh_0 (\delta_0 +i\rho)^{-1}\psi_j|^2\right)^{1/2} \|_{L^2(\bt^d, \omega_* d\bx)}\\ &\le C \| \omega_*\|_{F^p(\bt^d)} \| \left(\sum_{j=-\infty}^\infty |\psi_j|^2\right)^{1/2} \|_{L^2(\bt^d,\frac{d\bx}{\omega_*})}\\ &\le C \| \omega\|_{F^p(\bt^d)} \| \left(\sum_{j=-\infty}^\infty |\psi_j|^2\right)^{1/2} \|_{L^2(\bt^d,\frac{d\bx}{\omega_*})}\\ &\le C \| \omega\|_{F^p(\bt^d)} \| \psi\|_{L^2(\bt^d, \frac{d\bx}{\omega_*})}\\ &\le C \| \omega\|_{F^p(\bt^d)} \| \psi\|_{L^2(\bt^d, \frac{d\bx}{\omega})}. \endaligned $$ \enddemo We will give the estimate of $\bh_0(\delta_0+i\rho)^{-1} \psi_j$ for $j\ge 1$ in details. The case $j\le 0$ may be handled in the same manner. To this end, we note that, by (3.6)-(3.7), $$ \aligned &(\bn +\bk)A(\bn +\bk)^T\\ &=|(\bn +\bb)B|^2 +2\delta_0 (n_1 +b_1) s_0 +(\delta^2 -\rho^2) a_1 s_0 +2i\rho (n_1 +\frac12 )s_0 \endaligned \tag 6.5 $$ where $B=\sqrt{A}$. Fix $j\ge 1$, let $$ z_j=-\rho^2 a_1 s_0 +2i\rho \cdot 2^j \cdot s_0. \tag 6.6 $$ Since $\re\, \sqrt{z_j} \ge c\min (2^j,\sqrt{|\rho|2^j}) \ge c_0 >0 $ (see \cite{21}), by Theorem 4.3, we have $$ \aligned \| \sum_{\bn\in\z^d} & \frac{\hat{\psi}_j(\bn)e^{i\bn\bx}} {|(\bn+\bb)B|^2 +z_j} \|_{L^2(\bt^d, \omega\, d\bx)}\\ & \le C \| \omega\|_{F^p(\bt^d)} \|\psi_j\|_{L^2(\bt^d,\frac{d\bx}{\omega})}. \endaligned \tag 6.7 $$ It remains to show that $$ \aligned \| \bh_0(\delta_0 +\rho)^{-1}\psi_j & - \sum_{\bn\in \z^d} \frac{\hat{\psi}_j(\bn) e^{i\bn\bx}} { |(\bn +\bb)B|^2 +z_j} \|_{L^2(\bt^d, \omega d\bx)}\\ &\le C \| \omega\|_{F^p(\bt^d)} \|\psi_j\|_{L^2(\bt^d,\frac{d\bx}{\omega})}. \endaligned \tag 6.8 $$ Using the Minkowski's inequality, we can bound the left hand side of (6.8) by $$ \aligned & \sum_{M=1}^\infty \| \sum_{|\bn B|\in [M-1, M)} \frac{ \hat{\psi}_j(\bn) e^{i\bn\bx} \left\{ |(\bn +\bb)B|^2 +z_j -(\bn +\bk)A(\bn +\bk)^T\right\} } {[(\bn +\bk)A (\bn +\bk)^T]\cdot [|(\bn +\bb|)B|^2 +z_j]} \|_{L^2 (\bt^d, \omega d\bx)}\\ &\le C \| \omega\|_{F^p(\bt^d)}^{1/2} \sum_{M=1}^\infty M^{1/2}\cdot\\ &\ \cdot \| \sum_{|\bn B|\in [M-1, M)} \frac{ \hat{\psi}_j(\bn) e^{i\bn\bx} \left\{ |(\bn +\bb)B|^2 +z_j -(\bn +\bk)A(\bn +\bk)^T\right\} } {[(\bn +\bk)A (\bn +\bk)^T]\cdot [|(\bn +\bb|)B|^2 +z_j]} \|_{L^2 (\bt^d, d\bx)}\\ &\le C \| \omega\|_{F^p(\bt^d)}^{1/2} \sum_{M=1}^\infty M^{1/2}\cdot\\ &\ \ \ \ \ \ \ \ \ \cdot \sup \Sb \bn\in \z^d\\ n_1\in [2^{j-1}, 2^j-1]\\ |\bn B|\in [M-1, M) \endSb \left| \frac{ |(\bn +\bb)B|^2 +z_j -(\bn +\bk)A(\bn +\bk)^T } {[(\bn +\bk)A (\bn +\bk)^T]\cdot [|(\bn +\bb|)B|^2 +z_j]} \right|\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \|\sum\Sb \bn\in\z^d\\ |\bn B|\in [M-1, M) \endSb \hat{\psi}_j(\bn) e^{i\bn\bx} \|_{L^2(\bt^d, d\bx)}\\ &\le C \| \omega\|_{F^p(\bt^d)} \| \psi_j\|_{L^2(\bt^d, \frac{d\bx}{\omega})} \sum_{M=1}^\infty M\cdot\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot\sup \Sb \bn\in \z^d\\ n_1\in [2^{j-1}, 2^j-1]\\ |\bn B|\in [M-1, M) \endSb \left| \frac{ |(\bn +\bb)B|^2 +z_j -(\bn +\bk)A(\bn +\bk)^T } {[(\bn +\bk)A (\bn +\bk)^T]\cdot [|(\bn +\bb|)B|^2 +z_j]} \right|\\ & \le C \| \omega\|_{F^p(\bt^d)} \| \psi_j\|_{L^2(\bt^d, \frac{d\bx}{\omega})}\\ &\ \ \cdot \sum_{M=1}^\infty M \sup \Sb \bn\in \z^d\\ |\bn B|\in [M-1,M) \endSb \frac{ |\rho| 2^j}{ \left\{ \big| |(\bn+\bb)B|^2 -\rho^2 a_1 s_0\big| + 2^j |\rho|\right\}^2 } \endaligned $$ where we have used (5.2) and its dual $$ \aligned \| \sum_{|\bn B|\in [k,k+1)} & \hat{\psi}(\bn) e^{i\bn\bx} \|_{L^2(\bt^d, d\bx)}\\ &\le C (k+1)^{1/2} \left( \| \omega\|_{F^p(\bt^d)}\right)^{1/2} \|\psi\|_{L^2(\bt^d,\frac{d\bx}{\omega})} \endaligned \tag 6.9 $$ in the first and third inequalities respectively. 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