Hormander分析不等式

02-18

$Omega subset C^n$ 為擬凸域

$Omega=cup_{j=1}^{infty}K_j$ , $hat K_j=K_j subset K_{j+1}$ .

$eta_j in C_0^{infty}(Omega)$ , $eta_j |_{K_{j-1}}=1$ , $supp eta_j subset K_j$ .

$psi in C_0^{infty}(Omega)$

$|e^{psi}|^2=1+sum_{j=1}^{infty}sum_{k=1}^n |frac{partial eta_j}{partial ar z_k}|^2$ . $|partial eta_j|^2 leq |e^{psi}|^2$ .

$phiin C_0^{infty}(Omega) cap PSH(Omega)$ .

$phi_1=psi-2phi$ , $phi_2=psi-phi$ , $phi_3=psi$ .

$sum_{j=1}^n frac{partial^2 psi}{partial z_j partial ar z_k} xi_jar xi_k geq 2(|partial psi|^2+e^{psi})|xi|^2$ . $forall zin Omega$ , $forall xi in C^n$ .

$L^2_{(p,q)}(Omega,phi_1) longrightarrow L^2_{(p,q+1}(Omega,phi_2) longrightarrow L_{(p,q+2)}^2(Omega,phi_3)$ .

$H_1 longrightarrow H_2 longrightarrow H_3$

$H_i=L^2_{(p,q+1-i)}(Omega,phi_i)$ .

$||cdot ||_{H_j}=||cdot||_{phi_j}$ , $phi_j longrightarrow e^{-phi_j}d lambda$ . $dlambda$ lesbegue測度.

$forall fin D_{(p,q+1)}(Omega)$ , $D=C_0^{infty}$ .

$T^*: H_2 longrightarrow H_1$ .

$<f,T^*g>_{phi_1}=<Tf,g>_{phi_2}$ , $f in H_1,gin H_2$ .

$f(z)=sum_{|I|=p,|J|=q}f_{I,J} dz_{I}wedge dz_J$ .

$g(z)=sum_{I,J}sum_{j=1}^n g_{I,j,J}dz_I wedge dar z_j wedge d ar z_J$ .

$Tf(z)=arpartial f(z)=sum_{|I|=p,|J|=q}sum_{j=1}^n(-1)^pfrac{partial f_{I,J}}{partial ar z_j}dz_I wedge d ar z_j wedge ar z_J$ .

$<Tf,g>_{phi_2}=int_{Omega}sum_{j=1}^n(-1)^p frac{f_{I,J}}{partial ar z_j}ar g_{I,j,J}e^{-phi_2}dlambda$

$sum_{j=1}^n(-1)^pint_{Omega}frac{f_{I,J}}{partial ar z_j}(ar g_{I,j,J}e^{-phi_2})dlambda$

(integral by part) $=sum_{j=1}^n(-1)^{p-1}int_{Omega}f_{I,J}frac{partial}{partial ar z_j}(ar g_{I,j,J}e^{-phi_2})dlambda$ .

另一方面

$<f,T^*g>_{phi_1}=sum_{|I|=p,|J|=q}int_{Omega}f_{I,J}overline{(T^*g)_{I,J}}e^{-phi_1}dlambda$

$f(z)=sum_{|I|=p,|J|=q}f_{I,J}dz_Iwedge dar z_J$ .

$T^*g(z)=sum_{|I|=p,|J|=q}(T^*g)_{I,J}dz_I wedge dar z_J$ .

又由於 $<f,T^*g>_{phi_1}=<Tf,g>_{phi_2}$ ，從此式比較係數得：

$T^*g(z)=(-1)^{p-1}sum_{|I|=p,|J|=q}sum_{j=1}^ne^{phi} frac{partial}{partial z_j}(e^{-phi_2} g_{I,j,J})dz_I wedge d ar z_J$ .

進一步化簡：

$T^*g(z)=(-1)^{p-1}sum_{|I|=p,|J|=q}sum_{j=1}^ne^{phi} frac{partial}{partial z_j}(e^{-phi_2} g_{I,j,J})dz_I wedge d ar z_J$ $=(-1)^{p-1}e^{-phi}sum_{|I|=p,|J|=q}sum_{j=1}^n(frac{partial h_{I,j,J}}{partial z_j}-frac{partial phi}{partial z_j}g_{I,j,J})dz_Iwedge d ar z_J+(-1)^{p-1}e^{-phi}sum_{|I|=p,|J|=q}sum_{j=1}^nfrac{partial phi}{partial z_j}g_{I,j,J}dz_Iwedge d ar z_J$

$=X+Y$ .

因此： $T^*f=X+Y,X=T^*f-Y$

$||X||_{phi_1}leq ||T^*f||_{phi_1}+||Y||_{phi_1}$ .

$||X||_{phi_1}^2 leq 2(||T^*f||_{phi_1}^2+||Y||_{phi_1}^2)$ , $f=g$ .

$Y=(-1)^{p-1}e^{-phi}sum_{|I|=p,|J|=q}sum_{j=1}^nfrac{partial phi}{partial z_j}g_{I,j,J}dz_Iwedge d ar z_J$ .

$||Y||_{phi_1}^2=int_{Omega_1}|(-1)^{p-1}e^{-phi}sum_{j}frac{partial psi}{partial z_j} g_{I,j,J}|^2e^{-phi_1}dlambda$

$=int_{Omega}|sum_j frac{partial psi}{partial z_j }g_{I,jJ}|^2e^{-phi}dlambda$ $leq int_{Omega} |f|^2|arpartial phi|^2e^{-phi}dlambda$ $=||f||_{phi_1}^2$

$X =(-1)^{p-1}e^{-phi}sum_{|I|=p,|J|=q}sum_{j=1}^n(frac{partial h_{I,j,J}}{partial z_j}-frac{partial phi}{partial z_j}g_{I,j,J})dz_Iwedge d ar z_J$ .

$delta_j g=e^{phi}frac{partial}{partial z_j}e^{-phi}g=frac{partial g}{partial z_j}-frac{partial phi}{partial z_j}g$ .

$g,h in C_0^{infty}(Omega)$ .

$<g,frac{partial}{partial ar z_j}>_{phi}=-<delta_j g,h>_{phi}$ .

$int_{Omega}g overline{(frac{partial h}{partial ar z_j})}e^{-phi}dlambda=int_{Omega}overline{(frac{partial h}{partial ar z_j})}(ge^{-phi})dlambda$

$=int_{Omega}ar h e^{phi}frac{partial}{partial z_j}(g e^{-phi})dlambda$

$-<delta_j g,h>_{phi}$ .

$[delta_j,ar partial_k]=frac{partial^2 phi}{partial z_jpartial ar z_k}$ , $[A,B]=AB-BA$ .

$[delta_j,delta_k]g=[frac{partial}{partial z_j}-frac{partialphi}{partial z_j},frac{partial}{partial ar z_k}]g$

$=[-frac{partial phi}{partial z_j},frac{partial}{partial ar z_k}]g$

$=frac{partial^2 phi}{partial z_jpartial ar z_k}g$ .

$<delta_j g,delta_k h>_{phi}=<frac{partial g}{partial ar z_k},frac{partial h}{partial ar z_j}>_{phi}+<g,frac{partial^2 phi}{partial z_jpartial ar z_k}h>_{phi}$

實際上

$<g,ardelta_j h>_{phi}$

$=<g,ar delta_jdelta_k h>_{phi}=<g,delta_k(ardelta_j h)+frac{partial^2 phi}{partial z_jpartial ar z_k}h>_{phi}=<arpartial_k g,ar partial_j h>_{phi}+<g,frac{partial^2 phi}{partial z_jpartial ar z_k}h>_{phi}$ .

下面計算 $||X||^2_{phi}$

$X =(-1)^{p-1}e^{-phi}sum_{|I|=p,|J|=q}sum_{j=1}^n(frac{partial h_{I,j,J}}{partial z_j}-frac{partial phi}{partial z_j}g_{I,j,J})dz_Iwedge d ar z_J$ .

$phi_1=phi-2psi$

$||X||_{phi_1}^2=int_{Omega}sum_{I,J}|sum_j delta_j f_{I,j,J}|^2e^{-2psi} e^{-phi_1}dlambda$

$sum_{I,J}sum_{j,k}int_{Omega}<delta_jf_{I,j,J},delta_kf_{I,k,J}>e^{-psi}dlambda$

結合前面的計算告訴我們：

$||X||_{phi_1}^2leq ||T^*f||^2+int_{Omega}|f|^2|arpartial phi|^2e^{-psi}dlambda$ .

$||X||^2_{phi_1}=sum_{I,J}sum_{j,k}<delta_if_{I,i,J},delta_kf_{I,k,J}>_{phi}$

$=sum_{I,J}sum_{j,k}<frac{partial}{partial ar z_k}f_{I,j,J},frac{partial}{partial ar z_j}f_{I,k,J}>_{phi}+sum_{I,J}sum_{j,k}<f_{I,j,J},frac{partial^2 phi}{partial z_j partial ar z_k}f_{I,k,J}>_{phi}$

涉及到之後的計算，進一步，我們將會得到：

$||X||^2_{phi_1}=sum_{I,J}sum_{j,k}<delta_if_{I,i,J},delta_kf_{I,k,J}>_{phi}$

$=sum_{I,J}sum_{j,k}<frac{partial}{partial ar z_k}f_{I,j,J},frac{partial}{partial ar z_j}f_{I,k,J}>_{phi}+sum_{I,J}sum_{j,k}<f_{I,j,J},frac{partial^2 phi}{partial z_j partial ar z_k}f_{I,k,J}>_{phi}$ $=sum_{I,J}sum_jint_{Omega}|frac{partial f_{I,J}}{partial ar z_j}|^2e^{-phi}dlambda-||Sf||_{phi}^2+sum_{I,J}sum_{j,k}<f_{I,j,J},frac{partial^2 phi}{partial ar z_j partialar z_k f_{I,k,J}}>_{phi}$

關鍵是計算 $Sf$

我們有線性映射：

$L_{(p,q+1)}^2(Omega,phi_2) longrightarrow L^2_{(p,q+2)}(Omega,phi_3)$

$Sf=delta f=(-1)^{p}sum_{|I|=p,|J|=q+1}sum_{j=1}^n frac{partial f_{I,J}(z)}{partial ar z_j} dz_Iwedge dar z_j wedge dar z_J$

$=sum_{I,K}sum_{j}(-1)^pfrac{partial f_{I,J}(z)}{partial ar z_j}sigma_k^{j,J}dz_I wedge d ar z_k$ .

$||Sf||_{phi}^2=sum_{I,K}int_{Omega}|sum_{J}sum_j(-1)^pfrac{partial f_{I,J(z)}}{partial ar z_j}sigma_{k}^{I,J}|^2e^{-phi}dlambda$

$=sum_{I,K}int_{Omega}sum_{J,L}sum_{j,l}(frac{partial f_{I,J}(z)}{partial ar z_j}sigma_k^{i,J})(frac{partial f_{I,L}}{partial ar z_l}sigma_{l,L}^k)e^{-phi}dlambda$

$=sum_{I,J,L}sum_{j,l}frac{partial f_{I,J}}{partial ar z_j}frac{partial f_{I,L}}{partial ar z_l}sum_k sigma_k^{j,J}sigma_{lL}^k e^{-phi}dlambda$

(我們知道 $sum_k sigma_k^{j,J}sigma_{l,L}^k=sigma_{l,L}^{k,J}$ )

$=sum_{I,J,L}sum_{j,l}frac{partial f_{I,J}}{partial ar z_j}frac{partial f_{I,L}}{partial ar z_l}sum_k sigma_{l,L}^{k,J} e^{-phi}dlambda$ .

( $sigma^{j,J}_{l,L}=sigma^J_L$ $j eq l$ $J=j cup K$

$sigma_L^J=1$ $J=l$

$0 eq sigma_{lL}^{jJ} = sigma_{jlk}^{jJ}sigma_{ljk}^{jlk}sigma_{lL}^{lek}$

$=sigma_{lk}^Jsigma_L^{jk}$ )

$=sum_{I,J}sum_{j otin J}int_{Omega}|frac{partial f_{I,J}}{partial ar z_j}|^2e^{-phi}dlambda-sum_{I,K}sum_{j eq L}int_{Omega}frac{partial f_{I,J,k}}{partial z_j}frac{partial f_{I,l,K}}{partial z_l}e^{phi}dlambda$ .

$sum_{I,J}sum_{jin J}int_{Omega}|frac{partial f_{I,J}}{partial ar z_j}|^2e^{-phi}dlambda$

( $j=lk$ ) $=sum_{I,K}sum_{j=l}int_{Omega}frac{partial f_{I,j,k}}{partial ar z_j}overline{frac{partial f_{I,j,k}}{partial ar z_l}}e^{-phi}dlambda$ .

$||delta f||^2_{phi}=sum_{I,J}sum_jint_{Omega}|frac{partial f_{I,J}}{partial z_j}|^2e^{-phi}dlambda-sum_{I,K}sum_{j,l}int_{Omega}frac{partial f_{I,j,k}}{partial z_j}overline{frac{partial f_{J,j,k}}{partial z_l}}e^{-phi}dlambda$ .

結合這些計算我們得到了：

$||X||^2_{phi_1}=sum_{I,J}sum_{j,k}<delta_if_{I,i,J},delta_kf_{I,k,J}>_{phi}$

$=sum_{I,J}sum_{j,k}<frac{partial}{partial ar z_k}f_{I,j,J},frac{partial}{partial ar z_j}f_{I,k,J}>_{phi}+sum_{I,J}sum_{j,k}<f_{I,j,J},frac{partial^2 phi}{partial z_j partial ar z_k}f_{I,k,J}>_{phi}$

$=sum_{I,J}sum_jint_{Omega}|frac{partial f_{I,J}}{partial ar z_j}|^2e^{-phi}dlambda-||Sf||_{phi}^2+sum_{I,J}sum_{j,k}<f_{I,j,J},frac{partial^2 phi}{partial ar z_j partialar z_k f_{I,k,J}}>_{phi}$

這樣我們證明了：

Hormander $L^2$ inequality:

$sum_{|I|=p,|J|=q}sum_{j,k=1}^nint_{Omega}frac{partial^2 phi}{partial z_j partial ar z_k}f_{I,j,J}overline{f_{I,k,J}}e^{-phi}dlambda+sum_{|I|=p,|J|=q}sum_{j,k=1}^nint_{Omega}|frac{partial f_{I,J}}{partial ar z_j}|^2e^{-phi}dlambda$

$leq 2||T^*f||_{phi_1}^2+||delta f||_{phi_3}^2+2int_{Omega}|f|^2|delta phi|^2e^{-phi}dlambda$ .

此不等式加上levy-form的標準的不等式：

$L(phi,X)geq 2(|arpartial phi|^2+e^{-phi}2||X||^2)$ 其中 $phi$ 多重次調和函數

可得：

$forall f in D_{p,q+1}(Omega,e^{phi})$

$||f||_{phi_2}^2leq ||T^*f||_{phi_1}^2+||delta||_{phi_3}^2$ .