The resolution of $4$-dimensional massless field operators of higher spins was constructed by Eastwood-Penrose-Wells by using the twistor method. Recently physicists are interested in $6$-dimensional physics including the massless field operators of higher spins on Lorentzian space $\Bbb R^{ 5,1}$. Its Euclidean version $\mathscr{ D}_0$ and their function theory are discussed in \cite{ wangkang3}. In this paper, we construct an exact sequence of Hilbert spaces as weighted $L^2$ spaces resolving $\mathscr{ D}_0$: $$L^2_\varphi(\Bbb R^6, \mathscr{ V}_0)\overset{ \mathscr{ D}_0}\longrightarrow L^2_\varphi(\Bbb R^6,\mathscr{ V}_1)\overset{ \mathscr{ D}_1}\longrightarrow L^2_\varphi(\Bbb R^6, \mathscr{ V}_2)\overset{ \mathscr{ D}_2}\longrightarrow L^2_\varphi(\Bbb R^6, \mathscr{ V}_3)\longrightarrow 0,$$ with suitable operators $\mathscr{ D}_l$ and vector spaces $\mathscr{ V}_l$. Namely, we can solve $\mathscr{ D}_{ l}u=f$ in $L^2_\varphi(\Bbb R^6, \mathscr{ V}_{ l})$ when $\mathscr{ D}_{ l+1} f=0$ for $f\in L^2_{ \varphi}(\Bbb R^6, \mathscr{ V}_{ l+1})$. This is proved by using the $L^2$ method in the theory of several complex variables, which is a general framework to solve overdetermined PDEs under the compatibility condition. To apply this method here, it is necessary to consider weighted $L^2$ spaces, an advantage of which is that any polynomial is $L^2_{ \varphi}$ integrable. As a corollary, we prove that $$ P(\Bbb R^6, \mathscr{ V}_0)\overset{ \mathscr{ D}_0}\longrightarrow P(\Bbb R^6,\mathscr{ V}_1)\overset{ \mathscr{ D}_1}\longrightarrow P(\Bbb R^6, \mathscr{ V}_2)\overset{ \mathscr{ D}_2}\longrightarrow P(\Bbb R^6, \mathscr{ V}_3)\longrightarrow 0$$ is a resolution, where $P(\Bbb R^6, \mathscr{ V}_l)$ is the space of all $\mathscr{ V}_l$-valued polynomials. This provides an analytic way to construct a resolution of a differential operator acting on vector valued polynomials.
Categories: math-ph math.CV math.MP