The Resolution of Euclidean Massless Field Operators of Higher Spins on $${\mathbb {R}}^6$$ R 6 and the $$L^2$$ L 2 Method

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 $${\mathbb {R}}^{5,1}$$ . Its Euclidean version $$\mathscr {D}_0$$ and their function theory are discussed in (Complex Anal Oper Theory 12:1219–1235, 2018). In this paper, we construct an exact sequence of weighted $$L^2$$ spaces resolving $$\mathscr {D}_0$$ : $$\begin{aligned} L^2_\varphi ({\mathbb {R}}^6, \mathscr {V}_0)\overset{\mathscr {D}_0}{\longrightarrow }L^2_\varphi ({\mathbb {R}}^6, \mathscr {V}_1)\overset{\mathscr {D}_1}{\longrightarrow }L^2_\varphi ({\mathbb {R}}^6, \mathscr {V}_2)\overset{\mathscr {D}_2}{\longrightarrow }L^2_\varphi ({\mathbb {R}}^6, \mathscr {V}_3)\longrightarrow 0, \end{aligned}$$ with suitable operators $$\mathscr {D}_l$$ and vector spaces $$\mathscr {V}_l$$ . Namely, we can solve $$\mathscr {D}_{l}u=f$$ in $$L^2_\varphi ({\mathbb {R}}^6, \mathscr {V}_{l})$$ when $$\mathscr {D}_{l+1} f=0$$ for $$f\in L^2_{\varphi }({\mathbb {R}}^6, \mathscr {V}_{l+1})$$ . 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 $$\begin{aligned} P({\mathbb {R}}^6, \mathscr {V}_0)\overset{\mathscr {D}_0}{\longrightarrow }P({\mathbb {R}}^6,\mathscr {V}_1)\overset{\mathscr {D}_1}{\longrightarrow }P({\mathbb {R}}^6, \mathscr {V}_2)\overset{\mathscr {D}_2}{\longrightarrow }P({\mathbb {R}}^6, \mathscr {V}_3)\longrightarrow 0 \end{aligned}$$ is a resolution, where $$P({\mathbb {R}}^6, \mathscr {V}_l)$$ 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.