数学代写|复分析作业代写Complex function代考|KMA152

数学代写|复分析作业代写Complex function代考|4th-Order Higher Spin Operator D4

Now for the main result in the $4 t h$-order higher spin case.
Theorem 3 Up to a multiplicative constant, the unique 4th-order conformally invariant differential operator is $\mathcal{D}_4: C^{\infty}\left(\mathbb{R}^m, \mathcal{H}_k\right) \longrightarrow C^{\infty}\left(\mathbb{R}^m, \mathcal{H}_k\right)$, where
$$\mathcal{D}_4=\mathcal{D}_2^2-\frac{8}{(m+2 k-2)(m+2 k-4)} \mathcal{D}_2 \Delta_x .$$
Hereafter we may suppress the $k$ index for the operator since there is little risk of confusion. The strategy is similar to that used above. It is sufficient to show only invariance under inversion. We have the definition for harmonic inversion as follows.
Definition 3 Harmonic inversion is a (conformal) transformation defined as
$$\mathcal{J}_4: C^{\infty}\left(\mathbb{R}^m, \mathcal{H}_k\right) \longrightarrow C^{\infty}\left(\mathbb{R}^m, \mathcal{H}_k\right): f(x, u) \mapsto \mathcal{J}_4[f](x, u):=|x|^{4-m} f\left(\frac{x}{|x|^2}, \frac{x u x}{|x|^2}\right)$$
Note this inversion consists of the classical Kelvin inversion $\mathcal{J}$ on $\mathbb{R}^m$ in the variable $x$ composed with a reflection $u \mapsto \omega u \omega$ acting on the dummy variable $u$ (where $x=$ $|x| \omega$ ) and a multiplication by a conformal weight term $|x|^{4-m}$. It satisfies $\mathcal{J}_4^2=$ 1. Then a similar calculation as in Proposition A.1 in [6] provides the following lemma.

Lemma 8 The special conformal transformation is defined as
$$\mathcal{C}4:=\mathcal{J}_4 \partial{x_j} \mathcal{J}4=2\langle u, x\rangle \partial{u_j}-2 u_j\left\langle x, D_u\right\rangle+|x|^2 \partial_{x_j}-x_j\left(2 \mathbb{E}_x+m-4\right)$$
Proposition 2 The special conformal transformations $\mathcal{C}_4$, with $j \in{1,2, \ldots, m}$ are generalized symmetries of $\mathcal{D}_4$. More specifically,
$$\left[\mathcal{D}_4, \mathcal{C}_4\right]=-8 x_j \mathcal{D}_4 .$$
In particular, this shows $\mathcal{J}_4 \mathcal{D}_4 \mathcal{J}_4=|x|^8 \mathcal{D}_4$, which generalizes the case of the classical higher order Dirac operator $D_x^4$. This also implies $\mathcal{D}_4$ is invariant under inversion and hence conformally invariant.

If the previous proposition holds, then the conformal invariance of $\mathcal{D}_4$ can be summarized in the following theorem:
Theorem 4 The first-order generalized symmetries of $\mathcal{D}_4$ are given by:

1. The infinitesimal rotations $L_{i, j}^x+L_{i, j}^u$, with $1 \leq i<j \leq m$.
2. The shifted Euler operator $m+2 \mathbb{E}_x-4$.
3. The infinitesimal translations $\partial_{x_j}$, with $1 \leq j \leq m$.
4. The special conformal transformations $\mathcal{J}4 \partial{x_j} \mathcal{J}_4$, with $1 \leq j \leq m$.
These operators span a Lie algebra which is isomorphic to the conformal Lie algebra $\mathfrak{s o}(1, m+1)$, whereby the Lie bracket is the ordinary commutator.
Proof The proof is similar as in [14] via transvector algebras.

数学代写|复分析作业代写Complex function代考|Connection with Lower Order Conformally Invariant

To construct higher order conformally invariant operators, one possible method is by composing and combining lower order conformally invariant operators. In this section, we will rewrite our operators $\mathcal{D}3$ and $\mathcal{D}_4$ in terms of first-order and secondorder conformally invariant operators. This might help us to construct higher order conformally invariant differential operators by induction from the lower order ones. Recall $\mathcal{D}_3$ maps $C^{\infty}\left(\mathbb{R}^m, \mathcal{M}_k\right)$ to $C^{\infty}\left(\mathbb{R}^m, \mathcal{M}_k\right)$. If we fix $x \in \mathbb{R}^m$, then for any $f(x, u) \in \mathcal{M}_k$, we have $\mathcal{D}_3 f(x, u) \in \mathcal{M}_k$. In other words, $\mathcal{D}_3$ should be equal to the sum of contributions to $\mathcal{M}_k$ of all terms in $\mathcal{D}_3$. Notice that if we apply each term of $\mathcal{D}_3$ to $f(x, u) \in C^{\infty}\left(\mathbb{R}^m, \mathcal{M}_k\right)$, we will get a $k$-homogeneous polynomial in $u$ that is in the kernel of $\Delta_u^2$. Hence, we can decompose it by harmonic decomposition as follows: $$\mathcal{P}_k=\mathcal{H}_k \oplus u^2 \mathcal{H}{k-2},$$

where $\mathcal{P}k$ is the $k$-homogeneous polynomial space and $\mathcal{H}_k$ is the $k$-homogeneous harmonic polynomial space. The Almansi-Fischer decomposition provides further $$\mathcal{H}_k=\mathcal{M}_k \oplus u \mathcal{M}{k-1},$$
where $\mathcal{M}k$ is the $k$-homogeneous monogenic polynomial space; therefore, the contribution of each term to $\mathcal{M}_k$ can be written with two projections. For instance, the contribution of $u^3\left\langle D_u, D_x\right\rangle^3 f(x, u)$ to $\mathcal{M}_k$ is $P_k P_1 u^3\left\langle D_u, D_x\right\rangle^3 f(x, u)$, where $$\mathcal{P}_k \stackrel{P_1}{\rightarrow} \mathcal{H}_k \stackrel{P_k}{\rightarrow} \mathcal{M}_k,$$ and $$P_1=1+\frac{u^2 \Delta_u}{2(m+2 k-4)}, P_k=1+\frac{u D_u}{m+2 k-2} .$$ We also notice that for fixed $x \in \mathbb{R}^m$ and $f(x, u) \in \mathcal{M}_k$, $$u^3\left\langle D_u, D_x\right\rangle^3 f(x, u),|u|^2\left\langle D_u, D_x\right\rangle^2 D_x f(x, u) \in u^2 \mathcal{H}{k-2},$$
and $u\left\langle D_u, D_x\right\rangle D_x^2 \in u \mathcal{M}_{k-1}$. Hence, their contributions to $\mathcal{M}_k$ are all zero. Therefore,
$$\mathcal{D}_3=P_k P_1\left(D_x^3+\frac{4}{m+2 k}\left\langle u, D_x\right\rangle\left\langle D_u, D_x\right\rangle D_x-\frac{8 u\left\langle u, D_x\right\rangle\left\langle D_u, D_x\right\rangle^2}{(m+2 k)(m+2 k-2)}\right)$$

复分析代考

数学代写|复分析作业代写Complex function代考|4th-Order Higher Spin Operator D4

$$\left[\mathcal{D}_4, \mathcal{C}_4\right]=-8 x_j \mathcal{D}_4$$

1. 无穷小的旋转 $L_{i, j}^x+L_{i, j}^u$ ， 和 $1 \leq i<j \leq m$.
2. 移位的欧拉算子 $m+2 \mathbb{E}_x-4$.
3. 特殊共形变换 \$\mathcal{}}$4 \backslash$partial${x$j$} \backslash$mathcal{}}_4, with$1 \backslash l$eq$j \backslash l$leq 米 .TheseoperatorsspanaLiealgebrawhichisisomorphictothecon formalLiealgebra Imathfrak{so}(1, m+1)\$，其中李括号是普通的换向器。
证明证明与 [14] 中的类似，通过向量代数。

数学代写|复分析作业代写Complex function代考|Connection with Lower Order Conformally Invariant

$$\mathcal{D}_3=P_k P_1\left(D_x^3+\frac{4}{m+2 k}\left\langle u, D_x\right\rangle\left\langle D_u, D_x\right\rangle D_x-\frac{8 u\left\langle u, D_x\right\rangle\left\langle D_u, D_x\right\rangle^2}{(m+2 k)(m+2 k-2)}\right)$$

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