# Algebra of pseudo-differential operators

The class of pseudod-ifferential operators form an associative algebra of Fourier integral operators. Moreover, given symbols $a,b,c\in C^\infty$ (each associated to some pseudo differential operator), for the composition of symbols (#) there holds:

$$\text{op}(a)\circ \text{op}(b) = \text{op}(a\# b),$$

so, obviously, $\#$ should be an associative operation as well.

The latter is equivalent to prove for

$$(a \# b)(x,y)=\sum_{|\alpha|\geq 0} \frac 1 {\alpha!} \partial_y^\alpha a(x,y) D_x^\alpha b(x,y)\quad (1)$$

there holds $(a\#b)\# c = a\#(b\# c)$, where $D=-i\partial$.

To check this, I first defined $\circ_N$ which considers in $(1)$ only the multi-indexes up to length $N$, i.e.

$$(a \circ_N b)(x,y)=\sum_{|\alpha|\leq N} \frac 1 {\alpha!} \partial_y^\alpha a(x,y) D_x^\alpha b(x,y).$$

Now, using the general Leibniz product rule I get

$$LS := (a\circ_N b) \circ_N c = \sum_{|\alpha|\leq N} \frac 1 {\alpha!} \partial_\xi^\alpha (\sum_{|\beta|\leq N}) \partial_\xi^\beta a D_x^\beta b) D_x^\alpha c = \sum_{|\alpha|\leq N,\ |\beta|\leq N,\ \alpha_1+\alpha_2=\alpha} \frac{1} {\beta!\alpha_1!\alpha_2!} (\partial_\xi^{\alpha_1+\beta} a) (\partial_\xi^{\alpha_2}D_x^\beta b) (D_x^\alpha c).$$

Similarly, we see

$$RS := a \circ_N (b\circ_N c) = \sum_{|\alpha|\leq N,\ |\beta|\leq N,\ \alpha_1+\alpha_2=\alpha} \frac{1} {\beta!\alpha_1!\alpha_2!} (\partial_\xi^\alpha a) (\partial_\xi^{\beta}D_x^{\alpha_2} b) (D_x^{\alpha_1+\beta} c).$$

But unfortunately it is $LS\neq RS$; the larger $N$, the more different terms arise on each side and they will never cancel.

What's my mistake?

Why do you think the two are different?

If you don't limit the sum to length $N$ you have (I use $\gamma$ and $\delta$ for the multiindices in the summation for $RS$ to avoid confusion):

$$LS := (a\circ b) \circ c = \sum_{\alpha_1,\alpha_2,\beta} \frac{1} {\beta!\alpha_1!\alpha_2!} (\partial_\xi^{\alpha_1+\beta} a) (\partial_\xi^{\alpha_2}D_x^\beta b) (D_x^{\alpha_1+\alpha_2} c).$$

$$RS := a \circ (b\circ c) = \sum_{\gamma_1,\gamma_2,\delta} \frac{1} {\delta!\gamma_1!\gamma_2!} (\partial_\xi^{\gamma_1 + \gamma_2} a) (\partial_\xi^{\delta}D_x^{\gamma_2} b) (D_x^{\gamma_1+\delta} c).$$

and one sees that replacing $\alpha_1 = \gamma_1$, $\alpha_2 = \delta$ and $\beta = \gamma_2$ the two expressions are exactly identical.

So your mistake is a mistake of notation. Dummy variables in summations can be renamed.

You are correct, however, in saying that truncated to order $N$ the operation is not necessarily associative. But even there you should be able to see partial cancellations.