Then the pseudo-differential operator ${\operatorname{Op}}_t(a)$ is the linear and continuous operator on ${\mathcal S}_s ({\mathbf R^{d}})$, given by $$\label{e0.5}
({\operatorname{Op}}_t(a)f)(x)
=
(2\pi  ) ^{-d}\iint a((1-t)x+ty,\xi )f(y)e^{i{\langle x-y,\xi\rangle} }\,
dyd\xi .$$ For general $a\in {\mathcal S}_s'({\mathbf R^{2d}})$, the pseudo-differential operator ${\operatorname{Op}}_t(a)$ is defined as the continuous operator from ${\mathcal S}_s({\mathbf R^{d}})$ to ${\mathcal S}_s'({\mathbf R^{d}})$ with distribution kernel $$\label{atkernel}
K_{a,t}(x,y)=(2\pi )^{-d/2}({\mathscr F}_2^{-1}a)((1-t)x+ty,x-y).$$ Here ${\mathscr F}_2F$ is the partial Fourier transform of $F(x,y)\in
{\mathcal S}_s'({\mathbf R^{2d}})$ with respect to the $y$ variable. This definition makes sense, since the mappings $$\label{homeoF2tmap}
{\mathscr F}_2\quad \text{and}\quad F(x,y)\mapsto F((1-t)x+ty,y-x)$$ are homeomorphisms on ${\mathcal S}_s'({\mathbf R^{2d}})$. In particular, the map $a\mapsto K_{a,t}$ is a homeomorphism on ${\mathcal S}_s'({\mathbf R^{2d}})$.

The standard (Kohn-Nirenberg) representation, $a(x,D)={\operatorname{Op}}(a)$, and the Weyl quantization ${\operatorname{Op}}^w(a)$ of $a$ are obtained by choosing $t=0$ and $t=1/2$, respectively, in and .

\[BijKernelsOps\] By Fourier’s inversion formula, and the kernel theorem [@LozPerTask Theorem 2.2] for operators from Gelfand-Shilov spaces to their duals, it follows that the map $a\mapsto {\operatorname{Op}}_t(a)$ is bijective from ${\mathcal S}_s'({\mathbf R^{2d}})$ to the set of all linear and continuous operators from ${\mathcal S}_s({\mathbf R^{d}})$ to ${\mathcal S}_s'({\mathbf R^{2d}})$.

By Remark \[BijKernelsOps\], it follows that for every $a_1\in {\mathcal S}_s '({\mathbf R^{2d}})$ and $t_1,t_2\in 