I am learning some pseudo-differential operator and wondering is the Fourier transform of a compactly supported distribution a symbol, which gives a way to interpret the Fourier Inversion for those distributions.
By symbol I mean $|\partial^\alpha_x\partial^\beta_\xi a(x,\xi)|\leq C_{\alpha,\beta}(1+|\xi|)^{m-|\beta|}$ for some $m$ (in this case, our $a$ does not depend on $x$).