The relationship between regular distribution and tensor product of distribution?

Question

For a regular distribution $T_f$, where $f$ is defined on $\mathbb R^n$, $$ \langle T_f, \varphi\rangle := \int_{\mathbb R^n} f(x) \varphi(x) dx. $$ The tensor product of two distributions $S$ and $T$ is definded by $$ \langle T \otimes S, \varphi \rangle := \langle T, \langle S, \varphi_\cdot \rangle \rangle, $$ where $$ \varphi_\cdot(y) = \varphi(\cdot, y). $$ If $T_f$ and $T_g$ are two regular distributions, where $f$ is defined on $\mathbb R^n$ and $g$ on $\mathbb R^m$, then $$ \langle T_f \otimes T_g, \varphi \rangle = \int_{\mathbb R^n \times \mathbb R^m} f(x) g(y) \varphi(x, y) dx dy, $$ so that the tensor product of distribution is a special case of a regular distribution, ie. $T_f \otimes T_g = T_h$, where $h(x,y)=f(x)g(y)$. Is this correct?

Answer 1

This is correct. By the definition of the tensor product of distributions,

$$ \langle T_f \otimes T_g, \varphi \rangle = \langle T_f, x \mapsto \int_{\mathbb R^m} g(y) \varphi(x, y) dy \rangle = \int_{\mathbb R^m} f(x) \int_{\mathbb R^m} g(y) \varphi(x, y) dy dx, $$ which by Fubini's theorem equals the integral you describe in your question.