Let $x \in \mathbb{R}$ and define $x_+:=xH(x)$, where $H$ is the Heaviside function. By induction we can show that $E(x)=\dfrac{x_+^{k-1}}{(k-1)!}$ is a fundamental solution of $$P=\frac{d^{k}}{dx^{k}},$$ that is $$PE=\delta.$$ My problem is
Show that $$E_k(x)=\frac{(x_1)_+^{k-1}\ldots (x_n)_+^{k-1}}{(k-1)!^{n}}$$ is a fundamental solution of $L=\partial^{k}_{x_1}\ldots \partial_{x_n}^{k}$.
I really can't apply the Leibniz rule to solve the problem, because I don't have the product of a smooth function with a distribution.
Thank you for your help.