Two Results on the Commutative Product of Distributions and Functions - Fisher, Brian (Sarajevo Journal of Mathematics, 2024)
Let $f$ and $g$ be distributions and let $f_n=(f*\delta_n x)$ and $g_n =(g*\delta _n)(x)$, where $\delta _n(x)$ is a certain sequence converging to the Dirac delta-function. The product $f .g$ of $f$ and $g$ is defined to be the limit of the sequence $\{f_ng_n\}$, provided its limit $h$ exists in the sense that $$\lim _{n \to \infty} \langle f_n (x) g_n (x),\vphi (x) \rangle =\langle h(x), \vphi (x) \rangle $$ for all functions $\vphi$ in ${\mathcal D}.$ It is proved that \begin{align*}(\sgn x|x| ^{-r} \ln ^p |x|). ( |x| ^\mu \ln ^q|x|) &=\sgn x |x| ^{-r +\mu } \ln ^{p+q}|x| ,\\ (|x| ^{-r} \ln ^p |x|). (\sgn x |x| ^\mu \ln ^q|x| ) &=\sgn x |x| ^{-r +\mu } \ln ^{p+q}|x| \end{align*} for $-2<-r+\mu< -1,$ $r=1,2, \ldots$ and $p,q =0,1,2. \ldots .$ 2000 Mathematics Subject Classification. 46F10