(Mellin convolution in the group \(((0, \infty), \cdot)\) ) Define on \(((0, \infty), \mathscr{B}(0, \infty))\) the measure \(\mu(d x)=x^{-1} d x\). The Mellin convolution of measurable \(u, w:(0, \infty) ightarrow \mathbb{R}\) is given by

\[u \circledast w(x):=\int_{(0, \infty)} u\left(x y^{-1}ight) w(y) \mu(d y)\]

(i) If \(u, w \geqslant 0\), then \(u \circledast w=w \circledast u\) is measurable and

\[\int_{0}^{\infty} u \circledast w d \mu=\int_{0}^{\infty} u d \mu \int_{0}^{\infty} w d \mu\]

(ii) If \(u \in \mathcal{L}^{p}(\mu), p \in[1, \infty]\), and \(w \in \mathcal{L}^{1}(\mu)\), then \(u \circledast w \in \mathcal{L}^{p}(\mu)\) and

\[\|u \circledast w\|_{p} \leqslant\|u\|_{p} \cdot\|w\|_{1} .\]