Consider the following exerpt from the paper "Non-linear Quantum Processes" by Segal:
with the norm $\|F\|=\left(\int\|F(x)\|^p d x\right)^{1 / p}$$\|F\|=\left(\int\|F(x)\|^p \, d x\right)^{1 / p}$, then the operator $T_1^{\prime}: F \rightarrow G$, where $G(x)=\int K_1(x, y) F(y) d y$$G(x)=\int K_1(x, y) F(y) \, d y$, exists and is a contraction from $L_p\left(M_1, \mathbf{B}\right)$ to $L_q\left(M_1, \mathbf{B}\right)$. For the mapping $y \rightarrow K_1(x, y) F(y)$ is easily seen to be strongly measurable from $M_1$ to $\mathbf{B}$, for each $x$; and $$ \|G(x)\| \leqq \int K_1(x, y)\|F(y)\| d y, $$$$ \|G(x)\| \leqq \int K_1(x, y)\|F(y)\| \, d y, $$ i.e., $\|G(\cdot)\|=T_1(\|F(\cdot)\|)$, so that ||$G(\cdot)\left\|_{\mathbf{B}}\right\|_q \leqq\|\| F(\cdot)\left\|_{\mathbf{B}}\right\|_p$.
I'm unable to understand the equation $\|G(x)\| \leqq \int K_1(x, y)\|F(y)\| d y$$\|G(x)\| \leqq \int K_1(x, y)\|F(y)\|\, d y$. What does the norm in $\|F(y)\|$ mean? Are we just taking the absolute value $|F(y)|$? I remember being told that this inequality concerns Minkowski's inequality, but I'm unable to see this from the notation. Minkowski's inequality should have implied $$\|G(x)\|_p\leq \int \|K_1(x,y)F(y)\|_p\, dy$$ Any help with this would be great.
