I am working through the Graduate Studies in Mathematics Volume 29 text: Fourier Analysis and am struggling to understand a line in a particular proof. We are working with $f \in L^p$, $1 \leq p < \infty$ and define $$ \sigma_N f(x) = \int_0^1 f(t)F_N(x-t)dt, $$ where $F_N$ is the Fejer kernel. Indeed we have the familiar properties that $F_N(t) \geq 0$ and $\|F_N\|_1 = 1$. In the proof of Theorem 1.10, which claims that $\lim_{N \rightarrow \infty} \| \sigma_Nf - f\|_p = 0$, we have the following as the first line: $$ \| \sigma_Nf - f\|_p = \int_{-1/2}^{1/2}\|f(\cdot -t) - f(\cdot)\|_p F_N(t)dt.$$
In my attempt to understand this I have done the following \begin{align*} \sigma_Nf(x) - f(x) ={}& \int_0^1f(x-t)F_N(t)dt - f(x)\\ ={}& \int_0^1 f(x-t)F_N(t)dt - \int_0^1f(x)F_N(t)dt\\ ={}& \int_0^1 (f(x-t) - f(x))F_N(t)dt. \end{align*}
So I can write $\|\sigma_Nf - f\|_p = \| \int_{-1/2}^{1/2} (f(x-t) - f(x))F_N(t)dt \|_p. $ However I fail to see how we can "pass" the norm through the integral and around our Fejer kernel. I feel strongly it has to do with the integral of the Fejer Kernel being one, but cannot justify it.
Any help clarifying this is much appreciated!
There is a typo. Equality does not hold but there is an inequality. See Wikipedia for 'Minkowski's inequality for integrals' to get $\leq $ in place of $=$.
Ref: https://en.wikipedia.org/wiki/Minkowski_inequality