= L1 bound for a Volterra power-kernel integral
{c}
{title2=$\|V_\beta f\|_1\leq\beta^{-1}\|f\|_1$}
For nonnegative integrable $f$, <Tonelli's theorem> applied on the triangle $0<y<x<1$ gives the exact identity
$$
\|V_\beta f\|_1=\int_0^1f(y)\left(\int_y^1(x-y)^{\beta-1}\,dx\right)dy
=\frac1\beta\int_0^1(1-y)^\beta f(y)\,dy\leq\frac1\beta\|f\|_1.
$$
For signed or complex integrable $f$, first apply the same calculation to $|f|$ and use the <triangle inequality for an integral>. The defining <integral> then exists absolutely for <almost every> $x$, and the same <L1 norm> bound holds. The constant $1/\beta$ is sharp: use $f_n=n\mathbf1_{(0,1/n)}$, whose <L1 norm> is one and whose weighted <integrals> above tend to $1/\beta$.
Finiteness at every point is stronger than this <L1 space> assertion. For $0<\beta<1$, take $x_0=1/2$, $\gamma=(1+\beta)/2$, and $f(y)=(x_0-y)^{-\gamma}$ on $(0,x_0)$, zero otherwise. It is integrable because $\gamma<1$, but $(V_\beta f)(x_0)=\infty$ because the power near $y=x_0$ is $\beta-1-\gamma=(\beta-3)/2<-1$. This is compatible with an integrable output after changing values on a <null set>.
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