= Volterra power-kernel integral
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{title2=$(V_\beta f)(x)=\int_0^x(x-y)^{\beta-1}f(y)\,dy,\quad\beta>0$}
For nonnegative <Lebesgue measurable> $f$ on $(0,1)$, the displayed <Volterra power-kernel integral> is defined for every $0<x<1$ as a nonnegative extended <Lebesgue integral>. The factor is finite and continuous on the open integration <interval> $(0,x)$, and an endpoint value can be assigned arbitrarily. The result may initially be infinite; for example $f(y)=1/y$ makes every value infinite. At $\beta=1$, it reduces to the ordinary <Volterra operator>.
For joint <integration>, use the <integral kernel> $(x-y)^{\beta-1}$ on $0<y<x<1$ and zero elsewhere. This piecewise prescription avoids the singular diagonal and never raises a negative number to a noninteger power. The <L1 bound for a Volterra power-kernel integral> gives an integrable output, finite <almost everywhere>, whenever $f$ is integrable.
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