The Hardy Littlewood Sobole inequality says
If $1 < p, r < \infty$ and $ 0 < \alpha < n $ be such that $ \frac{1}{p} + \frac{ \alpha }{ n} = \frac{1}{r} + 1 $. Then for any $ f \in L^p ( \mathbb R ^n )$, we have $$|| I_ \alpha f ||_{ L^r ( \mathbb R ^n )} \le C_{p ,\alpha , n } ||f||_{ L^ p(\mathbb R^n )} $$ for some positive constant $ C_{ p ,\alpha , n }$ depending on $p,\alpha,n$ only, but is independent of $f$.
My question is: why is it often called "fractional integration" inequality? What is "fractional integration"?