  We rule out the existence of Leray's backward self-similar solutions to the
Navier--Stokes equations with profiles in $L^{12/5}(\R^3)$ or in the
Marcinkiewicz space $L^{q,\infty}(\R^3)$ for $q\in(12/5, 6)$. This follows from
a more general result formulated in terms of Morrey spaces and the first order
Riesz's potential.
