  We show local and global scale invariant regularity estimates for
subsolutions and supersolutions to the equation $-{\rm div}(A\nabla
u+bu)+c\nabla u+du=-{\rm div}f+g$, assuming that $A$ is elliptic and bounded.
In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$,
$d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we show that,
with the surprising exception of the reverse Moser estimate, scale invariant
estimates with "good" constants (that is, depending only on the norms of the
coefficients) do not hold in general. On the other hand, assuming a necessary
smallness condition on $b,d$ or $c,d$, we show a maximum principle and Moser's
estimate for subsolutions with "good" constants. We also show the reverse Moser
estimate for nonnegative supersolutions with "good" constants, under no
smallness assumptions when $q<\infty$, leading to the Harnack inequality for
nonnegative solutions and local continuity of solutions. Finally, we show that,
in the setting of Lorentz spaces, our assumptions are the sharp ones to
guarantee these estimates.
