  In this paper, we study maximum principles for Laplacian and fractional
Laplacian with critical integrability. We first consider the critical cases for
Laplacian with zero order term and first order term. It is well known that for
the Laplacian with zero order term $-\Delta +c(x)$ in $B_1$, $c(x)\in
L^p(B_1)$($B_1\subset \mathbf{R}^n$), the critical case for the maximum
principle is $p=\frac{n}{2}$. We show that the critical condition $c(x)\in
{L^{\frac{n}{2}}(B_1)}$ is not enough to guarantee the strong maximum
principle. For the Laplacian with first order term $-\Delta
+\vec{b}(x)$($\vec{b}(x)\in L^p(B_1)$), the critical case is $p=n$. In this
case, we establish the maximum principle and strong maximum principle for
Laplacian with first order term.
  We also extend some of the maximum principles above to the fractional
Laplacian. We replace the classical lower semi-continuous condition on
solutions for the fractional Laplacian with some integrability condition. Then
we establish a series of maximum principles for fractional Laplacian under some
integrability condition on the coefficients. These conditions are weaker than
the previous regularity conditions. The weakened conditions on the coefficients
and the non-locality of the fractional Laplacian bring in some new
difficulties. Some new techniques are developed.
