New maximum principles for linear elliptic equations Hung-ju KuoNeil Trudinger 35J15maximum principleslinear elliptic equations$k$-Hessian operatorGreen's functionlocal estimates We prove extensions of the estimates of Aleksandrov and Bakel'man for linear elliptic operators in Euclidean space $\mathbb{R}^n$ to inhomogeneous terms in $L^q$ spaces for $q < n$. Our estimates depend on restrictions on the ellipticity of the operators determined by certain subcones of the positive cone. We also consider some applications to local pointwise and $L^2$ estimates. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3073 10.1512/iumj.2007.56.3073 en Indiana Univ. Math. J. 56 (2007) 2439 - 2452 state-of-the-art mathematics http://iumj.org/access/