| Summary: | We use ``Hardy-type'' inequalities to derive $L^q$ estimates for solutions of equations containing the $p$-Laplacian with $p>1$. We begin by deriving some inequalities using elementary ideas from an early article [B3] which has been largely overlooked. Then we derive $L^q$ estimates of the boundary behavior of test functions of finite energy, and consequently of principal (positive) eigenfunctions of functionals containing the $p$-Laplacian. The estimates contain exponents known to be sharp when $p=2$. These lead to estimates of the effect of boundary perturbation on the fundamental eigenvalue. Finally, we present global $L^q$ estimates of solutions of the Cauchy problem for some initial-value problems containing the $p$-Laplacian.
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