4 problems

Optimal-exponent conjecture. In each of these estimates, the power of $\varepsilon$ is optimal.

Let $H$ be the operator and $c_0$ the constant appearing in Theorem 4, whose boundary estimate has the form … for normalized eigenfunctions $Hf=\lambda f$. Optimality conjecture. T…

Let $U$ be a bounded region in ${\bf R}^{N}$, let $H=-\Delta_{\mathrm{DIR}}$ act in $L^{2}(U)$, and let $d(x)$ be the distance from $x$ to a closed subset of…

Let $U$ be a region, let $H$ be a uniformly elliptic second-order operator on $L^{2}(U)$ with Dirichlet boundary conditions, and let $Q$ be its associated quadratic form. For…