arXiv subjects
Publications and source records attributed to A. -K. Gallagher.
It is shown that the $p$-Poincaré inequality holds on an open set $Ω$ in $\mathbb{R}^n$ if and only if the strict $p$-capacitary inradius of $Ω$ is finite. To that end, new upper and lower bounds for the infimum for the associated nonlinear Rayleigh quotients are derived.
We show that the Poincaré inequality holds on an open set $D\subset\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on $D$ is equivalent to the finiteness of the strict inradius of $D$ measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet--Laplacian.
Let $Ω\subset\mathbb{C}$ be an open set. We show that $\overline{\partial}$ has closed range in $L^{2}(Ω)$ if and only if the Poincaré-Dirichlet inequality holds. Moreover, we give necessary and sufficient potential-theoretic conditions for the $\overline{\partial}$-operator to have closed range in $L^{2}(Ω)$. We also give a new necessary and sufficient potential-theoretic condition for the Bergman space of $Ω$ to be infinite dimensional.
Let $Ω$ be an unbounded domain in $\mathbb{R}\times\mathbb{R}^{d}.$ A positive harmonic function $u$ on $Ω$ that vanishes on the boundary of $Ω$ is called a Martin function. In this note, we show that, when $Ω$ is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition $Ω$ is symmetric, then the maximum of any Martin function along a slice $Ω\cap (\{t\}\times\mathbb{R}^d)$ is attained at $(t,0).$
A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of D'Angelo.