# Analysis Seminar

#### Optimal Hardy-Type Inequality for Nonnegative Second-Order Elliptic Operator: An Answer to a Problem of Shmuel Agmon

Speaker: Yehuda Pinchover, Technion

Location: Warren Weaver Hall 1302

Date: Thursday, December 10, 2015, 11 a.m.

Synopsis:

We give a general answer to the following fundamental problem posed by Shmuel Agmon 30 years ago:

Given a (symmetric) linear elliptic operator $$P$$ of second-order in $$\mathbb{R}^n$$, find a nonnegative weight function $$W$$ which is "as large as possible", such that for some neighborhood of infinity $$G$$ the following inequality holds

$$(P - W) \geq 0$$ in the sense of the associated quadratic form on $$C_0^\infty(G)$$.

We construct, for a general subcritical second-order elliptic operator $$P$$ on a domain $$D$$ in $$\mathbb{R}^n$$ (or on a noncompact manifold), a Hardy-type weight $$W$$ which is optimal in the following natural sense:

1. For any $$\lambda \leq 1$$, the operator $$(P - \lambda W) \geq 0$$ on $$C_0^\infty(D)$$,
2. For $$\lambda = 1$$, the operator $$(P - \lambda W)$$ is null-critical in $$D$$,
3. For any $$\lambda > 1$$, and any neighborhood of infinity $$G$$ of $$D$$, the operator $$(P - \lambda W)$$ is not nonnegative on $$C_0^\infty(G)$$.
4. If $$P$$ is symmetric and $$W>0$$, then the spectrum and the essential spectrum of the operator $$W^{-1}P$$ are equal to $$[1,\infty)$$.

Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators on a general domain $$D$$ or on a noncompact manifold. Moreover, the results can be generalized to certain $$p$$-Laplacian type operators. The constructed weight $$W$$ is given by an explicit simple formula involving two positive solutions of the equation $$Pu=0$$.

This is a joint work with Baptiste Devyver and Martin Fraas.