Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon
Let \(P\) be a nonnegative second-order linear elliptic operator with real coefficients, which are defined on a domain or on a noncompact manifold in \(\mathbb{R}^n\). The authors study the problem of finding as large as possible (in an appropriate sense) weight functions \(W\geq 0\) for weighted Hardy-type inequalities involving the operator \(P\) and punctured domains of the type \(\Omega\setminus \{0\}\), with \(0\in \Omega\). When \(P\) is symmetric, the Hardy inequality with weight \(W\) on a domain \(\Omega\) has the following form \[ \displaystyle{q(\varphi)\geq \lambda \int_\Omega W(x)|\varphi(x)|^2\,dx} \text{ for all }\varphi\in C_0^\infty(\Omega),\tag{1} \] where \(\lambda\) is a positive constant and \(q\) is the quadratic form associated to \(P\). The operator \(P\) is said to be subcritical when inequality (1) holds with a nonzero weight \(W\), while \(P\) is said to be critical when inequality \((1)\) holds if and only if \(W=0\). For a subcritical operator \(P\) and a nonzero weight \(W\) satisfying \((1)\), the authors introduce the following further criticality condition: let \(\lambda_0>0\) be the best constant for inequality (1). The operator \(P-\lambda_0 W\) is said to be null-critical in \(\Omega\) if the corresponding Rayleigh-Ritz variational problem admits no minimizers. Then the authors extend the previous criticality conditions to a general (nonsymmetric) nonnegative operator \(P\) and give the definition of optimal Hardy-weight \(W\) in terms of the criticality properties of the operator \(P-\lambda W\), where \(\lambda\geq 1\). The main result of this paper states the existence of optimal Hardy-weights for nonnegative subcritical second-order linear elliptic operators. In particular, the authors prove that an optimal Hardy-weight for the operator \(Pu=-a_{ij}(x)\partial_i\partial_j u+b(x)\nabla u+c(x)u\) is \(W=\frac{1}{4}|\nabla \log(G/u)|_A^2\), where \(G\) is the minimal positive Green function of \(P\), \(u\) is a positive solution of \(Pu=0\) satisfying \(\lim_{x\rightarrow \infty}\frac{G(x)}{u(x)}=0\), and \(|\xi|_A^2=a^{ij}(x)\xi_i\xi_j\) for all \(\xi\in \mathbb{R}^n\).
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