[Paper Review] The best constant in a fractional Hardy inequality

Krzysztof Bogdan, Bartłomiej Dyda|ArXiv.org|Jul 11, 2008

Nonlinear Partial Differential Equations33 references4 citations

TL;DR

This paper establishes the optimal constant in a fractional Hardy inequality for the half-space using the censored stable process and Dirichlet forms. It proves that the best constant $\kappa_{d,\alpha}$, derived via a superharmonic function $w(x) = x_d^{(\alpha-1)/2}$, is sharp and cannot be improved, with explicit formula involving the Gamma and Beta functions, resolving an open problem in non-local analysis.

ABSTRACT

We prove an optimal Hardy inequality for the fractional Laplacian on the half-space.

Motivation & Objective

• To determine the best possible constant in a fractional Hardy inequality for the half-space $D = \{x_d > 0\}$.
• To establish sharpness of the constant $\kappa_{d,\alpha}$ in the inequality involving the fractional Laplacian and distance to the boundary.
• To connect the inequality to stochastic processes, specifically the censored $\alpha$-stable process, via Dirichlet forms and superharmonic functions.
• To generalize and refine prior results on Hardy-type inequalities with rough constants, particularly for $\alpha \neq 1$.
• To provide a complete characterization of the optimal weight $x_d^{-\alpha}$ in the right-hand side of the inequality.

Proposed method

• Use of Fitzsimmons' general framework linking superharmonic functions to Hardy-type inequalities via Dirichlet forms.
• Application of the censored $\alpha$-stable process on the half-space, whose Dirichlet form is explicitly given by a double integral with kernel $|x-y|^{-d-\alpha}$.
• Construction of a specific superharmonic function $w(x) = x_d^{(\alpha-1)/2}$, whose logarithmic derivative yields the optimal weight $\nu = -\mathcal{L}w/w = \kappa_{d,\alpha} x_d^{-\alpha}$.
• Proof of sharpness via a sequence of test functions supported in expanding annuli and level sets, with careful $L^2$-type estimates using homogeneity and symmetry.
• Use of the regional fractional Laplacian $\Delta^{\alpha/2}_D$ as the generator of the censored process, and comparison with the killed stable process to derive the final inequality.
• Explicit computation of the constant $\kappa_{d,\alpha}$ using the Beta and Gamma functions, involving $B\left(\frac{1+\alpha}{2}, \frac{2-\alpha}{2}\right)$ and normalization constants.

Experimental results

Research questions

• RQ1 What is the best constant $\kappa_{d,\alpha}$ in the fractional Hardy inequality on the half-space for $0 < \alpha < 2$?
• RQ2 Can the constant $\kappa_{d,\alpha}$ be improved, or is it sharp for all $u \in C_c(D)$?
• RQ3 How does the choice of superharmonic function $w(x) = x_d^{(\alpha-1)/2}$ lead to the optimal weight $x_d^{-\alpha}$ in the inequality?
• RQ4 What is the relationship between the censored stable process and the sharp fractional Hardy inequality?
• RQ5 Why does $\kappa_{d,\alpha} = 0$ when $\alpha = 1$, and how does this differ from $\alpha \neq 1$?

Key findings

• The best constant in the fractional Hardy inequality on the half-space is given by the explicit formula $\kappa_{d,\alpha} = \frac{\pi^{\frac{d-1}{2}}\Gamma(\frac{1+\alpha}{2})}{\Gamma(\frac{\alpha+d}{2})} \cdot \frac{B\left(\frac{1+\alpha}{2}, \frac{2-\alpha}{2}\right) - 2^\alpha}{\alpha 2^\alpha}$.
• The inequality $\frac{1}{2}\iint_D \frac{(u(x)-u(y))^2}{|x-y|^{d+\alpha}} \, dx\,dy \geq \kappa_{d,\alpha} \int_D u^2(x) x_d^{-\alpha} \, dx$ holds for all $u \in C_c(D)$, and fails if $\kappa_{d,\alpha}$ is replaced by any larger constant.
• $\kappa_{d,\alpha} = 0$ when $= 1$, but $\kappa_{d,\alpha} > 0$ for $\alpha \neq 1$, indicating a critical change in behavior at $\alpha = 1$.
• The constant $\kappa_{d,\alpha}$ is derived from the ratio $-\mathcal{L}w/w$ for $w(x) = x_d^{(\alpha-1)/2}$, which is superharmonic for the censored $\alpha$-stable process.
• The inequality implies a sharp constant $\frac{\Gamma^2(\frac{1+\alpha}{2})}{\pi}$ in the Dirichlet form of the censored stable process on $D$, which is optimal.
• The proof relies on a sequence of test functions supported in annular regions and level sets, with uniform $L^2$-type bounds that grow as $n^{2(d-1)}$, proving the sharpness of the constant.