Near-Critical Asymptotics and Radial Sharpness for a Two-Weight Hardy Inequality

Abstract

In this paper, we study the optimal constant $C(p,γ)$ in a planar two-weight Hardy inequality relating the gradient and Hessian of a scalar potential, posed by Maz&#39;ya in [14, Problem 3.11]. For every $1\le p<\infty$, we prove \[ \lim_{γ\toγ_c} 2^{p/2}|γ-γ_c|^p C(p,γ)=1, \qquad γ_c=1-\frac2p. \] The proof reduces the curl-free constraint to an angular minimization problem and yields qualitative rigidity of almost extremizers for $p>1$. Moreover, we determine the sharp constant in the radial class for every $1\le p<\infty$ and $γ\neγ_c$, thereby resolving the radial case of Maz&#39;ya&#39;s problem. We also show that the radial inequality remains valid at $γ=γ_c$, with the same sharp constant \[ C_{\rm rad}(p,γ) =\left[1+\left(γ-1+\frac2p\right)^2\right]^{-p/2}. \] For general, not necessarily radial, potentials, we identify explicit parameter regions where this constant remains sharp and establish two sufficient criteria for symmetry breaking. The exact transition remains open for $p\ne2$. In the quadratic case, we recover the known sharp constant and derive a decomposition of its deficit. At the critical exponent, we obtain an exact remainder identity for the logarithmic Hardy inequality, whose classical constant remains sharp among radial potentials. The logarithmic inequality fails at $p=1$.