We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.
We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}...
We study $L^p_x L^\infty_t$ maximal estimates for exponential sums associated to $C^2$ graph hypersurfaces, motivated by Schr\"odinger maximal estimates on $\mathbb{T}^d$. We show that the conjectured maximal estimate for the periodic Schr\"odinger equation fails when one allows small perturbations of the paraboloid, w...
We establish a quantitative stability estimate for the critical Hardy inequality on bounded domains containing the origin. Our result improves the existing quantitative stability estimate by reducing the exponent in the distance function from $N^{2}$ to $N$, replacing the Lorentz-Zygmund framework with the Luxemburg no...
In this paper, we investigate the analytic properties of the Hardy-type operator $-\Delta + \mu H_0$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, where $\mu \geq -1$ and $H_0$ is a critical Hardy potential at infinity--arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green...
Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homo...
A celebrated conjecture of P.-L. Lions concerns maximal regularity for viscous Hamilton--Jacobi equations. In this paper, we study the endpoint case. We consider normalized strong solutions of $$-\Delta u+|Du|^\gamma=f$$ in $\mathbb T^d$, where $d\geq 2$, $\gamma>2$, and $f\in L^{q_c}(\mathbb T^d)$ with $q_c=d(\gamma-1...
Fan-Ze Kong· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
