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 norm of the critical exponential Orlicz space $\operatorname{Exp}L^{\frac{N}{N-1}}(\Omega)$, and avoiding any cut-off modification of the virtual extremizers. As a consequence, we also obtain an improved quantitative stability estimate for the critical Hardy inequality with the logarithmic weight considered by Cianchi and Ferone, where the exponent is likewise reduced from $N^{2}$ to $N$ and the distance is measured directly from the virtual extremizer without truncation. The proof is completely rearrangement-free and relies on a critical Hardy inequality with a remainder term, scale-invariant Sobolev inequalities, and a refined dyadic summation argument. These ingredients yield stronger quantitative stability estimates for both forms of the critical Hardy inequality.
Let $n \ge 3$ and fix $p>n$. We establish a sharp quantitative stability result in $W^{1,p}$ for Liouville's theorem for nonconstant, sense-preserving quasiregular mappings whose outer distortion $K$ is sufficiently close to one. We first reorganize Reshetnyak's classical local argument \cite{R1976}, drawing systematic...
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 Uchiy...
Philipp Hake, Matthias Keller, Felix Pogorzelski· 0 citations
We study the weighted bilinear Hardy inequality for the same-direction product $H_I f H_I g$ on an arbitrary real interval $I$, with $0<q<\infty$, $1\le p_1,p_2\le\infty$, and measurable weights allowed to take the values $0$ and $+\infty$. The characterisation is formulated through directed compact restrictions $J\Sub...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
This survey paper provides a comprehensive study of unitarily invariant norms on the algebra of $n \times n$ matrices. This investigation leads naturally to the theory of symmetric gauge functions, a class of norms on $\mathbb{R}^n$ characterized by invariance and monotonicity properties. We develop the necessary frame...
S. Garcia, J. Mashreghi, M. Ptak et al.· 0 citations
We give a detailed proof of interior $W^{2,p}$ regularity for uniformly elliptic equations in nondivergence form \[ a^{ij}(x)D_{ij}u+b^i(x)D_i u=f. \] The leading matrix is assumed to have sufficiently small mean oscillation on the balls under consideration. The proof uses a modern real-variable argument: a constant-co...
L. D'Onofrio· 0 citations
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