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Register a new userTrace Hardy inequality for the Euclidean space with a cut and its applications
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We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Sigmasubsetmathbb R^d$, $d ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy weight is given in terms of the geodesic distance to the boundary of $Sigma$. We provide its applications to the heat equation on $mathbb R^d$ with an insulating cut at $Sigma$ and to the Schrodinger operator with a $delta$-interaction supported on $Sigma$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.
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We investigate the large-distance asymptotics of optimal Hardy weights on $mathbb Z^d$, $dgeq 3$, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar $frac{(d-2)^2}{4}|x|^{-2}$ as $|x|toinfty$. We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on $mathbb Z^d$: (1) averages over large sectors have inverse-square scaling, (2), for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3), for i.i.d. coefficients, there is a matching inverse-square lower bound on moments. The results imply $|x|^{-4}$-scaling for Rellich weights on $mathbb Z^d$. Analogous results are also new in the continuum setting. The proofs leverage Greens function estimates rooted in homogenization theory.
We establish an analog Hardy inequality with sharp constant involving exponential weight function. The special case of this inequality (for n=2) leads to a direct proof of Onofri inequality on S^2.
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There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $lambda=n-alpha$ (that is for the case of $alpha>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.
We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given by Hardy is improved by a non-negative potential term depending on properties of the magnetic field.
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