Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

By Rupert L. Frank, Elliott H. Lieb, Robert Seiringer. 2007

334 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/2abfbf60a24a573dc2cf88f73e5b336d15d7a26d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

The best constant in a fractional Hardy inequality (2008, background): “We suggest the former as a possible setup for studying Dirichlet boundary value problems for non-local integro-differential operators and the corresponding stochastic processes [30], [38] on subdomains of R [3], beyond the “convolutional” case of the whole of R [4], [21].”
Nonexistence results for a class of fractional elliptic boundary value problems (2012, background): “In our next result the linear term is related to the relativistic Hardy inequality, see [18] and [17].”
On the fractional p-Laplacian equations with weight and general datum (2016, background): “We refer [21], [6] and [1] for a complete discussion about this fact.”
Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies (2014, background): “The isoperimetry of balls in nonlocal isoperimetric problems has been addressed in [19].”
Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations (2013, background): “It is also known (see for instance [16]) that the map α 7→ λ(α) is continuous and monotone decreasing.”
Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of L\'{e}vy processes (2012, background): “There is an increasing literature studying these operators from both a probabilistic and an analytic point of view [18, 11, 12, 76, 26, 57, 34, 59, 46, 47, 60, 45, 20, 5, 56].”
Optimal results for the fractional heat equation involving the Hardy potential (2014, background): “RN u(x)− u(y) |x− y|N+2s dy, s ∈ (0, 1), where aN,s := 2 π N 2 Γ( 2 ) |Γ(−s)| , is the normalization constant so that the identity (−∆)u = F(|ξ|Fu), ξ ∈ R , s ∈ (0, 1), holds for every u ∈ S(R ), the Schwartz class (see [24]).”
Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians (2009, background): “Seiringer [22] showed Hardy-Lieb-Thirring inequality for such Schrödinger operators.”
Pitt's inequality and the fractional Laplacian: Sharp error estimates (2007, methodology): “The “ground state spectral representation” (5) appears in the recent paper by Frank, Lieb and Seiringer [19].”
Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory (2010, background): “This observation plays a crucial role in the proof of spectral estimates for the perturbed form h in [7].”
The extremal solution for the fractional Laplacian (2013, methodology): “was obtained by Herbst [24] in 1977; see also [18].”
The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian (2015, methodology): “The Hardy inequality (4) plays an important role, for instance, in a general proof of the stability of the relativistic matter, see [23].”
Hardy-Sobolev-Maz'ya inequalities for arbitrary domains (2011, background): “The papers [12] and [13] develop an approach how to deduce Lieb-Thirring-type inequalities from (a-priori weaker) Sobolev-type inequalities.”
Fractional Laplacian with Hardy potential (2017, methodology): “tical case >. In Section 5 we discuss the quadratic form E~ of ~pand apply Davies’ method for the upper bound of ~pin Section 5.1. Acknowledgements We thank Rupert Frank for detailed comments on [19]. We thank Rodrigo Banuelos,~ Bart lomiej Dyda, Jerome Goldstein, Martin Hairer, Panki Kim, Giorgi”