Journal of Statistical Physics

, Volume 154, Issue 3, pp 866–876 | Cite as

Relativistic Hardy Inequalities in Magnetic Fields



We deal with Dirac operators with external homogeneous magnetic fields. Hardy-type inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality with the same best constant as in the free case. This leaves naturally open an interesting question whether there exist magnetic fields for which a Hardy inequality with a better constant than the usual one, in connection with the well known diamagnetic phenomenon arising in non-relativistic models.


Dirac equation Electromagnetic potentials Hardy inequalities 

Mathematics Subject Classification

35J10 35L05 



The first and third authors were supported by the Italian Project FIRB 2012: “Dispersive dynamics: Fourier Analysis and Variational Methods”


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Dip. di Matematica “G. Castelnuovo”SAPIENZA Università di RomaRomeItaly
  2. 2.Dep. de MatemáticasUniversidad del País VascoBilbaoSpain
  3. 3.Basque Center for Applied Mathematics (BCAM)BilbaoSpain
  4. 4.Dip. di Matematica “L. Tonelli”Università Degli Studi di PisaPisaItaly

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