Hypoelliptic functional inequalities
Ruzhansky M. Yessirkegenov N.
June 2024Springer Science and Business Media Deutschland GmbH
Mathematische Zeitschrift
2024#307Issue 2
In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Sobolev, Rellich, Hardy–Littllewood–Sobolev, Gagliardo–Nirenberg, Caffarelli–Kohn–Nirenberg and Heisenberg–Pauli–Weyl type uncertainty inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. The approach developed in this paper relies on establishing integral versions of Hardy inequalities on homogeneous Lie groups, for which we also find necessary and sufficient conditions for the weights for such inequalities to be true. Consequently, we link such integral Hardy inequalities to different hypoelliptic inequalities by using the Riesz and Bessel kernels associated to the described hypoelliptic operators.
22E30 , 43A80 , 46E35 , Caffarelli–Kohn–Nirenberg inequality , Gagliardo–Nirenberg inequality , Graded Lie group , Hardy and critical Hardy inequalities , Hardy–Littlewood–Sobolev inequality , Rellich inequality , Rockland operator , Stratified Lie group
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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom
SDU University, Kaskelen, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Department of Mathematics: Analysis
School of Mathematical Sciences
SDU University
Institute of Mathematics and Mathematical Modeling
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