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Logarithmic Sobolev and interpolation inequalities on the sphere: constructive stability results

Giovanni Brigati, Jean Dolbeault, Nikita Simonov

Abstract

We consider Gagliardo-Nirenberg inequalities on the sphere which interpolate between the Poincaré inequality and the Sobolev inequality, and include the logarithmic Sobolev inequality as a special case. We establish explicit stability results in the subcritical regime using spectral decomposition techniques, and entropy and carré du champ methods applied to nonlinear diffusion flows.

Logarithmic Sobolev and interpolation inequalities on the sphere: constructive stability results

Abstract

We consider Gagliardo-Nirenberg inequalities on the sphere which interpolate between the Poincaré inequality and the Sobolev inequality, and include the logarithmic Sobolev inequality as a special case. We establish explicit stability results in the subcritical regime using spectral decomposition techniques, and entropy and carré du champ methods applied to nonlinear diffusion flows.
Paper Structure (13 sections, 16 theorems, 177 equations)

This paper contains 13 sections, 16 theorems, 177 equations.

Table of Contents

  1. Introduction and main results
  2. Improvements under orthogonality constraints
  3. Improvements by the carré du champ method
  4. A simple estimate based on the heat flow, below the Bakry--Emery exponent
  5. An estimate based on the fast diffusion flow, valid up to the critical exponent
  6. Comparison with other estimates
  7. Global stability results
  8. Improved Gaussian inequalities, hypercontractivity and stability
  9. Carré du champ method and improved inequalities
  10. Algebraic preliminaries
  11. Diffusion flow and monotonicity
  12. Interpolation
  13. Improved functional inequalities

Key Result

Theorem 1

Let $d\ge1$. For any $F\in\mathrm H^1({\mathbb S}^d,d\mu)$ such that we have with $\mathscr C_d=\frac{2}{d+2}$.

Theorems & Definitions (22)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Theorem 6
  • Theorem 7
  • proof : Proof of Theorem \ref{['StabSubcriticalSphere']}
  • Proposition 8
  • Proposition 9
  • ...and 12 more