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Weighted functional inequalities involving MFI for symmetric alpha-stable laws

Develop and prove functional inequalities of Log-Sobolev or Poincaré type, potentially with appropriate weights, that involve Mixed Fractional Information M_alpha for symmetric alpha-stable distributions across alpha in (0, 2].

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Background

The paper shows that a simple, unweighted Log-Sobolev inequality relating relative entropy and MFI fails in the Cauchy case, reflecting the heavy-tailed nature of stable laws and the absence of a spectral gap. Prior work has established weighted inequalities for heavy-tailed measures and functional-inequality frameworks using Fractional Fisher Information, suggesting the need for tailored, weighted inequalities in the MFI setting.

Consequently, the authors identify the formulation and proof of appropriate weighted Log-Sobolev or Poincaré-type inequalities that incorporate MFI as a significant open problem, which would be central to understanding concentration and convergence properties for symmetric alpha-stable measures.

References

While alternative approaches using Fractional Fisher Information have yielded candidate functional inequalities for SalphaS laws , finding and proving corresponding inequalities (potentially weighted LSI or Poincaré-type) involving MFI remains a significant open problem.

— Mixed Fractional Information: Consistency of Dissipation Measures for Stable Laws (2504.13423 - Cook, 18 Apr 2025) in Section 4.3 (Functional Inequalities and Concentration)