Revista de la
Unión Matemática Argentina
Large-scale homogeneity and isotropy versus fine-scale condensation: A model based on Muckenhoupt-type densities
Hugo Aimar and Federico Morana

Volume 68, no. 1 (2025), pp. 69–78    

Published online (final version): March 29, 2025

https://doi.org/10.33044/revuma.3670

Download PDF

Abstract

In this brief note we aim to provide, through a well-known class of singular densities in harmonic analysis, a simple approach to the fact that the homogeneity of the universe on scales of the order of a hundred million light years is entirely compatible with the fine- scale condensation of matter and energy. We give precise and quantitative definitions of homogeneity and isotropy on large scales. Then we show that Muckenhoupt densities have the ingredients required for a model of the large-scale homogeneity and the fine-scale condensation of the universe. In particular, these densities can take locally infinitely large values (black holes) and, at the same time, they are independent of location at large scales. We also show some locally singular densities that satisfy the large-scale isotropy property.

References

  1. H. Aimar, M. Carena, R. Durán, and M. Toschi, Powers of distances to lower dimensional sets as Muckenhoupt weights, Acta Math. Hungar. 143 no. 1 (2014), 119–137.  DOI  MR  Zbl
  2. T. C. Anderson, J. Lehrbäck, C. Mudarra, and A. V. Vähäkangas, Weakly porous sets and Muckenhoupt ${A}_p$ distance functions, J. Funct. Anal. 287 no. 8 (2024), article no. 110558, 34 pp.  DOI  MR  Zbl
  3. R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.  DOI  MR  Zbl
  4. R. G. Durán and F. López García, Solutions of the divergence and analysis of the Stokes equations in planar Hölder-$\alpha$ domains, Math. Models Methods Appl. Sci. 20 no. 1 (2010), 95–120.  DOI  MR  Zbl
  5. B. Dyda, L. Ihnatsyeva, J. Lehrbäck, H. Tuominen, and A. V. Vähäkangas, Muckenhoupt $A_p$-properties of distance functions and applications to Hardy–Sobolev-type inequalities, Potential Anal. 50 no. 1 (2019), 83–105.  DOI  MR  Zbl
  6. J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985.  MR  Zbl
  7. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman, San Francisco, CA, 1973.  MR
  8. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.  DOI  MR  Zbl
  9. F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, J. Funct. Anal. 73 no. 1 (1987), 179–194.  DOI  MR  Zbl
  10. E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, NJ, 1993.  DOI  MR  Zbl