Revista de la
Unión Matemática Argentina
Extrapolation of compactness for certain pseudodifferential operators
María J. Carro, Javier Soria, and Rodolfo H. Torres
Volume 66, no. 1 (2023), pp. 177–186    

https://doi.org/10.33044/revuma.4365

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Abstract

A recently developed extrapolation of compactness on weighted Lebesgue spaces is revisited and a new application to a class of compact pseudodifferential operators is presented.

References

  1. J. Alvarez and C. Pérez, Estimates with $A_\infty$ weights for various singular integral operators, Boll. Un. Mat. Ital. A (7) 8 no. 1 (1994), 123–133.  MR  Zbl
  2. A. Bényi and R. H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141 no. 10 (2013), 3609–3621.  DOI  MR  Zbl
  3. B. Bongioanni, A. Cabral, and E. Harboure, Extrapolation for classes of weights related to a family of operators and applications, Potential Anal. 38 no. 4 (2013), 1207–1232.  DOI  MR  Zbl
  4. A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.  DOI  MR  Zbl
  5. M. Cao, A. Olivo, and K. Yabuta, Extrapolation for multilinear compact operators and applications, Trans. Amer. Math. Soc. 375 no. 7 (2022), 5011–5070.  DOI  MR  Zbl
  6. A. Clop and V. Cruz, Weighted estimates for Beltrami equations, Ann. Acad. Sci. Fenn. Math. 38 no. 1 (2013), 91–113.  DOI  MR  Zbl
  7. H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Functional Analysis 18 (1975), 115–131.  DOI  MR  Zbl
  8. J. Duoandikoetxea, Extrapolation of weights revisited: new proofs and sharp bounds, J. Funct. Anal. 260 no. 6 (2011), 1886–1901.  DOI  MR  Zbl
  9. C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 no. 3-4 (1972), 137–193.  DOI  MR  Zbl
  10. J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies 116, North-Holland, Amsterdam, 1985.  MR  Zbl
  11. E. Harboure, R. A. Macías, and C. Segovia, Extrapolation results for classes of weights, Amer. J. Math. 110 no. 3 (1988), 383–397.  DOI  MR  Zbl
  12. T. Hytönen and S. Lappas, Extrapolation of compactness on weighted spaces: bilinear operators, Indag. Math. (N.S.) 33 no. 2 (2022), 397–420.  DOI  MR  Zbl
  13. T. Hytönen and S. Lappas, Extrapolation of compactness on weighted spaces, Rev. Mat. Iberoam. 39 no. 1 (2023), 91–122.  DOI  MR  Zbl
  14. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.  DOI  MR  Zbl
  15. C. Pérez and R. H. Torres, Sharp maximal function estimates for multilinear singular integrals, in Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), Contemp. Math. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 323–331.  DOI  MR  Zbl
  16. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993.  MR  Zbl
  17. C. B. Stockdale, P. Villarroya, and B. D. Wick, Sparse domination results for compactness on weighted spaces, Collect. Math. 73 no. 3 (2022), 535–563.  DOI  MR  Zbl
  18. R. H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 90 no. 442 (1991).  DOI  MR  Zbl
  19. A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J. (2) 30 no. 1 (1978), 163–171.  DOI  MR  Zbl
  20. Q. Xue, K. Yabuta, and J. Yan, Weighted Fréchet-Kolmogorov theorem and compactness of vector-valued multilinear operators, J. Geom. Anal. 31 no. 10 (2021), 9891–9914.  DOI  MR  Zbl