The regularity problem for degenerate elliptic operators in weighted spaces

• Pascal Auscher ^[1] ; Li Chen ^[2] ; José María Martell ^[3] ; Cruz Prisuelos-Arribas ^[4]
  1. [1] University of Paris-Saclay

     University of Paris-Saclay

     Arrondissement de Palaiseau, Francia

     
  2. [2] Louisiana State University

     Louisiana State University

     Estados Unidos

     
  3. [3] Consejo Superior de Investigaciones Científicas

     Consejo Superior de Investigaciones Científicas

     Madrid, España

     
  4. [4] Universidad de Alcalá

     Universidad de Alcalá

     Alcalá de Henares, España

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 2, 2023, págs. 563-610
• Idioma: inglés
• DOI: 10.4171/RMI/1357
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• Resumen

  • We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let Lw be a degenerate elliptic operator with degeneracy given by a fixed weight w 2 A2.dx/ in Rn, and consider the associated block second order degenerate elliptic problem in the upper-half space R nC1 C . We obtain non-tangential bounds for the full gradient of the solution of the block case operator given by the Poisson semigroup in terms of the gradient of the boundary data. All this is done in the spaces Lp.vdw/, where v is a Muckenhoupt weight with respect to the underlying natural weighted space .Rn;wdx/. We recover earlier results in the non-degenerate case (when w 1, and with or without weight v). Our strategy is also different and more direct thanks in particular to recent observations on change of angles in weighted square function estimates and non-tangential maximal functions. Our method gives as a consequence the (unweighted) L2 .dx/-solvability of the regularity problem for the block operator L˛u.x; t / D jxj ˛ divx jxj ˛ A.x/rxu.x; t / @ 2 t u.x; t / for any complex-valued uniformly elliptic matrix A and for all " < ˛ < 2n=.n C 2/, where " depends just on the dimension and the ellipticity constants of A.