Journal Article

Lp theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part

Li, Linhan, Pipher, Jill, Mayboroda, Svitlana, Hofmann, Steve

2022

Overview

We consider the operator L = - div ( A ∇ ) , where A is an n × n matrix of real coefficients and satisfies the ellipticity condition, with n ≥ 2 . We assume that the coefficients of the symmetric part of A are in L ∞ ( R n ) , and those of the anti-symmetric part of A only belong to the space B M O ( R n ) . We create a complete narrative of the L p theory for the square root of L and show that it satisfies the L p estimates L f L p ≲ ∇ f L p for 1 < p < ∞ , and ∇ f L p ≲ L f L p for 1 < p < 2 + ϵ for some ϵ > 0 depending on the ellipticity constant and the BMO semi-norm of the coefficients. Moreover, we prove the L p estimates for some vertical square functions associated to e - t L . In another article of the authors, these results are used to establish the solvability of the Dirichlet problem for elliptic equation div ( A ( x ) ∇ u ) = 0 in the upper half-space ( x , t ) ∈ R + n + 1 with the boundary data in L p ( R n , d x ) for some p ∈ ( 1 , ∞ ) .

• Publisher: Springer Berlin Heidelberg,Springer Nature B.V
• Subject: Boundary value problems / Coefficients / Decomposition / Dirichlet problem / Ellipticity / Estimates / Half spaces / Mathematics / Mathematics and Statistics / Operators (mathematics)