## Abstract

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 BMO(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 ∥Lf∥Lp≲∥∇f∥Lp for 1 < p< ∞, and ∥∇f∥Lp≲∥Lf∥Lp 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}, dx) for some p∈ (1 , ∞).

Original language | English (US) |
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Pages (from-to) | 935-976 |

Number of pages | 42 |

Journal | Mathematische Zeitschrift |

Volume | 301 |

Issue number | 1 |

DOIs | |

State | Published - May 2022 |

### Bibliographical note

Funding Information:S. Hofmann acknowledges support of the National Science Foundation (currently Grant Number DMS-1664047). S. Mayboroda is supported in part by the National Science Foundation Grants DMS 1344235, DMS 1839077 and Simons Foundation Grant 563916, SM.

Publisher Copyright:

© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

## Keywords

- Bounded mean oscillation (BMO)
- Elliptic operators
- L estimates
- Square root operator
- Unbounded coefficients
- Vertical square functions

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