The present paper pioneers the study of the Dirichlet problem with Lq boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in Ω:= Rn\Rd, with d < n − 1. Following results of David, Feneuil and Mayboroda, we introduce an appropriate degenerate elliptic operator and show that the Dirichlet problem is solvable for all q > 1, provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case d = n − 1, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first n−1 rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.
Bibliographical noteFunding Information:
The second author was supported by the Alfred P. Sloan Fellowship, the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089, the NSF RAISE-TAQ grant DMS 1839077, and the Simons Foundation grant 563916, SM. The third author was partially supported by NSF grant numbers DMS-1361823, DMS-1500098, DMS-1664867, DMS-1902756 and the Institute for Advanced Study.
© European Mathematical Society
- Carleson measures
- Complex coefficients
- Degenerate elliptic operators
- Dirichlet problem
- Moser-type estimates
- Non-tangential maximal function
- Square functionals