The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb{R}^n \setminus \mathbb{R}^d$, 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.

中文翻译：

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具有较低维边界的区域中的Dirichlet问题

本文率先研究了具有$ L ^ q $边界数据的Dirichlet问题，该数据用于具有较低维边界的域中具有复杂系数的二阶算子，例如，在$ \ Omega：= \ mathbb {R} ^ n \ setminus \ mathbb {R} ^ d $，其中$ d <n-1 $。根据David，Feneuil和Mayboroda的结果，我们引入了一个合适的退化椭圆算子，并证明只要系数满足小Carleson范数条件，Dirichlet问题对于所有$ q> 1 $都是可解的。

即使在经典情况$ d = n-1 $的情况下，（类似）我们的结果也是新的。系数的条件比以前已知的条件更宽松（最值得注意的是，我们对系数矩阵的前$ n-1 $行没有施加任何限制），并且结果更一般。我们在平方函数和非切向最大函数之间建立局部而不是全局估计，也许甚至更重要的是，我们在边界处建立新的Moser型估计并改善内部估计。