Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on $\partial_t A$
Martin Ulmer
TL;DR
The paper proves an equivalence principle for elliptic boundary value problems in the upper half-space: under an $L^1$ Carleson condition on the transversal derivative $\partial_t A$ and the bound $|\partial_t A|\le C/t$, solvability of the adjoint Dirichlet problem $(D)^*_{p'}$ implies solvability of the Regularity problem $(R)_p$ for some $p>1$. The approach hinges on establishing $L^p$ bounds for area functions associated with the approximation semigroup $\mathcal{P}_t=e^{-t^2 L_{||}^t}$ and its tangential derivatives, via a decomposition of $\partial_t\mathcal{P}_t f$ into $W_1 f$ and $W_2 f$ and a delicate real interpolation framework using tent spaces and Hardy–Sobolev atoms. The analysis splits into two main parts: $W_2$-driven estimates controlled by Carleson and Hardy–Sobolev bounds, and $\nabla_{||}\mathcal{P}_t$ with $W_1$-driven estimates, all culminating in $L^p$ area-function bounds for $\;T_t f\in\{\nabla_{||}\mathcal{P}_t f, \partial_t\mathcal{P}_t f, t\nabla_{||}\partial_t\mathcal{P}_t f\}$. Consequently, the Regularity problem becomes solvable in a broader operator class, including those with mixed $L^1-L^{\infty}$ and $L^1$-Carleson-type control on $|\partial_t A|$, thereby extending known results beyond the classical DKP or $t$-independent settings. The work also provides two corollaries in the 1D/upper-half-plane context, asserting solvability of the Regularity problem for operators in these broader regimes. The methodology blends heat semigroup techniques, area- and tent-space theory, and real interpolation to achieve robust $L^p$-solvability results for nonsymmetric elliptic operators.
Abstract
We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume an $L^1$-Carleson condition on only $|\partial_t A|$ the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this $L^1$-Carleson condition on $|\partial_t A|$.