Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n \Gamma \subset \mathbb {R}^n be an Ahlfors regular set of dimension d > n − 1 d>n-1 (not necessarily integer) and Ω = R n ∖ Γ \Omega = \mathbb {R}^n \setminus \Gamma . Let L = − div ⁡ A ∇ L = - \operatorname {div} A\nabla be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A A are bounded from above and below by a multiple of dist ⁡ ( ⋅ , Γ ) d + 1 − n \operatorname {dist}(\cdot , \Gamma )^{d+1-n} . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p L^p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L L , establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when Γ \Gamma is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L L for which the harmonic measure given here is absolutely continuous with respect to the d d -Hausdorff measure on Γ \Gamma and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1.To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n be an Ahlfors regular set of dimension d < n − 1 (not necessarily integer) and Ω = R n \ Γ. Let L = − div A∇ be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A are bounded from above and below by a multiple of dist(•, Γ) d+1−n . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L, establish its doubling property, non-degeneracy, changeof-the-pole formulas, and, finally, the comparison principle for local solutions.In another article to appear, we will prove that when Γ is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L for which the harmonic measure given here is absolutely continuous with respect to the d-Hausdorff measure on Γ and vice versa. It thus extends Dahlberg's theorem to some sets of codimension higher than 1.

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension n − 1 in R n , and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher codimension, on a Lipschitz graph Γ of dimension d in R n , d < n − 1, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ω L is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ω L and the Hausdorff measure.Résumé. Dans son célèbre théorème de 1977, B. Dahlberg a prouvé que pour les domaines de R n bornés par un graphe lipschitzien de dimension n − 1, la mesure harmonique est absolument continue par rapportà la mesure de surface, résultat qui a ensuiteétéétendu aux domaines avec accès non-tangentiel, et au delà.Dans ce papier on démontre le premier analogue de ce théorème pour le complémentaire d'un graphe lipschitzien Γ de dimension d < n−1 avec une petite constante de Lipschitz. On construit un opérateur linéaire elliptique dégénéré L dont la mesure harmonique associée ω L est absolument continue par rapportà la mesure de Hausdorff H d sur Γ. Plus généralement, on donne des conditions suffisantes sur la matrice des coefficients de L pour que ω L et H d |Γ soient mutuellement absolument continues.Key words/Mots clés. boundary with co-dimension higher than 1, degenerate elliptic operators, Dahlberg's theorem, harmonic measure in higher codimension.

We study the L p boundedness of Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on L p for 1 < p < 2, which shows that Gaussian estimates of the heat kernel are not a necessary condition for this. In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for 1 < p < 2. This yields a full picture of the ranges of p ∈ (1, +∞) for which respectively the Riesz transform is L p -bounded and the reverse inequality holds on L p on such manifolds and graphs. This picture is strikingly different from the Euclidean one.

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