Abstract. We show that, under very general conditions on the domain Ω and the Dirichlet part D of the boundary, the operator −∇·µ∇+1 1/2 with mixed boundary conditions provides a topological isomorphism between W 1,p D (Ω) and L p (Ω), if p ∈ ]1, 2]. IntroductionThe main purpose of this paper is to identify the domain of the square root of a divergence form operator(The subscript D indicates the subspace of W 1,p (Ω) whose elements vanish on the boundary part D.) Our focus lies on non-smooth geometric situations in R d for d ≥ 2. So, we allow for mixed boundary conditions and, additionally, deviate from the Lipschitz property of the domain Ω in the following spirit: the boundary ∂Ω decomposes into a closed subset D (the Dirichlet part) and its complement, which may share a common frontier within ∂Ω. Concerning D, we only demand that it satisfies the well-known Ahlfors-David condition (equivalently: is a (d − 1)-set in the sense of Jonsson/Wallin [37, II.1]), and only for points from the complement we demand bi-Lipschitzian charts around. As special cases, the pure Dirichlet (D = ∂Ω) and pure Neumann case (D = ∅) are also included in our considerations. Finally the coefficient function µ is just supposed to be real, measurable, bounded and elliptic in general, cf. Assumption 4.2. Together, this setting should cover nearly everything that occurs in real-world problems -as long as the domain does not have irregularities like cracks meeting the Neumann boundary part ∂Ω \ D.The identification of the domain for fractional powers of elliptic operators, in particular that of square roots, has a long history. Concerning Kato's square root problem -in the Hilbert space L 2 -see e.g. [10], [24], [6] (here only the non-selfadjoint case is of interest). Early efforts, devoted to the determination of domains for fractional powers in the non-Hilbert space case seem to culminate in [48]. In recent years the problem has been investigated in the case of L p (p = 2) for instance in [5], [8], [35], [36], [33], [9]; but only the last three are dedicated to the case of a nonsmooth Ω = R d . In [9] the domain is a strong Lipschitz domain and the boundary conditions are either pure Dirichlet or pure Neumann. Our result generalizes this to a large extent and, at the same time, gives a new proof for these special cases, using more 'global' arguments. Since, in the case of a non-symmetric coefficient function µ, for the nonsmooth constellations described above no general condition is known that assures (−∇ · µ∇ + 1) 1/2 : W 1,2 D (Ω) → L 2 (Ω) to be an isomorphism, this is supposed as one of our assumptions. This serves then as our starting point to show the corresponding isomorphism property of (−∇ · µ∇ + 1)

Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by P f (x) = y p(x, y)f (y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ∇f p and (I − P ) 1/2 f p uniformly in f for 1 < p < +∞. These conditions are different for p < 2 and p > 2. The proofs rely on recent techniques developed to handle operators beyond the class of Calderón-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest.

We prove that W 1 p is a real interpolation space between W 1for p > q 0 and 1 ≤ p 1 < p < p 2 ≤ ∞ on some classes of manifolds and general metric spaces, where q 0 depends on our hypotheses.