Weighted Nonlinear Potential Theory

Abstract

Abstract. The potential theoretic idea of the " thinness of a set at a given point" is extended to the weighted nonlinear potential theoretic setting-the weights representing in general singularities/degeneracies-and conditions on these weights are given that guarantee when two such notions are equivalent at the given point. When applied to questions of boundary regularity for solutions to (degenerate) elliptic second-order partial differential equations in bounded domains, this result relates the boundary Wiener criterion for one operator to that of another, and in the linear case gives conditions for boundary regular points to be the same for various operators. The methods also yield two weight norm inequalities for Riesz potentials'<<:(//'*«&) ', 1 < p < q < oo, which at least in the first-order case (a = 1) have found some use in a number of places in analysis.1. Introduction. In recent years a number of papers have appeared pointing out the connection between the theory of weights a la Muckenhoupt-Wheeden-FeffermanCoifman et al. and the pointwise behavior of solutions to certain elliptic second order partial differential equations in bounded domains of Euclidean yV-space. In particular, and especially of interest to us in this paper, is the question of the boundary regularity of such solutions to Dirichlet problems for certain degenerate divergence form hnear and nonlinear operators as well as for linear nondivergence uniformly elliptic operators. See [16,34,9]. The condition that must be met at the boundary is, of course, some analogue of the Wiener criteria. These criteria can be expressed as a Dini type integral involving a capacity set function built out of the differential operator. In the simplest case, these capacities can be viewed as a slight modification of the Newtonian capacity (associated with the Laplace operator) using the structure coefficients of the differential operator. In the nondivergence linear equation case, these capacities may well involve the use of the Green's function for the differential operator on some fixed ball. However, in any case the idea is to study these capacities and their corresponding Wiener criteria in order to obtain useful sufficient conditions that will imply when two differential operators have the same boundary regular points.

Untitled

“…Proof. (1) Assume first that f ∈ Z; we may assume f Z = 1. Then there exists u with u = Au + f. Then u q ≥ (Au) q + f q and so Au ≥ A 2 u + Af so that Af Z ≤ 1.…”

“…(ii) For some constant C we have K(Kω) q ≤ C Kω < ∞ a.e. (iii) ω satisfies (1) and (3). (iv) ω satisfies (2) and (3).…”

“…where we stress that we require Kg(x) ≥ 1 for every (not almost every) x ∈ E. The theory of these capacities and the corresponding potential theory (which is usually called nonlinear potential theory in the nonclassical case p = 2) is due to B. Fuglede, N. G. Meyers, V. G. Maz ya and V. P. Havin, Yu. G. Reshetnyak; a weighted theory, for dσ = v dx, where v ∈ A ∞ , was developed by D. Adams [1] (see also [2], [24], [35], [57], and the literature cited there).…”

“…The proof of Theorem 1.4 involves a quantity which under mild assumptions is equivalent to the capacity of a ball. The two-sided estimates of Cap B are obtained in Theorem 5.4 without any restrictions on the underlying measure σ for a wide class of K. (This is a generalization of D. Adams's formula for the weighted capacity of a ball proved in [1] in the case of Riesz potentials for σ ∈ A ∞ ; an upper estimate for arbitrary σ can be found in [51].) This should be compared to a similar estimate of ||χ B || Zq,K in Section 4 which clarifies the role of the infinitesimal inequality.…”

“…After this paper was finished, we learned that H. Brezis and X. Cabre [9] considered very recently another special case of (1.19) where the inhomogeneous term ω is a positive uniformly bounded function on Ω. 1 In particular, they proved in a different way the following "nonsolvability" result (for bounded domains Ω with smooth boundary): (1.19) with bounded ω has no solutions unless G(δ q σ) ∈ L ∞ (Ω). Note that in this case Gω(x) δ(x), and our characterization (1.20) boils down to a sharper necessary and sufficient condition…”