Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces

Abstract

We prove the nonlinear fundamental convergence theorem for superharmonic functions on metric measure spaces. Our proof seems to be new even in the Euclidean setting. The proof uses direct methods in the calculus of variations and, in particular, avoids advanced tools from potential theory. We also provide a new proof for the fact that a Newtonian function has Lebesgue points outside a set of capacity zero, and give a sharp result on when superharmonic functions have L q -Lebesgue points everywhere.

The Fundamental Convergence Theorem for p(·)-Superharmonic Functions

“…In nonlinear potential theory, the theorem was first proved in [22]. Apparently, the most recent version of the fundamental convergence theorem appears in [4], which gives a new proof based on the direct methods in the calculus of variations. This proof, which works also in metric spaces, does not require as advanced tools of potential theory as the previous proofs, which is an advantage for us.…”

Nonlinear balayage on metric spaces

“…Superharmonic functions are ess lim infregularized, and a function in N 1,p loc (Ω) is superharmonic if and only if it is an ess lim inf-regularized superminimizer. However, there are superharmonic functions not belonging to N 1,p loc (Ω), and thus they are not superminimizers, see also a discussion in Björn-Björn-Parviainen [8]. A superharmonic function u satisfies the strong minimum principle: If u attains its minimum in Ω at some point x ∈ Ω, then u is constant in the component containing x.…”

“…We will need two results for superminimizers and superharmonic functions from Björn-Björn-Parviainen [8]. For the second result, called the fundamental convergence theorem, we first need to define the lim inf-regularization of a function f : Ω → R aŝ…”

“…First, we use the fundamental convergence theorem from Björn-Björn-Parviainen [8] to show that regularizing changes the infimum only on a set of capacity zero and that the resulting function is superharmonic. This makes it possible for us to develop the theory of balayage in a way different from Heinonen-Kilpeläinen-Martio [24], where a substantial part of the balayage theory was developed before proving that the infimum only needs to be regularized on a set of capacity zero.…”

The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in R^n and metric spaces

“…The following two results from Björn-Björn-Parviainen [7] (Lemma 3.2 and Corollary 3.3), following from Mazur's lemma (see, e.g., Theorem 3.12 in Rudin [29]), will play a major role in the existence proof for the obstacle problem. …”