On extensions of Sobolev functions defined on regular subsets of metric measure spaces

We characterize the restrictions of first order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator. Let (X, d, µ) be a metric space (X, d) equipped with a Borel measure µ, which is non-negative and outer regular, and is finite on every bounded subset. In this paper we describe the restrictions of first order Sobolev functions to measurable subsets of X which have a certain regularity property. There are several known ways of defining Sobolev spaces on abstract metric spaces, where of course we cannot use the notion of derivatives. Of particular interest to us, among these definitions, is the one introduced by Haj lasz [14]. But let us first consider a classical characterization of classical Sobolev spaces due to Calderón. Since it does not use derivatives, it can lead to yet another way of defining Sobolev spaces on metric spaces. In [2] (see also [3]) Calderón characterizes the Sobolev spaces W k,p (R n) in terms of L p-properties of sharp maximal functions. To generalize this characterization to the setting of a metric measure space (X, d, µ), let f be a locally integrable real valued function on X and let α be a positive number. Then the fractional sharp maximal function of f , is defined by f ♯ α (x) := sup r>0 r −α µ(B(x, r)) B(x,r) |f − f B(x,r) | dµ. Here B(x, r) := {y ∈ X : d(y, x) < r} denotes the open ball centered at x with radius r, and, for every Borel set A ⊂ X with µ(A) < ∞, f A denotes the average value of f over A f A := 1 µ(A) A f dµ.