On extensions of Sobolev functions defined on regular subsets of metric measure spaces
This paper deals with extension operators in Sobolev spaces defined in metric measure spaces. Two Sobolev-type spaces are considered, the Hajłasz--Sobolev space \(M^{1,p}\) and what the author calls Calderón--Sobolev space given by the norm \[ \| u\|_{L^p} + \| u^\#\|_{L^p}, \] where \(u^\#\) is the sharp maximal function \[ u^\#(x) := \sup_{r>0}\frac{1}{r\mu(B(x,r))}\int_{B(x,r)} | u(y)-u_{B(x,r)}| \, d\mu(y), \] and \(u_{B(x,r)}\) is the integral average of \(u\) in the ball \(B(x,r)\). Under doubling and reverse doubling conditions on the measure \(\mu\), it is shown that \(u\in L^p(S)\) can be extended from \(S\subset X\) to the Calderón--Sobolev space on the whole space \(X\) if the set \(S\) is regular, i.e., \[ \mu(B(x,r))\leq C\mu(B(x,r)\cap S)\text{ for every }x\in S,\;r\in (0,\delta_S), \] and the sharp maximal function in \(S\), \[ u^\#_S(x) := \sup_{r>0} \frac{1}{r\mu(B(x,r))}\int_{B(x,r)\cap S} | u(y)-u_{B(x,r)}| \, d\mu(y), \] belongs to \(L^p(S)\) (Theorem~1.2). A similar result, Theorem~1.3, is also given for Hajłasz--Sobolev spaces.
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