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We show the density of smooth Sobolev functions W k,∞ (Ω) ∩ C ∞ (Ω) in the Orlicz-Sobolev spaces L k,Ψ (Ω) for bounded simply connected planar domains Ω and doubling Young functions Ψ.

We obtain symmetrization inequalities in the context of Fractional Hajłasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure.