Let [Formula: see text] be a complex variety in a bounded domain [Formula: see text] in [Formula: see text]. We are interested in finding sufficient conditions on [Formula: see text] so that plurisubharmonic functions which are bounded from above on [Formula: see text] can be approximated from above by continuous functions on [Formula: see text] and plurisubharmonic on [Formula: see text] Next, we discuss the possibility to extend a given real valued continuous function on [Formula: see text] to a maximal plurisubharmonic on [Formula: see text] which is continuous up to the boundary.

We establish a comparison principle for locally bounded plurisubharmonid functions on complex varieties (possibly with singularities) in bounded domains in C n .(dd c u) n .An analogous comparison principle was also obtained by Bedford (see Theorem 4.3 in [Be]) for bounded plurisubharmonic functions on open subsets of complex spaces. This result is the first inspiration for our work. The other one comes from the following sharper form of Theorem 1.1 that was obtained a few years later byThen for any constant r ≥ 0 and w 1 , • • • , w n ∈ P SH(D) with −1 ≤ w j < 0 we

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