q-Plurisubharmonicity and q-pseudoconvexity in cn

Publ. Mat. 50 (2006), 349–369 q-PLURISUBHARMONICITY AND q-PSEUDOCONVEXITY IN Cn Nguyen Quang Dieu Abstract We generalize classical results for plurisubharmonic functions and hyperconvex domain to q-plurisubharmonic functions and q-hyperconvex domains. We show, among other things, that Bq-regular domains are q-hyperconvex. Moreover, some smoothing results for q-plurisubharmonic functions are also given. According to Andreotti and Grauert (see [AG]), a smooth C2function uon an open subset Ω of Cnis called q-plurisubharmonic (q-psh. for short, 0 ≤q≤n−1) if its complex Hessian has at least (n−q) nonnegative eigenvalues at each point of Ω, or equivalently the Levi form of uat every point of Ω is positive definite when restricted to some complex linear subspace of codimension q. Later on, Hunt and Murray (see [HM]) found a natural extension of this notion to the class of upper semicontinuous functions. The set of q-psh. functions has most of the properties of usual plurisubharmonic functions e.g., invariance under holomorphic maps, satisfy the maximum principle, etc. However, this class is not closed under addition for q > 0 and thus standard smoothing techniques (e.g., by convolving with an approximation of identity) available for plurisubharmonic functions do not apply, a fact which hampered early work on the subject. In fact Diederich and Fornæss [DF] constructed examples showing the impossibility of smoothing continuous q-psh. function by C∞smooth q-psh. ones. See also [Sl3, p. 154] for a related example. On the positive side, an approximation result was obtained by Slodkowski in [Sl1], where he shows that every q-psh. function is pointwise limit of a sequence of q-psh. functions whose second order derivatives exist almost everywhere. 2000 Mathematics Subject Classification. 32U95, 32F10. Key words. q-plurisubharmonic, q-convex, q-pseudoconvex, q-hyperconvex, Bq-regular domain. 350 Nguyen Quang Dieu The aim of the present paper is to investigate analogues of plurisubharmonicity and pseudoconvexity in the context of q-plurisubharmonicity and q-pseudoconvexity. Now we outline the organization of the paper. After recalling in Section 2 some background on q-psh. functions, in Section 3 we begin studying some properties of Bq-regular domains. Here we recall that a bounded domain Ω in Cnis said to be Bq-regular if every continuous function on ∂Ω can be extended continuously to a q-psh. function on Ω. When q= 0, those are precisely the B-regular domains introduced by Sibony in [Si]. The main result of the section is Theorem 3.6, which gives some connections between Bq-regularity of a domain and of its boundary. Here we encounter some difficulty in generalizing Theorem 2.1 of [Si] to the context of q-psh. functions. The main reason is, as said before, the non-additivity of the class of q-psh. functions, thus we do not know, for instance, whether a smoothly bounded Bq-regular domain should have aBq-regular boundary. In this section we also introduce the concept of q-hyperconvexity. This is the true analogue of classical hyperconvexity. Moreover, the new class enjoys most of the properties of hyperconvexity, e.g., q-hyperconvexity is purely a local concept (Proposition 3.2). The last section is devoted to studying smoothing results for q-psh. functions. More precisely, we show in Theorem 4.1 that on a q-pseudoconvex domains every q-psh. function is the pointwise limit of a decreasing sequence of piecewise smooth strictly q-psh. functions. This result may be considered as an analogue of a well known approximation theorem due to Fornæss and Narasimhan. In view of the above mentioned example of Diederich and Fornæss, piecewise smoothness seems to be the best possible regularity of the approximating sequence. The paper ends up with another approximation theorem (Theorem 4.3), in which we deal with approximation of bounded from above q-psh. functions. In particular, the theorem says that on a Bq-regular domain every bounded from above q-psh. function is the pointwise limit of a decreasing sequence of q-psh. functions which are continuous up to the boundary. This result, in the case q= 0 has been proved in slightly more general form in [NW] (see also Theorem 4.1 of [Wi]). 2. Preliminaries on q-plurisubharmonic functions In this section, we will collect some known facts about q-psh. functions. For more background, the reader may consult [HM], [Sl1], [Bu]. Definition 2.1. Let Ω be an open set in Cnand u: Ω →[−∞,∞) be an upper semicontinuous function and qbe an integer, 0 ≤q≤n−1. q-plurisubharmonicity. . . 351 (i) uis said to be q-plurisubharmonic on Ω if for every complex linear subspace of dimension q+ 1 intersecting Ω, for every closed ball B (in L), and for every smooth plurisuperharmonic function gdefined in a neighbourhood of B(in L) satisfying u≤gon ∂B we have u≤g on B. (ii) If for every point z0∈Ω we can find a neighbourhood Uand ε > 0 such that u(z)−ε|z|2is q-psh. on Ω then we say that uis strictly q-psh. (iii) uis said to be q-plurisuperharmonic if −uis q-psh. (iv) If uis q-psh. and (n−q−1)-plurisuperharmonic then we say that uis q-Bremermann. (v) If uis locally the maximum of a finite number of C2smooth q-psh. functions, then we say that uis piecewise q-psh. Remarks. (i) Definition 2.1 (a) is given by Hunt and Murray in [HM]. The definition of q-plurisubharmonicity makes sense also for q≥n. However, in this case every upper semicontinuous function is q-psh. Observe that the function identically −∞ is allowed to be q-psh. (ii) According to Lemma 2.6 in [HM], if u∈ C2(Ω) then uis q-psh. if and only if for every z∈Ω the complex Hessian ∂2u ∂zj∂zk1≤j,k≤nhas at least (n−q) nonnegative eigenvalues, or equivalently the Levi form hL(u, z)λ, λi= n X j,k=1 ∂2u ∂zj∂zk (z)λjλk, where λ= (λ1,...,λn), is positive definite on a complex linear subspace of codimension qin Cn. Thus, in the case of smooth functions, the concept of q-psh. functions introduced by Hunt and Murray coincides with the original one given by Andreotti and Grauert at the beginning of this paper. (iii) The concept of q-Bremermann functions has been introduced first by Hunt and Murray in [HM] where they are called q-complex MongeAmp`ere instead. Here we follow the terminology of Slodkowski in [Sl1]. If q= 0 then 0-Bremermann functions are precisely maximal plurisubharmonic function (see [Kl]). Likewise, piecewise smooth q-psh. functions are also called (q+1)-convex with corners by Diederich and Fornæss in [DF]. We now list basic properties of q-psh. functions that will be frequently referred to. 352 Nguyen Quang Dieu Proposition 2.2. Let Ωbe an open set of Cnand 0≤q≤n−1. Then (i) If uis q-psh. on Ωthen so are λu and u+vfor every λ > 0and every 0-psh. function v. (ii) For every family {uα}α∈Aof locally uniformly bounded from above q-psh. function on Ω, the function (sup{uα:α∈A})∗is also q-psh. on Ω. (iii) The limit of a decreasing sequence of q-psh. functions is q-psh. (iv) (Maximum principle) If uis q-psh. on Ωthen for every relatively compact open set Uof Ωwe have supUu≤sup∂U u. (v) If uis an upper semicontinuous function on Ωsuch that for every z0∈Ω, we can find a neighbourhood Uof z0such that u|Uis q-psh., then uis q-psh. on Ω. (vi) If uis q-psh. on Ωand f: Ω′→Ωis a holomorphic mapping, where Ω,Ω′are open subsets of Cnand Ck, respectively, then u◦fis q-psh. on Ω′. (vii) If uis q-psh. function on Ω, then the function ˜u= k X j=1 χj(u+vj) is q-psh. on Ω, where χj:R→Rare convex increasing functions, vjare 0-psh. functions on Ω. In particular, χ◦uis q-psh. for every increasing convex function χ:R→R. (viii) If uand vare q- (resp. r-) psh. functions on Ωthen max(u, v)is max(q, r)-psh. on Ω,u+vis (q+r)-psh. on Ωand min(u, v)is (q+r+ 1)-psh. on Ω. (ix) (Gluing Lemma) If Ω′⊂Ω, u is q-psh. on Ωand u′is q-psh. on Ω′. Assume that lim sup z′→z u′(z′)≤u(z)for all z∈Ω∩∂Ω′, then the function v(z) = (u(z)z∈Ω\Ω′ max(u(z), u′(z)) z∈Ω′ is q-psh. on Ω. Here by u∗we mean the upper regularization of a function u:X→ [−∞,∞), where Xis a subset of Cni.e., u∗(x) = lim sup z→x u(z),∀x∈X. q-plurisubharmonicity. . . 353 For the proof, we require the following generalization of Richberg’s approximation theorem for 0-psh. functions. Bungart’s Approximation Theorem (Theorem 5.3 in [Bu]).Assume uis a continuous stricitly q-psh. function on an open set Wof Cn and ga continuous function such that u < g on W. Then there exists a piecewise smooth strictly q-psh. function ˜uon Wsatisfying u < ˜u < g. In particular, there is a monotone decreasing sequence of piecewise smooth strictly q-psh. function that converges to uuniformly on W. Proof of Proposition 2.2: The properties (i)–(iii) follow quickly from the definition of q-plurisubharmonicity. For the more subtle ones, (iv) is proved in Lemma 2.7 in [HM] (see also [Sl1, p. 307]), the last two assertions of (viii) are deep theorems of Slodkowski (see Theorems 5.1 and 6.1 in [Sl1]). Note that (v) is contained in the remark following Lemma 2.7 in [HM]. Here is a brief proof of this fact. With no loss of generality we may assume that q=n−1. Assume that uis not q-psh., then we can find a ball Bcompactly belonging to Ω and a continuous function von Bwhich is 0-psh. on Bsuch that maxB(u+v)>max∂B(u+ v). Then there is ε > 0 so small that M:= max B ˜u > max ∂B ˜u, where ˜u(z) = u(z) + v(z) + 2ε|z|2. Choose z∗∈Bsuch that ˜u(z∗) = M and set f(z) = 2ε|z|2−M−ε|z−z∗|2. Then we have (u+v+f)(z∗) = 0,(u+v+f)(z)≤ −ε|z−z∗|2,∀z∈B. Choose a small ball B′about z∗such that B′⊂Band u|B′is q-psh. Since v+fis plurisubharmonic on Cn, we have 0 = (u+v+f)(z∗)≤ sup∂B′(u+v+f)<0,which is absurd. Now (vi) is undoubtedly wellknown, but due to the lack of an explicit reference we give a proof. First we check that u◦fis q-psh. Assume that uis of class C2on Ω. Then using the chain rule and the holomorphicity of fwe get n X j.k=1 ∂2v ∂zj∂zk (z)λjλk= n X l,m=1 ∂2u ∂wl∂wm (f(z))λ′ lλ′m,∀z∈Ω, where λ′ l=Pn j=1 λj∂fl ∂zj(z), f= (f1,...,fk). Since the Levi form of uis positive definite on a complex linear subspace Eof codimension q, we infer, from the last expression, that the Levi form of vis positive definite on some complex linear subspace E′of codimension no larger than q. 354 Nguyen Quang Dieu Thus vis q-psh. This implies that vis also q-psh. if uis piecewise smooth q-psh. Now suppose that uis continuous on Ω. Using Bungart’s Approximation Theorem, we deduce that ucan be locally uniformly approximated by piecewise smooth q-psh. functions. Thus vis again qpsh. The general case now follows from the preceding facts and the fact that ucan be approximated locally from above by a decreasing sequence of continuous q-psh. functions (Theorem 2.9 in [Sl1]). Finally we deal with (vii). Using the same reasonings as above, we may reduce to the case u,χj,vjare C2smooth functions. Now a direct computation using convexity of χand plurisubharmonicity of vjgives hL(˜u, z)λ, λi ≥   k X j=1 χ′ j(u(z) + vj(z)) hL(u, z)λ, λi. It follows that ˜uis q-psh. Finally, for (ix) we define for each k≥1 the function vk(z) = (u(z) + 1/k z ∈Ω\Ω′ max(u(z) + 1/k, u′(z)) z∈Ω′. Since lim sup z′→z u′(z′)≤u(z) for all z∈Ω∩∂Ω′, we deduce that vk≡ u+ 1/k on a neighbourhood of Ω ∩∂Ω′. Thus, by (v) we infer that vkis q-psh. on Ω. Observe that vk↓von Ω. Therefore vis q-psh. on Ω. The proof is thereby completed. As q-psh. functions do not have the additive property, the following operator introduced by Slodkowski, seems to be a good substitute for the usual convolution. Definition 2.3 (see [Sl1, p. 309]).Let uand gbe two functions defined on Cnwith values in [−∞,∞). The supremum-convolution of uand g, denoted by u∗sgis defined by (u∗sg)(z) := sup{u(x)g(z−x) : x∈Cn},∀z∈Cn. If uis defined only on a subset Uof Cn,u∗sgis understood as ˜u∗sg, where ˜u=uon Uand 0 on Cn\U. Here by B(z, r) we mean the open ball with center zand radius r. The most useful properties of the supremum-convolution are summarized in the following q-plurisubharmonicity. . . 355 Proposition 2.4. Let ube a q-psh. function on an open set Ωof Cnand gbe a continuous function on Cn,0≤g≤1,g(0) = 1,supp g⊂B(0, r), r > 0. Assume that the set {z∈Ω, u(z) = −∞} has empty interior. Then we have (i) u∗sgis continuous q-psh. on Ωr:= {z∈Ω : B(z, r)⊂Ω}. (ii) u∗sgrconverges pointwise to uon Ωas rtends to 0. Proof: (i) This part is essentially contained in [Sl1]. However, for the reader’s convenience we sketch some details. First we check the continuity of u∗sgon Ωr. As gis continuous and {z:u(z) = −∞} has empty interior we infer u∗sgis real valued and lower semicontinuous. Now supp gis contained in B(0, r) so (u∗sg)(z) = sup{u(z+x) + g(x) : x∈B(0, r)},∀z∈Ωr. Now assume that u∗sgis not upper semicontinuous on Ωr. Then there are z∗∈Ωr,ε > 0 and a sequence {zj}tending to z∗such that (u∗sg)(z∗) + ε < (u∗sg)(zj),∀j≥1. Choose a sequence {xjε} ⊂ Ωrso that (u∗sg)(zj)≤u(zj+xjε)g(xjε) + ε/3,∀j≥1. Passing to a subsequence we may assume that {xjε}converges to xε∈ B(0, r). Observe that uis upper semicontinuous so there is j0≥1 satisfying u(zj+xjε)≤u(z∗+xε) + ε/3 for j≥j0. It implies that u(z∗+xε)g(xε) + ε≤(u∗sg)(z∗) + ε≤(u∗sg)(zj) ≤u(zj+xjε)g(xjε) + ε/3 ≤u(z∗+xε)g(xjε) + 2ε/3,∀j≥j0. Letting jtend to ∞, we obtain a contradiction to the continuity of g. Thus u∗sgis continuous on Ur. So it follows from Proposition 2.2 (ii) that u∗sgis q-psh. on Ωr. (ii) We have (u∗sgr)(z) = sup{u(z+x)gr(x) : x∈B(0, r)} = sup{u(z+x)g(x/r) : x∈B(0, r)} ≥ u(z)g(0) = u(z). Since 0 ≤g≤1 we also have (u∗sgr)(z)≤sup{u(z+x) : x∈B(0, r)}. Now the upper semicontinuity of uimplies that limr→0(u∗sgr)(z) = u(z). For the ease of exposition, we will say that a subset Eof a domain Ω is q-pluripolar (in Ω) if there is a q-psh. function uon Ω such that u6≡ −∞ and u≡ −∞ on E. It should be pointed out that the structure 356 Nguyen Quang Dieu of q-pluripolar set may be very “wild” when q > 0. This can be seen by considering the singular locus of upper semicontinuous functions depending only on qvariables. More interesting examples are provided by the next result which is a consequence of Theorem 2.5 and Proposition 5.2 in [Sl2]. Proposition 2.5. Let Ωbe an open set in Cnand Xbe a complex analytic subset of codimension qin Ω. Then the function identically 0on X and −∞ elsewhere is q-psh. on Ω. In particular, Ω\Xis q-pluripolar. It follows from the above result that for q > 0 the union of two q-pluripolar set is in general not q-pluripolar. The next result is an analogue of the removable singularities for bounded plurisubharmonic function Proposition 2.6. Let Ω,Ω′be open subsets of Cn,Ω′⋐Ω. Let vbe a q-psh. function on Ωsuch that v6≡ −∞ and E={z∈Ω′:v(z) = −∞} is closed in Ω′. Then every q-psh. function uon Ω′\E, which is locally bounded from above near every point of Ecan be extended through Eto a(q+r)-psh. function on Ω. Proof: Since Ω′is bounded, by subtracting a positive constant, we may assume v < 0 on Ω′. For ε > 0 we set uε=(u+εv on Ω\E −∞ on E. By Proposition 2.2 (ix) we have uεis (q+r)-psh. on Ω. Set ˜u= (sup{uε: ε > 0})∗. Then ˜uis (q+r)-psh. on Ω, in view of Proposition 2.2 (viii). Since uε≤uon Ω\E, we deduce that ˜u≤uon Ω\E. Observe that limε→0uε=uon Ω\E. Therefore ˜u=uon Ω\E. Proposition 2.7. Let Ωbe a bounded domain in Cn. Assume that uand vare two continuous functions on Ωwhich are q-Bremermann function on Ω. Then sup Ω |u−v| ≤ sup ∂Ω |u−v|. Proof: Let α= sup∂Ω|u−v|, we only need to show that u−v≤αon Ω. As uand −vare qand (n−q−1)-psh. respectively, we have u−vis (n−1)-psh. by Proposition 2.2 (ix). Now the desired conclusion follows from the maximum principle (Proposition 2.2 (v)). We also need Choquet’s Topological Lemma (see Lemma 2.3.4 in [Kl]). q-plurisubharmonicity. . . 357 Choquet’s Lemma. Let {uα}α∈Abe a family of functions on an open set Ω⊂Cn, which are locally bounded from above. Then there exists a countable subfamily {αj} ⊂ Asuch that (sup{uα:α∈A})∗= (sup{uαj:j≥1})∗. Moreover, if uαis lower continuous for every α∈A, then we can choose {αj}such that sup{uα:α∈A}= sup{uαj:j≥1}. 3. q-pseudoconvex domains and Dirichlet problem We first recall, according to Slodkowski (see [Sl2, p. 121]), that a domain Ω is said to be q-pseudoconvex if there is a neighbourhood U of ∂Ω so that the function −log d(z) is q-psh. on U∩Ω, where d(z) = dist(z, ∂Ω).It follows from Theorem 4.3 in [Sl2] that Ω is q-psh. if and only if there exists a neighbourhood Uof ∂Ω and a q-psh. function u on U∩Ω such that limz→∂Ωu(z) = ∞. As in the proof of Theorem 2.6.10 in [H¨o], if Ω is bounded, by gluing the function −log d(z) with a suitable convex increasing function of |z|2we get a continuous q-psh. exhaustion function for Ω. Adding |z|2to this function we obtain a continuous strictly q-psh. exhaustion function ϕfor Ω. By Bungart’s Approximation Theorem, we can even assume that this function is piecewise smooth strictly q-psh. Definition 3.1. A bounded domain Ω is said to be q-hyperconvex if it admits a negative continuous q-psh. exhaustion function. Remarks. (a) As in the 0-hyperconvex case, it is easy to check that every q-hyperconvex domain is q-pseudoconvex. On the other hand, not every bounded q-pseudoconvex domain is q-hyperconvex. Indeed, consider Ω = D\E, where Dis a q-pseudoconvex domain in Cnand Eis the zero set of a holomorphic function fon D,f6≡ 0. Let ube a q-psh. exhaustion function for Ω, it is clear that u−log |f|is a q-psh. exhaustion function for D\E. Thus Ω is q-pseudoconvex. On the other hand, Ω is not q-hyperconvex in view of Proposition 2.6 and the maximum principle (Proposition 2.2 (iv)). (b) We do not know if a bounded domain Ω is q-hyperconvex if it admits a negative q-psh. exhaustion (not necessarily continuous) function. This is true when q= 0 (see Theorem 1.6 in [Bl]). Concerning q-hyperconvexity we have the following results which are analogous to the well known facts for hyperconvexity. 364 Nguyen Quang Dieu for some sufficiently fast decreasing sequence εj↓0, we may achieve that ϕj> ϕj+1 on Ω. Now we consider two cases. Case 1. uis bounded from below. Fix j≥1, then there is a sequence {δj,m}decreasing to 0 so fast that uj,m := u∗sρδj,m +δj,m(1 + |z|2)> u on Ωj+2 and uj,m < ϕjon ∂Ωj+1. Applying Bungart’s Approximation Theorem to the sequence uj,m we can find a sequence of piecewise smooth strictly r-psh. functions vj,m on Ωj+2 such that vj,m ↓u on Ωj+2 as m→ ∞,vj,m > u on Ωj+2 and vj,m < ϕjon ∂Ωj+1. The Gluing Lemma implies that ˜uj,m =(max(vj,m, ϕj) on Ωj+1 ϕjon Ω\Ωj+1 defines a piecewise smooth, strictly s-psh. function on Ω. Moreover, ˜uj,m ↓max(u, ϕj) on Ωj+1 as mtends to ∞. It follows that given j there is p(j) so large that ˜uj+1,p <˜uj,m on Ωjfor p≥p(j) and ˜uj+1,p < ˜ur,m on Ωrfor r≤jand p≥p(j). Thus we can choose a sequence {m(j)}tending to ∞fast enough so that ˜uj+1,m(j+1) <˜uj,m(j)on Ωj and ˜uj+1,m(j+1) <˜ur,j on Ωrfor r≤j. Finally we define uj= max p≥j˜up,m(p). Observe the maximum is locally taken by a finite number of piecewise smooth strictly s-psh. functions. This implies that ujis piecewise smooth strictly s-psh. and uj↓u. Case 2. General u. From the first case, we deduce that for each N≥1 there is a sequence uN,k of piecewise smooth strictly s-psh. functions that decreases to max(u, −N) as ktends to ∞. For each m, choose p(m)> m so large that um,p(m)< uj,m +1 m2on Ωj,1≤j≤m. This is possible because by Dini’s Theorem max(um,l, uj,m) converges uniformly to uj,m on Ωjas lgoes to ∞. Set uj= max m≥jum,p(m)+1 m. Observe that for z∈Ωland m≥p(l) we have um,p(m)(z) + 1 m< ul,m(z) + 1 m2+1 m< ul,p(l)(z) + 1 l. q-plurisubharmonicity. . . 365 It implies that locally ujis maximum of a finite number of piecewise smooth strictly s-psh. functions. Thus ujis piecewise smooth strictly s-psh. and uj↓u. The theorem is completely proved. Corollary 4.2. Let Ωbe a q-pseudoconvex domain in Cn, let Kbe a compact subset of Ωand ωan open neighbourhood of the q-plurisubharmonic hull ˆ KP Ωof Ki.e., ˆ KP Ω={z∈Ω : u(z)≤sup K u, u is q-psh. on Ω}. Then there is a piecewise smooth strictly q-psh. function uon Ωsuch that (a) u < 0on Kand u > 0on Ω\ω. (b) {z:u(z)< c}is relatively compact in Ωfor every c∈R. Proof: Using a compactness argument and Proposition 2.4, as in the proof of Theorem 2.6.11 in [H¨o], we can find a continuous q-psh. function von Ω satisfying (a) and (b). Applying Theorem 4.1 we get a piecewise smooth strictly q-psh. function usatisfying (a) and (b). We are done. Theorem 4.3. Let Ωbe a bounded q-hyperconvex domain in Cn. Assume that there is a compact set Pin ∂Ωhaving the following properties. (a) For every p∈(∂Ω)\P, there is a q-psh. barrier at p. (b) There is a negative, locally bounded q′-psh. function gon Ωsuch that P={z∈Ω : g∗(z) = −∞}. Then for every bounded from above r-psh. function uon Ωand every compact set Kof (∂Ω)\Pthere is a sequence of bounded from above strictly s-psh. functions {uj}on Ωwhich are continuous on Ω∪Ksuch that uj↓u∗on Ω∪Kwhere s= max{r+q′,2q}. Proof: We follow the lines of Theorem 3.2 in [NW]. Choose a negative continuous q-psh. exhaustion function vfor Ω. Since u∗is upper semicontinuous on ∂Ω, there is a sequence {ϕj}of continuous functions on ∂Ω that decreases to u∗on ∂Ω. Now for each jwe set Φj= sup{ϕ:ϕis q-psh. on Ω,continuous on Ω, ϕ ≤ϕjon ∂Ω}. Then Φjis bounded and lower semicontinuous on Ω. It follows from (a) that Φj=ϕjon (∂Ω)\P. It also follows from Choquet’s Topological Lemma that there is a sequence {ϕk,j}k≥1of q-psh. function on Ω, continuous on Ω that increases to Φjon Ω. It follows from the hypothesis (a) that ϕk,j ↑ϕjon (∂Ω)\P. 366 Nguyen Quang Dieu Fix a function ρ∈ C(Cn) such that ρ(0) = 1, 0 ≤ρ≤1 and supp ρ⊂ B(0,1). Fix j≥1, we claim that there are δj∈(0,1/j), aj≥1 such that (u∗sρδj)−1 j+1 j(g∗sρδj)≤jv +ϕaj,j on Ωδj. To see this, we argue by contradiction. Assume otherwise, then there are sequences {δm}↓0, {bm}↑∞and a sequence of points {xm}, xm∈Ωδmsuch that (u∗sρδm)(xm)−1 j+1 j(g∗sρδm)(xm)≥jv(xm) + ϕbm,j(xm). Passing to a subsequence we may assume that {xm}converges to x∗∈ ∂Ω. Using the upper semicontinuity of u∗and g∗on Ω and the definition of supremum-convolution, we have lim sup m→∞ (u∗sρδm)(xm) + 1 j(g∗sρδm)(xm)≤u∗(x∗) + 1 jg∗(x∗). Since limz→∂Ωv(z) = 0, we deduce that lim sup m→∞ (jv(xm) + ϕbm,j(xm)) ≥lim sup m→∞ ϕbm,j(xm). Putting all this together, we obtain u∗(x∗)≥u∗(x∗) + 1 jg∗(x∗)≥lim sup m→∞ ϕbm,j(xm). Combining this inequality with (b) we infer x∗6∈ P. Thus x∗∈(∂Ω)\P. But then we have ϕbm,j(x∗)↑ϕj(x∗)> u∗(x∗). A contradiction! The claim is therefore proved. This implies, in view of the Gluing Lemma and Proposition 2.2 (viii), that the function vj=     max (u∗sρδj)−1 j+1 j(g∗sρδj), jv +ϕaj,jon Ωδj jv +ϕaj,j on Ω\Ωδj is s-psh. on Ω and continuous on Ω. Moreover vjconverges pointwise to u∗on Ω ∪K. Now set uj(z) = sup m≥j {vm(z)}+|z|2/j. q-plurisubharmonicity. . . 367 It is clear that uj↓u∗on Ω ∪K. Fix j≥1, we will show that ujis continuous on Ω ∪K. Since each ujis continuous on Ω we have ujis lower semicontinuous there. It remains to check that ujis upper semicontinuous on Ω ∪K. Assume otherwise, then there is a∈Ω∪K,ε > 0 and sequences jk↑ ∞,zjk→asuch that vjk(zjk)> vjk(a) + ε, zjk∈Ω∪K, ∀k≥1. Consider two cases. Case 1. a∈Ω. Then we may assume that zjk∈Ωδjk,∀k. Since gis bounded near aand v < 0, from the definition of vjwe infer that for all klarge enough the following inequalities hold vjk(a)≥(u∗sρδjk)(a)−ε/2, vjk(zjk)≤(u∗sρδjk)(zjk). This is absurd, in view of the upper semicontinuity of uat a. Case 2. a∈K. Since vj≡ϕaj,j on Kfor all j, we may assume that zjk∈Ω, ∀k. Since v < 0 on Ω we have vjk< ϕajk,jkon Ω\Ωδjkand vjk<max(ϕajk,jk,(u∗sρδjk)) on Ωδjk. Since the ϕajk,jkare continuous on Ω and u∗is upper semicontinuous at a, we get a contradiction. Thus ujis continuous on Ω ∪K. The desired conclusion now follows. Remark. Let Ω be the Hartogs triangle {(z, w)∈C2: 0 <|z|<|w|<1}. Then Ω is pseudoconvex and 1-strictly pseudoconvex, since it can be defined as {(z, w)∈C2: max(|z|2− |w|2,|w|2−1) <0}. In particular Ω is B1-regular. Consider the bounded psh. function u(z, w) = |z/w|. By the maximum principle we can check that ucan not be approximated from above by continuous functions on Ω which are 0-psh. on Ω. (For details see Section 4 of [Wi]). 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Wikstr¨ om, Jensen measures and boundary values of plurisubharmonic functions, Ark. Mat. 39(1) (2001), 181–200. Department of Mathematics Hanoi Univerrsity of Education (Dai Hoc Su Pham Hanoi) Hanoi Vietnam E-mail address:dieu−[email protected] Primera versi´o rebuda el 15 de juliol de 2005, darrera versi´o rebuda el 2 de desembre de 2005.