We show here a  weak  Hölder regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equation with data in the Lp space and Ω satisfying an f-property. The f-property is a potential-theoretical condition that holds for all pseudoconvex domains of finite type and many examples of infinite-type ones

Baracco, Luca

Khanh, Tran Vu

Pinton, Stefano

arXiv

Luca Baracco

Tran Vu Khanh

Stefano Pinton

Crossref

Comptes Rendus Mathematique

The complex Monge–Ampère equation on weakly pseudoconvex domains

Archive of European Integration

• • VOLUME 11, NUMBER 1 LONG GESTATION, BUT EASY BIRTH: THE EURO ARRIVES SMOOTHLY "In the end, it turned out to be almost a formality, different from what many ofus expected just a year ago." So described European Central Bank (ECB) President Wllll Duisenberg the tranquil yet historic birth of the euro which, at 12:00 AM on January 1, 1999, became the sions between NCUs of the euro, a monetary amount in one NCU must first be converted into euro ( via its fixed conversion rate) and then converted to the other NCU (also via its euro rate). This method, know as "triangulation", is the only legally enforceable conversion method ( unless others yield the official currency of 11 EU member states, with an irrevocably fixed conversion rate against the participating national currencies ( see table). EURO CONVERSION RATES exact same result). Although euro notes and coins will not appear until January 1, 2002, the new cur-rency can now be used by consumers, retail-ers, companies and public administrations in the form of ''written ( or "book") money", i.e. personal checks, traveler's' checks, bank transfers, credit and debit cards and elec-tronic purses. And from now on, the euro's value against all other currencies, including the US dollar and those of the four non-par-ticipating EU countries ( the UK, Denmark, Sweden and Greece), will fluctuate accord-ing to market conditions. However, the Dan-ish krone and the Greek drachma have While the successful launch was largely seamless, it belied the years of intense and thorough preparation - and political com-mitment - that went behind it. In terms of scale and complexity, EMU has no historical precedent. The 11-country euro zone corre-sponds to an economic area of 292 million people, 19.4% of world GDP and 18.6% of total world trade. And, over time, it will most certainly expand. European Commission President Jacques Santer hailed the smooth start of 1 EUR = Country 40.3399 BEF ............... Belgium 1.95583 DEM ............... Germany 166.386 ESP .................. Spain 6.55957 FRF ................. France . 787564 IEP ................. Ireland 1936.27 ITL .................... Italy 40.3399 LUF ............ Luxembourg 2.20371 NLG ........... . Netherlands 13.7603 ATS ................. Austria 200.482 PTE ................ Portugal 5.94573 FIM ................ Finland the euro on the international financial markets as evidence of its solid status. Founded in an economic culture of stability, the euro will be a strong currency and will help create the right conditions for prosperity and employment, he said. joined the ERM II (see EURECOM, Decem-ber 1998) to limit fluctuations vis-a-vis the euro to ±15% of their central euro rates (which are in line with their previous ecu cen-tral rates). Exchange rates between third country currencies and the NCUs are derived from the latest euro exchange rate, similar to the triangulation method referred to above. Based on a proposal from the European Commission ( and in consultation with the ECB), the EU Council of Ministers adopted the euro conversion rates on the afternoon of December 31, 1998, exactly as outlined in a May 2, 1998 Joint Communique (see EURE-COM, May 1998). Accordingly, the national currency units (NCUs) have ceased to be currencies in their own right: they are now non-decimal denom-inations of the euro - and the only legal tender for retail cash transactions during a three-year transition period - until they are phased out by July 2002 (at the latest). With regard to conver-In terms of the single monetary policy and the European Sys-tem of Central Banks (i.e. the ECB and the 11 participating national central banks), the ECB's Governing Council expressed its satisfaction with the changeover to one integrated euro area money market and with the functioning of TARGET, the EU-wide, real-time gross settlement system for euro payments. To keep abreast of euro matters, check out the ECB's website (www.ecb.int), as well as the Commission's (europa.eu.inf/ eurol). US PREPARES SANCTIONS AGAINST EU OVER BANANAS After the US Administration's publica-tion last month of a list of retaliatory tariff increases aimed at the EU over the banana dispute (see EURECOM, Novem-ber 1998), European Commission Vice President Sir Leon Brittan made the fol-lowing statement: "I very much regret that the US has gone ahead and published this list of sanctions ... and hope that even now wiser counsels will prevail and the US will find a way of responding to * * * * * * * * * *** ----------------~-------------------~~~~~~~~==1 our desire to resolve this issue lawfully and amicably. We have given the United States every opportunity to get the banana dispute resolved rapidly in the proper place - the World Trade Organization. They could at any time have requested a panel to deter-mine the issue. We had no desire or ability to prevent them. Indeed, when they failed to do so we launched our own request for a panel, so determined were we that this should be an independent, binding ruling. However, the US has regrettably cho-sen not to take this multilateral path, but rather to embark on unilateral trade sanc-tions. These are specifically condemned by WTO rules. Indeed it is difficult to think of a more obvious breach of the multilat-eral trade system than to take the law into your own hands in defiance of the rules. This list is arbitrary in what it covers ( e.g. certain cheeses, candles, greeting cards, bed linens, cashmere sweater, cer-tain handbags, sweet biscuits, chande-liers) and arbitrary in its value. It threatens European firms and jobs which have no link with the banana dispute whatsoever. Put simply, it is an example of unilateralism at its worst. This dispute is no longer about bananas - it is about respect for the mul-tilateral trade system. We believe our new banana rules ... are entirely WTO-compati-ble. If US disagrees they have a very clear path open to them - that is to challenge it within the WTO. What is unacceptable is to announce illegal sanctions." As a result of the US action, the EU will accelerate its moves in the WTO for con-sultations leading to a panel against US unilateral measures under Section 301 of the US Trade Act (see EURECOM, Decem-ber 1998). At the time of writing, the US had just formally announced it would seek WTO authority to impose retaliatory sanctions on the list of targeted EU products at a January 25 meeting of the WTO's Disputes Settlement Body, despite the WTO dispute panel already established (by the US' co-complainant Ecuador) to examine whether the EU's banana policy modifica-tions comply with the WTO ruling. E1JRE£OM, JANIJARY 1999, PG. 2 EU SPURNS US' "3G" STANDARDS CLAIM In response to a December 19 letter signed by US Secretary of State Madeleine Albright, US Trade Representative Char-lene Barshefsky, US Secretary of Com-merce William Daley and Federal Communications Commissioner William Kennard, EU Telecommunications Com-missioner Martin Bangemann firmly rejected US allegations that the EU is rais-ing barriers to US firms in the third gener-ation (3G) mobile communications market. EU policy, answered Commis-sioner Bangemann, is to have market demand met by a broad competitive offer-ing of mobile multimedia services, fully in line with the EU regulatory framework and its WTO obligations. 3G mobile communications will offer, in addition to mobile telephony and messag-ing services, a new dimension of wireless communications, including access to improved mobile Internet access and fast data transmissions for multimedia services. Bangemann underlined that the EU fully supports global harmonization of 3G technology standards via the Interna-tional Telecommunications Union (ITU), but that the decisions on this must be an industry-led process of technical and com-mercial considerations. The ITU, a United Nations body, is scheduled to select a global standard for 3G mobile phones by March 31 this year. A recent EU decision on a Universal Mobile Telecommunications System (UMTS) does not define any technological content nor does it establish UMTS as an exclusive standard. It does foresee the issuance of at least one UMTS license per member state (as of January 1, 2000) to ensure an interoperable service across the EU and to facilitate pan-European "roam-ing" for consumers. Still, the UTMS con-cept and its eventual standards will be decided by industry-with non-European participation - and will not limit other 3G technologies in the European market. Commissioner Bangemann expressed surprise that the US government seems to be asking for regulatory intervention in an industry-led process. Further, he said that "we would not deem it appropriate to impose backwards compatibility between second and third generation systems, like the US seems to want, nor to impose con-vergence of 3G standards towards a single standard, let alone a particular 3G stan-dard, be it at the EU or global level." SWITZERLAND, EU STRIKE TRADE DEAL After over four years of tough talks, the EU and Switzerland have reached a pre-liminary trade accord covering bilateral agreements in seven sectors, providing an important basis for the EU to deepen rela-tions with an important partner in the center of Europe. Ever since Swiss voters rejected the European Economic Area (EEA) in a December 1992 referendum ( see EURE-COM, December 1992), relations with the EU have been more or less stagnant, while the rest of the continent - either through EU membership, membership candidacy or association agreement - has drawn closer to the Union. With the new agree-ment, this deadlock has been overcome, opening new possibilities for strengthened EU-Swiss bilateral cooperation. EU Commissioner Hans van den Broek, who negotiated on the EU's behalf, termed the accord a "balanced" yet interim step, adding that "it was up to the Swiss to decide what it is a step towards," especially as regards possible EU membership. The accord augments a longstanding EU-Switzerland free trade agreement, and covers the following (mainly) non-tariff barrier dossiers: the free movement of peo-ple, road transport, air transport, mutual recognition of product standards, public procurement, joint scientific research pro-jects and agriculture. However, like the failed EEA treaty just over six years ago, the EU-Switzerland trade pact must go in front of Swiss voters in a referendum -always a formidable stumbling block. "Geographically no neighbor can be closer to us than Switzerland. (This) deal brings the contractual and economic rela-tionship between us somewhat more into line with this fact," remarked van den Broek. • • * * * * * * * * * ~:;:~===============:::~=========================~*** • • • COMMISSION SURVIVES EP CENSURE VOTE Although not big news on this side of the Atlantic, the Commission has been embroiled in a serious dispute with the European Parliament (EP) over the Com-mission's financial management of the EU budget and, in some cases, budgetary fraud. In addition, there were calls from MEPs for the resignation of two Commis-sioners, Edith Cresson and Manuel Marin, who were responsible for the services where fraud was involved. Ultimately, this growing conflict led to an EP vote to censure the Commission on January 14 which, had it passed, would have forced the entire 20-member Com-mission to resign. However, based on an eight-point plan to strengthen financial and personnel management put forward by Commission President Jacques Santer - with a "zero tolerance" for fraud - the EP spared the Commission, rejecting the censure motion by a vote of 293 to 232 (27 abstentions). EP resolutions for the indi-vidual resignations were also defeated. President Santer promised proposals to modernize and restructure the admin-istration of the Commission by the end of January, and to keep MEPs informed of their progress. According to Santer, the vote showed that the EU institutions func-tion in a democratic manner. Under the present Treaty, the Commis-sion is a single "college", and individual Commissioners cannot be targeted for maladministration - it's either all or none. This, however, MEPs are determined to change. Said former Commissioner and current MEP Willy de Clerq: "The indivd-ual responsibility of commissioners has been put on the agenda. We will not take it off again." .. .IN BRIEF ... "I warmly welcome President Clin-ton's acceptance of the call for a new Round of global trade negotiations," said Sir Leon Brittan in response to US Presi-dent Clinton's State of the Union address on January 19. After nearly three years of QUOTES "It shoul,d have happened a lot ear-lier.'" Former German Chancellor and an acknowledged ''godfather" of EMU Helmut Schmid~ commenting on the euro. 'Prices are going to fall ( due to EMU). That will increase demand and put more money in consumers' hands. That's going to have a positive effect on the European economy, and that will be good for the United States." US Undersecretary of Commerce for international trade David L.Aaron. "I believe the euro 'is going to be a rival to the dollar. That will impose some d'is-cipline on the US economy, which we EU campaigning for a Millennium Round, Sir Leon found it heartening "to find our efforts meeting success in such a crucial quarter." With the Transatlantic Eco-nomic Partnership (TEP) in train ( see EURECOM, November 1998), the EU and US have a ready means for discussing what subjects should be in the Round ( and how). Sir Leon cautioned, however, that efforts to convince the rest of the world of a new Round's benefits, particularly in some developing countries, will need redoubling. "But our ability to do this will also depend on the proof we give of our commitment to a fair, rules-based multi-lateral system," remarked Sir Leon. "I renew my appeal to the US and President Clinton to draw back from present threats of unilateral sanctions against its main trading partners and supporters of the new Round - the EU, Japan and Canada." ... For all those EU consumers thirsting away for excellent US ( mainly California) wines, there is good news: the EU has agreed to extend until December 31, 2003 a derogation permitting US wine imports which may "have undergone oenological practices not allowed by Community rules". EU wine legislation, the world's most developed, defined and strict, often presents difficulties to third-country have not been used to." Clyde Prest.owitz, president, Economic StrafJJgy InstitufJJ. 'The dollar and the euro are eyeing each other warily. A promiS'ing adver .. sary to · the privileged US currency has emerged." Cuban President Fidel C(JJJt,o. "Some have argued that a Europe with a singl,e number in the global direc-tory might ultimately pose a threat to the US. But in a global economy, the US has infinitely less to fear from an open and integrated Europe ... than a Europe that turns inward and seeks insulation." US Deputy Treasury Secretary Lawrence Sum11rers. imports that are subject to less exacting vinification rules and practices. Neverthe-less, the EU is the world's foremost importer of wine, and its citizens quaff almost 60% of global wine consumption. (It is the largest exporter as well). This extension enables negotiations on an EU-US agreement on trade in wine to continue ( the previous deadline for agree-ment was December 31, 1998). At issue in the talks are import conditions, oenological practices of each party and the protection of appellation as to origin, a particularly important area for the terroir-oriented Europeans. Both parties have indicated that a "satisfactory" agreement can be expected within the given timeframe. ... The number of insurance companies in the EU fell by 3.4% between 1996 and 1997, according to a new Eurostat report. In 1997, the EU had 2,995 such enter-prises, down from 3,099 in 1996 ( all fig-ures exclude the UK). The report holds that the decrease was likely due to com-panies merging as a result of growing com-petition in the EU's single market. The euro's introduction will eliminate a fur-ther barrier to competition, it says. ... The prestigious Salzburg Seminar is convening a new session, The Transat-EIJRECOM, JANIJABY 1999, PG. 3 * * * * * * * * * * * * lantic Agenda at the Turn of the Century, at the Schloss Leopoldskron in Salzburg, Austria from March 20-27, 1998. This ses-sion will examine transatlantic relations against the changing backdrop of the post-Cold War order, and identify the underly-ing values and goals of the relationship as Europe and the US approach the next cen-tury. Areas of focus will include, inter alia: the continuing evolution of the EU and NATO; burden-sharing in response to global challenges; economic security and competitiveness; multilateral trade rela-tions; and the transatlantic business agenda. Salzburg Seminar Fellows are generally mid-career professionals who possess significant experience in fields relevant to the theme of the session and represent a cross-section of the private and public sectors. Proficiency in English is also necessary. Places are limited. For more information, please contact Shanta Rau at ( 43) 662-839-83-14 7 or srau@ salsem. ac. at. ... Germany took over the EU Council presidency on January 1, and will guide the work of the Council for the first six months of 1999. Its agenda will focus on creating jobs for a Europe that faces worldwide competition; internal consoli-dation and further development of the Union (i.e. Agenda 2000); and strengthen-ing the EU's profile in its relations with other countries. In addition to the cus-tomary European Council summit toward the end of the presidency ( which will take place June 3-4, 1998 in Cologne), an extra-ordinary European Council is slated for March 24-25 in Brussels, with the aim of reaching agreement on Agenda 2000 reforms (see EURECOM, December 1998). The German presidency's website is www.eu-presidency.de/. ... Fordham University Law School will hold its Seventh Annual Conference on International Intellectual Property Law * * * * * * * * * *** on April 9 and 10, 1999 in Manhattan (140 W. 62nd St.). The two-day conference will analyze international developments in copyright, patent and trademark law with regard to the EU, Asia, the US, the World Intellectual Property Organization (WIPO) and the Information Society. It will feature speakers from the European Commission, the US government, WIPO, academia, the judiciary and the US and other bars. For more information, please contact T. Scott Lilly at: tel (212) 636-6777, fax (212) 636-6984 or e-mail Slilly@mail. lawnetfordham.edu. MONTHLY BULLETIN OF EUROPEAN UNION ECONOMIC AND FINANCIAL NEWS For additional information on any article in this issue, please write or telephone Christopher Matthews or Kerstin Erickson at (212) 371-3804. EURECOM is also online at http://www.eurunion.org/news Our e-mail address is: eurecom@usany.eudel Please send a regular free copy of EURECOM to: Position----------------Organization ----------------Address----------------Special Interests ______________ _ * * * * * * * * * * * * EURECOM is published by Wouter Wilton, Director of Press & Public Affairs, the European Commission, 3 Dag Hammarsltjold Plaza, 305 East47th Street, New York, NY 10017. It is edited by Christopher Matthews. The contents of EURECOM do not necessarily reflect the views of the European Union's institutions. Any article may be reproduced without express permission. However, acknowledgement of the source and a copy of any material published would be appreciated. Printed on Recycled Paper. • • v08E· TLE (irn) auoqda1a.1 • L I001 !N '}[JO,\ M.aN '·1s llllv we::;i 908 '"eZ'llJd PJO~SJ'ellllll'eH 8-ea E UOJSSJUIUIOCJ U"eadom:3: 

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Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 355 (2017) 411–414Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comComplex analysis/Partial differential equationsThe complex Monge–Ampère equation on weakly pseudoconvex domains ✩L’équation de Monge–Ampère complexe sur les domaines faiblement pseudo-convexesLuca Baracco a, Tran Vu Khanh b, Stefano Pinton aa Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italyb School of Mathematics and Applied Statistics, University of Wollongong, NSW, 2522, Australiaa r t i c l e i n f o a b s t r a c tArticle history:Received 29 July 2016Accepted after revision 22 February 2017Available online 11 March 2017Presented by Jean-Pierre DemaillyWe show here a “weak” Hölder regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge–Ampère equation with data in the Lp space and  satisfying an f -property. The f -property is a potential-theoretical condition that holds for all pseudoconvex domains of finite type and many examples of infinite-type ones.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous montrons ici une régularité de Hölder « faible » jusqu’au bord d’une solution du problème de Dirichlet pour l’équation de Monge–Ampère complexe, de donnée dans l’espace Lp , sur un domaine satisfaisant une f -propriété. Cette f -propriété est une condition de théorie du potentiel qui est satisfaite par tous les domaines pseudo-convexes de type fini et de nombreux exemples de type infini.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionFor a C2, bounded, pseudoconvex domain  ⊂⊂ Cn , the Dirichlet problem for the Monge–Ampère equation consists of{ u ∈ P S H() ∩ L∞loc(),(ddcu)n = ψ dV in ,u = ϕ on b.(1.1)✩ The research of T.V. Khanh was supported by the Australian Research Council DE160100173.E-mail addresses: baracco@math.unipd.it (L. Baracco), tkhanh@uow.edu.au (T.V. Khanh), pinton@math.unipd.it (S. Pinton).http://dx.doi.org/10.1016/j.crma.2017.02.0041631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.412 L. Baracco et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 411–414A great deal of work has been done for the case where  is strongly pseudoconvex. Within this domain, we can divide the literature into three kinds of data ψ .– The Hölder data: Bedford–Taylor prove in [2] that u ∈ C α2 (̄) if ϕ ∈ Cα(b), ψ 1n ∈ C α2 (̄) for 0 < α ≤ 2.– The smooth data: Caffarelli, Kohn and Nirenberg prove in [4] that u ∈ C∞(̄), for ϕ ∈ C∞(b) and ψ ∈ C∞(̄), in case ψ > 0 in ̄ and b is smooth.– The Lp data: Guedj, Kolodziej and Zeriahi prove in [6] that if ψ ∈ Lp() with p > 1 and ϕ ∈ C1,1(b) then u ∈ Cγ (̄)for any γ < γp := 2qn+1 where 1q + 1p = 1.When  is no longer strongly pseudoconvex but has a certain “finite type”, there are some known results for this problem due to Blocki [3], Coman [5], and Li [11]. Recently, Ha and the second author gave a general related result to a Hölder data under the hypothesis that  satisfies an f -property (see Definition 2.1 below). The f -property is a consequence of the geometric “type” of the boundary. All pseudoconvex domains of finite type satisfy the f -property as well as many classes of domains of infinite type (see [9,7,8] for discussion on the f -property). Using the f -property, a “weak” Hölder regularity for the solution to the Dirichlet problem of the complex Monge–Ampère equation is obtained in [9]. Coming back to the case of  of finite type, in a recent paper with Zampieri [1], we prove the Hölder regularity for ψ ∈ Lp , with p > 1. The purpose of the present paper is to generalize the result in [1] to a pseudoconvex domain satisfying an f -property. For this purpose, we recall the definition of a weak Hölder space in [9,7]. Let f be an increasing function such that limt→+∞ f (t) = +∞, f (t)  t . For a subset A of Cn , define the f -Hölder space on A by f (A) = {u : ‖u‖L∞(A) + supz,w∈A,z 	=wf (|z − w|−1) · |u(z) − u(w)| < ∞}and set‖u‖ f (A) = ‖u‖L∞(A) + supz,w∈A,z 	=wf (|z − w|−1) · |u(z) − u(w)|.Note that the notion of the f -Hölder space includes the standard Hölder space α by taking f (t) = tα (so that f (|h|−1) =|h|−α) with 0 < α ≤ 1. Here is our resultTheorem 1.1. Let  ⊂ Cn be a bounded, pseudoconvex domain admitting the f -property. Suppose that ∞∫1daaf (a)< ∞ and denote by g(t) :=⎛⎝ ∞∫tdaaf (a)⎞⎠−1for t ≥ 1. If 0 < α ≤ 2, ϕ ∈ tα (b), and ψ ≥ 0 on  with ψ ∈ Lp with p > 1, then the Dirichlet problem for the complex Monge–Ampère equation (1.1) has a unique plurisubharmonic solution u ∈ gβ (). Here β = min(α, γ ), for any γ < γp = 2nq+1 where 1p + 1q = 1.The proof follows immediately from Theorem 2.2 and 2.5 below. Throughout the paper we use  and  to denote an estimate up to a positive constant, and ≈ when both of them hold simultaneously. Finally, the indices p, α, β , γ and γponly take ranges as in Theorem 1.1.2. Hölder regularity of the solutionWe start this section by defining the f -property as in [7,8].Definition 2.1. For a smooth, monotonic, increasing function f : [1, +∞) → [1, +∞) with f (t)t−1/2 decreasing, we say that  has the f -property if there exist a neighborhood U of b and a family of functions {ϕδ} such that(i) the functions ϕδ are plurisubharmonic, C2 on U , and satisfy −1 ≤ ϕδ ≤ 0,(ii) i∂∂̄ϕδ  f (δ−1)2 Id and |Dϕδ |  δ−1 for any z ∈ U ∩ {z ∈  : −δ < r(z) < 0}, where r is a C2-defining function of .In [7], using the f -property, the second author constructed a family of plurisubharmonic peak functions with good esti-mates. This family of plurisubharmonic peak functions yields the existence of a defining function ρ which is uniformly strictly plurisubharmonic and weakly Hölder (see [9]).Theorem 2.2 (Khanh [7] and Ha–Khanh [9]). Assume that  is a bounded, pseudoconvex domain admitting the f -property as in Theorem 1.1. Then there exists a uniformly strictly-plurisubharmonic defining function of  that belongs to the g2-Hölder space of , which means thatL. Baracco et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 411–414 413ρ ∈ g2(̄),  = {ρ < 0} and i∂∂̄ρ ≥ Id. (2.1)The existence and uniqueness of the solution u ∈ L∞() to the equation (1.1) need a weaker condition, in particular, one only need ρ ∈ C0(̄), as shown by [10].Theorem 2.3 (Kolodziej [10]). Let  be a bounded domain in Cn. Assume that there exists a function ρ such thatρ ∈ C0(̄),  = {ρ < 0} and i∂∂̄ρ ≥ Id.Then, for any ϕ ∈ C0(b), ψ ∈ Lp(), there is a unique plurisubharmonic solution u(, ϕ, ψ) ∈ C0(̄).To improve the smoothness of u, we increase the smoothness of ρ and ψ .Theorem 2.4 (Ha–Khanh [9]). Let ρ satisfy (2.1). If ϕ ∈ tα (b) and ψ 1n ∈ gα (), then the Dirichlet problem for the complex Monge–Ampère equation (1.1) has a unique plurisubharmonic solution u(, ϕ, ψ) ∈ gα ().Now we focus on lowering the smoothness of ψ and prove the following theorem.Theorem 2.5. Let ρ satisfy (2.1). If ϕ ∈ tα (b) and ψ ∈ Lp(), then the Dirichlet problem for the complex Monge–Ampère equation (1.1) has a unique plurisubharmonic solution u(, ϕ, ψ) ∈ gβ ().In order to prove this theorem, we need to construct a subsolution with L p data. Here, v is a subsolution to (1.1) in the sense that v is plurisubharmonic, v|b = ϕ and (ddc v)n ≥ ψ dV in .Proposition 2.6. Let ρ satisfy (2.1). Then there is a subsolution v ∈ gβ (̄) to (1.1) for ϕ ∈ Cα(b) and ψ ∈ Lp().Proof. For a large ball B containing , we set ψ̃(z) :={ψ(z) if z ∈ ,0 if z ∈ B \ . First, we apply Theorem 1 in [6] on B with ψ̃ ∈ Lp(B) and zero-valued boundary condition; it follows u1 = u(B, 0, ψ̃) ∈ tγ (B̄). Second, we apply Theorem 2.4 on twice: first for u2 := u(, −u1|b, 0) ∈ gγ , since u1|b ∈ tγ , and second for u3 := u(, ϕ, 0) ∈ gα by the hypothesis ϕ ∈ tα . Finally, taking the summation v = u1 + u2 + u3, we have the conclusion. Proof of Theorem 2.5. Keeping the notation of Theorem 2.3, let u(, ϕ, ψ) ∈ C0(̄) be the solution to (1.1). What follows is dedicated to showing that this C0 plurisubharmonic solution u(, ϕ, ψ) is in fact in gβ (̄). By Theorem 2.4 we have that w := u(, ϕ, 0) is in gα (̄). Let v be as in Proposition 2.6 then the comparison principle yields at oncev ≤ u(,ϕ,ψ) ≤ w. (2.2)By (2.2) and the gβ -Hölder regularity of v and w , we get|u(z) − u(ζ )|  [g(|z − ζ |−1)]−β z ∈ ̄, ζ ∈ b,and therefore for δ suitably small|u(z) − u(z′)|  [g(δ−1)]−β, z, z′ ∈  \ δ and |z − z′| < δ (2.3)where δ := {z ∈ Cn : r(z) < −δ} and r is the C2 defining function for  with |∇r| = 1 on b. We have to prove that (2.3)also holds for z, z′ ∈ δ . For z ∈ ̄δ , we use the notationu δ2(z) := sup|ζ |< δ2u(z + ζ ), ũ δ2(z) := 1σ2n−1(δ2)2n−1∫bB(z, δ2 )u(ζ )dS(ζ ),andû δ2(z) := 1σ2n(δ2 )2n∫B(z, δ2 )u(ζ )dV (ζ ),where σ2n−1(δ2)2n−1 = Vol(bB(z, δ2 )) and σ2n( δ2 )2n = Vol(B(z, δ2 )). It is obvious thatû δ2≤ ũ δ2≤ u δ2in δ. (2.4)Furthermore, we have an L1 estimate of the difference between u and ũ δ2and of the stability estimate in the following theorems (2.7 and 2.8).414 L. Baracco et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 411–414Theorem 2.7 (Baracco–Khanh–Pinton–Zampieri [1]). For any 0 < ε < 1, we have‖ũ δ2− u‖L1(δ)  δ1−ε . (2.5)Theorem 2.8 (Guedj–Kolodziej–Zeriahi [6]). Fix 0 ≤ f ∈ Lp(), p > 1. Let U , W be two bounded plurisubharmonic functions in such that (ddcU )n = f dV in  and let U ≥ W on ∂. Fix s ≥ 1 and 0 ≤ η < snq+s , 1p + 1q = 1. Then there exists a uniform constant C = C(η, ‖ f ‖Lp()) > 0 such thatsup(W − U ) ≤ C‖(W − U )+‖ηLs(),where (W − U )+ := max(W − U , 0).By (2.3), we haveũ δ2≤ u δ2≤ u + c[g(δ−1)]−β, on bδ for suitable constant c.Thus, we can apply Theorem 2.8 for δ with U := u + c[g(δ−1)]−β , W := ũ δ2and s := 1; thus we getsupδ(ũ δ2− (u + c[g(δ−1)]−β))Theorem 2.8‖(ũ δ2− (u + c[g(δ−1)]−β))+ ‖ηL1(δ) ‖ũ δ2− u‖ηL1(δ)Theorem 2.7δ(1−ε)η, (2.6)for any η < 12 γp = 1nq+1 where 1q + 1p = 1. Taking γ < γp , β = min(α, γ ), ε = γp−γγp+γ > 0 and η = 14 (γp + γ ) < 12 γp so that (1 − ε)η = γ2 , it followssupδ(ũ δ2− u) δ(1−ε)η + [g(δ−1)]−β  δ γ2 + [g(δ−1)]−β  [g(δ−1)]−β, (2.7)where the last inequality of (2.7) follows by g(δ−1)  δ− 12 (by the conditions on f in the f -property).Similarly to [6, Lemma 4.2] by using the fact that g(cδ−1)) ≈ g(δ−1) for any constant c > 0, one can state the equivalence betweensupδ(uδ − u)  [g(δ−1)]−β and supδ(ûδ − u)  [g(δ−1)]−β .Using this equivalence together with the inequalities in (2.4), it follows that (2.7) is equivalent tosupδ(u δ2− u)  [g(δ−1)]−β . (2.8)From (2.3) and (2.8), it is easy to prove that|u(z) − u(z′)|  [g(|z − z′|−1)]−β for any z, z′ ∈ ̄. References[1] L. Baracco, Tran Vu Khanh, S. Pinton, G. Zampieri, Hölder regularity of the solution to the complex Monge–Ampère equation with Lp density, Calc. Var. Partial Differ. Equ. 55 (2016) 74.[2] E. Bedford, B.A. Taylor, The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math. 37 (1) (1976) 1–44.[3] Z. Blocki, The complex Monge–Ampère operator in hyperconvex domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 23 (4) (1997) 721–747, 1996.[4] L. Caffarelli, J.J. Kohn, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge–Ampère, and uniformly elliptic, equations, Commun. Pure Appl. Math. 38 (2) (1985) 209–252.[5] D. Coman, Domains of finite type and Hölder continuity of the Perron–Bremermann function, Proc. Amer. Math. Soc. 125 (12) (1997) 3569–3574.[6] V. Guedj, S. Kolodziej, A. Zeriahi, Hölder continuous solutions to Monge–Ampère equations, Bull. Lond. Math. Soc. 40 (6) (2008) 1070–1080.[7] Tran Vu Khanh, Lower bounds on the Kobayashi metric near a point of infinite type, J. Geom. Anal. 26 (1) (2016) 616–629.[8] Tran Vu Khanh, G. Zampieri, Regularity of the ∂-Neumann problem at point of infinite type, J. Funct. Anal. 259 (11) (2010) 2760–2775.[9] L. Kim Ha, Tran Vu Khanh, Boundary regularity of the solution to the complex Monge–Ampère equation on pseudoconvex domains of infinite type, Math. Res. Lett. 22 (2) (2015) 467–484.[10] S. Kolodziej, The complex Monge–Ampère equation, Acta Math. 180 (1) (1998) 69–117.[11] S.-Y. Li, On the existence and regularity of Dirichlet problem for complex Monge–Ampère equations on weakly pseudoconvex domains, Calc. Var. Partial Differ. Equ. 20 (2) (2004) 119–132.

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L'équation de Monge–Ampère complexe sur les domaines faiblement pseudo-convexes

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EURECOM:Monthly Bulletin of European Community Economic and Financial News. January 1999 Vol. 11, No. 1

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