* (45) Latora V, Nicosia V, Russo G. 2017 Complex Networks: Principles, Methods and Applications. Cambridge University Press.
  * (46) Barabási AL. 2016 Network science. Cambridge University Press.
  * (47) Newman M. 2010 Networks: An Introduction. Oxford: Oxford University Press.
  * (48) Lambiotte R, Schaub M. 2021 Modularity and Dynamics on Complex Networks. Cambridge University Press.
  * (49) Macdonald CB, Ruuth SJ. 2010 The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput. 31, 4330–4350.
  * (50) Woolley TE, Baker RE, Maini PK. 2017 Turing’s theory of morphogenesis: where we started, where we are and where we want to go. The Incomputable: Journeys Beyond the Turing Barrier pp. 219–235.
  * (51) Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C. 2008 Synchronization in complex networks. Phys. Rep. 469, 93–153.
  * (52) Masuda N, Porter MA, Lambiotte R. 2017 Random walks and diffusion on networks. Phys. Rep. 716-717, 1–58.
  * (53) Cencetti G, Battiston F, Carletti T, Fanelli D. 2020 Generalized patterns from local and non local reactions. Chaos Solit. Fractals 134, 109707.
  * (54) May RM. 1972 Will a large complex system be stable?. Nature 238, 413.
  * (55) Prigogine I, Nicolis G. 1967 Symmetry breaking instabilities in dissipative systems. J. Chem. Phys. 46, 3542.
  * (56) Watts DJ, Strogatz SH. 1998 Collective dynamics of ‘small-world’networks. Nature 393, 440–442.
  * (57) Ide Y, Izuhara H, Machida T. 2016 Turing instability in reaction–diffusion models on complex networks. Physica A 457, 331–347.
  * (58) Pastor-Satorras R, Vespignani A. 2010 Patterns of complexity. Nat. Phys. 6, 480.
  * (59) Bianconi G. 2022 Multilayer netwokrs, Structure and Function. Oxford University Press.
  * (60) Holme P, Saramäki J. 2012 Temporal networks. Phys. Rep. 519, 97–125.
  * (61) Asllani M, Challenger J, Pavone F, Sacconi L, Fanelli D. 2014 The theory of pattern formation on directed networks. Nat. Comm. 5.
  * (62) Rovinsky AB, Menzinger M. 1992 Chemical instability induced by a differential flow. Phys. Rev. Lett. 69, 1193.
  * (63) He L, Su H. 2024 Spatiotemporal patterns of reaction–diffusion systems with advection mechanisms on large-scale regular networks. Chaos Solit. Fractals 178, 114314.
  * (64) Fruchart M, Hanai R, Littlewood PB, Vitelli V. 2021 Non-reciprocal phase transitions. Nature 592, 363–369.
  * (65) Asllani M, Carletti T, Fanelli D, Maini P. 2020 A universal route to pattern formation in multicellular systems. Eur. Phys. J. B 93.
  * (66) Ritchie J. 2023 Turing instability and pattern formation on directed networks. Commun. Nonlinear Sci. Numer. Simul. 116, 106892.
  * (67) Carletti T, Muolo R. 2022 Non-reciprocal interactions enhance heterogeneity. Chaos Solit. Fractals 164, 112638.
  * (68) Newman ME, Watts DJ. 1999 Scaling and percolation in the small-world network model. Physical review E 60, 7332.
  * (69) Trefethen L, Embree M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton Univ. Press.
  * (70) Trefethen L, Trefethen A, Reddy S, Driscoll T. 1993 Hydrodynamic Stability Without Eigenvalues. Science 261, 578–584.
  * (71) Asllani M, Carletti T. 2018 Topological resilience in non-normal networked systems. Phys. Rev. E 97.
  * (72) Neubert M, Caswell H. 1997 Alternatives to resilience for measuring the responses of ecological systems to perturbations. Math. Biosci. 78, 653–665.
  * (73) Murphy BK, Miller KD. 2009 Balanced amplification: a new mechanism of selective amplification of neural activity patterns. Neuron 61, 635–648.
  * (74) Hennequin G, Vogels TP, Gerstner W. 2014 Optimal control of transient dynamics in balanced networks supports generation of complex movements. Neuron 82, 1394–1406.
  * (75) Ferraz LB, Muolo R, Caudano Y, Carletti T. 2023 Revisiting weak values through non-normality. J. Phys. A: Math. Theor. 56, 475303.
  * (76) Ganguli S, Huh D, Sompolinsky H. 2008 Memory traces in dynamical systems. Proc. Natl. Acad. Sci. 105, 18970–18975.
  * (77) Baggio G, Rutten V, Hennequin G, Zampieri S. 2020 Efficient communication over complex dynamical networks: The role of matrix non-normality. Sci. Adv. 6, eaba2282.
  * (78) Lindmark G, Altafini C. 2020 Centrality measures and the role of non-normality for network control energy reduction. IEEE Control Syst. Lett. 5, 1013–1018.
  * (79) Neubert M, Caswell H, Murray J. 2002 Transient Dynamics and Pattern Formation: Reactivity is Necessary for Turing Instabilities. Math. Biosci. 175(1), 1–11.
  * (80) Klika V. 2017 Significance of non-normality-induced patterns: Transient growth versus asymptotic stability. Chaos 27, 073120.
  * (81) Biancalani T, Jafarpour F, Goldenfeld N. 2017 Giant amplification of noise in fluctuation-induced pattern formation. Phys. Rev. Lett. 118, 018101.
  * (82) Muolo R, Asllani M, Fanelli D, Maini P, Carletti T. 2019 Patterns of non-normality in networked systems. Journal Theoretical Biology 480, 81.
  * (83) Nicoletti S, Fanelli D, Zagli N, Asllani M, Battistelli G, Carletti T, Chisci L, Innocenti G, Livi R. 2019 Resilience for stochastic systems interacting via a quasi-degenerate network. Chaos 29, 083123.
  * (84) Dorchain M, Muolo R, Carletti T. 2023 Pattern reconstruction through generalized eigenvectors on defective networks. EPL 144, 11004.
  * (85) Nishikawa T, Motter A. 2006 Synchronization is optimal in nondiagonalizable networks. Phys. Rev. E 73, 065106(R).
  * (86) Asllani M, Lambiotte R, Carletti T. 2018 Structure and dynamical behavior of non-normal networks. Sci. Adv. 4, Eaau9403.
  * (87) MacKay RS, Johnson S, Sansom B. 2020 How directed is a directed network?. R. Soc. Open Sci. 7, 201138.
  * (88) Pilgrim C, Guo W, Johnson S. 2020 Organisational social influence on directed hierarchical graphs, from tyranny to anarchy. Sci. Rep. 10, 4388.
  * (89) Nartallo-Kaluarachchi R, Asllani M, Goriely A, Lambiotte R. 2024 Broken detailed balance and entropy production in directed networks. arXiv preprint arXiv:2402.19157.
  * (90) Duan C, Nishikawa T, Eroglu D, Motter AE. 2022 Network structural origin of instabilities in large complex systems. Sci. Adv. 8, eabm8310.
  * (91) Busiello DM, Carletti T, Fanelli D. 2018 Homogeneous-per-layer patterns in multiplex networks. Eur. Phys. Letters 121, 48006.
  * (92) Asllani M, Carletti T, Fanelli D. 2016 Tune the topology to create or destroy patterns. Eur. Phys. J. B 89, 1–10.
  * (93) Kurant M, Thiran P. 2006 Layered complex networks. Phys. Rev. Lett. 96, 138701.
  * (94) Zou SR, Zhou T, Liu AF, Xu XL, He DR. 2010 Topological relation of layered complex networks. Phys. Lett. A 374, 4406–4410.
  * (95) Bullmore E, Sporns O. 2009 Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10, 186–198.
  * (96) Yang J, Wang W, Chen G. 2009 A two-level complex network model and its application. Physica A 388, 2435–2449.
  * (97) Wasserman S, Faust K. 1994 Social Network Analysis: Methods and Applications. Cambridge University Press.
  * (98) Cozzo E, Ferraz de Arruda G, Aparecido Rodrigues F, Moreno Y. 2018 Multiplex Networks Basic Formalism and Structural Properties. Springer.
  * (99) Boccaletti S, Bianconi G, Criado R, Del Genio CI, Gómez-Gardenes J, Romance M, Sendina-Nadal I, Wang Z, Zanin M. 2014 The structure and dynamics of multilayer networks. Phys. Rep. 544, 1–122.
  * (100) Battiston F, Nicosia V, Latora V. 2014 Structural measures for multiplex networks. Phys. Rev. E 89, 032804.
  * (101) Krause AL, Klika V, Halatek J, Grant PK, Woolley TE, Dalchau N, Gaffney EA. 2020 Turing patterning in stratified domains. Bull. Math. Biol. 82, 136.
  * (102) Duda FP, Forte Neto FS, Fried E. 2023 Modelling of surface reactions and diffusion mediated by bulk diffusion. Phil. Trans. R. Soc. A 381, 20220367.
  * (103) Diez A, Krause AL, Maini PK, Gaffney EA, Seirin-Lee S. 2024 Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry. Bull. Math. Biol. 86, 13.
  * (104) Gomez S, Diaz-Guilera A, Gomez-Gardenes J, Perez-Vicente CJ, Moreno Y, Arenas A. 2013 Diffusion dynamics on multiplex networks. Phys. Rev. Lett. 110, 028701.
  * (105) Asllani M, Busiello D, Carletti T, Fanelli D, Planchon G. 2014 Turing patterns in multiplex networks. Phys. Rev. E 90, 042814.
  * (106) Kouvaris N, Hata S, Diaz-Guilera A. 2015 Pattern formation in multiplex networks. Sci. Rep. 5, 10840.
  * (107) Lei L, Yang J. 2021 Patterns in coupled FitzHugh–Nagumo model on duplex networks. Chaos Solit. Fractals 144, 110692.
  * (108) Song M, Gao S, Liu C, Bai Y, Zhang L, Xie B, Chang L. 2023 Cross-diffusion induced Turing patterns on multiplex networks of a predator–prey model. Chaos Solit. Fractals 168, 113131.
  * (109) Masuda N, Lambiotte R. 2016 A guide to temporal networks. World Scientific.
  * (110) Ghosh D, Frasca M, Rizzo A, Majhi S, Rakshit S, Alfaro-Bittner K, Boccaletti S. 2022 The synchronized dynamics of time-varying networks. Phys. Rep. 949, 1–63.
  * (111) Li A, Cornelius SP, Liu YY, Wang L, Barabási AL. 2017 The fundamental advantages of temporal networks. Science 358, 1042–1046.
  * (112) Stilwell D, Bollt E, Robertson D. 2006 Sufficient Conditions for Fast Switching Synchronization in Time-Varying Network Topologies. SIAM J. Appl. Dyn. 5, 140–156.
  * (113) Petit J, Lauwens B, Fanelli D, Carletti T. 2017 Theory of Turing Patterns on Time Varying Networks. Phys. Rev. Lett. 119, 148301.
  * (114) Lucas M, Fanelli D, Carletti T, Petit J. 2018 Desynchronization induced by time-varying network. EPL 121, 58008.
  * (115) Chavez M, Hwang DU, Amann A, Hentschel HGE, Boccaletti S. 2005 Synchronization is Enhanced in Weighted Complex Networks. Phys. Rev. Lett. 94, 218701.
  * (116) Amritkar R, Hu CK. 2006 Synchronized state of coupled dynamics on time-varying networks. Chaos 16, 015117.
  * (117) Zhang Y, Strogatz SH. 2021 Designing temporal networks that synchronize under resource constraints. Nat. Comm. 12, 3273.
  * (118) Van Gorder R. 2021 A theory of pattern formation for reaction-diffusion systems on temporal networks. Proc. R. Soc. A 477, 20200753.
  * (119) Carletti T, Fanelli D. 2022 Theory of synchronisation and pattern formation on time varying networks. Chaos Solit. Fractals 159, 112180.
  * (120) Guo L, Shi X, Cao J. 2021 Turing patterns of Gierer–Meinhardt model on complex networks. Nonlinear Dyn. 105, 899–909.
  * (121) Ji Y, Shen J. 2020 Turing instability of Brusselator in the reaction-diffusion network. Complexity 2020, 1–12.
  * (122) FitzHugh R. 1961 Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445.
  * (123) Nagumo J, Arimoto S, Yoshizawa S. 1962 An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061.
  * (124) Zheng Q, Shen J. 2020 Turing instability induced by random network in FitzHugh-Nagumo model. Appl. Math. Comput. 381, 125304.
  * (125) Asllani M, Di Patti F, Fanelli D. 2012 Stochastic Turing patterns on a network. Phys. Rev. E 86, 046105.
  * (126) Asllani M, Biancalani T, Fanelli D, McKane AJ. 2013 The linear noise approximation for reaction-diffusion systems on networks. Eur. Phys. J. B 86, 1–10.
  * (127) Xiao R, Gao Q, Azaele S, Sun Y. 2023 Effects of noise on the critical points of Turing instability in complex ecosystems. Phys. Rev. E 108, 014407.
  * (128) Carletti T, Muolo R. 2021 Finite propagation enhances Turing patterns in reaction–diffusion networked systems. J. Phys. Complex. 2, 045004.
  * (129) Petit J, Carletti T, Asllani M, Fanelli D. 2015 Delay-induced Turing-like waves for one-species reaction-diffusion model on a network. EPL 111, 58002.
  * (130) Petit J, Asllani M, Fanelli D, Lauwens B, Carletti T. 2016 Pattern formation in a two-component reaction–diffusion system with delayed processes on a network. Physica A 462, 230–249.
  * (131) Chang L, Liu C, Sun G, Wang Z, Jin Z. 2019 Delay-induced patterns in a predator–prey model on complex networks with diffusion. New J. Phys. 21, 073035.
  * (132) Tian C, Ling Z, Zhang L. 2020 Delay-driven spatial patterns in a network-organized semiarid vegetation model. Appl. Math. Comput. 367, 124778.
  * (133) Carletti T, Nakao H. 2020 Turing patterns in a network-reduced FitzHugh-Nagumo model. Phys. Rev. E 101, 022203.
  * (134) Zheng Q, Wang Z, Shen J. 2017 Pattern dynamics of network-organized system with cross-diffusion. Chin. Phys. B 26, 020501.
  * (135) Kuehn C, Soresina C. 2024 Cross-diffusion induced instability on networks. J. Complex Netw. 12, cnad052.
  * (136) Bramburger J, Holzer M. 2023 Pattern formation in random networks using graphons. SIAM J. Math. Anal. 55, 2150–2185.
  * (137) Siebert B, Hall C, Gleeson J, Asllani M. 2020 Role of modularity in self-organization dynamics in biological networks. Phys. Rev. E 102, 052306.
  * (138) van der Kolk J, García-Pérez G, Kouvaris NE, Serrano MÁ, Boguñá M. 2023 Emergence of geometric turing patterns in complex networks. Phys. Rev. X 13, 021038.
  * (139) Asllani M, Busiello DM, Carletti T, Fanelli D, Planchon G. 2015 Turing instabilities on Cartesian product networks. Sci. Rep. 5, 12927.
  * (140) Goras L, Ungureanu P, Chua LO. 2017 On turing instability in nonhomogeneous reaction-diffusion CNN’s. IEEE Trans. Circuits Syst. I: Regul. Pap. 64, 2748–2760.
  * (141) Nicolaides C, Juanes R, Cueto-Felgueroso L. 2016 Self-organization of network dynamics into local quantized states. Sci. Rep. 6, 21360.
  * (142) Hata S, Nakao H. 2017 Localization of Laplacian eigenvectors on random networks. Sci. Rep. 7, 1121.
  * (143) Wolfrum M. 2012 The Turing bifurcation in network systems: Collective patterns and single differentiated nodes. Physica D 241, 1351–1357.
  * (144) McCullen N, Wagenknecht T. 2016 Pattern formation on networks: from localised activity to Turing patterns. Sci. Rep. 6, 27397.
  * (145) Mimar S, Juane MM, Park J, Munuzuri AP, Ghoshal G. 2019 Turing patterns mediated by network topology in homogeneous active systems. Phys. Rev. E 99, 062303.
  * (146) Hütt MT, Armbruster D, Lesne A. 2022 Predictable topological sensitivity of Turing patterns on graphs. Phys. Rev. E 105, 014304.
  * (147) Chang L, Guo L, Liu C, Wang Z, Sun G. 2022 The qualitative and quantitative relationships between pattern formation and average degree in networked reaction-diffusion systems. Chaos 32, 093129.
  * (148) Pranesh S, Jaiswal D, Gupta S. 2024 Effect of clustering on Turing instability in complex networks. https://arxiv.org/abs/2406.17440.
  * (149) Estrada E, Estrada-Rodriguez G, Gimperlein H. 2020 Metaplex networks: Influence of the exo-endo structure of complex systems on diffusion. SIAM Rev. 62, 617–645.
  * (150) Yeakel JD, Moore JW, Guimarães Jr PR, de Aguiar M. 2014 Synchronisation and stability in river metapopulation networks. Ecol. Lett. 17, 273–283.
  * (151) Gross T, Allhoff KT, Blasius B, Brose U, Drossel B, Fahimipour AK, Guill C, Yeakel JD, Zeng F. 2020 Modern models of trophic meta-communities. Phil. Trans. R. Soc. Lond. B 375, 20190455.
  * (152) Hata S, Nakao H, Mikhailov A. 2014 Dispersal-induced destabilization of metapopulations and oscillatory Turing patterns in ecological networks. Sci. Rep. 4, 3585.
  * (153) Brechtel A, Gramlich P, Ritterskamp D, Drossel B, Gross T. 2018 Master stability functions reveal diffusion-driven pattern formation in networks. Phys. Rev. E 97, 032307.
  * (154) Baron JW, Galla T. 2020 Dispersal-induced instability in complex ecosystems. Nat. Comm. 11, 6032.
  * (155) Fernandes LD, De Aguiar M. 2012 Turing patterns and apparent competition in predator-prey food webs on networks. Phys. Rev. E 86, 056203.
  * (156) Liu C, Chang L, Huang Y, Wang Z. 2020 Turing patterns in a predator–prey model on complex networks. Nonlinear Dyn. 99, 3313–3322.
  * (157) Ghorai S, Poria S. 2016 Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity. Chaos Solit. Fractals 91, 421–429.
  * (158) Buscarino A, Corradino L, Fortuna L, Frasca M, Chua L. 2016 Turing patterns in memristive cellular nonlinear networks. IEEE Trans. Circuits Syst. I Regul. Pap. 63, 1222–1230.
  * (159) Buscarino A, Corradino C, Fortuna L, Frasca M. 2019 Turing patterns via pinning control in the simplest memristive cellular nonlinear networks. Chaos 29, 103145.
  * (160) Hata S, Nakao H, Mikhailov A. 2012 Global feedback control of Turing patterns in network-organized activator-inhibitor systems. Eur. Phys. Lett. 98, 64004.
  * (161) Gao S, Chang L, Romić I, Wang Z, Jusup M, Holme P. 2022 Optimal control of networked reaction–diffusion systems. J. R. Soc. Interface 19, 20210739.
  * (162) Liu C, Gao S, Song M, Bai Y, Chang L, Wang Z. 2022 Optimal control of the reaction-diffusion process on directed networks. Chaos 32, 063115.
  * (163) Cencetti G, Clusella P, Fanelli D. 2018 Pattern invariance for reaction-diffusion systems on complex networks. Sci. Rep. 8, 16226.
  * (164) Nicoletti S, Carletti T, Fanelli D, Battistelli G, Chisci L. 2020 Generating directed networks with prescribed Laplacian spectra. J. Phys. Complex. 2, 015004.
  * (165) Ali SH, Hütt MT. 2022 Inferring missing edges in a graph from observed collective patterns. Phys. Rev. E 105, 064610.
  * (166) Duan M, Chang L, Jin Z. 2019 Turing patterns of an SI epidemic model with cross-diffusion on complex networks. Physica A 533, 122023.
  * (167) Chang L, Duan M, Sun G, Jin Z. 2020 Cross-diffusion-induced patterns in an SIR epidemic model on complex networks. Chaos 30, 013147.
  * (168) Zheng Q, Shen J, Xu Y, Pandey V, Guan L. 2022 Pattern mechanism in stochastic SIR networks with ER connectivity. Physica A 603, 127765.
  * (169) Zhou J, Ye Y, Arenas A, Gómez S, Zhao Y. 2023 Pattern formation and bifurcation analysis of delay induced fractional-order epidemic spreading on networks. Chaos Solit. Fractals 174, 113805.
  * (170) He L, Su H. 2023 Turing pattern of an SIRI model on large-scale homogeneous and heterogeneous networks. Nonlinear Dyn. 111, 16605–16626.
  * (171) Sun GQ, He R, Hou LF, Gao S, Luo X, Liu Q, Zhang Y, Chang L. 2024 Dynamics of diseases spreading on networks in the forms of reaction-diffusion systems. to appear in EPL.
  * (172) He L, Su H. 2024 Parameter estimation for network-organized Turing system based on convolution neural networks. Commun. Nonlinear Sci. Numer. Simul. 130, 107781.
  * (173) Chang L, Wang X, Sun G, Wang Z, Jin Z. 2024 A time independent least squares algorithm for parameter identification of Turing patterns in reaction–diffusion systems. J. Math. Biol. 88, 5.
  * (174) Matas-Gil A, Endres RG. 2024 Unraveling biochemical spatial patterns: Machine learning approaches to the inverse problem of stationary Turing patterns. iScience 27, 109822.
  * (175) Zheng MM, Shao B, Ouyang Q. 2016 Identifying network topologies that can generate turing pattern. J. Theor. Biol. 408, 88–96.
  * (176) Diego X, Marcon L, Müller P, Sharpe J. 2018 Key features of Turing systems are determined purely by network topology. Phys. Rev. X 8, 021071.
  * (177) Challenger J, Burioni R, Fanelli D. 2015 Turing-like instabilities from a limit cycle. Phys. Rev. E 92, 022818.
  * (178) Nakao H, Mikhailov AS. 2009 Diffusion-induced instability and chaos in random oscillator networks. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 79, 036214.
  * (179) Pikovsky A, Kurths J, Rosenblum M. 2001 Synchronization: a universal concept in nonlinear sciences vol. 12. Cambridge University Press.
  * (180) Winfree AT. 1967 Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42.
  * (181) Kuramoto Y. 1975 Self-entrainment of a population of coupled non-linear oscillators. In Araki H, editor, International Symposium on Mathematical Problems in Theoretical Physics pp. 420–422 Berlin, Heidelberg. Springer Berlin Heidelberg.
  * (182) Fujisaka H, Yamada T. 1983 Stability Theory of Synchronized Motion in Coupled-Oscillator Systems. Prog. Theor. Phys. 69, 32–47.
  * (183) Pecora L, Carroll T. 1998 Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109.
  * (184) Muolo R. 2023 Symmetry breaking and Turing patterns on networks and higher-order structures: nonlinear beasts and where to find them. PhD Thesis, University of Namur.
  * (185) Van Gorder R. 2020 Turing and Benjamin-Feir instability mechanisms in non-autonomous systems. Proc. R. Soc. A 476, 20200003.
  * (186) Muolo R, Carletti T, Gleeson J, Asllani M. 2021 Synchronization dynamics in non-normal networks: the trade-off for optimality. Entropy 23, 36.
  * (187) O’Brien J, Oliveira K, Gleeson J, Asllani M. 2021 Hierarchical route to the emergence of leader nodes in real-world networks. Phys. Rev. Res. 3, 023117.
  * (188) Muolo R, O’Brien JD, Carletti T, Asllani M. 2024 Persistence of chimera states and the challenge for synchronization in real-world networks. Eur. Phys. J. B 97, 6.
  * (189) Nakao H. 2014 Complex Ginzburg-Landau equation on networks and its non-uniform dynamics. Eur. Phys. J.: Spec. Top. 223, 2411–2421.
  * (190) Patania A, Petri G, Vaccarino F. 2017 The shape of collaborations. EPJ Data Sci. 6, 18.
  * (191) Andrew R, Fagan M, Ballyk B, Rosen A. 2021 Seizure susceptibility and the osmotic state. Brain Res. 498, 175–180.
  * (192) Wang M, Arteaga D, He B. 2013 Brain mechanisms for simple perception and bistable perception. Proc. Natl. Acad. Sci. U.S.A. 110, E3350–E3359.
  * (193) Petri G, Expert P, Turkheimer F, Carhart-Harris R, Nutt D, Hellyer P, Vaccarino F. 2014 Homological scaffolds of brain functional networks. J. Royal Soc. Interface 11, 20140873.
  * (194) Sizemore A, Giusti C, Kahn A, Vettel J, Betzel R, Bassett D. 2018 Cliques and cavities in the human connectome. J. Comp. Neurosci. 44, 115–145.
  * (195) Grilli J, Barabás G, Michalska-Smith M, Allesina S. 2017 Higher-order interactions stabilize dynamics in competitive network models. Nature 548, 210–213.
  * (196) Centola D, Becker J, Brackbill D, Baronchelli A. 2018 Experimental evidence for tipping points in social convention. Science 360, 1116–1119.
  * (197) Iacopini I, Foote JR, Fefferman NH, Derryberry EP, Silk MJ. 2024 Not your private tête-à-tête: leveraging the power of higher-order networks to study animal communication. Phil. Trans. R. Soc. Lond. B 379, 20230190.
  * (198) Tanaka T, Aoyagi T. 2011 Multistable attractors in a network of phase oscillators with three-body interactions. Phys. Rev. Lett. 106, 224101.
  * (199) Krawiecki A. 2014 Chaotic synchronization on complex hypergraphs. Chaos Solit. Fractals 65, 44–50.
  * (200) Ashwin P, Rodrigues A. 2016 Hopf normal form with SN symmetry and reduction to systems of nonlinearly coupled phase oscillators. Physica D: Nonlinear Phenomena 325, 14–24.
  * (201) Bick C, Ashwin P, Rodrigues A. 2016 Chaos in generically coupled phase oscillator networks with nonpairwise interactions. Chaos 26, 094814.
  * (202) Berge C. 1973 Graphs and hypergraphs. North-Holl Math. Libr. Amsterdam: North-Holland.
  * (203) Aleksandrov P. 1998 Combinatorial topology vol. 1. Courier Corporation.
  * (204) Kuramoto Y, Nakao H. 2019 On the concept of dynamical reduction: the case of coupled oscillators. Phil. Trans. R. Soc. A 377, 20190041.
  * (205) Kuramoto Y. 1984 Chemical oscillations, waves, and turbulence. Springer-Verlag, New York.
  * (206) Kori H, Kuramoto Y, Jain S, Kiss IZ, Hudson JL. 2014 Clustering in globally coupled oscillators near a Hopf bifurcation: theory and experiments. Phys. Rev. E 89, 062906.
  * (207) Contemori S, Di Patti F, Fanelli D, Miele F. 2016 Multiple-scale theory of topology-driven patterns on directed networks. Phys. Rev. E 93, 032317.
  * (208) León I, Pazó D. 2019 Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation. Phys. Rev. E 100, 012211.
  * (209) Nakao H. 2016 Phase reduction approach to synchronisation of nonlinear oscillators. Contemp. Phys. 57, 188–214.
  * (210) Schaub M, Benson A, Horn P, Lippner G, Jadbabaie A. 2020 Random walks on simplicial complexes and the normalized Hodge 1-Laplacian. SIAM Rev. 62, 353–391.
  * (211) Carletti T, Battiston F, Cencetti G, Fanelli D. 2020 Random walks on hypergraphs. Phys. Rev. E 101, 022308.
  * (212) Carletti T, Fanelli D, Lambiotte R. 2021 Random walks and community detection in hypergraphs. J. Phys. Complex. 2, 015011.
  * (213) Neuhäuser L, Mellor A, Lambiotte R. 2020 Multibody interactions and nonlinear consensus dynamics on networked systems. Phys. Rev. E 101, 032310.
  * (214) Neuhäuser L, Lambiotte R, Schaub M. 2021 Consensus dynamics on temporal hypergraphs. Phys. Rev. E 104, 064305.
  * (215) Skardal P, Arenas A. 2019 Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes. Phys. Rev. Lett. 122, 248301.
  * (216) Lucas M, Cencetti G, Battiston F. 2020 Multiorder Laplacian for synchronization in higher-order networks. Phys Rev. Res. 2, 033410.
  * (217) León I, Muolo R, Hata S, Nakao H. 2023 Higher-order interactions induce anomalous transitions to synchrony. Chaos 34, 013105.
  * (218) Kuehn C, Bick C. 2021 A universal route to explosive phenomena. Sci. Adv. 7, eabe3824.
  * (219) Skardal P, Arenas A. 2020 Higher-order interactions in complex networks of phase oscillators promote abrupt synchronization switching. Comm. Phys. 3.
  * (220) Gambuzza L, Di Patti F, Gallo L, Lepri S, Romance M, Criado R, Frasca M, Latora V, Boccaletti S. 2021 Stability of synchronization in simplicial complexes. Nat. Comm. 12, 1–13.
  * (221) de Arruda GF, Tizzani M, Moreno Y. 2021 Phase transitions and stability of dynamical processes on hypergraphs. Comm. Phys. 4, 1–9.
  * (222) St-Onge G, Sun H, Allard A, Hébert-Dufresne L, Bianconi G. 2021 Universal Nonlinear Infection Kernel from Heterogeneous Exposure on Higher-Order Networks. Phys. Rev. Lett. 127, 158301.
  * (223) Iacopini I, Petri G, Barrat A, Latora V. 2019 Simplicial models of social contagion. Nat. Comm. 10, 2485.
  * (224) de Arruda G, Petri G, Moreno Y. 2020 Social contagion models on hypergraphs. Phys. Rev. Res. 2, 023032.
  * (225) Carletti T, Fanelli D, Nicoletti S. 2020 Dynamical systems on hypergraphs. J. Phys. Complex. 1, 035006.
  * (226) Gao S, Chang L, Perc M, Wang Z. 2023 Turing patterns in simplicial complexes. Phys. Rev. E 107, 014216.
  * (227) Muolo R, Gallo L, Latora V, Frasca M, Carletti T. 2023 Turing patterns in systems with high-order interaction. Chaos Solit. Fractals 166, 112912.
  * (228) Wang W, Liu S. 2023 Spatio-temporal patterns of non-autonomous systems on hypergraphs: Turing and Benjamin–Feir mechanisms. New J. Phys. 25, 023008.
  * (229) Sun H, Radicchi F, Kurths J, Bianconi G. 2023 The dynamic nature of percolation on networks with triadic interactions. Nature Communications 14, 1308.
  * (230) Millán AP, Sun H, Torres JJ, Bianconi G. 2023 Triadic percolation induces dynamical topological patterns in higher-order networks. arXiv preprint arXiv:2311.14877.
  * (231) Battiston F, Cencetti G, Iacopini I, Latora V, Lucas M, Patania A, Young JG, Petri G. 2020 Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92.
  * (232) Battiston F, Amico E, Barrat A, Bianconi G, de Arruda G, Franceschiello B, Iacopini I, Kéfi S, Latora V, Moreno Y, Murray M, Peixoto T, Vaccarino F, Petri G. 2021 The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098.
  * (233) Boccaletti S, De Lellis P, Del Genio C, Alfaro-Bittner K, Criado R, Jalan S, Romance M. 2023 The structure and dynamics of networks with higher order interactions. Phys. Rep. 1018, 1–64.
  * (234) Gao Z, Ghosh D, Harrington H, Restrepo J, Taylor D. 2023 Dynamics on networks with higher-order interactions. Chaos 33, 040401.
  * (235) Bick C, Gross E, Harrington HA, Schaub MT. 2023 What are higher-order networks?. SIAM Rev. 65, 686–731.
  * (236) Bianconi G. 2021 Higher-Order Networks:An introduction to simplicial complexes. Cambridge University Press.
  * (237) Battiston F, Petri G. 2022 Higher-Order Systems. Springer.
  * (238) Nakahara M. 2003 Geometry, topology and physics. CRC Press.
  * (239) Lek-Heng L. 2020 Hodge Laplacians on Graphs. SIAM Rev. 62, 685–715.
  * (240) Linne ML, Aćimović J, Saudargiene A, Manninen T. 2022 Neuron-Glia Interactions and Brain Circuits. In Computational Modelling of the Brain pp. 87–103. Springer.
  * (241) Faskowitz J, Betzel R, Sporns O. 2022 Edges in brain networks: Contributions to models of structure and function. Netw. Neurosci. 6, 1–28.
  * (242) Santoro A, Battiston F, Petri G, Amico E. 2023 Higher-order organization of multivariate time series. Nat. Phys. 19, 221–229.
  * (243) Katifori E, Szöllősi G, Magnasco M. 2010 Damage and fluctuations induce loops in optimal transport networks. Phys. Rev. Lett. 104, 048704.
  * (244) Rocks J, Liu A, Katifori E. 2021 Hidden topological structure of flow network functionality. Phys. Rev. Lett. 126, 028102.
  * (245) Witthaut D, Hellmann F, Kurths J, Kettemann S, Meyer-Ortmanns H, Timme M. 2022 Collective nonlinear dynamics and self-organization in decentralized power grids. Rev. Mod. Phys. 94, 015005.
  * (246) Barbarossa S, Sardellitti S. 2020 Topological signal processing over simplicial complexes. IEEE Trans. Signal Process. 68, 2992–3007.
  * (247) Sardellitti S, Barbarossa S. 2021 Topological signal representation and processing over cell complexes. In 2021 55th Asilomar Conference on Signals, Systems, and Computers pp. 1558–1562.
  * (248) Schaub M, Segarra S. 2018 Flow smoothing and denoising: Graph signal processing in the edge-space. In 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP) pp. 735–739. IEEE.
  * (249) Schaub M, Zhu Y, Seby JB, Roddenberry T, Segarra S. 2021 Signal processing on higher-order networks: Livin’on the edge… and beyond. Signal Process. 187, 108149.
  * (250) Millán A, Torres J, Bianconi G. 2020 Explosive higher-order Kuramoto dynamics on simplicial complexes. Phys. Rev. Lett. 124, 218301.
  * (251) Millán A, Restrepo J, Torres J, Bianconi G. 2022 Geometry, topology and simplicial synchronization. In Higher-Order Systems pp. 269–299. Springer.
  * (252) Torres J, Bianconi G. 2020 Simplicial complexes: higher-order spectral dimension and dynamics. J. Phys. Complex. 1, 015002.
  * (253) Ghorbanchian R, Restrepo J, Torres J, Bianconi G. 2021 Higher-order simplicial synchronization of coupled topological signals. Comm. Phy. 4, 1–13.
  * (254) Calmon L, Restrepo JG, Torres JJ, Bianconi G. 2022 Dirac synchronization is rhythmic and explosive. Comm. Phys. 5, 1–17.
  * (255) Arnaudon A, Peach RL, Petri G, Expert P. 2022 Connecting Hodge and Sakaguchi-Kuramoto through a mathematical framework for coupled oscillators on simplicial complexes. Comm. Phys. 5, 211.
  * (256) DeVille L. 2021 Consensus on simplicial complexes: Results on stability and synchronization. Chaos 31, 023137.
  * (257) Reitz M, Bianconi G. 2020 The higher-order spectrum of simplicial complexes: a renormalization group approach. J. Phys. A 53, 295001.
  * (258) Ziegler C, Skardal P, Dutta H, Taylor D. 2022 Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes. Chaos 32, 023128.
  * (259) Bodnar C, Frasca F, Wang Y, Otter N, Montufar GF, Lio P, Bronstein M. 2021 Weisfeiler and lehman go topological: Message passing simplicial networks. In International Conference on Machine Learning pp. 1026–1037. PMLR.
  * (260) Roddenberry T, Segarra S. 2019 HodgeNet: Graph neural networks for edge data. In 2019 53rd Asilomar Conference on Signals, Systems, and Computers pp. 220–224. IEEE.
  * (261) Hajij M, Istvan K, Zamzmi G. 2020 Cell complex neural networks. In NeurIPS 2020.
  * (262) Skardal P, Arenas A. 2020 Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching. Comm. Phys. 3, 1–6.
  * (263) Kovalenko K, Dai X, Alfaro-Bittner K, Raigorodskii A, Perc M, Boccaletti S. 2021 Contrarians synchronize beyond the limit of pairwise interactions. Phys. Rev. Lett. 127, 258301.
  * (264) Alvarez-Rodriguez U, Battiston F, de Arruda G, Moreno Y, Perc M, Latora V. 2021 Evolutionary dynamics of higher-order interactions in social networks. Nat. Hum. Behav. 5, 586–595.
  * (265) Lee Y, Lee J, Oh S, Lee D, Kahng B. 2021 Homological percolation transitions in growing simplicial complexes. Chaos 31, 041102.
  * (266) Tang Y, Shi D, Lü L. 2022 Optimizing higher-order network topology for synchronization of coupled phase oscillators. Comm. Phys. 5, 1–12.
  * (267) Zhang Y, Latora V, Motter A. 2021 Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions. Comm. Phys. 4, 1–9.
  * (268) Chutani M, Tadić B, Gupte N. 2021 Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with competing simplex-encoded couplings. Phys. Rev. E 104, 034206.
  * (269) Horak D, Jost J. 2013 Spectra of combinatorial Laplace operators on simplicial complexes. Adv. Math. 244, 303–336.
  * (270) Bianconi G. 2021 The topological Dirac equation of networks and simplicial complexes. J. Phys. Complex. 2, 035022.
  * (271) Lloyd S, Garnerone S, Zanardi P. 2016 Quantum algorithms for topological and geometric analysis of data. Nat. Comm. 7, 1–7.
  * (272) Ameneyro B, Siopsis G, Maroulas V. 2022 Quantum persistent homology for time series. In 2022 IEEE/ACM 7th Symposium on Edge Computing (SEC) pp. 387–392. IEEE.
  * (273) Knill O. 2013 The Dirac operator of a graph. arXiv preprint arXiv:1306.2166.
  * (274) Giambagli L, Calmon M, Muolo R, Carletti T, Bianconi G. 2022 Diffusion-driven instability of topological signals coupled by the Dirac operator. Phys. Rev. E 106.
  * (275) Muolo R, Carletti T, Bianconi G. 2024 The three way Dirac operator and dynamical Turing and Dirac induced patterns on nodes and links. Chaos Solit. Fractals 178, 114312.
  * (276) Giambagli L. 2024 A Journey Through Reciprocal Space: from Deep Spectral Learning to Topological Signals. PhD Thesis, University of Florence and University of Namur.
  * (277) Miranda M, Estrada-Rodriguez G, Estrada E. 2023 What Is in a Simplicial Complex? A Metaplex-Based Approach to Its Structure and Dynamics. Entropy 25, 1599–1599.
  * (278) Carletti T, Giambagli L, Bianconi G. 2023 Global topological synchronization on simplicial and cell complexes. Phys. Rev. Lett. 130, 187401.
  * (279) Baccini F, Geraci F, Bianconi G. 2022 Weighted simplicial complexes and their representation power of higher-order network data and topology. Phsy. Rev. E 106, 034319.
  * (280) Wang R, Muolo R, Carletti T, Bianconi G. 2024 Global topological synchronization of weighted simplicial complexes. to appear in Phys. Rev. E.
  * (281) Gallo L, Muolo R, Gambuzza L, Latora V, Frasca M, Carletti T. 2022 Synchronization induced by directed higher-order interactions. Comm. Phys. 5, 236.
  * (282) Della Rossa F, Liuzza D, Iudice FL, De Lellis P. 2023 Emergence and control of synchronization in networks with directed many-body interactions. Phys. Rev. Lett. 131, 207401.
  * (283) Von Der Gracht S, Nijholt E, Rink B. 2023 Hypernetworks: Cluster Synchronization Is a Higher-Order Effect. SIAM J. Appl. Math. 83, 2329–2353.
  * (284) Gong X, Higham D, Zygalakis KC, Bianconi G. 2024 Higher-order Connection Laplacians for Directed Simplicial Complexes. J. Phys. Complex.
  * (285) De Lellis P, Della Rossa F, Iudice FL, Liuzza D. 2022 Pinning control of hypergraphs. IEEE Control Syst. Lett. 7, 691–696.
  * (286) Li X, He R, Xi Y, Xue Y, Wang Y, Luo X. 2024 The increasing strength of higher-order interactions may homogenize the distribution of infections in Turing patterns. Chaos Solit. Fractals 178, 114369.
  * (287) Ardizzone V, Lewandowski P, Luk MH, Tse YC, Kwong NH, Lücke A, Abbarchi M, Baudin E, Galopin E, Bloch J et al.. 2013 Formation and control of Turing patterns in a coherent quantum fluid. Sci. Rep. 3, 3016.
  * (288) Krause AL, Ellis MA, Van Gorder RA. 2019 Influence of curvature, growth, and anisotropy on the evolution of Turing patterns on growing manifolds. Bull. Math. Biol. 81, 759–799.
  * (289) Sánchez-Garduno F, Krause AL, Castillo JA, Padilla P. 2019 Turing–Hopf patterns on growing domains: the torus and the sphere. J. Theor. Biol. 481, 136–150.
  * (290) Rogov K, Pogromsky A, Steur E, Michiels W, Nijmeijer H. 2018 Pattern prediction in networks of diffusively coupled nonlinear systems. IFAC-PapersOnLine 51, 62–67.
  * (291) Rogov K, Pogromsky A, Steur E, Michiels W, Nijmeijer H. 2019 Pattern analysis in networks of diffusively coupled Lur’e systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 29, 1950200.
  * (292) Chang X, Cai CR, Wang CY, Liu XS, Zhang JQ, Jin K, Yang WL. 2023 Combined effect of simplicial complexes and interlayer interaction: An example of information-epidemic dynamics on multiplex networks. Phys. Rev. Res. 5, 013196.
  * (293) Krishnagopal S, Bianconi G. 2023 Topology and dynamics of higher-order multiplex networks. Chaos Solit. Fractals 177, 114296.
  * (294) Ye Y, Zhou J, Zhao Y. 2023 Pattern formation in reaction-diffusion information propagation model on multiplex simplicial complexes. preprint https://www.researchsquare.com/article/rs-3024570/v1.


# A remark on the Hölder regularity of solutions to the complex Hessian
equation

Sławomir Kołodziej and Ngoc Cuong Nguyen  Faculty of Mathematics and Computer
Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
<EMAIL_ADDRESS>Department of Mathematical Sciences, KAIST,
291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea<EMAIL_ADDRESS>
###### Abstract.

We prove that the Dirichlet problem for the complex Hessian equation has the
Hölder continuous solution provided it has a subsolution with this property.
Compared to the previous result of Benali-Zeriahi and Charabati-Zeriahi we
remove the assumption on the finite total mass of the measure on the right
hand side.

##  1\. Introduction

The theory of weak solutions of the complex Monge-Ampère equations started
with the fundamental works of Bedford and Taylor [2, 3]. It has found numerous
profound applications in complex geometry (see e.g. the survey [31]). The
Hölder continuous solutions are interesting because such functions often
appear in complex dynamic in higher dimension as potentials of Green currents
associated to dominating holomorphic maps [16] and [32], as well as in
approximation theory [29].

Thanks to a series of works [1], [2], [8, 9], [18], [27, 28], [30] and [34]
the Hölder regularity of the solution to the complex Monge-Ampère equation is
rather well-understood. Notably, it was shown in [1] that if the measure on
the right hand side has a density in $L^{p}$, $p>1$ and the boundary data is
Hölder continuous, then the solution is Hölder continuous. Such a regularity
was also proved for the volume form of a generic smooth compact submanifold in
$\mathbb{C}^{n}$ in [30] and [34]. These measures are singular with respect to
the Lebesgue measure. A necessary and sufficient condition for the existence
of a Hölder continuous solution via Hölder continuous subsolution was proved
in [27, 28], which was inspired by [19].

Recently, potential theory has been developed very successfully for the study
of weak solutions to complex Hessian equations (of which the Monge-Ampère
equation is a special case) both in domains in $\mathbb{C}^{n}$ and on compact
Kähler manifolds by Błocki [6], Dinew and the first author [13, 14], and Lu
[23, 24, 25]. The study of Hölder regularity of $m$-subharmonic solutions to
the complex Hessian equation is recently very active [4], [7], [10], [17],
[21], [22], [26] and [33]. Although the statements are often similar to the
ones for the complex Monge-Ampère equation, the proofs are substantially
different and usually the Hölder exponent of the solutions is probably far
from optimal.

We focus now on the Hölder regularity of $m$-subharmonic solutions to the
complex Hessian equation assuming the existence of a Hölder continuous
subsolution. Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$. Let $1\leq
m<n$ be an integer. Let us denote by $\beta=dd^{c}|z|^{2}$ the standard Kähler
form and $SH_{m}(\Omega)$ the set of all $m$-subharmonic functions in $\Omega$
defined in [6]. For $v\in SH_{m}(\Omega)\cap L^{\infty}(\Omega)$ its complex
Hessian measure is given by

$H_{m}(v):=(dd^{c}v)^{m}\wedge\beta^{n-m}.$

Let $\varphi\in SH_{m}(\Omega)\cap C^{0,\alpha}(\bar{\Omega})$ with
$0<\alpha\leq 1$ and let $\mu$ be a positive measure in $\Omega$ satisfying

(1.1) $\mu\leq H_{m}(\varphi).$

To study the Dirichlet problem we assume that $\Omega$ is a bounded strictly
$m$\- pseudoconvex domain in $\mathbb{C}^{n}$. Let $\rho\in SH_{m}(\Omega)\cap
C^{2}(\bar{\Omega})$ be a strictly $m$-subharmonic defining function of
$\Omega$. By adding $\varphi$ to an $m$-subharmonic envelope whose boundary
value is equal to $-\varphi$ as in [26, Theorem 4.7], we obtain a new Hölder
continuous subsolution (with possibly smaller Hölder exponent) with zero
boundary value. Therefore, without loss of generality we may assume that

(1.2) $\varphi=0\quad\text{on }\partial\Omega.$

Given the boundary data $\phi\in C^{0,\alpha}(\partial\Omega)$ we wish to
solve the Hessian equation

(1.3) $u\in SH_{m}(\Omega)\cap C^{0}(\bar{\Omega}),\quad
H_{m}(u)=\mu,\quad\text{and }u=\phi\quad\text{on }\partial\Omega.$

The main result of the paper is as follows.

######  Theorem 1.1.

Let $\mu$ be the measure satisfying (1.1). Then, the Dirichlet problem (1.3)
is solvable and there exists $0<\alpha^{\prime}\leq 1$ such that

(1.4) $u\in C^{0,\alpha^{\prime}}(\bar{\Omega}).$

This result was proved in [21] for a compactly supported measure $\mu$ in
$\Omega$. Later, Benali-Zeriahi [4, Theorem B] (in a special case) and
Charabati and Zeriahi [10, Theorem 4] solved the Dirichlet problem (1.3) $\&$
(1.4) for any $\mu$ having finite total mass, i.e., $\mu(\Omega)<+\infty$.
More precisely, it was a consequence of a general statement [10, Theorem 3]
for the modulus of continuity under such an assumption. In other words, they
obtained the Hölder continuous subsolution theorem for finite mass measures.
Here, we are able to remove the finite mass assumption. The method to prove
the theorem is inspired by [4] and [21]. As in [21] we do not invoke neither
the viscosity solutions nor the rooftop envelopes. However, the Hölder
exponent of the solution is worse than the one in [4] and [10] (see also [5]).

In general the complex Hessian measure of a Hölder continuous $m$-subharmonic
function may have unbounded total mass. Thus, for applications it is desirable
to prove the Hölder continuous subsolution theorem for possibly unbounded
total mass measures. Let us consider specific examples.

######  Example 1.2.

Let $\Omega$ be a strictly $m$-pseuconvex domain in $\mathbb{C}^{n}$ and
$\rho$ is a $C^{2}$ strictly $m$-subharmonic defining function of $\Omega$ as
above.

  * (a)

Consider the function $-(-\rho)^{\alpha}$ for some $0<\alpha<1$. Then, it is
Hölder continuous on $\bar{\Omega}$ and $m$-subharmonic in $\Omega$. It is
easy to see that for any $x\in\partial\Omega$ and $U_{x}$ a neighborhood of
$x$,

$\int_{\Omega\cap U_{x}}H_{m}(-|\rho|^{\alpha})=+\infty.$

It follows that the measure $\mu={\bf 1}_{U_{x}}\cdot H_{m}(-|\rho|^{\alpha})$
has unbounded mass on $\Omega$ and $\mu\leq H_{m}(-|\rho|^{\alpha})$.

  * (b)

Take $G\subset\Omega$ be such that $\mu:={\bf 1}_{G}\cdot
H_{m}(-|\rho|^{\alpha})$ satisfies $\mu(\Omega)<+\infty$. Then, it is not
clear how to find $\varphi\in\mathcal{E}^{0}_{m}(\Omega)\cap
C^{0,\alpha}(\bar{\Omega})$ such that $\mu\leq H_{m}(\varphi)$, where
$\mathcal{E}^{0}_{m}(\Omega)$ is the Cegrell class of bounded $m$-subharmonic
functions whose boundary values are zero and whose total masses of complex
Hessian measures are finite.

######  Example 1.3.

Take $\varphi\in SH_{m}(\Omega)\cap C^{0,\alpha}(\bar{\Omega})$ such that
$\varphi\neq 0$ on $\partial\Omega$. Assume $\mu\leq H_{m}(\varphi)$ in
$\Omega$ and $\int_{\Omega}H_{m}(\varphi)<+\infty$. Then, it is not clear if
one can find $\varphi^{\prime}$ Hölder continuous $m$-subharmonic function
such that $\varphi^{\prime}=0$ on $\partial\Omega$ and
$\int_{\Omega}H_{m}(\varphi^{\prime})<+\infty$.

The Hölder regularity of solutions to complex Hessian equations or fully non-
linear elliptic equations on compact Kähler manifold is also very active field
of research see e.g., [11], [12], [15], [20] and [35]. It is worth to point
out that the Hölder regularity of quasi $m$-subharmonic solutions to the
equation on a compact Kähler manifold, under similar hypothesis as the ones
mentioned above, is widely open. There is a partial progress made by Cheng and
Yu [11] in case nonnegative holomorphic bisectional curvature of the metric.

Acknowledgement. The paper has been completed while the second author was
visiting VIASM (Vietnam Institute for Advanced Study in Mathematics). He would
like thank to the institute for the great hospitality and excellent working
conditions. The first author is partially supported by grant no.
2021/41/B/ST1/01632 from the National Science Center, Poland.

##  2\. Hessian measures dominated by capacity

In this section we consider the positive Radon measure $\mu$ which is
dominated by the complex Hessian measure associated to a Hölder continuous
$m$-subharmonic function $\varphi$ satisfying (1.1) and (1.2). Our goal is to
prove that there exist uniform constants $C,\alpha_{0}>0$ depending only on
$\varphi$ and $\Omega$ such that for every compact set $K\subset\Omega$,

$\mu(K)\leq C\left[cap_{m}(K,\Omega)\right]^{1+\alpha_{0}},$

where the $m$-capacity defined as in [13], namely

$cap_{m}(K,\Omega)=\sup\left\\{\int_{K}H_{m}(w):w\in SH_{m}(\Omega),\;-1\leq
w\leq 0\right\\}.$

This inequality was proved under the assumption of the compact support of the
measure in [21]. Later it was proved without this extra hypothesis by Benali
and Zeriahi [4] and a more general formulation was given by Charabati and
Zeriahi [10], where they employed the viscosity solution. Here we give an
alternative proof with a smaller exponent but it does not invoke neither
viscosity solutions nor the rooftop envelope. This is done by slightly
extending the Hölder continuous subsolution to a neighborhood of
$\bar{\Omega}$ using the defining function $\rho$. The cost is the growth of
the Hölder norm of the new subsolution and we need to carefully analyze it.

Since $\rho$ is a $C^{2}$ defining function of $\Omega$, we can assume that
$\rho$ is a strictly $m$-subharmonic function in a neighborhood
$\widetilde{\Omega}$ of $\bar{\Omega}$. Let us define for $\varepsilon>0$ the
sets

$\Omega_{\varepsilon}=\\{z\in\Omega:{\rm
dist}(z,\partial\Omega)>\varepsilon\\},$

and

$D_{\varepsilon}=\\{z\in\Omega:\rho(z)<-\varepsilon\\}.$

These two domains are comparable as, by the Hopf lemma, there exist constants
$c_{0},\varepsilon_{0}>0$ such that

(2.1) $\Omega_{\varepsilon}\subset D_{c_{0}\varepsilon}\quad\text{for every
}0<\varepsilon\leq\varepsilon_{0}.$

Let us consider the following extension of the subsolution $\varphi$

(2.2)
$\psi:=\max\left\\{\varphi-\varepsilon,\frac{M\rho}{c_{0}\varepsilon}\right\\},$

where $M=1+\|\varphi\|_{L^{\infty}(\Omega)}$. Clearly, $\psi$ is defined on
$\widetilde{\Omega}$. Furthermore, $\psi=\varphi-\varepsilon$ on
$\Omega_{\varepsilon}$ and $\psi=M\rho/(c_{0}\varepsilon)$ near
$\partial\Omega$. Thus, $\psi$ is $\alpha$-Hölder continuous and
$m$-subharmonic in the neighborhood $\widetilde{\Omega}$ of $\bar{\Omega}$ and

(2.3) $H_{m}(\varphi)=H_{m}(\psi)\quad\text{on }\Omega_{\varepsilon}.$

Let us fix an $\varepsilon_{0}>0$ depending only on $\Omega$ such that for
every $0<\varepsilon\leq\varepsilon_{0}$,

(2.4) $\bar{\Omega}\subset\\{z\in\widetilde{\Omega}:{\rm
dist}(z,\partial\widetilde{\Omega})>\varepsilon\\}.$

As $\varphi-\varepsilon\leq\psi\leq 0$,

(2.5) $\|\psi\|_{L^{\infty}(\bar{\Omega})}\leq M.$

Now we are ready to give a slightly different proof of the following result
due to Benali and Zeriahi [4].

######  Theorem 2.1.

There exist constants $A_{1}>0$ and $\alpha_{0}>0$ such that for any compact
set $K\subset\Omega$

$\int_{K}H_{m}(\varphi)\leq A_{1}\left[cap_{m}(K)\right]^{1+\alpha_{0}}.$

Precisely,
$A_{1}=C\|\varphi\|_{L^{\infty}(\Omega)}^{m}\|\varphi\|_{C^{0,\alpha}(\bar{\Omega})}^{m}$
with $C$ a uniform constant depending only on $\Omega$ and $\alpha_{0}$ which
is explicitly computable in terms of $\alpha,m$ and $n$.

Following the idea in [4] we divide a compact $K$ into two parts
$K\cap\Omega_{\varepsilon}$ and $K\setminus\Omega_{\varepsilon}$. To deal with
the first part we need

######  Lemma 2.2.

Fix $1<p<n/m$ and $0<\varepsilon\leq\varepsilon_{0}$, where $\varepsilon_{0}$
is given in (2.1). Let $E\subset\Omega_{2\varepsilon}$ be a Borel set. Then,

$\int_{E}H_{m}(\varphi)\leq
A_{2}\left(\varepsilon^{\alpha}+\varepsilon^{-2m}[cap_{m}(E)]^{p-1}\right)cap_{m}(E),$

where
$A_{2}=C\|\varphi\|_{L^{\infty}(\Omega)}^{m}\|\varphi\|_{C^{0,\alpha}(\bar{\Omega})}$.

###### Proof.

Without loss of generality we may assume that $E$ is a regular compact set.
This means that the relative extremal function

$u_{E}(z)=\sup\\{v(z):v\in SH_{m}(\Omega):v\leq 0,\;v\leq-1\text{ on }E\\}$

is a continuous $m$-subharmonic function on $\bar{\Omega}$. For the general
$E$ one can use the fact that $\mu$ is a Radon measure and $m$-capacity enjoys
the inner regularity property. Recall that $\psi$ is the extension defined in
(2.2). Let us denote by $\psi_{\varepsilon}=\psi*\chi_{\varepsilon}$, where
$\\{\chi_{\varepsilon}\\}_{\varepsilon>0}$ is the radial smoothing kernels,
the standard regularization of $\psi$ in $\widetilde{\Omega}$ for
$0<\varepsilon\leq\varepsilon_{0}$. We have by (2.4) that the domain of
$\psi_{\varepsilon}$ contains $\bar{\Omega}$. Hence, $\psi_{\varepsilon}\in
SH_{m}(\Omega)\cap C^{\infty}(\bar{\Omega})$. Using the Hölder continuity of
$\psi=\varphi-\varepsilon$ we have

$0\leq\psi_{\varepsilon}-\psi\leq
c_{1}\varepsilon^{\alpha}:=\delta\quad\text{on }\bar{\Omega}_{2\varepsilon},$

where $c_{1}=\|\varphi\|_{C^{0,\alpha}(\bar{\Omega})}$. The compactness of $E$
gives $u_{E}=0$ on $\partial\Omega$. Also $\psi_{\varepsilon}\geq\psi$ on
$\bar{\Omega}$ by the subharmonicity of $\psi$. Consider the set
$E^{\prime}:=\\{3\delta u_{E}+\psi_{\varepsilon}<\psi-2\delta\\}$. Then,

$E\subset E^{\prime}\subset\\{u_{E}<-1/2\\}\subset\subset\Omega,$

where we used the fact $E\subset\Omega_{2\varepsilon}$ for the first inclusion
and $\psi_{\varepsilon}\geq\psi$ for the second one. In what follows we write
for any Borel set $G\subset\Omega$,

(2.6) $cap_{m}(G):=cap_{m}(G,\Omega)$

Clearly, $cap_{m}(E)\leq cap_{m}(E^{\prime})$. On the other hand, by the
comparison principle [13, Theorem 1.4],

(2.7) $\displaystyle cap_{m}(E^{\prime})$ $\displaystyle\leq
cap_{m}\left(\\{u_{E}<-1/2\\}\right)$ $\displaystyle\leq
2^{m}\int_{\\{u_{E}<0\\}}(dd^{c}u_{E})^{m}\wedge\beta^{n-m}$
$\displaystyle=2^{m}cap_{m}(E).$

Next, notice from (2.5) that on $\bar{\Omega}$,

$dd^{c}\psi_{\varepsilon}\leq\frac{C\|\psi\|_{L^{\infty}(\Omega)}\beta}{\varepsilon^{2}}\leq\frac{CM\beta}{\varepsilon^{2}},\quad\|\psi_{\varepsilon}+u_{E}\|_{L^{\infty}(\Omega)}\leq
M.$

Since $E\subset\Omega_{\varepsilon}$,
$\int_{E}H_{m}(\varphi)=\int_{E}H_{m}(\psi)$. The comparison principle and the
above inequalities give us that

$\displaystyle\int_{E^{\prime}}H_{m}(\psi)$
$\displaystyle\leq\int_{E^{\prime}}H_{m}(3\delta u_{E}+\psi_{\varepsilon})$
$\displaystyle\leq\int_{E^{\prime}}3\delta
H_{m}(u_{E}+\psi_{\varepsilon})+\int_{E^{\prime}}H_{m}(\psi_{\varepsilon})$
$\displaystyle\leq 3\delta
M^{m}cap_{m}(E^{\prime})+CM^{m}\varepsilon^{-2m}\left[cap_{m}(E^{\prime})\right]^{p},$

where we also used [13, Proposition 2.1] for the last inequality with
$1<p<n/m$. Combining with the equivalence (2.7) of $cap_{m}(E)$ and
$cap_{m}(E^{\prime})$ we get that for every Borel set
$E\subset\Omega_{\varepsilon}$,

(2.8) $\int_{E}H_{m}(\varphi)\leq
CM^{m}\left(c_{1}\varepsilon^{\alpha}+\varepsilon^{-2m}[cap_{m}(E)]^{p-1}\right)cap_{m}(E).$

Since we may assume $c_{1}=\|\varphi\|_{C^{0,\alpha}(\bar{\Omega})}\geq 1$,
the desired inequality follows. ∎

The next lemma is due to Benali and Zeriahi [4, Lemma 4.1] which takes care of
the second part in the decomposition.

######  Lemma 2.3.

Let $F\subset\Omega\setminus\Omega_{2\varepsilon}$ be a compact set. Then,

(2.9) $\int_{F}H_{m}(\varphi)\leq A_{3}\;\varepsilon^{m\alpha}cap_{m}(F),$

where $A_{3}=2^{m\alpha}\|\varphi\|_{C^{0,\alpha}(\bar{\Omega})}^{m}$.

###### Proof.

Without loss of generality we can assume that $F$ is regular in the sense that
the relative extremal function $u_{F}$ with respect to $\Omega$ is continuous
as in the proof of Lemma 2.2. Since $\varphi$ is Hölder continuous on
$\bar{\Omega}$ and $\varphi=0$ on $\partial\Omega$, we have for $z\in F$

$-\varphi(z)\leq c_{1}{\rm
dist}(z,\partial\Omega)^{\alpha}<2^{\alpha}c_{1}\varepsilon^{\alpha}=:\delta,$

where $c_{1}=\|\varphi\|_{C^{0,\alpha}(\bar{\Omega})}$. Consider the set
$F^{\prime}=\\{\delta u_{F}<\varphi\\}$. Since $u_{F}=-1$ on $F$, we have
$F\subset F^{\prime}$. Note that $u_{F}=0$ on $\partial\Omega$, hence
$F^{\prime}\subset\subset\Omega$. By the comparison principle

$\displaystyle\int_{F^{\prime}}H_{m}(\varphi)$
$\displaystyle\leq\int_{F^{\prime}}H_{m}(\delta u_{F})$
$\displaystyle=\delta^{m}\int_{F^{\prime}}H_{m}(u_{F})$
$\displaystyle=\delta^{m}cap_{m}(F).$

The desired inequality follows. ∎

###### End of proof of Theorem 2.1.

Now for any compact set $K\subset\Omega$ we write
$K=(K\cap\Omega_{\varepsilon})\cup(K\setminus\Omega_{\varepsilon}).$ Let us
denote by $E$ and $F$ the two sets of this union. Then, both of them have
$m$-capacity less than $cap_{m}(K)$. It follows from (2.8) and (2.9) that

(2.10) $\displaystyle\int_{K}H_{m}(\varphi)$
$\displaystyle=\int_{E}H_{m}(\varphi)+\int_{F}H_{m}(\varphi)$
$\displaystyle\leq
A_{2}\left(\varepsilon^{\alpha}+\varepsilon^{-2m}\;[cap_{m}(E)]^{p-1}\right)cap_{m}(E)+A_{3}\;\varepsilon^{m\alpha}cap_{m}(F)$
$\displaystyle\leq
A_{4}\left(\varepsilon^{\alpha}+\varepsilon^{-2m}[cap_{m}(K)]^{p-1}+\varepsilon^{m\alpha}\right)cap_{m}(K),$

where $A_{4}=\max\\{A_{2},A_{3}\\}$. Notice that this inequality holds for
every $0<\varepsilon\leq\varepsilon_{0}$, where $\varepsilon_{0}$ is fixed and
defined in (2.4).

The proof of the theorem will follow by choosing a suitable $\varepsilon>0$.
To this end we define $\alpha_{1}>0$ by

$p-1-2m\alpha_{1}=\alpha_{1}\alpha\Leftrightarrow\alpha_{1}=\frac{p-1}{2m+\alpha}.$

If $[cap_{m}(K)]^{\alpha_{1}}\geq\varepsilon_{0}$, then by (2.10)

$\displaystyle\int_{K}H_{m}(\varphi)$ $\displaystyle\leq
A_{4}\left(\varepsilon^{\alpha}_{0}+\varepsilon^{-2m}_{0}cap_{m}(K)^{p-1}+\varepsilon^{m\alpha}_{0}\right)cap_{m}(K)$
$\displaystyle\leq
A_{4}\left(\left[cap_{m}(K)\right]^{1+\alpha_{1}\alpha}+\varepsilon_{0}^{-2m}[cap_{m}(K)]^{p}+[cap_{m}(K)]^{1+m\alpha_{1}\alpha}\right).$

So, we get the desired inequality in this case. Otherwise, if
$[cap_{m}(K)]^{\alpha_{1}}<\varepsilon_{0}$, then we can choose
$\varepsilon=[cap_{m}(K)]^{\alpha_{1}}$. It follows from (2.10) that
$\int_{K}H_{m}(\varphi)$ is less than

$A_{4}\left(\left[cap_{m}(K)\right]^{1+\alpha_{1}\alpha}+\left[cap_{m}(K)\right]^{p-2m\alpha_{1}}+[cap_{m}(K)]^{1+m\alpha_{1}\alpha}\right).$

Therefore, the inequality

$\int_{K}H_{m}(\varphi)\leq A_{1}\left[cap_{m}(K)\right]^{1+\alpha_{0}}$

holds for $\alpha_{0}=\alpha_{1}\alpha>0$ and $A_{1}=3A_{4}$. The proof of the
theorem is completed. ∎

##  3\. The Hölder regularity of the solution

Let us proceed to prove Theorem 1.1. Thanks to the proof in [21, Theorem 1.2]
and Theorem 2.1 we can find $u\in SH_{m}(\Omega)\cap C^{0}(\bar{\Omega})$
satisfying

(3.1) $(dd^{c}u)^{m}\wedge\beta^{n-m}=\mu,\quad u=\phi\quad\text{on
}\partial\Omega.$

The continuous solution has been obtained by Charabati and Zeriahi [10] for a
more general family of measures: those which are dominated by a complex
Hessian measure associated to a continuous $m$-subharmonic function whose
modulus of continuity satisfies the Dini-type condition.

Under the Hölder continuity of the boundary data $\phi\in
C^{0,\alpha}(\partial\Omega)$ we wish to show that

$u\in C^{0,\alpha^{\prime}}(\bar{\Omega})\quad\text{for some
}\alpha^{\prime}>0.$

Here, we may assume that the Hölder exponent of $\phi$ is the same as the one
of $\varphi$, otherwise we can decrease $\alpha$. We will see in the proof
that the exponent $\alpha^{\prime}$ is explicitly computable in terms of
$\alpha,m$ and $n$. We follow closely the argument in [28] which deals with
the complex Monge-Ampère equation. Consider the sequence of sets
$\Omega_{\varepsilon}$ with $\varepsilon>0$ small. Recall that
$M=1+\|\varphi\|_{L^{\infty}(\Omega)}$ and denote

$\widetilde{\varphi}_{\varepsilon}=\max\\{\varphi-\varepsilon,M\rho/\varepsilon\\}.$

This is similar to $\psi$ but we need to keep track of the dependence on
$\varepsilon$ more carefully this time. Notice that
$\widetilde{\varphi}_{\varepsilon}$ is $\alpha-$Hölder continuous and
$m$-subharmonic in a neighborhood of $\bar{\Omega}$, its value on the boundary
$\partial\Omega$ is zero. Moreover, for $\varepsilon>0$ small

(3.2)
$\|\widetilde{\varphi}_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq\frac{A}{\varepsilon},\quad\|\widetilde{\varphi}_{\varepsilon}\|_{C^{0,\alpha}(\bar{\Omega})}\leq\frac{A}{\varepsilon},$

where
$A=M(\|\varphi\|_{C^{0,\alpha}(\bar{\Omega})}+\|\rho\|_{C^{1}(\bar{\Omega})})$.
Next, we can follow the lines of the proofs of [28, Lemma 4.4, Remark 4.5,
Corollary 4.6] to get the following two lemmata with the constant $C>0$
depending only on $\Omega$ and $\rho$.

######  Lemma 3.1.

We have for $k=1,...,m$

$\int_{\Omega}(dd^{c}\widetilde{\varphi}_{\varepsilon})^{k}\wedge\beta^{n-m+k}\leq\frac{CM^{k}}{\varepsilon^{k}}.$

Moreover,

${\bf 1}_{D_{\varepsilon}}\cdot\mu\leq
H_{m}(\widetilde{\varphi}_{\varepsilon})\quad\text{ on }D_{\varepsilon}.$

Since the norms of $\widetilde{\varphi}_{\varepsilon}$ satisfy (3.2) we can
apply Theorem 2.1 for this Hölder continuous subsolution to get

######  Lemma 3.2.

Let $\alpha_{0}$ be defined in Theorem 2.1. Then, for every compact set
$K\subset\Omega$,

$\int_{K}H_{m}(\widetilde{\varphi}_{\varepsilon})\leq\frac{CA^{2m}}{\varepsilon^{2m}}[cap_{m}(K)]^{1+\alpha_{0}}.$

We are in the position to state an analogue of [28, Proposition 4.7] for the
complex Hessian equation.

######  Proposition 3.3.

Let $u$ be the solution in (3.1). Let $v\in SH_{m}\cap L^{\infty}(\Omega)$ be
such that $v=u$ on $\Omega\setminus\Omega_{\varepsilon}$. Then, there is
$0<\alpha_{2}<1$ such that

$\sup_{\Omega}(v-u)\leq\frac{C}{\varepsilon^{2m}}\left(\int_{\Omega}\max\\{v-u,0\\}d\mu\right)^{\alpha_{2}},$

where $C>0$ is a uniform constant independent of $\varepsilon$.

###### Proof.

Without loss of generality we may assume that $\sup_{\Omega}(v-u)>0$. Set

(3.3) $s_{0}:=\inf_{\Omega}(u-v).$

We know that for $0<s<|s_{0}|$,

(3.4) $U(s):=\\{u<v+s_{0}+s\\}\subset\subset\Omega_{\varepsilon}.$

The only difference from [21, Proposition 2.5] is that now the constant
$C/\varepsilon^{m}$ appeared on the right hand side. As shown in [28] for the
Monge-Ampère equation the proof of the proposition will follow once we have
the following inequality.

######  Lemma 3.4.

Let $\alpha_{0}>0$ be as in Theorem 2.1. Then, for every
$0<s,t<\frac{|s_{0}|}{2}$,

(3.5)
$t^{m}cap_{m}(U(s))\leq\frac{C(\alpha_{0})}{\varepsilon^{2m}}\left[cap_{m}(U(s+t))\right]^{1+\alpha_{0}}.$

###### Proof of Lemma 3.4.

Let $0\leq w\leq 1$ be a $m$-subharmonic function in $\Omega$. By the
comparison principle [13, Theorem 1.4],

$\int_{\\{u<v+s_{0}+s+tw\\}}H_{m}(v+tw)\leq\int_{\\{u<v+s_{0}+s+tw\\}}H_{m}(u),$

Since $U(s)\subset\\{u<v+s_{0}+s+tw\\}\subset U(s+t)$, it follows that

$\displaystyle\int_{U(s)}H_{m}(v+tw)$
$\displaystyle\leq\int_{\\{u<v+s_{0}+s+tw\\}}H_{m}(v+tw)$
$\displaystyle\leq\int_{\\{u<v+s_{0}+s+tw\\}}H_{m}(u),$
$\displaystyle\leq\int_{U(s+t)}d\mu.$

As $t^{m}\int_{U(s)}H_{m}(w)\leq\int_{U(s)}H_{m}(v+tw)$, taking supremum over
all such $w$ we get

(3.6) $t^{m}cap_{m}(U(s))\leq\int_{U(s+t)}d\mu.$

On the other hand, by Lemma 3.1

${\bf 1}_{D_{c_{0}\varepsilon}}\cdot d\mu\leq
H_{m}(\widetilde{\varphi}_{c_{0}\varepsilon})\quad\text{in }\Omega$

as two measures, where $c_{0}$ is the constant defined in (2.1). Combining
this with $U(s+t)\subset\Omega_{\varepsilon}\subset D_{c_{0}\varepsilon}$ we
have

(3.7) $\displaystyle\int_{U(s+t)}d\mu$
$\displaystyle\leq\int_{U(s+t)}H_{m}(\widetilde{\varphi}_{c_{0}\varepsilon})$
$\displaystyle\leq\frac{C(\alpha_{0})}{(c_{0}\varepsilon)^{2m}}\left[cap_{m}(U(s+t))\right]^{1+\alpha_{0}},$

where the last inequality follows from Lemma 3.2. Thus, the proof of the lemma
follows from (3.6) and (3.7). ∎

Then, the rest of the proof of the proposition is the same as in [18, Theorem
1.1] with obvious change of notations. ∎

An important step is to derive the $L^{1}-L^{1}$ (with respect to the measure
$\mu$ and the Lebesgue measure) stability estimate of solutions. The one
stated below is an analogue of [28, Theorem 4.9] for the complex Hessian
equation. For the sake of completeness we give the detailed proof.

######  Theorem 3.5.

Let $u$ be the solution in (3.1). Let $v\in SH_{m}\cap L^{\infty}(\Omega)$ be
such that $v=u$ on $\Omega\setminus\Omega_{\varepsilon}$. Then, there is
$0<\alpha_{3}<1$ such that

$\int_{\Omega}|v-u|d\mu\leq\frac{C}{\varepsilon^{m+1}}\left(\int_{\Omega}|v-u|dV_{2n}\right)^{\alpha_{3}},$

where $C>0$ is a uniform constant independent of $\varepsilon$.

###### Proof.

Put

(3.8)
$S_{k,\varepsilon}:=(dd^{c}\widetilde{\varphi}_{\varepsilon})^{k}\wedge\beta^{n-k},\quad
k=0,...,m.$

where
$\widetilde{\varphi}_{\varepsilon}=\max\\{\varphi-\varepsilon,M\rho/\varepsilon\\}$.
Since $\mu\leq S_{m,\varepsilon}$ on $\Omega_{\varepsilon}$, it is enough to
show that there is $0<\tau\leq 1$ satisfying

(3.9)
$\int_{\Omega}(v-u)S_{m,\varepsilon}\leq\frac{C}{\varepsilon^{m+1}}\|v-u\|_{1}^{\tau}.$

for $v\geq u$ on $\Omega$, where here and below we denote by $\|\cdot\|_{1}$
the $L^{1}$-norm with respect to the Lebesgue measure. (In the general case we
use the identity

$|v-u|=(\max\\{v,u\\}-u)+(\max\\{v,u\\}-v)$

and apply twice the inequality (3.9) to get the theorem.)

Now we can use the inductive arguments in $0\leq k\leq m$. Clearly it holds
for $k=0$ with $\alpha_{2}=1$. Assume the inequality holds for $k<m-1$. Let us
denote $\\{\chi_{t}\\}_{t>0}$ the family of standard radial smoothing kernels.
To pass from the $k$-th step to the step number $(k+1)\leq m$ we need the
following inequality (with the notation
$S_{\varepsilon}:=(dd^{c}\widetilde{\varphi}_{\varepsilon})^{k}\wedge\beta^{n-k-1}$)

(3.10)
$\displaystyle\int_{\Omega}(v-u)dd^{c}\widetilde{\varphi}_{\varepsilon}\wedge
S_{\varepsilon}$
$\displaystyle\leq\left|\int_{\Omega}(v-u)dd^{c}\widetilde{\varphi}_{\varepsilon}*\chi_{t}\wedge
S_{\varepsilon}\right|$
$\displaystyle\quad+\left|\int_{\Omega}(v-u)dd^{c}(\widetilde{\varphi}_{\varepsilon}*\chi_{t}-\varphi_{\varepsilon})\wedge
S_{\varepsilon}\right|$ $\displaystyle=:I_{1,\varepsilon}+I_{2,\varepsilon}.$

Notice that $\widetilde{\varphi}_{\varepsilon}$ is well-defined on the
neighborhood $\widetilde{\Omega}$ of $\bar{\Omega}$, so is
$\widetilde{\varphi}_{\varepsilon}*\chi_{t}$ for $t>0$ small. Moreover,
$\|\widetilde{\varphi}_{\varepsilon}\|_{C^{0,\alpha}(\widetilde{\Omega})}\leq
C/\varepsilon.$ Since for $t>0$,

(3.11)
$\left|\frac{\partial^{2}\widetilde{\varphi}_{\varepsilon}*\chi_{t}}{\partial
z_{j}\partial\bar{z}_{k}}(z)\right|\leq\frac{C\|\widetilde{\varphi}_{\varepsilon}\|_{L^{\infty}(\Omega)}}{t^{2}}\leq\frac{C(1+\|\varphi\|_{L^{\infty}(\Omega)})}{t^{2}},$

and by the induction hypothesis at $k$-th step, there exists $0<\tau_{k}\leq
1$ such that

$\int_{\Omega}(v-u)S_{\varepsilon}\wedge\beta\leq\frac{C}{\varepsilon^{k+1}}\|v-u\|_{1}^{\tau_{k}},$

we conclude that

(3.12) $\displaystyle I_{1,\varepsilon}$
$\displaystyle\leq\frac{C(1+\|\varphi\|_{L^{\infty}(\Omega)})}{t^{2}}\int_{\Omega}(v-u)S_{\varepsilon}\wedge\beta$
$\displaystyle\leq\frac{C(1+\|\varphi\|_{L^{\infty}(\Omega)})}{\varepsilon^{k+1}\,t^{2}}\|v-u\|_{1}^{\tau_{k}}.$

By integration by parts, using $u=v$ on $\Omega\setminus\Omega_{\varepsilon}$,
and

$\displaystyle\left|\widetilde{\varphi}_{\varepsilon}*\chi_{t}(z)-\widetilde{\varphi}_{\varepsilon}(z)\right|$
$\displaystyle\leq\frac{Ct^{\alpha}}{\varepsilon},$

it follows that

(3.13) $\displaystyle I_{2,\varepsilon}$
$\displaystyle\leq\frac{Ct^{\alpha}}{\varepsilon}\int_{\Omega_{\varepsilon}}(dd^{c}v+dd^{c}u)\wedge
S_{\varepsilon}.$

At this point the total mass of the right hand side may go to infinity as
$\varepsilon\to 0^{+}$. However, we can control it uniformly in terms of a
negative exponent of $\varepsilon$. The argument is inspired by [28, Lemma
4.3]. Namely, we obtain

(3.14) $\int_{\Omega_{\varepsilon}}(dd^{c}u+dd^{c}v)\wedge
S_{\varepsilon}\leq\frac{C\|u+v\|_{L^{\infty}(\Omega)}(1+\|\varphi\|_{L^{\infty}(\Omega)})^{k}}{\varepsilon^{k+1}}.$

Indeed, we first have

$\displaystyle\int_{\Omega_{\varepsilon}}dd^{c}(u+v)\wedge(dd^{c}\widetilde{\varphi}_{\varepsilon})^{k}\wedge\beta^{n-k-1}$
$\displaystyle\leq\frac{2}{\varepsilon^{\prime}}\int_{\Omega}\left(\max\\{\rho,-\varepsilon^{\prime}/2\\}-\rho\right)\wedge
dd^{c}(u+v)\wedge(dd^{c}\widetilde{\varphi}_{\varepsilon})^{k}\wedge\beta^{n-k-1}$
$\displaystyle\leq\frac{C}{\varepsilon}\|u+v\|_{L^{\infty}(\Omega)}\int_{\Omega}(dd^{c}\rho)\wedge(dd^{c}\widetilde{\varphi}_{\varepsilon})^{k}\wedge\beta^{n-k-1},$

where $\varepsilon^{\prime}=c_{0}\varepsilon$ with $c_{0}$ defined by (2.1).
The desired inequality (3.14) follows from Lemma 3.1. Now, combining (3.13)
and (3.14) we get that

(3.15) $I_{2,\varepsilon}\leq\frac{Ct^{\alpha}}{\varepsilon^{k+2}}.$

Next, it is easy to see (from Lemma 3.1) that

$\int_{\Omega}(v-u)S_{m,\varepsilon}\leq\frac{C\|u\|_{L^{\infty}(\Omega)}(1+\|\varphi\|_{L^{\infty}(\Omega)})^{m}}{\varepsilon^{m}}.$

Therefore, we can assume that $0<\|v-u\|_{1}<0.01$. Thanks to (3.12) and
(3.15) we have

$\int_{\Omega}(v-u)dd^{c}\widetilde{\varphi}_{\varepsilon}\wedge
S_{\varepsilon}\leq\frac{C}{\varepsilon^{k+1}\,t^{2}}\|v-u\|_{1}^{\tau_{k}}+\frac{Ct^{\alpha}}{\varepsilon^{k+2}}.$

If we choose
$t=\|v-u\|_{1}^{\frac{\tau_{k}}{3}},\quad\tau_{k+1}=\frac{\alpha\tau_{k}}{3},$
then

$\int_{\Omega}(v-u)S_{\varepsilon}\wedge
dd^{c}\widetilde{\varphi}_{\varepsilon}\leq\frac{C}{\varepsilon^{k+2}}\|v-u\|_{1}^{\tau_{k+1}}.$

Thus, the induction argument is completed, and the theorem follows. ∎

The next result is from [28, Lemma 4.10] and its proof used only the
subharmonicity. Therefore, it is also valid for $m$-subharmonic functions.

######  Lemma 3.6.

Let $u$ be the solution in (3.1) and define for $z\in\Omega_{\delta}$,

$\hat{u}_{\delta}(z):=\frac{1}{\sigma_{2n}\delta^{2n}}\int_{|\zeta|\leq\delta}u(z+\zeta)dV_{2n}(\zeta),$

where $\sigma_{2n}$ is the volume of the unit ball. Then, for $\delta>0$
small,

(3.16) $\int_{\Omega_{\delta}}|\hat{u}_{\delta}-u|dV_{2n}\leq C\delta.$

###### Proof.

Since $\Omega_{\delta}\subset D_{c_{0}\delta}$, we have

(3.17) $\displaystyle\int_{\Omega_{\delta}}dd^{c}u(z)\wedge\beta^{n-1}$
$\displaystyle\leq\frac{2}{c_{0}\delta}(\max\\{\rho,-c_{\delta}/2\\}-\rho)dd^{c}u\wedge\beta^{n-1}$
$\displaystyle\leq\frac{C\|u\|_{L^{\infty}(\Omega)}}{\delta}\int_{\Omega}dd^{c}\rho\wedge\beta^{n-1}.$

Therefore, using the classical Jensen formula for subharmonic functions (see
e.g., [18, Lemma 4.3]),

(3.18) $\displaystyle\int_{\Omega_{\delta}}|\hat{u}_{\delta}-u|dV_{2n}$
$\displaystyle\leq\int_{\Omega_{2\delta}}|\hat{u}_{\delta}-u|dV_{2n}+\|u\|_{L^{\infty}(\Omega)}\int_{\Omega_{\delta}\setminus\Omega_{2\delta}}dV_{2n}$
$\displaystyle\leq C\delta^{2}\int_{\Omega_{\delta}}\Delta u+c_{6}\delta$
$\displaystyle\leq C\delta.$

This is the required inequality. ∎

We are ready to prove the Hölder continuity of the solution $u$ in (3.1).

###### End of Proof of Theorem 1.1.

Let us fix $\delta$ such that $0<\delta<\delta_{0}$ small and let
$\varepsilon$ be such that $\delta\leq\varepsilon<\delta_{0}$ which is to be
determined later. Recall that

(3.19) $\Omega_{\delta}:=\\{z\in\Omega:dist(z,\partial\Omega)>\delta\\};$

and for $z\in\Omega_{\delta}$ we define

(3.20) $\displaystyle u_{\delta}(z):=\sup_{|\zeta|\leq\delta}u(z+\zeta).$

Directly from definitions and the Hölder continuity of the boundary data as in
[28, Lemma 4.1] we have the following inequalities.

######  Lemma 3.7.

We have for $z\in\bar{\Omega}_{\delta}\setminus\Omega_{\varepsilon}$,

(3.21) $u_{\delta}(z)\leq u(z)+c_{5}\varepsilon^{\alpha},$

where
$c_{5}=\|\varphi\|_{C^{0,\alpha}(\bar{\Omega})}+\|\phi\|_{C^{0,\alpha}(\bar{\Omega})}$.
In particular,

(3.22)
$\sup_{\Omega_{\delta}}(\hat{u}_{\delta}-u)\leq\sup_{\Omega_{\varepsilon}}(\hat{u}_{\delta}-u)+c_{5}\varepsilon^{\alpha}.$

######  Remark 3.8.

It is important to keep in mind that the uniform constants $C,c_{i}>0$ which
appear in the lemma, and several times below are independent of $\delta$ and
$\varepsilon$.

Thanks to Lemma 3.7 and $\hat{u}_{\delta}\leq u_{\delta}$ we have
$\hat{u}_{\delta}-c_{5}\varepsilon^{\alpha}\leq u$ on
$\partial\Omega_{\varepsilon}$. Therefore, the function

(3.23)
$\tilde{u}:=\begin{cases}\max\\{\hat{u}_{\delta}-c_{5}\varepsilon^{\alpha},u\\}\quad&\mbox{
on }\Omega_{\varepsilon},\\\ u\quad&\mbox{ on
}\Omega\setminus\Omega_{\varepsilon},\end{cases}$

belongs to $SH_{m}(\Omega)\cap C^{0}(\bar{\Omega})$. Notice that
$\tilde{u}\geq u$ in $\Omega$, and

(3.24) $\tilde{u}=u\quad\mbox{ on }\Omega\setminus\Omega_{\varepsilon}.$

Again, by the second part of Lemma 3.7 we have that

(3.25) $\displaystyle\sup_{\Omega_{\delta}}(\hat{u}_{\delta}-u)$
$\displaystyle\leq\sup_{\Omega_{\varepsilon}}(\hat{u}_{\delta}-u)+c_{5}\varepsilon^{\alpha}$
$\displaystyle\leq\sup_{\Omega}(\tilde{u}-u)+2c_{5}\varepsilon^{\alpha}.$

By the stability estimate (Proposition 3.3) there exists $0<\alpha_{2}\leq 1$
such that

(3.26) $\displaystyle\sup_{\Omega}(\tilde{u}-u)$
$\displaystyle\leq\frac{C}{\varepsilon^{2m}}\left(\int_{\Omega}\max\\{\tilde{u}-u,0\\}d\mu\right)^{\alpha_{2}}$
$\displaystyle\leq\frac{C}{\varepsilon^{2m}}\left(\int_{\Omega}|\tilde{u}-u|d\mu\right)^{\alpha_{2}},$

where we used the fact that $\tilde{u}=u$ outside $\Omega_{\varepsilon}$.
Using Theorem 3.5, there is $0<\alpha_{3}\leq 1$ such that

(3.27) $\displaystyle\sup_{\Omega}(\tilde{u}-u)$
$\displaystyle\leq\frac{C}{\varepsilon^{2m+(m+1)\alpha_{2}}}\left(\int_{\Omega}|\tilde{u}-u|dV_{2n}\right)^{\alpha_{2}\alpha_{3}}$
$\displaystyle\leq\frac{C}{\varepsilon^{3m+1}}\left(\int_{\Omega_{\delta}}|\hat{u}_{\delta}-u|dV_{2n}\right)^{\alpha_{2}\alpha_{3}},$

where we used $0\leq\tilde{u}-u\leq{\bf
1}_{\Omega_{\varepsilon}}\cdot(\hat{u}_{\delta}-u)$ and
$\Omega_{\varepsilon}\subset\Omega_{\delta}$ for the second inequality. It
follows from (3.25), (3.27) and Lemma 3.6 that

(3.28) $\sup_{\Omega_{\delta}}(\hat{u}_{\delta}-u)\leq
C\varepsilon^{\alpha}+\frac{C\delta^{\alpha_{2}\alpha_{3}}}{\varepsilon^{3m+1}}.$

Now, we choose $\alpha_{4}=\alpha\alpha_{2}\alpha_{3}/(3m+1+\alpha)$ and

$\varepsilon=\delta^{\frac{\alpha_{2}\alpha_{3}}{3m+1+\alpha}}.$

Then, $\sup_{\Omega_{\delta}}(\hat{u}_{\delta}-u)\leq C\delta^{\alpha_{4}}.$
Finally, thanks to [18, Lemma 4.2] (see also [36]) we infer that
$\sup_{\Omega_{\delta}}(u_{\delta}-u)\leq C\delta^{\alpha_{4}}.$ The proof of
the theorem is finished. ∎

## References

  * [1] L. Baracco, T.-V. Khanh, S. Pinton, G. Zampieri, Hölder regularity of the solution to the complex Monge-Ampère equation with $L^{p}$ density, Calc. Var. Partial Differential Equations 55 (2016) no. 4. art 74, 8pp.
  * [2] E. Bedford, B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère operator, Invent. math. 37 (1976), 1-44.
  * [3] E. Bedford, B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40.
  * [4] A. Benali and A. Zeriahi, The Hölder continuous subsolution theorem for complex Hessian equations, J. Éc. polytech. Math., 7, 2020, 981–1007.
  * [5] A. Benali and A. Zeriahi, Erratum to: “The Hölder continuous subsolution theorem for complex Hessian equations”, J. Éc. polytech. Math., 8 (2021), 779–789.
  * [6] Z. Błocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble) 55 (2005), 1735-1756.
  * [7] M. Charabati, Modulus of continuity of solutions to complex Hessian equations, Internat. J. Math. 27 (2016), no. 1.
  * [8] M. Charabati, Hölder regularity for solutions to complex Monge-Ampère equations, Ann. Polon. Math. 113 (2015) no. 2, 109–127.
  * [9] M. Charabati, Regularity of solutions to the Dirichlet problem for Monge-Ampère equations, Indiana Univ. Math. J., 66 (2017), No 6, 2187–2204.
  * [10] M. Charabati, A. Zeriahi, The continuous subsolution problem for complex Hessian equations,  arXiv:2007.10194. To appear in Indiana University Mathematics Journal.
  * [11] J. Cheng, Y. Xu, Regularization of $m$-subharmonic functions and Hölder continuity, preprint arXiv:2208.14539. To appear in Mathematical Research Letters.
  * [12] J.-P. Demailly, S. Dinew, V. Guedj, S. Kołodziej, H.-H. Pham, A. Zeriahi, Hölder continuous solutions to Monge-Ampère equations, J. Eur. Math. Soc. (JEMS) 16 (2014) no. 4, 619–647.
  * [13] S. Dinew, S. Kołodziej, A priori estimates for complex Hessian equations, Anal. PDE 7 (2014), no. 1, 227–244.
  * [14] S. Dinew, S. Kołodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Amer. J. Math, 139 (2017), No 2, 403–415.
  * [15] T.-C. Dinh, V.-A. Nguyen, Characterization of Monge-Ampère measures with Hölder continuous potentials. J. Funct. Anal. 266 (2014), no. 1, 67–84
  * [16] T.-C. Dinh, V.-A. Nguyen, N. Sibony, Exponential estimates for plurisubharmonic functions and stochastic dynamics, J. Differential Geom. 84 (2010), no. 3, 465-488.
  * [17] H. Elaini and A. Zeriahi, Weighted Green functions for complex Hessian operators, preprint arXiv:2206.05527.
  * [18] V. Guedj, S. Kołodziej, A. Zeriahi, Hölder continuous solutions to Monge-Ampère equations, Bull. Lond. Math. Soc. 40 (2008), 1070-1080.
  * [19] S. Kołodziej, The range of the complex Monge-Ampère operator. II, Indiana Univ. Math. J.44 (1995) no. 3, 765–782.
  * [20] S. Kołodziej, Hölder continuity of solutions to the complex Monge-Ampère equation with the right hand side in $L^{p}$. The case of compact Kähler manifolds, Math. Ann. 342 (2008), 379-386.
  * [21] S. Kołodziej and N.-C. Nguyen, An inequality between complex Hessian measures of Hölder continuous $m$-subharmonic functions and capacity, Geometric analysis—in honor of Gang Tian’s 60th birthday, Progr. Math., 333, 157–166, Birkhäuser/Springer, Cham 2020.
  * [22] S.-Y. Li, On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian, Asian J. Math. 8 (2004), 87-106.
  * [23] C.H. Lu, Équation hessiennes complexes, Thère Doctorat. (2012) Toulouse.
  * [24] H. C. Lu, Viscosity solutions to complex Hessian equations, J. Funct. Anal. 264 (2013), no. 6, 1355–1379.
  * [25] C. H. Lu, A variational approach to complex Hessian equations in $\mathbb{C}^{n}$, J. Math. Anal. Appl. 431 (2015), no. 1, 228–259.
  * [26] N.-C. Nguyen, Hölder continuous solutions to complex Hessian equations, Potential Anal. 41 (2014), no. 3, 887–902.
  * [27] N.-C. Nguyen, On the Hölder continuous subsolution problem for the complex Monge-Ampère equation, Calc. Var. Partial Differential Equations 57 (2018), no. 1, Art. 8, 15 pp.
  * [28] N.-C. Nguyen, On the Hölder continuous subsolution problem for the complex Monge-Ampère equation II, Anal. PDE, 13 (2020), No 2, 435–453.
  * [29] W. Pawłucki and W. Pleśniak, Markov’s inequality and $C^{\infty}$ functions on sets with polynomial cusps, Math. Ann. 275 (1986), no. 3, 467–480.
  * [30] H.-H. Pham, Hölder continuity of solutions to the Monge-Ampère equations on compact Kähler manifolds. Ann. Inst. Fourier (Grenoble) 60 (2010), no. 5, 1857–1869.
  * [31] D.H. Phong, J. Song and J. Sturm, Complex Monge-Ampère equations, Surveys in differential geometry. Vol. XVII, Surv. Differ. Geom., 17 327–410, Int. Press, Boston, MA, 2012\.
  * [32] N. Sibony, Dynamique des applications rationnelles de $\mathbb{P}^{k}$, Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthèses, 8, ix–x, xi–xii, 97–185, Soc. Math. France, Paris, 1999.
  * [33] A. Vinacua, Nonlinear elliptic equations and the complex Hessian, Communications in partial differential equations 13.12 (1988), 1467-1497.
  * [34] D.-V. Vu, Complex Monge-Ampère equation for measures supported on real submanifolds, Math. Ann., 372 (2018), No 1-2, 321–367.
  * [35] J. Wang and B. Zhou, Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations, Sci. China Math., 67 (2024), No 3, 557–576.
  * [36] A. Zeriahi, Remarks on the modulus of continuity of subharmonic functions, arXiv: 2007.08399.