MAKSIMAL m-PLYURISUBGARMONIK FUNKSIYALAR

Ismoilov Muxiddin Baxrom o‘g‘li   Ahmedov Qurbonbay Yuldashbayevich  Jumaniyozova Xayriniso Xamidbek qizi

Arxiv / №6 · 2026 / Maqola

MAKSIMAL m-PLYURISUBGARMONIK FUNKSIYALAR

Annotatsiya

O'ZBEKCHA

Maqolada m-plyurisubgarmonik va m-subgarmonik funksiyalar sinflari va ularning o‘zaro aloqasi tahlil qilingan. Maksimal m-plyurisubgarmonik funksiyalar uchun maksimum prinsipi aso-sida fundamental teoremalarning analoglari isbot qilingan. Bu teoremalarni isbot qilishda m-plyurisub-garmonik funksiyalar orasidagi bog‘lanishdan, ya’ni, haqiqiy fazoni kompleks fazoga kengaytirishdan foy-dalanilgan. Bundan tashqari, maksimallikni ta’minlovchi yetarli shartlar keltirilib, nazariy natijalarga aniq misol keltirilgan

Kalit so'zlar: m- o‘lchamli tekisliklar, garmonik funksiyalar, subgarmonik funksiyalar, plyurisub-garmonik funksiyalar, m-subgarmonik funksiyalar, m-plyurisubgarmonik funksiyalar, maksimal m-plyurisubgarmonik funksiyalar.

РУССКИЙ

статье исследуются классы m-плюрисубгармонических и m-субгармони-ческих функций, а также анализируется их взаимосвязь. Для максимальных m-плюрисубгармони-ческих функций доказаны аналоги фундаментальных теорем на основе принципа максимума. При доказательстве данных теорем была использована взаимосвязь между рассматриваемыми функ-циями, а именно вложения вещественного пространства в комплексное пространство. Кроме то-го, сформулированы достаточные условия, обеспечивающие максимальность, и приведён пример, иллюстрирующий полученные теоретические результаты.

Ключевые слова: m- мерная плоскость, гармонические функции, субгармонические функции, плюрисубгармонические функции, m- плюрисубгармонические функции, максимальные m- плюри-субгармонические функции.

ENGLISH

The article analyzes the classes of m-plurisubharmonic and m-subharmonic func-tions and their interrelations. Analogues of fundamental theorems for maximal m-plurisubharmonic functions are proved based on the maximum principle. In proving these theorems, the relationship between these functions is used, namely, the embedding of real space into a complex space. In addition, sufficient conditions ensuring maximality are given and an example of theoretical results is given.

Keywords: m- measure plane, harmonic functions, subharmonic functions, plurisubharmonic func-tions, m-subharmonic functions, m-plurisubharmonic functions, maximalm-plurisubharmonic func-tions.

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Foydalanilgan adabiyotlar

Садуллаев А. Теория плюрипотенциала. Применения. Palmarium Akademic Publishing, 2012, 316 с.
Абдуллаев Б.И., Шарипов Р.А., 𝑚𝑚−субгармонические функции во всем пространстве ℂ𝑛𝑛. Функции Грина. “Узбекский математический журнал”, Ташкент, 2013, № 3, c. 3 – 8.
Abdullaev B.I., Imomkulov S.A., Sharipov R.A. Removable singular sets of m-subharmonic functions. Algebra, Complex Analysis, and Pluripotential Theory, Springer Proceedings in Mathematics&Statistics. V. 264, 2018, p. 1–11.
Abdullayev B.I., Subharmonic functions on complex Hyperplanes of n.Journal of Siberian Federal University, Mathematics and Physics, Krasnoyarsk, 2013, № 6(4), p. 409–416.
Sadullaev A. Further developments of the pluripotential theory (survey) // in Algebra, complex analysis, and pluripotential theory, Springer Proceedings in Mathematics & Statistics, 264(2018), p. 167–182.
Drnovšek B., Forstnerič F. Minimal hulls of compact sets in 3//Trans. Amer. Math. Soc., 368(2016), pp. 7477–7506.
Harvey F.R. and Lawson H.B. Jr., An introduction to potential theory in calibrated geometry. Amer. J. Math. 131, 4(2009), pp. 893–944.
Harvey F.R. and Lawson H.B Jr. Duality of positive currents and plurisubharmonic functions in calibrated geometry. Amer. J.Math. 131. 5(2009), pp. 1211–1240.
Harvey F.R. and Lawson H.B. Jr., Geometric plurisubharmonicity and convexity – an introduction. Advances in Mathematics 230 (2012), pp. 2428–2456.
Harvey F.R. and Lawson H.B. Jr., Plurisubharmonicity in a general geometric context. Geometry and Analysis, 1(2010), pp. 363–401.

← Songa qaytish

[1] Садуллаев А. Теория плюрипотенциала. Применения. Palmarium Akademic Publishing, 2012, 316 с.

[2] Абдуллаев Б.И., Шарипов Р.А., 𝑚𝑚−субгармонические функции во всем пространстве ℂ𝑛𝑛. Функции Грина. “Узбекский математический журнал”, Ташкент, 2013, № 3, c. 3 – 8.

[3] Abdullaev B.I., Imomkulov S.A., Sharipov R.A. Removable singular sets of m-subharmonic functions. Algebra, Complex Analysis, and Pluripotential Theory, Springer Proceedings in Mathematics&Statistics. V. 264, 2018, p. 1–11.

[4] Abdullayev B.I., Subharmonic functions on complex Hyperplanes of n.Journal of Siberian Federal University, Mathematics and Physics, Krasnoyarsk, 2013, № 6(4), p. 409–416.

[5] Sadullaev A. Further developments of the pluripotential theory (survey) // in Algebra, complex analysis, and pluripotential theory, Springer Proceedings in Mathematics & Statistics, 264(2018), p. 167–182.

[6] Drnovšek B., Forstnerič F. Minimal hulls of compact sets in 3//Trans. Amer. Math. Soc., 368(2016), pp. 7477–7506.

[7] Harvey F.R. and Lawson H.B. Jr., An introduction to potential theory in calibrated geometry. Amer. J. Math. 131, 4(2009), pp. 893–944.

[8] Harvey F.R. and Lawson H.B Jr. Duality of positive currents and plurisubharmonic functions in calibrated geometry. Amer. J.Math. 131. 5(2009), pp. 1211–1240.

[9] Harvey F.R. and Lawson H.B. Jr., Geometric plurisubharmonicity and convexity – an introduction. Advances in Mathematics 230 (2012), pp. 2428–2456.

[10] Harvey F.R. and Lawson H.B. Jr., Plurisubharmonicity in a general geometric context. Geometry and Analysis, 1(2010), pp. 363–401.