TRACES AND DISTANCES IN ANALYTIC FUNCTION SPACES IN Cn AND MARTINELLY — BOCHNER INTEGRALS
https://doi.org/10.22405/2226-8383-2015-16-2-254-272
Abstract
In this note we provide some analogues of our numerous recent results on traces and distances in terms of Martinelly — Bochner integrals and kernels. These are first results of this type in terms of such kernels. Some assertions for Martinelly — Bochner integrals related with Holder classes and Lebegues points will be also discussed. In recent years various new sharp results on traces and distances were provided in a big series of papers of the first author. In all these papers properties of Bergman-type kernels and Bergman-type integral representations are playing a critical role. The intension of this paper to find some analogues of these results in terms of or with the help of more general integral representations and more general kernels in analytic function spaces in higher dimension so-called Martinelly — Bochner integral representations and Martinelly — Bochner kernels in Cn. Our work consists of three parts. In the first part we partially generalize our results on traces. In the second part we provide estimates of distance function in terms of Martinelly — Bochner kernels and Martinelly — Bochner integrals. In the third part we present results on Martinelly — Bochner integrals related with Holder classes and Lebegues points. This type of issues arise naturally in view of recent series of papers and new results of the first author on multifunctional analytic spaces and related issues. In our proofs we modify the methods of earlier results and theorems for the case of Martinelli-Bochner integrals and kernels.
Keywords
Martinelly — Bochner integrals and kernels,
analytic function,
traces,
distances,
pseudoconvex domains
Supporting agencies: This work was supported by the Russian Foundation for Basic Research (grant 13- 353 01-97508) and by the Ministry of Education and Science of the Russian Federation (grant 1.1704.2014K).
About the Authors
R. Shamoyan
Bryansk State University
Russian Federation
Russian Federation
S. Kurilenko
Bryansk State University
Russian Federation
Russian Federation
References
1. Ortega, J. & Fabrega, J. 1994, "Mixed-norm spaces and interpolation" , Studia Math., vol. 109, no. 3, pp. 234–254.
2. Shamoyan, R. 2009, "On some characterizations of Carleson type measure in the unit ball" , Banach J. Math. Anal., vol. 3, no. 2, pp. 42–48.
3. Shamoyan, R. & Mihic, O. 2009, "On traces of holomorphic functions on the unit polyball" , Appl. Anal. Disorete Math, vol. 3, pp. 198–211.
4. Shamoyan, R. & Mihic, O. 2010, "On traces of Qp type spaces and mixed norm analytic function spaces on polyballs" , Siauliau. Math. Seminar, vol. 5, issue 13, p. 101.
5. Kytmanov, A. 2002, "Martinelly — Bochner integral and it is applications" , Nauka, Novosibirsk, 240 p. (in Russian)
6. Chirka, E. 1975, "Analytic representation of CR functions" , Mat. Sbornik, vol. 98(140), № 4(12), pp. 591-–623.
7. Shamoyan, R. & Mihic, O. 2009, "On traces of holomorphic functions on the unit polyball" , Appl. Anal. Discrete Math., vol. 3, pp. 198—211.
8. Arsenovic, M. & Shamoyan, R. 2013, "On embeddings, traces and multipliers in harmonic function spaces" , Kragujevac Math. Journal, vol. 37, no. 1, pp. 45–64.
9. Shamoyan, R. & Arsenovic, M. 2015, "On distance estimates and atomic decompositions in spaces of analytic functions on strictly pseudoconvex domains" , Bulletin of the Korean Mathematical Society, vol. 52, no. 1, pp. 85-—103.
10. Shamoyan, R. F. & Kurilenko, S. M. 2014, "On a sharp estimate for a distance function in Bergman type analytic spaces in Siegel domains of second type" , Mathematica Montisnigri, vol. XXX, pp. 5–16. 11. Shamoyan, R. F. & Kurilenko, S. M. 2014, "On extremal problems in tubular domains over symmetric cones" , Issues of Analysis, vol. 3(21), no. 1, pp. 44–65.
11. Shamoyan, R. & Mihic, O. 2015, "On traces of Bergman spaces in pseudoconvex domains, preprint" , Jornal of Function Spaces, vol. 2015, 10 p.
12. Shamoyan, R. & Kurilenko, S. 2015, "On traces of Herz type and Bloch type spaces on pseudoconvex domains" , Issues of Analysis, Petrozavodsk.
13. Shamoyan, R. & Mihic, O. 2010, "On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball" , Boletin de la Asociacion Matematica Venezolana, vol. XVII, no. 2, pp. 89–103.
14. Shamoyan, R. & Mihic, O. 2009, "On new estimates for distances in analytic function spaces in unit disk, polydisk, and unit ball" , Siberian Mathematical Reports, vol. 6, pp. 514–517.
15. Shamoyan, R. & Povprits, E. 2013, "Sharp theorems on traces in Bergman type spaces in tubular domains over symmetric cones " , Journal Siberian Federal University, vol. 6(4), pp. 527–538.
16. Arsenovic, M. & Shamoyan, R. 2011, "Trace theorems in harmonic function spaces on (Rn+1 )m + , multipliers theorems and related problems" , Kragujevac Journal of Mathematics, pp. 411–430.
17. Ehsani, Dariush 2007, "The ∂-Neumann problem on product domains in Cn" , Math. Ann., vol. 337, no. 4, pp. 797–816.
18. Charabarta, Debraj & Shaw, Mei-Chi 2011, "The Cauchy-Riemann equations on product domains" , Math. Ann., vol. 349, issue 4, pp. 977–998.
19. Bekolle, D., Bonami, D., Garrigos, G., Nana, C., Peloso, M. & Ricci, F. 2001, "Lecture notes on Bergman projectors in tube domain over cones, an analytic and geometric viewpoint" , Proceeding of the International Workshop on Classical Analysis, Yaounde, 75 p.
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For citations:
Shamoyan R., Kurilenko S. TRACES AND DISTANCES IN ANALYTIC FUNCTION SPACES IN Cn AND MARTINELLY — BOCHNER INTEGRALS. Chebyshevskii Sbornik. 2015;16(2):254-272. (In Russ.) https://doi.org/10.22405/2226-8383-2015-16-2-254-272
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