A sharp inequality for holomorphic functions on the polydisc
HTML articles powered by AMS MathViewer
- by Marijan Marković PDF
- Proc. Amer. Math. Soc. 141 (2013), 1693-1704 Request permission
Abstract:
In this paper we prove an isoperimetric inequality for holomorphic functions in the unit polydisc $\mathbf U^n$. As a corollary we derive an inclusion relation between weighted Bergman and Hardy spaces of holomorphic functions in the polydisc which generalizes the classical Hardy–Littlewood relation $H^p\subseteq A^{2p}$. Also, we extend some results due to Burbea.References
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
- E. F. Beckenbach and T. Radó, Subharmonic functions and surfaces of negative curvature, Trans. Amer. Math. Soc. 35 (1933), no. 3, 662–674. MR 1501708, DOI 10.1090/S0002-9947-1933-1501708-X
- Catherine Bénéteau and Dmitry Khavinson, The isoperimetric inequality via approximation theory and free boundary problems, Comput. Methods Funct. Theory 6 (2006), no. 2, 253–274. MR 2291136, DOI 10.1007/BF03321614
- Jacob Burbea, Inequalities for holomorphic functions of several complex variables, Trans. Amer. Math. Soc. 276 (1983), no. 1, 247–266. MR 684506, DOI 10.1090/S0002-9947-1983-0684506-0
- Jacob Burbea, Sharp inequalities for holomorphic functions, Illinois J. Math. 31 (1987), no. 2, 248–264. MR 882113
- Viktor Blåsjö, The isoperimetric problem, Amer. Math. Monthly 112 (2005), no. 6, 526–566. MR 2142606, DOI 10.2307/30037526
- Torsten Carleman, Zur Theorie der Minimalflächen, Math. Z. 9 (1921), no. 1-2, 154–160 (German). MR 1544458, DOI 10.1007/BF01378342
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- B. A. Fuks, Spetsial′nye glavy teorii analiticheskikh funktsiĭ mnogikh kompleksnykh peremennykh, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0174786
- T. W. Gamelin and D. Khavinson, The isoperimetric inequality and rational approximation, Amer. Math. Monthly 96 (1989), no. 1, 18–30. MR 979592, DOI 10.2307/2323251
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2
- G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), no. 1, 403–439. MR 1545260, DOI 10.1007/BF01180596
- Lars Hörmander, An introduction to complex analysis in several variables, Second revised edition, North-Holland Mathematical Library, Vol. 7, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0344507
- S. Jacobs, An Isoperimetric Inequality for Functions Analytic in Multiply Connected Domains, Mittag-Leffler Institute Report 5 (1972).
- David Kalaj, Isoperimetric inequality for the polydisk, Ann. Mat. Pura Appl. (4) 190 (2011), no. 2, 355–369. MR 2786177, DOI 10.1007/s10231-010-0153-2
- David Kalaj and Romeo Meštrović, Isoperimetric type inequalities for harmonic functions, J. Math. Anal. Appl. 373 (2011), no. 2, 439–448. MR 2720695, DOI 10.1016/j.jmaa.2010.08.009
- M. Keldysh and M. Lavrentiev, Sur la représentation conforme des domaines limités par des courbes rectifiables, Ann. Sci. École Norm. Sup. 54 (1937), no. 3, 1–38.
- Hong Oh Kim, On a theorem of Hardy and Littlewood on the polydisc, Proc. Amer. Math. Soc. 97 (1986), no. 3, 403–409. MR 840619, DOI 10.1090/S0002-9939-1986-0840619-X
- Clinton J. Kolaski, Isometries of Bergman spaces over bounded Runge domains, Canadian J. Math. 33 (1981), no. 5, 1157–1164. MR 638372, DOI 10.4153/CJM-1981-087-1
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- Miodrag Mateljević, The isoperimetric inequality and some extremal problems in $H^{1}$, Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), Lecture Notes in Math., vol. 798, Springer, Berlin, 1980, pp. 364–369. MR 577467
- M. Mateljević, The isoperimetric inequality in the Hardy class $H^{1}$, Mat. Vesnik 3(16)(31) (1979), no. 2, 169–178. MR 613907
- M. Mateljević and M. Pavlović, New proofs of the isoperimetric inequality and some generalizations, J. Math. Anal. Appl. 98 (1984), no. 1, 25–30. MR 728515, DOI 10.1016/0022-247X(84)90276-2
- M. Mateljević and M. Pavlović, Some inequalities of isoperimetric type concerning analytic and subharmonic functions, Publ. Inst. Math. (Beograd) (N.S.) 50(64) (1991), 123–130. MR 1252165
- M. Mateljević and M. Pavlović, Some inequalities of isoperimetric type for the integral means of analytic functions, Mat. Vesnik 37 (1985), no. 1, 78–80 (English, with Serbo-Croatian summary). International symposium on complex analysis and applications (Arandjelovac, 1984). MR 791569
- Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
- Miroslav Pavlović, Introduction to function spaces on the disk, Posebna Izdanja [Special Editions], vol. 20, Matematički Institut SANU, Belgrade, 2004. MR 2109650
- Miroslav Pavlović and Milutin R. Dostanić, On the inclusion $H^2(U^n)\subset H^{2n}(B_n)$ and the isoperimetric inequality, J. Math. Anal. Appl. 226 (1998), no. 1, 143–149. MR 1646501, DOI 10.1006/jmaa.1998.6061
- I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765
- L. I. Ronkin, Introduction to the theory of entire functions of several variables, Translations of Mathematical Monographs, Vol. 44, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by Israel Program for Scientific Translations. MR 0346175, DOI 10.1090/mmono/044
- Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
- Saburou Saitoh, The Bergman norm and the Szegő norm, Trans. Amer. Math. Soc. 249 (1979), no. 2, 261–279. MR 525673, DOI 10.1090/S0002-9947-1979-0525673-8
- Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423, DOI 10.1007/978-3-662-02414-0
- Dragan Vukotić, The isoperimetric inequality and a theorem of Hardy and Littlewood, Amer. Math. Monthly 110 (2003), no. 6, 532–536. MR 1984405, DOI 10.2307/3647909
- S. E. Warschawski, On differentiability at the boundary in conformal mapping, Proc. Amer. Math. Soc. 12 (1961), 614–620. MR 131524, DOI 10.1090/S0002-9939-1961-0131524-8
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- Marijan Marković
- Affiliation: Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b., 81000, Podgorica, Montenegro
- Email: marijanmmarkovic@gmail.com
- Received by editor(s): May 27, 2011
- Received by editor(s) in revised form: August 9, 2011, and September 9, 2011
- Published electronically: November 16, 2012
- Communicated by: Franc Forstneric
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 1693-1704
- MSC (2010): Primary 30H10, 32A36, 30A10
- DOI: https://doi.org/10.1090/S0002-9939-2012-11446-7
- MathSciNet review: 3020856