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Results in Mathematics

, 74:27 | Cite as

Sectorial Extensions, via Laplace Transforms, in Ultraholomorphic Classes Defined by Weight Functions

  • Javier Jiménez-Garrido
  • Javier SanzEmail author
  • Gerhard Schindl
Article
  • 9 Downloads

Abstract

We prove several extension theorems for Roumieu ultraholomorphic classes of functions in sectors of the Riemann surface of the logarithm which are defined by means of a weight function or weight matrix. Our main aim is to transfer the results of V. Thilliez from the weight sequence case to these different, or more general, frameworks. The technique rests on the construction of suitable kernels for a truncated Laplace-like integral transform, which provides the solution without resorting to Whitney-type extension results for ultradifferentiable classes. As a byproduct, we obtain an extension in a mixed weight-sequence setting in which assumptions on the sequence are minimal.

Keywords

Ultraholomorphic classes weight sequences functions and matrices Legendre conjugates Laplace transform extension operators indices of O-regular variation 

Mathematics Subject Classification

46E10 30E05 26A12 44A05 

Notes

Acknowledgements

The first two authors are partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under the Project MTM2016-77642-C2-1-P. The first author is partially supported by the University of Valladolid through a Predoctoral Fellowship (2013 call) co-sponsored by the Banco de Santander. The third author is supported by FWF-Project J 3948-N35, as a part of which he is an external researcher at the Universidad de Valladolid (Spain) for the period October 2016-September 2018.

The authors wish to express their gratitude to Prof. Óscar Blasco, from the Universidad de Valencia (Spain), for his helpful comments regarding Proposition 7.3.

References

  1. 1.
    Bari, K.N., Stečkin, S.B.: Best approximations and differential properties of two conjugate functions (in Russian). Trudy Moskov. Mat. Obšč. 5, 483–522 (1956)MathSciNetGoogle Scholar
  2. 2.
    Beurling, A.: Analytic continuation across a linear boundary. Acta Math. 128, 153–182 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blasco, O.: Operators on Fock-type and weighted spaces of entire functions. Funct. Approx. Comment. Math.  https://doi.org/10.7169/facm/1708
  4. 4.
    Bonet, J., Braun, R.W., Meise, R., Taylor, B.A.: Whitney’s extension theorem for nonquasianalytic classes of ultradifferentiable functions. Studia Math. 99(2), 155–184 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bonet, J., Meise, R., Taylor, B.A.: On the range of the Borel map for classes of nonquasianalytic functions. North-Holland Mathematics Studies. Prog. Funct. Anal. 170, 97–111 (1992)zbMATHGoogle Scholar
  7. 7.
    Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Results Math. 17(3–4), 206–237 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chaumat, J., Chollet, A.-M.: Surjectivité de l’application restriction à un compact dans les classes de fonctions ultradifférentiables. Math. Ann. 298, 7–40 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    de Roever, J.W.: Hyperfunctional singular support of ultradistributions. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 31, 585–631 (1985)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Debrouwere, A.: Analytic representations of distributions and ultradistributions. Master Thesis, Universiteit Gent. http://lib.ugent.be/fulltxt/RUG01/002/163/702/RUG01-002163702_2014_0001_AC.pdf (2014). Last Accessed 27 Oct 2017
  11. 11.
    Haraoka, Y.: Theorems of Sibuya-Malgrange type for Gevrey functions of several variables. Funkcial. Ekvac. 32, 365–388 (1989)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Jiménez-Garrido, J., Sanz, J., Schindl, G.: Indices of O-regular variation for weight functions and weight sequences (submitted). arxiv:1806.01605. Last Accessed 23 Aug 2018
  13. 13.
    Jiménez-Garrido, J., Sanz, J., Schindl, G.: Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes. J. Math. Anal. Appl. 469, 136–168 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jiménez-Garrido, J., Sanz, J.: Strongly regular sequences and proximate orders. J. Math. Anal. Appl. 438(2), 920–945 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jiménez-Garrido, J., Sanz, J., Schindl, G.: Sectorial extensions in ultraholomorphic classes defined by weight functions (submitted). arxiv:1805.09685. Last Accessed 23 Aug 2018
  16. 16.
    Jiménez-Garrido, J., Sanz, J., Schindl, G.: Log-convex sequences and nonzero proximate orders. J. Math. Anal. Appl. 448(2), 1572–1599 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kellay, K., Zarrabi, M.: Normality, non-quasianalyticity and invariant subspaces. J. Oper. Theory 46, 221–250 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Komatsu, H.: Ultradistributions. I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20, 25–105 (1973)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lastra, A., Malek, S., Sanz, J.: Continuous right inverses for the asymptotic Borel map in ultraholomorphic classes via a Laplace-type transform. J. Math. Anal. Appl. 396, 724–740 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lastra, A., Malek, S., Sanz, J.: Summability in general Carleman ultraholomorphic classes. J. Math. Anal. Appl. 430, 1175–1206 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mandelbrojt, S.: Séries adhérentes, régularisation des suites, applications. Gauthier-Villars, Paris (1952)zbMATHGoogle Scholar
  22. 22.
    Matsumoto, W.: Characterization of the separativity of ultradifferentiable classes. J. Math. Kyoto Univ. 24(4), 667–678 (1984)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Meise, R., Taylor, B.A.: Whitney’s extension theorem for ultradifferentiable functions of Beurling type. Ark. Mat. 26(2), 265–287 (1988)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Meise, R., Taylor, B.A.: Linear extension operators for ultradifferentiable functions of Beurling type on compact sets. Am. J. Math. 111(2), 309–337 (1989)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Petzsche, H.-J.: On E. Borel’s theorem. Math. Ann. 282(2), 299–313 (1988)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Petzsche, H.-J., Vogt, D.: Almost analytic extension of ultradifferentiable functions and the boundary values of holomorphic functions. Math. Ann. 267, 17–35 (1984)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rainer, A., Schindl, G.: Composition in ultradifferentiable classes. Studia Math. 224(2), 97–131 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rainer, A., Schindl, G.: Extension of Whitney jets of controlled growth. Math. Nachr. 290(14–15), 2356–2374 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rainer, A., Schindl, G.: On the extension of Whitney ultrajets. Studia Math. 245, 255–287 (2019)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ramis, J.P.: Les séries \(k-\)sommables et leurs applications. Lecture Notes in Physics, 126. Springer, Berlin (1980)Google Scholar
  31. 31.
    Ramis, J.P.: Dévissage Gevrey. Asterisque 59–60, 173–204 (1978)zbMATHGoogle Scholar
  32. 32.
    Sanz, J.: Linear continuous extension operators for Gevrey classes on polysectors. Glasg. Math. J. 45(2), 199–216 (2003)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sanz, J.: Flat functions in Carleman ultraholomorphic classes via proximate orders. J. Math. Anal. Appl. 415, 623–643 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Schindl, G.: Spaces of smooth functions of Denjoy–Carleman-type. Diploma Thesis, Universität Wien. http://othes.univie.ac.at/7715/1/2009-11-18_0304518.pdf (2009). Last Accessed 27 Oct 2017
  35. 35.
    Schindl, G.: Exponential laws for classes of Denjoy-Carleman differentiable mappings. PhD Thesis, Universität Wien. http://othes.univie.ac.at/32755/1/2014-01-26_0304518.pdf (2014). Last Accessed 27 Oct 2017
  36. 36.
    Schindl, G.: Characterization of ultradifferentiable test functions defined by weight matrices in terms of their Fourier transform. Note Mat. 36(2), 1–35 (2016)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Schmets, J., Valdivia, M.: Extension maps in ultradifferentiable and ultraholomorphic function spaces. Studia Math. 143(3), 221–250 (2000)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Thilliez, V.: Extension Gevrey et rigidité dans un secteur. Studia Math. 117, 29–41 (1995)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Thilliez, V.: Division by flat ultradifferentiable functions and sectorial extensions. Results Math. 44, 169–188 (2003)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Thilliez, V.: Smooth solutions of quasianalytic or ultraholomorphic equations. Monatsh. Math. 160(4), 443–453 (2010)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Tikhonov, S.: On generalized Lipschitz classes and Fourier series. Z. Anal. Anwend. 23(4), 745–764 (2004)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Tougeron, J.C.: An introduction to the theory of Gevrey expansions and to the Borel–Laplace transform with some applications. Lecture notes, Toronto University (1990)Google Scholar

Copyright information

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Authors and Affiliations

  1. 1.Departamento de Álgebra, Análisis Matemático, Geometría y Topología, Facultad de CienciasUniversidad de ValladolidValladolidSpain
  2. 2.Instituto de Investigación en Matemáticas IMUVAUniversidad de ValladolidValladolidSpain

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