Publication: Some properties of differentiable p-adic functions
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Abstract
In this paper, using the tools from the lineability theory, we distinguish certain subsets of p-adic differentiable functions. Specifically, we show that the following sets of functions are large enough to contain an infinite dimensional algebraic structure: (i) continuously differentiable but not strictly differentiable functions, (ii) strictly differentiable functions of order r but not strictly differentiable of order r + 1, (iii) strictly differentiable functions with zero derivative that are not Lipschitzian of any order α > 1, (iv) differentiable functions with unbounded derivative, and (v) continuous functions that are differentiable on a full set with respect to the Haar measure but not differentiable on its complement having cardinality the continuum.
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1] G. Ara´ujo, L. Bernal-Gonz´alez, G. A. Mu˜noz-Fern´andez, and J. B. Seoane-Sep´ulveda, Line�ability in sequence and function spaces, Studia Math. 237 (2017), no. 2, 119–136.
[2] R. M. Aron, L. Bernal Gonz´alez, D. M. Pellegrino, and J. B. Seoane Sep´ulveda, Lineability:
the search for linearity in mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016.
[3] R. M. Aron, V. I. Gurariy, and J. B. Seoane-Sep´ulveda, Lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795–803.
[4] R. M. Aron, D. P´erez-Garc´ıa, and J. B. Seoane-Sep´ulveda, Algebrability of the set of non�convergent Fourier series, Studia Math. 175 (2006), no. 1, 83–90.
[5] M. Balcerzak, A. Bartoszewicz, and M. Filipczak, Nonseparable spaceability and strong al�gebrability of sets of continuous singular functions, J. Math. Anal. Appl. 407 (2013), no. 2, 263–269
[6] A. Bartoszewicz, M. Bienias, M. Filipczak, and S. G l¸ab, Strong c-algebrability of strong
Sierpi´nski-Zygmund, smooth nowhere analytic and other sets of functions, J. Math. Anal.Appl. 412 (2014), no. 2, 620–630.
[7] A. Bartoszewicz, M. Bienias, and S. G l¸ab, Independent Bernstein sets and algebraic con�structions, J. Math. Anal. Appl. 393 (2012), no. 1, 138–143.
[8] A. Bartoszewicz, M. Filipczak, and M. Terepeta, Lineability of Linearly Sensitive Functions,
Results Math. 75 (2020), no. 2, Paper No. 64.
[9] A. Bartoszewicz and S. G l¸ab, Strong algebrability of sets of sequences and functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 827–835.
[10] F. Bayart, Linearity of sets of strange functions, Michigan Math. J. 53 (2005), no. 2, 291–303.
[11] F. Bayart and L. Quarta, Algebras in sets of queer functions, Israel J. Math. 158 (2007),285–296.
[12] L. Bernal-Gonz´alez, A. Bonilla, J. L´opez-Salazar, and J. B. Seoane-Sep´ulveda, Nowhere h¨olderian functions and Pringsheim singular functions in the disc algebra, Monatsh. Math. 188 (2019), no. 4, 591–609.
[13] L. Bernal-Gonz´alez, H. J. Cabana-M´endez, G. A. Mu˜noz-Fern´andez, and J. B. Seoane�Sep´ulveda, On the dimension of subspaces of continuous functions attaining their maximum finitely many times, Trans. Amer. Math. Soc. 373 (2020), no. 5, 3063–3083.
[14] L. Bernal-González, G. A. Muñóz-Fernández, D. L. Rodrí guez-Vidanes, and J. B. Seoane�Sep´ulveda, Algebraic genericity within the class of sup-measurable functions, J. Math. Anal. Appl. 483 (2020), no. 1, 123–576.
[15] L. Bernal-Gonz´alez and M. Ordóñez Cabrera, Lineability criteria, with applications, J. Funct.
Anal. 266 (2014), no. 6, 3997–4025.
[16] L. Bernal-González, D. Pellegrino, and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear
sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 71–130.
[17] N. Biehler, V. Nestoridis, and A. Stavrianidi, Algebraic genericity of frequently universal
harmonic functions on trees, J. Math. Anal. Appl. 489 (2020), no. 1, 124132, 11.
[18] P. Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical
Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication.
[19] M. C. Calder´on-Moreno, P. J. Gerlach-Mena, and J. A. Prado-Bassas, Lineability and modes
of convergence, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 114 (2020),
no. 1, Paper No. 18, 12.
[20] K. C. Ciesielski and J. B. Seoane-Sepúlveda, Differentiability versus continuity: restriction
and extension theorems and monstrous examples, Bull. Amer. Math. Soc. (N.S.) 56 (2019),
no. 2, 211–260.
[21] P. H. Enflo, V. I. Gurariy, and J. B. Seoane-Sep´ulveda, Some results and open questions on
spaceability in function spaces, Trans. Amer. Math. Soc. 366 (2014), no. 2, 611–625.
[22] J. Fernández-Sánchez, S. Maghsoudi, D. L. Rodríguez-Vidanes, and J. B. Seoane-Sepúlveda,
Classical vs. non-Archimedean analysis. An approach via algebraic genericity, Preprint
(2021).
[23] J. Fernández-Sánchez, M. E. Martínez-Gómez, and J. B. Seoane-Sepúlveda, Algebraic gener�icity and special properties within sequence spaces and series, Preprint (2019).
[24] G. Fichtenholz and L. Kantorovich, Sur les op´erations dans l’espace des functions bornées, Studia Math. 5 (1934), 69-98.
[25] G. B. Folland, A course in abstract harmonic analysis, 2nd ed., Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2016.
[26] F. J. García-Pacheco, N. Palmberg, and J. B. Seoane-Sep´ulveda, Lineability and algebrability of pathological phenomena in analysis, J. Math. Anal. Appl. 326 (2007), no. 2, 929–939.
[27] F. Q. Gouvˆea, p-adic numbers, An introduction, 2nd ed., Universitext, Springer-Verlag, Berlin, 1997.
[28] V. I. Gurariı, Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR
167 (1966), 971–973 (Russian).
[29] F. Hausdorff, Uber zwei Satze von G. Fichtenholz und L. Kantorovich, Studia Math. 6 (1936), 18–19 (German).
[30] P. Jim´enez-Rodr´ıguez, G. A. Muñóz-Fernández, and J. B. Seoane-Sepúlveda, Non-Lipschitz
functions with bounded gradient and related problems, Linear Algebra Appl. 437 (2012), no. 4, 1174–1181.
[31] S. Katok, p-adic analysis compared with real, Student Mathematical Library, vol. 37, Ameri�can Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, Univer�sity Park, PA, 2007.
[32] J. Khodabendehlou, S. Maghsoudi, and J. B. Seoane-Sepúlveda, Algebraic genericity and
summability within the non-Archimedean setting, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 115 (2021), no. 21.
[33] , Lineability and algebrability within p-adic function spaces, Bull. Belg. Math. Soc. Simon Stevin 27 (2020), 711–729.
[34] , Lineability, continuity, and antiderivatives in the non-Archimedean setting, Canad.Math. Bull. 64 (2021), no. 3, 638–650.
[35] B. Levine and D. Milman, On linear sets in space C consisting of functions of bounded variation, Comm. Inst. Sci. Math. M´ec. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] (4) 16
(1940), 102–105 (Russian, with English summary).
[36] K. Mahler, p-adic numbers and their functions, Second, Cambridge Tracts in Mathematics,
vol. 76, Cambridge University Press, Cambridge-New York, 1981.
[37] , An interpolation series for continuous functions of a p-adic variable, J. Reine Angew. Math. 199 (1958), 23–34.
[38] T. K. S. Moothathu, Lineability in the sets of Baire and continuous real functions, Topology Appl. 235 (2018), 83–91.
[39] T. Natkaniec, On lineability of families of non-measurable functions of two variable, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 115 (2021), no. 1, Paper No. 33, 10.
[40] A. M. Robert, A course in p-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer�Verlag, New York, 2000.
[41] W. H. Schikhof, Ultrametric calculus, An introduction to p-adic analysis, Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, Cambridge, 1984.
[42] J. B. Seoane-Sepúlveda, Chaos and lineability of pathological phenomena in analysis, Pro�Quest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–Kent State University.
[43] A. C. M. van Rooij, Non-Archimedean functional analysis, Monographs and Textbooks in Pure and Applied Math., vol. 51, Marcel Dekker, Inc., New York, 1978.