Obecná teorie derivování funkcí a měr na katedře matematické analýzy MFF UK

Zajíček, Luděk

Pokroky matematiky, fyziky a astronomie, Tome 45 (2000) no. 3, pp. 188-207

Détail

Classification : 01A73, 26-02, 26-03, 26A24, 26A27, 26A30, 26B05

Bibliographie

[ABBM] Agronsky, S. J., Biskner, R., Bruckner, A. M., Mařík, J.: Representations of functions by derivatives. Trans. Amer. Math. Soc. 263 (1981), 493–500. | MR

[AP] Aversa, V., Preiss, D.: Hearts density theorems. Real Anal. Exchange 13 (1987/88), 28–32.

[ALP] Aversa, V., Laczkovich, M., Preiss, D.: Extension of differentiable functions. Comment. Math. Univ. Carolinae 26 (1986), 597–609. | MR

[BBM] Blažek, J., Borák, E., Malý, J.: On Köpcke and Pompeiu functions. Časopis Pěst. Mat. 103 (1978), 53–61.

[BCEH] Belna, C. L., Cargo, G. T., Evans, M. J., Humke, P. D.: Analogues of the Denjoy-Young-Saks theorem. Trans. Amer. Math. Soc. 271 (1982), 253–260. | MR | Zbl

[Be] Berman, S. M.: Gaussian process with stationary increments: local times and sample function properties. Ann. Math. Statist. 41 (1970), 1260–1272. | MR

[BEH] Belna, C. L., Evans, M. J., Humke, P. D.: Symmetric and ordinary differentiation. Proc. Amer. Math. Soc. 72 (1978), 261–267. | MR | Zbl

[BL] Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Vol. 1: Colloq. publications (American Mathematical Society); v. 48: Providence, Rhode Island 2000. | MR | Zbl

[BP] Borwein, J. M., Preiss, D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Amer. Math. Soc. 303 (1987), 517–527. | MR | Zbl

[Br] Bruckner, A. M.: Differentiation of Real Functions. CRM Monographs Series, vol. 5, Providence 1994. | MR | Zbl

[Bu] Buczolich, Z.: The $n$-dimensional gradient has the 1-dimensional DenjoyC̄larkson property. Real Anal. Exchange 18 (1992–93), 221–224. | MR

[Fa] Fabian, M.: Gâteaux differentiability of convex functions and topology — weak Asplund spaces. John Wiley and Sons, Interscience 1997. | MR | Zbl

[Fe] Federer, H.: Geometric Measure Theory. Springer-Verlag, Berlin–New York 1969. | MR | Zbl

[FM] Fonseca, I., Malý, J.: Relaxation of multiple integrals below the growth exponent. Anal. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 309–338. | MR

[Fo] Foran, J.: The symmetric and ordinary derivative. Real Anal. Exchange 2 (1977), 105–108. | Zbl

[Fr] Freiling, Ch.: On the problem of characterizing derivatives. Real Anal. Exchange 23 (1997/98), 805–812. | MR | Zbl

[Gu] Guzmán, M. de: Differentiation of integrals in ${\mathbb {R}}^n$. Lecture Notes in Math. 481, Springer-Verlag, Berlin, New York 1975. | MR

[HMWZ] Holický, P., Malý, J., Weil, C. E., Zajíček, L.: A note on the gradient problem. Real Anal. Exchange 22 (1996/97), 225–235. | MR

[HŠZ] Holický, P., Šmídek, M., Zajíček, L.: Convex functions with nonmeasurable set of Gâteaux differentiability points. Comment. Math. Univ. Carolinae 39 (1998), 469–482. | MR

[Hu] Hunt, B. R.: The prevalence of continuous nowhere differentiable functions. Proc. Amer. Math. Soc. 122 (1994), 711–717. | MR | Zbl

[Chl] Chlebík, M.: Geometrická teória miery. Diplomová práce, MFF UK 1984.

[Cho] Choquet, Ch.: Application des propriétés descriptives de la fonction contingent a la théorie des fonctions de variable réelle et a la géometrie différentielle des variétés cartésiennes (These). J. Math. Pures Appl. 26 (1947), 115–256.

[Ke] Kelar, V.: On the first and the fifth class of Zahorski. Real Anal. Exchange 9 (1983/84), 233–250. | MR

[Ki1] Kirchheim, B.: Uniformly distributed measures, tangent measures and analytic varieties. Collection: Proceedings of the Conference: Topology and Measure V (Binz, 1987), 54–60. | MR

[Ki2] Kirchheim, B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc. 121 (1994), 113–123. | MR | Zbl

[KKM] Kauhanen, J., Koskela, P., Malý, J.: On functions with derivatives in a Lorentz space. Manuscripta Math. 100 (1999), 87–101. | MR

[KM] Kilpeläinen, T., Malý, J.: Sobolev inequalities on sets with irregular boundaries. Z. Anal. Angew. 19 (2000), 369–380. | MR

[Ko] Kolář, J.: Nowhere approximately differentiable and nowhere Hölder continuous functions — Porous sets and Haar null sets. Proc. Amer. Math. Soc. (v tisku).

[KP] Kowalski, O., Preiss, D.: Besicovitch-type properties of measures and submanifolds. J. Reine Angew. Math. 379 (1987), 115–151. | MR | Zbl

[LMZ] Lukeš, J., Malý, J., Zajíček, L.: Fine Topology Methods in Real Analysis and Potential Theory. Lecture Notes in Math. 1189, Springer–Verlag, Berlin, New York 1986.

[Mal1] Malý, J.: The Darboux property for gradients. Real Anal. Exchange 22 (1996/97), 167–173. | MR

[Mal2] Malý, J.: Where the continuous functions without unilateral derivatives are typical. Trans. Amer. Math. Soc. 283 (1984), 169–175. | MR

[Mal3] Malý, J.: Hölder type quasicontinuity. Potential Anal. 2 (1993), 249–254. | MR

[Mal4] Malý, J.: Absolutely continuous functions of several variables. J. Math. Anal. Appl. 231 (1999), 492–508. | MR

[Mal5] Malý, J.: A simple proof of the Stepanov theorem on differentiability almost everywhere. Expo. Math. 17 (1999), 59–61. | MR

[Mal6] Malý, J.: Sufficient conditions for change of variables in integral. Preprint MFF UK, KMA-2000-19, Proceedings “Analysis and Geometry”, Novosibirsk 1999 (v tisku).

[Max] Maximoff, I.: Sur la transformation continue de quelques fonctions en dérivées exactes. Bull. Soc. Phys. Math. Kazan (3) 12 (1940), 57–81. | MR | Zbl

[MM] Malý, J., Martio, O.: Lusin’s condition (N) and mappings of the class $W^{1,n}$. J. Reine Angew. Math. 458 (1995), 19–36. | MR

[MP] Malý, J., Pick, L.: An elementary proof of sharp Sobolev embeddings. Preprint MFF UK, MATH-KMA-2000/33, Proc. Amer. Math. Soc. (v tisku). | MR

[MPZ] Malý, J., Preiss, D., Zajíček, L.: An unusual monotonicity theorem with applications. Proc. Amer. Math. Soc. 102 (1988), 925-932. | MR

[MPT] Mejlbro, L., Preiss, D., Tišer, J.: Differentiation and determination of measures. (připravovaná kniha).

[MZ] Malý, J., Zajíček, L.: Approximate differentiation: Jarník points. Fund. Math. 140 (1991), 87–97. | MR

[Ph] Phelps, R. R.: Convex functions, monotone operators and differentiability. Lecture Notes in Math. 1364, Springer-Verlag, Berlin 1993. | MR

[Pr1] Preiss, D.: Gaussian measures and covering theorems. Comment. Math. Univ. Carolinae 20 (1979), 95–99. | MR | Zbl

[Pr2] Preiss, D.: Maximoff’s theorem. Real Anal. Exchange 5 (1979–80), 92–104. | MR

[Pr3] Preiss, D.: Gaussian measures and the density theorem. Comment. Math. Univ. Carolinae 22 (1981), 181–193. | MR | Zbl

[Pr4] Preiss, D.: Level sets of derivatives. Trans. Amer. Math. Soc. 272 (1982), 161–184. | MR | Zbl

[Pr5] Preiss, D.: Algebra generated by derivatives. Real Anal. Exchange 8 (1982–83), 208–216. | MR

[Pr6] Preiss, D.: Dimension of metrics and differentiation of measures. In collection: General topology and its relations to modern analysis and algebra, V (Prague, 1981). Sigma Ser. Pure Math. 3, Heldermann, Berlin 1983, 565–568. | MR

[Pr7] Preiss, D.: Geometry of measures in ${\mathbb {R}}^n$: distribution, rectifiability, and densities. Ann. Math. 125 (1987), 537–643. | MR

[Pr8] Preiss, D.: Differentiability of Lipschitz functions in Banach spaces. J. Funct. Anal. 91 (1990), 312–345. | MR

[Pr9] Preiss, D.: The work of Professor Jarník in Real Analysis. In collection: Life and work of Vojtěch Jarník, B. Novák ed., Prometheus 1999, 55–65.

[PPN] Preiss, D., Phelps, R. R., Namioka, I.: Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings. Israel J. Math. 72 (1990), 257–279. | MR | Zbl

[PŠ] Preiss, D., Šverák, V.: Derivatives of type $1$. Real Anal. Exchange (1986/87), 354–360. | MR

[PZ1] Preiss, D., Zajíček, L.: On the symmetry of approximate Dini derivates of arbitrary functions. Comment. Math. Univ. Carolinae 23 (1982), 691–697. | MR

[PZ2] Preiss, D., Zajíček, L.: Stronger estimates of smallness of sets of Fréchet nondifferentiability of convex functions. Rend. Circ. Mat. Palermo (2) Suppl. no. 3 (1984), 219–223.

[PZ3] Preiss, D., Zajíček, L.: On Dini and approximate Dini derivates of typical continuous functions. Preprint MFF UK, KMA-1999-13, Real Anal. Exchange (v tisku). | MR

[PZ4] Preiss, D., Zajíček, L.: Directional derivatives of Lipschitz functions. Preprint MFF UK, KMA-2000-28. | MR

[S1] Saks, S.: On the functions of Besicovitch in the space of continuous functions. Fund. Math. 19 (1932), 211–219. | Zbl

[S2] Saks, S.: Theory of the Integral. Monogr. Matematyczne 7, New York 1937. | Zbl

[Th1] Thomson, B. S.: Real Functions. Lecture Notes in Math. 1170, Springer-Verlag, Berlin 1985. | MR

[Th2] Thomson, B. S.: Symmetric properties of real functions. M. Dekker, New York 1994. | MR | Zbl

[Ti] Tišer, J.: Differentiation theorem for Gaussian measures on Hilbert space. Trans. Amer. Math. Soc. 308 (1988), 655–666. | MR

[U1] Uher, J.: Symmetrically differentiable functions are differentiable almost everywhere. Real Anal. Exchange 8 (1982/83), 253–261. | MR

[U2] Uher, J.: Symmetric semicontinuity implies continuity. Trans. Amer. Math. Soc. 293 (1986), 421–429. | MR | Zbl

[Zh] Zahorski, Z.: Sur la première dérivée. Trans. Amer. Math. Soc. 69 (1950), 1–54. | MR | Zbl

[Zj1] Zajíček, L.: On the differentiation of convex functions in finite and infinite dimensional spaces. Czechoslovak Math. J. 29 (1979), 340–348. | MR

[Zj2] Zajíček, L.: On the symmetry of Dini derivates of arbitrary functions. Comment. Math. Univ. Carolinae 22 (1981), 195–209. | MR

[Zj3] Zajíček, L.: Porosity and $\sigma $-porosity. Real Anal. Exchange 13 (1987/88), 314–350. | MR

[Zj4] Zajíček, L.: The differentiability structure of typical functions in $C[0,1]$. Real Anal. Exchange 13 (1987/88), 119, 113–116, 93.

[Zj5] Zajíček, L.: Unpublished results of K. Pekár and H. Zlonická on preponderant derivatives and $M_4$ sets. Real Anal. Exchange 15 (1989/90), 413–418. | MR

[Zj6] Zajíček, L.: On preponderant differentiability of typical continuous functions. Proc. Amer. Math. Soc. 124 (1996), 789–798. | MR

[Zj7] Zajíček, L.: Ordinary derivative via symmetric derivative and Lipschitz condition via symmetric Lipchitz condition. Real Anal. Exchange 23 (1997/98), 653–669. | MR

[Zj8] Zajíček, L.: On results of Jan Mařík in the theory of derivatives. Math. Bohem. 121 (1996), 385–395. | MR

[Zj9] Zajíček, L.: Differentiability properties of typical continuous functions. Real Anal. Exchange 25 (1999/2000), 149–157.

[Zm] Ziemer, W.: Weakly differentiable functions. Graduate Texts in Mathematics 120, Springer-Verlag, New York 1989. | MR | Zbl

[ZP] Zelený, M., Pelant, J.: The structure of the $\sigma $-ideal of $\sigma $-porous sets. Preprint MFF UK, KMA-1999-01. | MR