On the differentiability of Lipschitz-Besov functions
HTML articles powered by AMS MathViewer
- by José R. Dorronsoro PDF
- Trans. Amer. Math. Soc. 303 (1987), 229-240 Request permission
Abstract:
${L^r}$ and ordinary differentiability is proved for functions in the Lipschitz-Besov spaces $B_a^{p,q},\;1 \leqslant p < \infty ,\;1 \leqslant q \leqslant \infty ,\;a > 0$, using certain maximal operators measuring smoothness. These techniques allow also the study of lacunary directional differentiability and of tangential convergence of Poisson integrals.References
-
D. Adams, Lectures on ${L^p}$ potential theory, Umeȧ Univ. Reports, No. 2, 1981.
—, The classification problem for the capacities associated with the Besov and Triebel-Lizorkin spaces (preprint).
- N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385–475 (English, with French summary). MR 143935
- Thomas Bagby and William P. Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129–148. MR 344390, DOI 10.1090/S0002-9947-1974-0344390-6
- Alberto-P. Calderón and Ridgway Scott, Sobolev type inequalities for $p>0$, Studia Math. 62 (1978), no. 1, 75–92. MR 487419, DOI 10.4064/sm-62-1-75-92
- A.-P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171–225. MR 136849, DOI 10.4064/sm-20-2-181-225
- Calixto P. Calderón, Lacunary spherical means, Illinois J. Math. 23 (1979), no. 3, 476–484. MR 537803
- Calixto P. Calderón, Lacunary differentiability of functions in $\textbf {R}^{n}$, J. Approx. Theory 40 (1984), no. 2, 148–154. MR 732696, DOI 10.1016/0021-9045(84)90024-8
- C. P. Calderon, E. B. Fabes, and N. M. Rivière, Maximal smoothing operators, Indiana Univ. Math. J. 23 (1973/74), 889–898. MR 341058, DOI 10.1512/iumj.1974.23.23073
- Daniel J. Deignan and William P. Ziemer, Strong differentiability properties of Bessel potentials, Trans. Amer. Math. Soc. 225 (1977), 113–122. MR 422645, DOI 10.1090/S0002-9947-1977-0422645-7
- Ronald A. DeVore and Robert C. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc. 47 (1984), no. 293, viii+115. MR 727820, DOI 10.1090/memo/0293
- José R. Dorronsoro, Poisson integrals of regular functions, Trans. Amer. Math. Soc. 297 (1986), no. 2, 669–685. MR 854092, DOI 10.1090/S0002-9947-1986-0854092-3
- Herbert Federer and William P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p$-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139–158. MR 435361, DOI 10.1512/iumj.1972.22.22013
- Norman G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255–292 (1971). MR 277741, DOI 10.7146/math.scand.a-10981
- Yoshihiro Mizuta, On the boundary limits of harmonic functions with gradient in $L^{p}$, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 99–109 (English, with French summary). MR 743623
- Alexander Nagel, Walter Rudin, and Joel H. Shapiro, Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. of Math. (2) 116 (1982), no. 2, 331–360. MR 672838, DOI 10.2307/2007064
- Alexander Nagel and Elias M. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 (1984), no. 1, 83–106. MR 761764, DOI 10.1016/0001-8708(84)90038-0
- C. J. Neugebauer, Strong differentiability of Lipschitz functions, Trans. Amer. Math. Soc. 240 (1978), 295–306. MR 489599, DOI 10.1090/S0002-9947-1978-0489599-X
- C. J. Neugebauer, Smoothness of Bessel potentials and Lipschitz functions, Indiana Univ. Math. J. 26 (1977), no. 3, 585–591. MR 440359, DOI 10.1512/iumj.1977.26.26046
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Britt-Marie Stocke, Differentiability properties of Bessel potentials and Besov functions, Ark. Mat. 22 (1984), no. 2, 269–286. MR 765414, DOI 10.1007/BF02384383
- Mitchell H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean $n$-space. I. Principal properties, J. Math. Mech. 13 (1964), 407–479. MR 0163159
- H. Triebel, Theory of function spaces, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 38, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1983. MR 730762, DOI 10.1007/978-3-0346-0416-1
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 303 (1987), 229-240
- MSC: Primary 46E35; Secondary 26B05
- DOI: https://doi.org/10.1090/S0002-9947-1987-0896019-5
- MathSciNet review: 896019