singular integral operator; differentiable function; Boyd functions; pointwise regularity
Abstract :
[en] We generalize the T_u^p spaces introduced by Calderón and Zygmund and show that most of
the results obtained in their study of the pointwise estimates for solutions of elliptic partial
differential equations and systems can be generalized in this framework with Lp-conditions.
Disciplines :
Mathematics
Author, co-author :
Loosveldt, Laurent ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes
Nicolay, Samuel ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes
Language :
English
Title :
Generalized T_u^p spaces: On the trail of Calderón and Zygmund
Alternative titles :
[fr] Espace T_u^p spaces: sur la piste de Calderón et Zygmund
Publication date :
October 2020
Journal title :
Dissertationes Mathematicae
ISSN :
0012-3862
eISSN :
1730-6310
Publisher :
Institute of Mathematics, Polish Academy of Sciences, Varsovie, Poland
Volume :
554
Peer reviewed :
Peer reviewed
Funders :
F.R.S.-FNRS - Fonds de la Recherche Scientifique [BE]
C. Bennett and R. C. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, 1988.
J. Björn, Lq-differentials for weighted Sobolev spaces, Michigan Math. J. 47 (2000), 151–161.
V. Bogachev, Measure Theory, Springer, 2007.
D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice: With Smile, Inflation and Credit, 2nd ed., Springer, 2006.
V. Burenkov, Extension theorems for Sobolev spaces, in: J. Rossmann et al. (eds.), The Maz’ya Anniversary Collection. Vol. 1, 109 Operator Theory, Adv. Appl. 109, Birkhäuser, 1999, 187–200.
A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309.
A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79 (1957), 901–921.
A. P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 181–225.
F. Cobos and D. L. Fernandez, Hardy–Sobolev spaces and Besov spaces with a function parameter, in: M. Cwikel et al. (eds.), Function Spaces and Applications (Lund, 1986), Springer, 1988, 158–170.
D. E. Edmunds, P. Gurka, and B. Opic, On embeddings of logarithmic Bessel potential spaces, J. Funct. Anal. 146 (1997), 116–150.
D. E. Edmunds and D. Haroske, Spaces of Lipschitz type, embeddings and entropy numbers, Dissertationes Math. 380 (1999), 43 pp.
D. E. Edmunds and H. Triebel, Logarithmic Sobolev spaces and their applications to spectral theory, Proc. London Math. Soc. 71 (1995), 333–371.
L. Grafakos, Classical Fourier Analysis, 2nd ed., Grad. Texts in Math. 249, Springer, 2008.
E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Colloq. Publ. 31, Amer. Math. Soc., 1996.
J. C. Hull, Options, Futures, and Other Derivatives, 9th ed., Prentice-Hall, 2014.
S. Jaffard and B. Martin, Multifractal analysis of the Brjuno function, Invent. Math. 212 (2018), 109–132.
G. A. Kalyabin and P. I. Lizorkin, Spaces of generalized smoothness, Math. Nachr. 133 (1987), 7–32.
T. Kleyntssens and S. Nicolay, A refinement of the Sν-based multifractal formalism: Toward the detection of the law of the iterated logarithm, submitted, 2017.
D. Kreit and S. Nicolay, Some characterizations of generalized Hölder spaces, Math. Nachr. 285 (2012), 2157–2172.
D. Kreit and S. Nicolay, Characterizations of the elements of generalized Hölder–Zygmund spaces by means of their representation, J. Approx. Theory 172 (2013), 23–36.
D. Kreit and S. Nicolay, Generalized pointwise Hölder spaces defined via admissible sequences, J. Funct. Spaces 2018, art. 8276258, 11 pp.
H.-G. Leopold, Embeddings and entropy numbers in Besov spaces of generalized smoothness, in: Function Spaces (Poznań, 1998), Lecture Notes in Pure Appl. Math. 213, Dekker, New York, 2000, 323–336.
L. Loosveldt and S. Nicolay, Generalized spaces of pointwise regularity: To a general framework for the WLM, submitted, 2019.
L. Loosveldt and S. Nicolay, Some equivalent definitions of Besov spaces of generalized smoothness, Math. Nachr. 292 (2019), 2262–2282.
C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, in: M. Cwikel and J. Peetre (eds.), Interpolation Spaces and Allied Topics in Analysis (Lund, 1983), Springer, 1983, 183–201.
S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965.
C. Müller, Spherical Harmonics, Lecture Notes in Math. 17, Springer, 1966.
S. Nicolay and D. Kreit, A refined method for estimating the global Hölder exponent, in: S. Latifi (ed.), Information Technology: New Generations, Adv. Intelligent Systems Computing 448, Springer, 2016, 949–958.
P. J. Olver, Introduction to Partial Differential Equations, Undergrad. Texts in Math., Springer, 2014.
B. Opic and W. Trebels, Bessel potentials with logarithmic components and Sobolev-type embeddings, Anal. Math. 26 (2000), 299–319.
S. Seuret and A. Ubis, Local L2-regularity of Riemann’s Fourier series, Ann. Inst. Fourier (Grenoble) 67 (2017), 2237–2264.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, NJ, 1970.
W. A. Strauss, Partial Differential Equations: An Introduction, 2nd ed., Wiley, 2008.
L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of Un. Mat. Ital. 3, Springer, 2007.
W. P. Ziemer, Weakly Differentiable Functions, Grad. Texts in Math. 120, Springer, 1989.
