Abstract
Weighted spaces of continuous functions are introduced such that Weyl fractional integrals with orders from any finite nonnegative interval define equicontinuous sets of continuous linear endomorphisms for which the semigroup law of fractional orders is valid. The result is obtained from studying continuity and boundedness of convolution as a bilinear operation on general weighted spaces of continuous functions and measures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Berberian, On the convolution of a measure and a function. Proc. Amer. Math. Soc. 53 (1975), 399–403.
N. Bourbaki, Elements of Mathematics: Integration I.. Springer, Berlin (2004).
N. Bourbaki, Elements of Mathematics: Integration II.. Springer, Berlin (2004).
B. Davey, H. Priestley, Introduction to Lattices and Order2 Cambridge University Press, Cambridge (2002).
R. Edwards, The stability of weighted Lebesgue spaces. Trans. Amer. Math. Soc. 93 (1959), 369–394.
H. Feichtinger, Gewichtsfunktionen auf lokalkompakten Gruppen. Sitzungsberichte der österreichischen Akademie der Wissenschaften Mathem.-naturw. Klasse, Abt.II 188 (1979), 451–471.
G. Gaudry, Multipliers of weighted Lebesgue and measure spaces. Proc. London Math. Soc. 19 (1969), 327–340.
R. Hilfer, Applications of Fractional Calculus in Physics.. World Scientific Publ. Co., Singapore (2000), 10.1142/3779; https://www.worldscientific.com/worldscibooks/10.1142/3779.
R. Hilfer, Y. Luchko, Desiderata for fractional derivatives and integrals. Mathematics 7 (2019), ID 149. 10.3390/math7020149; https://www.mdpi.com/2227-7390/7/2/149.
E. Hille, R. Phillips, Functional Analysis and Semi-Groups.. American Mathematical Society, Providence (1957).
H. Hogbe-Nlend, Bornologies and Functional Analysis.. North-Holland Publ. Co., Amsterdam (1977).
T. Kleiner, R. Hilfer, Convolution operators on weighted spaces of continuous functions and supremal convolution. Annali di Matematica Pura ed Applicata 198 (2019).
M.L. Mazure, Equations de convolution et formes quadratiques. Annali di Matematica Pura ed Applicata CLVIII (1991), 75–97.
K. Miller, The Weyl fractional calculus. In: B. Ross, Fractional Calculus and its Applications, Lecture Notes in Math. 457 (1975), 80–91, Springer–Verlag.
J. Moreau, Fonctionelles convexes. Seminaire Jean Leray 2 (1967), 1–108.
L. Nachbin, Elements of Approximation Theory.. Instituto de Matematica Pura e Aplicada de Conselho Nacional de Pesquisas, Rio de Janeiro (1965).
J. Prolla, Weighted spaces of vector-valued continuous functions. Annali di Matematica Pura ed Applicata 89 (1971), 145–157.
H. Reiter, Über den Satz von Weil-Cartier. Monatshefte für Mathematik 86 (1978), 13–62.
H. Rockafellar, R. Wets, Variational Analysis. Grundlehren der mathematischen Wissenschaften 317 Springer, Berlin (2009).
K. Rosenthal, Quantales and Their Applications.. Longman Group, London (1990).
S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives.. Gordon and Breach, Berlin (1993).
W. Summers, Dual spaces of weighted spaces. Trans. of Amer. Math. Soc. 151 (1970), 323–333.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Kleiner, T., Hilfer, R. Weyl Integrals on Weighted Spaces. FCAA 22, 1225–1248 (2019). https://doi.org/10.1515/fca-2019-0065
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2019-0065