Abstract
What follows is a survey of recent results in several areas of the theory of function spaces which have been intensively studied in last years. In particular, we concentrate on the boundedness and compactness of the Hardy and imbedding operators and on the Poincaré and Friedrichs inequalities in weighted spaces, taking the general setting of Banach function spaces as a common framework. We do not attempt to present an exhaustive survey of the topics. Our aim is to show the current state of the subjects in question with an emphasis on our contribution to their solution.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W.A.J. Luxemburg: Banach function spaces ( Thesis: Technische Hogeschool to Delft, 1955 ).
P.R. Halmos: Measure theory. Graduate Texts in Math. 18. Springer-Verlag, New York-Heidelberg-Berlin, 1974.
C. Bennett and R. Sharpley: Interpolation of operators. Pure Appl. Math. 129, Academic Press, Inc., 1988.
S. Bradley: Hardy inequalities with mixed norms. Canad. Math. Bull. 21 (1978), no. 4, 405–408.
V. G. Maz’ya: Sobolev Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1985. (An extended and improved version of the books Einbettungssätze für Sobolevsche Räume 1, 2; Zur Theorie der Sobolevsche Räume. Teubner-Texte zur Mathematik, Vols. 21, 28, 38. Teubner, Leipzig, 1979, 1980, 1981.)
G. J. Sinnamon: Weighted Hardy and Opial-type inequalities. J. Math. Anal. Appl. 160 (1991), 434–445.
L. Pick: Weighted modular estimates for the Hardy operator. Preprint.
S. Bloom and R. Kerman: Weighted L4,integral inequalities for operators of Hardy type. Preprint, Brock University.
E. Sawyer: Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans. Amer. Math. Soc. 281 (1984), no. 1, 329–337.
E.I. Berezhnoi: Inequality with weights for Hardy operators in function spaces. Function spaces, Teubner—Texte zur Mathematik, Band 120. B.G. Teubner Verlagsgesellschaft, Stuttgart—Leipzig, 1991, 75–79.
K. Knopp: Über Reihen mit positiven Gliedern. J. London Math. Soc. 3 (1928), 205–211.
H. P. Heinig: Weighted inequalities in Fourier analysis. Nonlinear Analysis, Function Spaces and Applications, 4. Teubner—Texte zur Mathematik, Band 119, Leipzig, 1990, 42–85.
H.P. Heinig, R. Kerman, M. Krbec: Weighted exponential inequalities. Preprint no. 79, Math. Inst. Czech. Acad. Sci., Prague, 1992, 30 pp.
B. Opic and P. Gurka: Weighted inequalities for geometric means. Preprint no. 75 Math. Inst. Czech. Acad. Sci., Prague, 1992, 11 pp. (To appear in Proc. Amer. Math. Soc.)
L. Pick and B. Opic: On the geometric mean operator. Preprint no. 78 Math. Inst. Czech. Acad. Sci., Prague, 1992, 9 pp. (An extended version to appear in J. Math. Anal. Appl.)
Q. Lai and L. Pick: The Hardy operator, L oo ,and BMO. (To appear in J. London Math. Soc.)
B. Opic and A. Kufner: Hardy-type inequalities. Pitman Research Notes in Mathematics Series, 211. Longman Scientific and Technical, Harlow, 1990.
B. Opic: On the distance of the Riemann—Liouville operator from compact operators. Preprint no. 80 Math. Inst. Czech. Acad. Sci., Prague, 1992, 10 pp. (To appear in Proc. Amer. Math. Soc.)
F. J. Martin-Reyes and E. Sawyer: Weighted inequalities for Riemann—Liouville fractional integrals of order one and greater. Proc. Amer. Math. Soc. 106 (1989), 727–733.
V. D. Stepanov: Weighted inequalities for a class of Volterra convolution operators. J. London Math. Soc. (2) 45 (1992), 232–242.
V. D. Stepanov: Two-weighted estimates for Riemann—Liouville integrals. Preprint no. 39, Math. Inst. Czech. Acad. Sci., Prague, 1988, 28 pp.
S. Bloom and R. Kerman: Weighted norm inequalities for operators of Hardy type. Proc. Amer. Math. Soc. 113 (1991), 135–141.
R. K. Juberg: The measure of non-compactness in LP for a class of integral operators. Indiana Univ. Math. J. 23 (1974), 925–936.
Lit. Mat. Sb. 21 (1981), 143–148.
A. Avantaggiati: On compact embedding theorems in weighted Sobolev spaces. Czechoslovak Math. J. 29 (104) (1979), no. 4, 635–648.
M. Krbec and L. Pick: On imbeddings between weighted Orlicz spaces. Z. Anal. Anwend. 10 (1991), no. 1, 105–116.
L. Pick: A remark on continuous imbeddings between Banach function spaces. Colloquia Mathematica Societatis Janos Bolyai, Vol. 58. Approximation theory, Kecskemét (Hungary), 1990, 571–581.
J. A. Cavanati and F. Galaz—Fontes: Compactness of imbeddings between Banach spaces and applications to Sobolev spaces. J. London Math. Soc. (2) 41 (1990), 511–525.
P. Gurka and B. Opic: Continuous and compact imbeddings of weighted Sobolev spaces I, II, III. Czechoslovak Math. J. 38 (133) (1988), 730–744; ibid. 39 (114) (1989), 78–94; ibid. 41 (116) (1991), 317–341.
B. Opic and J. Râkosnik: Estimates for mixed derivatives of functions from anisotropic Sobolev—Slobodeckij spaces with weights. Quart. J. Math. Oxford (2) 42 (1991), 347–363.
M. Krbec, B. Opic and L. Pick: Imbedding theorems for weighted Orlicz—Sobolev spaces. J. London Math. Soc. 46 (3) (1992), 543–556.
D. E. Edmunds and B. Opic: Weighted Poincaré and Friedrichs inequalities. (To appear in J. London Math. Soc.)
D. E. Edmunds, B. Opic and L. Pick: Poincaré and Friedrichs inequalities in abstract Sobolev spaces. (To appear in Math. Proc. Camb. Phil. Soc.)
D. E. Edmunds, B. Opic and J. Râkosnik: Poincaré and Friedrichs inequalities in abstract Sobolev spaces II. (To appear in Math. Proc. Camb. Phil. Soc.)
C. J. Amick: Some remarks on Rellichs theorem and the Poincaré inequality. J. London Math. Soc. (2) 18 (1978), 81–93.
D. E. Edmunds and W. D. Evans: Spectral theory and embeddings of Sobolev spaces. Quart. J. Math. Oxford (2) 30 (1979), 431–453.
D.E. Edmunds and W. D. Evans: Spectral theory and differential operators. Oxford Mathematical Monographs. Clarendon Press, Oxford, 1987.
W. D. Evans and D. J. Harris: Sobolev embeddings for generalized ridged domains. Proc. London Math. Soc. (3) 54 (1987), 141–175.
G.H. Hardy, J.E. Littlewood and G. Plya: Inequalities. University Press, Cambridge, 1952.
A. Kufner, O. John and S. Fucik: Function spaces. Academia, Prague, 1977.
V.M. Kokilashvili: On Hardy’s inequalities in weighted spaces (Russian).Soobshch. Akad. Nauk Gruzin. SSR 96 (1979), no. 1, 37–40.
B. Muckenhoupt: Hardy’s inequality with weights. Studia Math. 44 (1972), 31–38.
G. Tomaselli: A class of inequalities. Boll. Un. Mat. Ital. (4) 2 (1969), 622–631.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Fachmedien Wiesbaden
About this chapter
Cite this chapter
Krbec, M., Opic, B., Pick, L., Rákosnik, J. (1993). Some Recent Results on Hardy Type Operators in Weighted Function Spaces and Related Topics. In: Schmeisser, HJ., Triebel, H. (eds) Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte zur Mathematik, vol 133. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-11336-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-663-11336-2_4
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-8154-2045-4
Online ISBN: 978-3-663-11336-2
eBook Packages: Springer Book Archive