Abstract
We give an analytic description for the completion of \(C_0^\infty (\mathbb {R}_+)\), where \(\mathbb {R}_+= (0,\infty )\), in Dirichlet space \(D^{1,p}(\mathbb {R}_+, \omega ):= \{ u \, :\mathbb {R}_+\rightarrow {{\mathbb {R}}}: u\ \) is locally absolutely continuous on \(\mathbb {R}_+\, and \, \Vert u^{'}\Vert _{L^p(\mathbb {R}_+, \omega )}<\infty \}\), for given continuous positive weight \(\omega \) defined on \(\mathbb {R}_+\), where \(1<p<\infty \). The conditions are described in terms of the modified variants of the \(B_p\) conditions due to Kufner and Opic from 1984, which in our approach are focusing on the integrability of \(\omega ^{-p/(p-1)}\) near zero or near infinity. Moreover, we propose applications of our results to: obtaining new variants of Hardy inequality, interpretation of boundary value problems in ODE’s defined on the half-line with solutions in \(D^{1,p}(\mathbb {R}_+, \omega )\), new results from complex interpolation theory dealing with interpolation spaces between weighted Dirichlet spaces, and for deriving new Morrey type embedding theorems for our Dirichlet space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Availability of data and materials
All of the material in the manuscript is owned by the authors and no permissions are required.
References
Adams, R., Fournier, J. F.: Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam (2003)
Cwikel, M., Einav, A.: Interpolation of weighted Sobolev spaces. J. Funct. Anal. 277(7), 2381–2441 (2019)
Dacorogna, B.: Direct methods in the calculus of variations. Second edition. Applied Mathematical Sciences 78. Springer, New York (2008)
Dhara, R.N., Kałamajska, A.: On equivalent conditions for the validity of Poincaré inequality on weighted Sobolev space with applications to the solvability of degenerated PDEs involving p-Laplacian. J. Math. Anal. Appl. 432(1), 463–483 (2015)
Doktorskiǐ, R. Ya.: Reiterative relations of the real interpolation method. (Russian) Dokl. Akad. Nauk SSSR 321(2) (1991), 241–245; translation in Soviet Math. Dokl. 44(3), 665–669 (1992)
Gogatishvili, A., Opic, B., Trebels, W.: Limiting reiteration for real interpolation with slowly varying functions. Math. Nachr. 278(1–2), 86–107 (2005)
Hajłasz, P., Kałamajska, A.: Polynomial asymptotics and approximation of Sobolev functions. Studia Math. 113(1), 55–64 (1995)
Kaczmarek, R., Kałamajska, A.: Density results and trace operator in weighted Sobolev Spaces defined on the half line equipped with power weights. J. Approx. Theory 291 (2023), Paper No. 105896. https://doi.org/10.1016/j.jat.2023.105896
Kałamajska, A., Pietruska-Pałuba, K.: On a variant of the Hardy inequality between weighted Orlicz spaces. Studia Math. 193(1), 1–28 (2009)
Kufner, A.: Weighted Sobolev Spaces. Translated from the Czech. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, (1985)
Kufner, A., John, O., Fucik, S.: Function spaces. Analysis. Noordhoff International Publishing, Leyden; Academia, Prague, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics (1977)
Kufner, A., Opic, B.: Hardy-type inequalities. Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Harlow (1990)
Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Commentationes Mathematicae Universitatis Carolinae 25(3), 537–554 (1984)
Maz’ya, V.G.: On weak solutions of the Dirichlet and Neumann problems. Trans. Moscow Math. Soc. 20, 135–172 (1969)
Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)
Maz’ya, V.G.: Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Contemp. Math. 338, 307–340 (2003)
Meerschaert, M.M., Sikorskii, A.: Stochastic models for fractional calculus. Second edition. De Gruyter Studies in Mathematics, 43. De Gruyter, Berlin (2019)
Martin, J., Milman, M.: Pointwise symmetrization inequalities for Sobolev functions and applications. Adv. Math. 225(1), 121–199 (2010)
Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Transactions of the American Mathematical Society 165, 207–226 (1972)
Rudin, W.: Functional analysis, 2nd edn. International Series in Pure and Applied Mathematics. McGraw-Hill Inc, New York (1991)
Sinnamon, G.J.: Weighted Hardy and Opial-type inequalities. J. Math. Anal. Appl. 160(2), 434–445 (1991)
Sobolev, S.L.: The density of compactly supported functions in the space \(L_p^{(m)}(E)\). ( in Russian), Sibirsk. Mat. Ž. 4, 673–682 (1963)
Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3(4), 697–718 (1976)
Acknowledgements
A.K. wish to thank several very pleasant stays at Istituto per le Applicazioni del Calcolo “Mauro Picone” Consiglio Nazionale delle Ricerche, where our collaboration originated. We also would like to thank the anonymous reviewer for useful comments.
Funding
The work of CC has been partially supported by Istituto Nazionale di Alta Matematica/Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this work.
Corresponding author
Ethics declarations
Ethical Approval
Not applicable.
Competing interests
The authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Claudia Capone and Agnieszka Kałamajska contributed equally to this work.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Capone, C., Kałamajska, A. Asymptotics, Trace, and Density Results for Weighted Dirichlet Spaces Defined on the Half-line. Potential Anal 60, 1301–1331 (2024). https://doi.org/10.1007/s11118-023-10089-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-023-10089-2