Abstract
We consider spatially periodic Laplace and Stokes resolvent problems in the whole space and show corresponding weighted resolvent estimates with weights in the Muckenhoupt class. A main tool is the use of Fourier techniques on the Schwartz-Bruhat space and on the tempered distributions together with a weighted transference principle à la Andersen and Mohanty (Proc Amer Math Soc 137(5):1689–1697, 2009) and a splitting of the function spaces into a mean-value free part and a nonperiodic part of functions defined on a lower-dimensional whole space, which then enables us to make use of a weighted Mikhlin multiplier theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, R.A., Fournier, J.J.F.: Sobolev spaces. In: Pure and Applied Mathematics (Amsterdam), vol. 140. 2nd edn. Elsevier/Academic Press, Amsterdam (2003)
Alfsen, E.M.: A simplified constructive proof of the existence and uniqueness of Haar measure. Math. Scand. 12, 106–116 (1963)
Andersen, K.F., Mohanty, P.: Restriction and extension of Fourier multipliers between weighted \(L^p\) spaces on \(\mathbb{R}^n\) and \(\mathbb{T}^n\). Proc. Amer. Math. Soc. 137(5), 1689–1697 (2009)
Boulmezaoud, T.Z., Razafison, U.: On the steady Oseen problem in the whole space. Hiroshima Math. J. 35(3), 371–401 (2005)
Bruhat, F.: Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes \(\wp \)-adiques. Bull. Soc. Math. France 89, 43–75 (1961)
Cartan, H.: Sur la mesure de Haar. C. R. Acad. Sci. Paris 211, 759–762 (1940)
de Leeuw, K.: On \(L_{p}\) multipliers. Ann. Math. 2(81), 364–379 (1965)
Denk, R., Hieber, M., Prüß, J.: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166(788), viii+114 (2003)
Edwards, R.E., Gaudry, G.I.: Littlewood-Paley and multiplier theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90. Springer, Berlin (1977)
Farwig, R., Sohr, H.: Weighted \(L^{q}\)-theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan 49(2), 251–288 (1997)
Fröhlich, A.: The Stokes operator in weighted \(L^{q}\)-spaces. II. Weighted resolvent estimates and maximal \(L^{p}\)-regularity. Math. Ann. 339(2), 287–316 (2007)
García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. In: North-Holland Mathematics Studies, vol. 116. North-Holland Publishing Co, Amsterdam (1985)
Haar, A.: Der Maßbegriff in der Theorie der kontinuierlichen Gruppen. Ann. Math. (2) 34(1), 147–169 (1933)
Iooss, G.: Théorie non linéaire de la stabilité des écoulements laminaires dans le cas de “l’échange des stabilités”. Arch. Rational Mech. Anal. 40, 166–208 (1970/1971)
Kunstmann, P.C., Weis, L.: Maximal \(L_{p}\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^{\infty }\)-functional calculus. In: Functional Analytic Methods for Evolution Equations. Lecture Notes in Mathematics, vol. 1855, pp. 65–311. Springer, Berlin (2004)
Kyed, M.: Maximal regularity of the time-periodic Navier-Stokes system. J. Math. Fluid. Mech. 16(3), 523–538 (2014)
Tobias, N.: \(\text{L}^{\rm p}\)-theory of cylindrical boundary value problems. Springer Spektrum, Wiesbaden, 2012. An operator-valued Fourier multiplier and functional calculus approach, Dissertation, University of Konstanz, Konstanz (2012)
Osborne, M.S.: On the Schwartz-Bruhat space and the Paley-Wiener theorem for locally compact abelian groups. J. Funct. Anal. 19, 40–49 (1975)
Pontryagin, L.S.: Nepreryvnye gruppy (in Russian), 2nd edn. Gosudarstv. Izdat. Tehn.-Teor. Lit, Moscow (1954)
Rudin, W.: Fourier analysis on groups. In: Interscience Tracts in Pure and Applied Mathematics, vol. 2. Interscience Publishers, New York-London (1962)
Sauer, J.: Extrapolation Theorem on Locally Compact Abelian Groups. Fachbereich Mathematik, TU Darmstadt, preprint no. 2688 (2014)
Sauer, J.: Maximal Regularity of the Spatially Periodic Stokes Operator and Application to Nematic Liquid Crystal Flows. Fachbereich Mathematik, TU Darmstadt, preprint no. 2690 (2014)
Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. In: Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton (1971)
Weil, A.: L’intégration dans les groupes topologiques et ses applications. Actual. Sci. Ind., no. 869. Hermann et Cie., Paris (1940)
Weis, L.: A new approach to maximal \(L_{p}\)-regularity. In: Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998). Lecture Notes in Pure and Applied Mathematics, vol. 215, pp. 195–214. Dekker, New York (2001)
Acknowledgments
The author has been supported by the International Research Training Group 1529.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sauer, J. Weighted resolvent estimates for the spatially periodic Stokes equations. Ann Univ Ferrara 61, 333–354 (2015). https://doi.org/10.1007/s11565-014-0221-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11565-014-0221-4