Abstract
In this paper we establish Sobolev type compact embedding theorems for Hörmander classes of pseudodifferential operators \(OpS^{-\alpha }_{1,\delta }\) on localizable Hardy space. Our work include new optimal boundedness results. As application, we obtain compact embeddings for compactly supported distributions with respect to the space variables in the nonhomogeneous localizable Hardy-Sobolev spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Álvarez, J., Hounie, J.: Estimates for the kernel and continuity properties of pseudodifferential operators. Ark. Mat. 28, 1–22 (1990)
Auscher, P., Russ, E., Tchamitchian, P.: Hardy Sobolev spaces on strongly Lipschitz domains of \(\mathbb {R}^{n}\). J. Funct. Anal. 218, 54–109 (2005)
Badr, N., Dafni, G.: An atomic decomposition of the Hajlasz Sobolev space \({m^{1}_{1}}\) on manifolds. J. Funct. Anal. 259(6), 1380–1420 (2010)
Brezis, H.: Functional Analysis. In: Sobolev spaces and Partial Differential Equations. Springer (2011)
Cho, Y. -K.: Inequalities related to hp smoothness of Sobolev type. Integr Equ Oper Theory 35, 471–484 (1999)
Dafni, G.: Nonhomogeneus div-curl lemmas and local hardy spaces. Adv. Diff. Eq. 10(5), 505–526 (2005)
Fefferman, C., Stein, E.: Hp spaces of several variables. Acta Math. 129, 137–193 (1972)
Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46, 27–42 (1979)
Hoepfner, G., Hounie, J., Picon, T.: Div curl type estimates for elliptic systems of complex vector fields. J. Math. Anal. Appl. 429, 774–799 (2015)
Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations, (Mathématiques et Applications). Springer, Berlin (1997)
Hounie, J.: Introdução aos Operadores Pseudo-Diferenciais. IMPA, Rio de Janeiro (1987)
Hounie, J., Kapp, R.: Pseudodifferential operators on local hardy spaces. J. Fourier Anal. Appl. 15, 153–178 (2009)
Johnsen, J.: Lp-theory of type 1, 1 −operators. Math. Nachr. 286(7), 712–729 (2013)
Koskela, P., Saksman, E.: Pointwise characterization of Hardy-Sobolev functions. Math. Res. Lett. 15(4), 727–744 (2008)
Krantz, S.: Fractional integration on Hardy Spaces. Studia Math. 63, 87–94 (1982)
Miyachi, A.: Hardy-Sobolev spaces maximal functions. J. Math. Soc. Japan 42(1), 73–90 (1990)
Moonens, L., Picon, T.: Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields. J. Funct. Anal. 275 (5), 1073–1099 (2018)
Päivärinta, L., Somersalo, E.: A generalization of the Calderon-Vaillancourt́ theorem to Lp and hp. Math. Nachr. 138, 145–156 (1988)
Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, New Jersey (1970)
Stein, E.: Harmonic Analysis: Real-Variable Methods, orthogonality, and Oscillatory Integrals. Princeton University Press, New Jersey (1993)
Stein, E., Weiss, G.: On the theory of harmonic functions of several variables I. Acta Math. 103, 25–62 (1960)
Taylor, M.: Tools for PDE. American Mathematical Society, Providence (2000)
Taylor, M.: Partial Differential Equations II, Qualitative Studies of Linear Equations, Applied Mathematical Sciences, 2nd edn. Springer, Berlin (2011)
Triebel, H.: Theory of function spaces, Monographs in Math. 78 Birkhauser (1983)
Acknowledgment
We wish to thank Prof. Jorge Hounie (Universidade Federal de São Carlos) and the referee for helpful discussions and suggestions concerning this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Work supported in part by CNPq (grant number 308826/2018-3, 311430/2018-0 and 409306/2016-9) and FAPESP (grant numbers 2018/14316-3, 2019/04995-3 and 2018/15484-7).
Rights and permissions
About this article
Cite this article
Hoepfner, G., Kapp, R. & Picon, T. On the Continuity and Compactness of Pseudodifferential Operators on Localizable Hardy Spaces. Potential Anal 55, 491–512 (2021). https://doi.org/10.1007/s11118-020-09866-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-020-09866-0