Abstract
In this work we establish the weak (1,1) continuity for pseudo-differential operators with symbols in toroidal \((1,\delta )\) classes.
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Agranovich, M.S.: Spectral properties of elliptic pseudodifferential operators on a closed curve. Funct. Anal. Appl. 13, 279–281 (1971)
Ashino, R., Nagase, M., Vaillancourt, R.: Pseudodifferential operators on \(L^p(\mathbb{R}^n)\) spaces. Cubo 6(3), 91–129 (2004)
Cardona, D.: Estimativos \(L^2\) para una clase de operadores pseudodiferenciales definidos en el toro. Rev. Integr. Temas Mat. 31(2), 147–152 (2013)
Delgado, J.: \(L^p\) bounds for pseudo-differential operators on the torus. Oper. Theory Adv. Appl. 231, 103–116 (2012)
Duoandikoetxea, J.: Fourier Analysis. American Mathematical Society, Providence (2000)
Folland, G.B.: Harmonic analysis in phase space. Princeton University Press, Princeton (1989)
Hörmander, L.: Pseudo-differential operators and Hypo-elliptic equations. Proc. symposium on singular Integrals. Am. Math. Soc. 10, 138–183 (1967)
Hörmander, L.: The Analysis of the linear partial differential operators, vol. 3. Springer, Berlin (1985)
Mclean, M.: Local and global description of periodic pseudo-differential operators. Math. Nachr. 150, 151–161 (1991)
Molahajloo, S., Wong, M.W.: Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on \(\mathbb{S}^1.\). J. Pseudo Differ. Oper. Appl. 1, 183–205 (2010)
Molahajloo, S.: A characterization of compact pseudo-differential operators on \(\mathbb{S}^1\) Oper. Theory Adv. Appl, vol. 213, pp. 25–29. Birkhüser, Boston (2011)
Molahajloo, S., Wong, M.W.: Pseudo-differential Operators on \(\mathbb{S}^1\). In: Rodino, L., Wong, M.W. (eds.) New developments in pseudo-differential operators, pp. 297–306 (2008)
Nagase, M.: The \(L^p\) boundedness of pseudo-differential operators with non-regular symbols. Commun. Partial Diff. Equ. 2, 1045–1061 (1977)
Nagase, M.: On a class of \(L^p-\)bounded pseudo-differential operators. Sci. Rep. College Gen. Ed., Osaka Univ. 33 1–7 (1984)
Stein, E.: Harmonic Analysis. Princeton University Press, Princeton (1993)
Ruzhansky, M., Turunen, V.: Pseudo-differential operators and symmetries: background analysis and advanced topics. Birkhaüser, Boston (2010)
Ruzhansky, M., Turunen, V.: Quantization of pseudo-differential operators on the Torus. J Fourier Anal. Appl. 16, 943–982 (2010)
Taylor, M.: Pseudo-differential operators. Four Lectures at MRSI . http://www.unc.edu/math/Faculty/met/msripde.pdf September (2008)
Taylor, M.: Pseudo-differential operators. Princeton University Press, Princeton (1991)
Turunen, V., Vainikko, G.: On symbol analysis of periodic pseudodifferential operators Z. Anal. Anwendungen. 17, 9–22 (1998)
Wong, M.W.: An introduction to pseudo-differential operators, 3rd edn. In: Series on Analysis, Applications and Computation, vol. 6. World Scientific, Singapore (2014)
Wong, M.W.: Discrete fourier analysis. Birkhäuser, Boston (2011)
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Cardona, D. Weak type (1,1) bounds for a class of periodic pseudo-differential operators . J. Pseudo-Differ. Oper. Appl. 5, 507–515 (2014). https://doi.org/10.1007/s11868-014-0101-9
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DOI: https://doi.org/10.1007/s11868-014-0101-9