Abstract
We introduce the Hölder-Zygmund spaces with a variable smoothness exponent, define equivalent norms, and prove a theorem on boundedness of pseudodifferential operators with Hörmander symbols in these spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hörmander L., “Pseudo-differential operators and hypoelliptic equations,” Singular Integrals. Proc. Symp. Pure Math., 10, 138–183 (1966).
Beals R., “Lp and Hölder estimates for pseudodifferential operators: Sufficient conditions,” Ann. Inst. Fourier, Grenoble, 29, No. 3, 239–260 (1979).
Karapetians N. K. and Ginzburg A. I., “Fractional integrals and singular integrals in the Hölder classes of variable order,” Integral Transforms Spec. Funct., 2, No. 2, 91 (1994).
Edmunds D. E. and Rákosnik J., “Sobolev embeddings with variable exponent,” Studia Math., 143, No. 3, 267–293 (2000).
Fan X., Shen J., and Zhao D., “Sobolev embeddings theorems for W k,p(·)(Ω),” J. Math. Anal. Appl., 262, No. 2, 749–760 (2001).
Zhikov V. V., “Solvability of the three-dimensional thermistor problem,” Proc. Steklov Inst. Math., 261, 98–111 (2008).
Zhikov V. V., “On the technique for passing to the limit in nonlinear elliptic equations,” Funct. Anal. Appl., 43, No. 2, 96–112 (2009).
Rabinovich V. S. and Samko S. G., “Singular integral operators on weighted variable exponent Lebesgue spaces on composed Carleson curves,” Funct. Anal. Appl., 46, No. 1, 73–76 (2012).
Stein E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton (1993).
Taylor M. E., Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Amer. Math. Soc., Providence, RI (2000) (Math. Surv. Monogr.; V. 81).
Rabinovich V. S., “Fredholm property of pseudo-differential operators on weighted Hölder-Zygmund spaces,” Oper. Theory, Adv. Appl., 164, No. 2, 95–114 (2006).
Kryakvin V. D., “Criteria for the compactness and the Fredholm property of pseudodifferential operators in Holder-Zygmund weighted spaces,” Differential Equations, 45, No. 1, 102–112 (2009).
Kryakvin V. D., “Characterization of pseudodifferential operators in Hölder-Zygmund spaces,” Differential Equations, 49, No. 3, 306–312 (2013).
Golovkin K. K., “On equivalent normalizations of fractional spaces,” Trudy Mat. Inst. Steklova, LXVI, 364–383 (1962).
Marchaud A., “Sur les dérivées et sur le différences des fonctions de variables rèeles,” J. Math. Pures Appl., 6, 337–425 (1927).
Jaffard S. and Meyer Y., “Wavelet methods for pointwise regularity and local oscillations of functions,” Mem. Amer. Math. Soc., 123, No. 587, 1–110 (1996).
Meyer Y., Wavelets and Operators, Cambridge Univ. Press, Cambridge (1992).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2014 Kryakvin V.D.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 6, pp. 1315–1327, November–December, 2014.
Rights and permissions
About this article
Cite this article
Kryakvin, V.D. Boundedness of pseudodifferential operators in Hölder-Zygmund spaces of variable order. Sib Math J 55, 1073–1083 (2014). https://doi.org/10.1134/S0037446614060093
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446614060093