Characterizations for the Fractional Integral Operators

A. Eroglu ; V. S. Guliev ; J. V. Azizov

Matematičeskie zametki, Tome 102 (2017) no. 5, pp. 789-804.

Résumé

In this paper, we study the boundedness of the fractional integral operator $I_{\alpha}$ on Carnot group $\mathbb{G}$ in the generalized Morrey spaces $M_{p,\varphi}(\mathbb{G})$. We shall give a characterization for the strong and weak type boundedness of $I_{\alpha}$ on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian $\mathcal{L}$ on $\mathbb{G}$, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.

Détail

Mots-clés : Carnot group
Keywords: fractional integral operator, generalized Morrey space.

Bibliographie

[1] A. Kaplan, “Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratics forms”, Trans. Amer. Math. Soc., 258:1 (1980), 147–153 | DOI | MR | Zbl

[2] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, Berlin, 2007 | MR | Zbl

[3] E. M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Math. Ser., 43, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[4] G. B. Folland, E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes, 28, Princeton Univ. Press, Princeton, 1982 | MR | Zbl

[5] G. B. Folland, E. M. Stein, “Estimates for the $\partial_{b}$-complex and analysis on the Heisenberg group”, Comm. Pure Appl. Math., 27 (1974), 429–522 | DOI | MR | Zbl

[6] N. Th. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math., 100, Cambridge Univ. Press, Cambridge, 1992 | MR | Zbl

[7] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud., 105, Princeton Univ. Press, Princeton, NJ, 1983 | MR | Zbl

[8] C. B. Morrey, Jr., “On the solutions of quasi-linear elliptic partial differential equations”, Trans. Amer. Math. Soc., 43:1 (1938), 126–166 | DOI | MR | Zbl

[9] Eridani, M. I. Utoyo, H. Gunawan, “A characterization for fractional integrals on generalized Morrey spaces”, Anal. Theory Appl., 28:3 (2012), 263–268 | MR | Zbl

[10] V. S. Guliyev, A. Akbulut, Y. Y. Mammadov, “Boundedness of fractional maximal operator and their higher order commutators in generalized Morrey spaces on Carnot groups”, Acta Math. Sci. Ser. B Engl. Ed., 33:5 (2013), 1329–1346 | DOI | MR | Zbl

[11] V. S. Guliev, “Obobschennye lokalnye prostranstva tipa Morri i drobnye integraly s grubym yadrom”, Probl. matem. anal., 71 (2013), 59–72 | Zbl

[12] F. Deringoz, V. S. Guliyev, S. G. Samko, “Boundedness of maximal and singular operators on generalized Orlicz–Morrey spaces”, Operator Theory, Operator Algebras and Applications, Oper. Theory Adv. Appl., 242, Birkhäuser Verlag, Basel, 2014, 1–24 | MR | Zbl

[13] V. S. Guliev, A. Eroglu, Ya. Ya. Mammadov, “Potentsial Rissa v obobschennykh prostranstvakh tipa Morri na gruppe Geizenberga”, Probl. matem. anal., 68 (2013), 29–44

[14] V. S. Guliev, Integralnye operatory v prostranstvakh funktsii na odnorodnykh gruppakh i na oblastyakh v $\mathbb R^n$, Dis. $\dots$ dokt. fiz.-matem. nauk, In-t. matem. im. V. A. Steklova RAN, M., 1994

[15] V. S. Guliev, Funktsionalnye prostranstva, integralnye operatory i dvukhvesovye otsenki na odnorodnykh gruppakh. Nekotorye prilozheniya, Chashyogly, Baku, 1999

[16] V. S. Guliyev, “Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces”, J. Inequal. Appl., 2009, 503948 | MR | Zbl

[17] D. R. Adams, “A note on Riesz potentials”, Duke Math. J., 42:4 (1975), 765–778 | DOI | MR | Zbl

[18] L. Capogna, D. Danielli, S. Pauls, J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progr. Math., 259, Birkhäuser, Basel, 2007 | MR | Zbl