Geodesic
On boundedness of the Hardy operator in Morrey-type spaces
Eurasian mathematical journal, Tome 2 (2011) no. 1, pp. 52-80.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we study the boundedness of the Hardy operator $H_\alpha$ in local and global Morrey-type spaces $LM_{p\theta,w(\cdot)}$, $GM_{p\theta,w(\cdot)}$ respectively, characterized by numerical parameters $p,\theta$ and a functional parameter $w$. We reduce this problem to the problem of a continuous embedding of one local Morrey-type space to another one. This allows obtaining, for all admissible values of the numerical parameters $\alpha,p_1,p_2,\theta_1,\theta_2$, sufficient conditions on the functional parameters $w_1$ and $w_2$ ensuring the boundedness of $H_\alpha$ from $LM_{p_1\theta_1,w_1(\cdot)}$ to $LM_{p_2\theta_2,w_2(\cdot)}$ and from $GM_{p_1\theta_1,w_1(\cdot)}$ to $GM_{p_2\theta_2,w_2(\cdot)}$. Moreover, for a certain range of the numerical parameters and under certain a priori assumptions on $w_1$ and $w_2$ these sufficient conditions coincide with the necessary ones. 
@article{EMJ_2011_2_1_a2,
     author = {V. I. Burenkov and P. Jain and T. V. Tararykova},
     title = {On boundedness of the {Hardy} operator in {Morrey-type} spaces},
     journal = {Eurasian mathematical journal},
     pages = {52--80},
     publisher = {mathdoc},
     volume = {2},
     number = {1},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2011_2_1_a2/}
}
TY  - JOUR
AU  - V. I. Burenkov
AU  - P. Jain
AU  - T. V. Tararykova
TI  - On boundedness of the Hardy operator in Morrey-type spaces
JO  - Eurasian mathematical journal
PY  - 2011
SP  - 52
EP  - 80
VL  - 2
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2011_2_1_a2/
LA  - en
ID  - EMJ_2011_2_1_a2
ER  - 
%0 Journal Article
%A V. I. Burenkov
%A P. Jain
%A T. V. Tararykova
%T On boundedness of the Hardy operator in Morrey-type spaces
%J Eurasian mathematical journal
%D 2011
%P 52-80
%V 2
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2011_2_1_a2/
%G en
%F EMJ_2011_2_1_a2
V. I. Burenkov; P. Jain; T. V. Tararykova. On boundedness of the Hardy operator in Morrey-type spaces. Eurasian mathematical journal, Tome 2 (2011) no. 1, pp. 52-80. http://geodesic.mathdoc.fr/item/EMJ_2011_2_1_a2/

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

[2] V. I. Burenkov, Sobolev Spaces on Domains, B. G. Teubner, Stuttgart–Leipzig, 1998 | MR | Zbl

[3] V. I. Burenkov, “Sharp estimates for integrals over small intervals for functions possessing some smoothness”, Progress in Analysis. Proceedings of the 3rd ISAAC Congress, World Scientific, New Jersey–London–Singapore–Hong Kong, 2003, 45–56 | DOI | MR

[4] V. I. Burenkov, A. Gogatishvili, V. S. Guliyev, R. Mustafaev, “Boundedness of the fractional maximal operator in local Morrey-type spaces”, Complex Analysis and Elliptic Equations, 55:8–10 (2010), 739–758 | DOI | MR | Zbl

[5] V. I. Burenkov, A. Gogatishvili, V. S. Guliyev, R. Mustafaev, Boundedness of the Riesz potential in local Morrey-type spaces (to appear)

[6] V. I. Burenkov, H. V. Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in the local Morrey-type spaces”, Studia Math., 163:2 (2004), 157–176 | DOI | MR | Zbl

[7] V. I. Burenkov, H. V. Guliyev, V. S. Guliyev, “Necessary and sufficient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces”, J. Comput. Appl. Math., 208 (2007), 280–301 | DOI | MR | Zbl

[8] V. I. Burenkov, V. S. Guliyev, “Necessary and sufficient conditions for boundedness of the Riesz potential in local Morrey-type spaces”, Potential Anal., 30 (2009), 211–249 | DOI | MR | Zbl

[9] V. I. Burenkov, V. S. Guliyev, A. Serbetchi, T. V. Tararykova, “Necessary and sufficient conditions for boundedness of the genuine singular integral operators in the local Morrey-type spaces”, Eurasian Mathematical Journal, 1:1 (2010), 32–53 | MR | Zbl

[10] F. Chiarenza, M. Frasca, “Morrey spaces and Hardy–Littlewood maximal function”, Rend. Math., 7 (1987), 273–279 | MR | Zbl

[11] V. S. Guliyev, Function spaces, integral operators and two weighted inequalities on homogeneous groups. Some applications., Elm, Baku, 1999 (Russian)

[12] V. S. Guliyev, R. Ch. Mustafayev, “Integral operators of potential type in spaces of homogeneous type”, Doklady Ross. Akad. Nauk, 354:6 (1997), 730–732 (Russian) | MR

[13] V. S. Guliyev, R. Ch. Mustafayev, “Fractional integrals in spaces of functions defined on spaces of homogeneous type”, Anal. Math., 24:3 (1998), 181–200 (Russian) | DOI | MR

[14] B. Kuttner, “Some theorems on fractional derivatives”, Proc. London Math. Soc., 3:12 (1953), 480–497 | DOI | MR | Zbl

[15] V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985 | MR | Zbl

[16] T. Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces”, Harmonic Analysis, ICM-90 Satelite Proceedings, Springer, Tokyo, 1991, 183–189 | DOI | MR

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

[18] E. Nakai, “Hardy–Littlewood maximal operator, singular integral operators and Riez potentials on generalized Morrey spaces”, Math. Nachr., 166 (1994), 95–103 | DOI | MR | Zbl

[19] V. D. Stepanov, “The weighted Hardy's inequalities non-increasing functions”, Trans. Amer. Math. Soc., 338:1 (1993), 173–186 | DOI | MR | Zbl

[20] G. Talenti, “Asservazioni sopra una classe di disuguaglianze”, Rend. Semin. Mat. e Fis. Milano, 39 (1969), 171–185 | DOI | MR | Zbl

[21] G. Tomaselli, “A class of inequalities”, Bull. Unione Mat. Ital., 2:6 (1969), 622–631 | MR | Zbl