# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Weighted Hardy and potential operators in the generalized Morrey spaces. (English) Zbl 1211.42018
The authors study the weighted $p\to q$-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces ${ℒ}^{p,\varphi }\left({ℝ}^{n},w\right)$ defined by an almost increasing function $\varphi \left(r\right)$ and radial type weight $w\left(|x|\right)$ from the Bary-Stechkin-type class. Sufficient conditions are obtained, in terms of some integral inequalities imposed on $\varphi$ and $w$, for such a $p\to q$-boundedness. In the case of local spaces the obtained conditions are also necessary. These results are applied to derive a similar weighted $p\to q$-boundedness of the Riesz potential operator.
##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 47B38 Operators on function spaces (general) 47G40 Potential operators
##### References:
 [1] Adams, D. R.: A note on Riesz potentials, Duke math. J. 42, No. 4, 765-778 (1975) · Zbl 0336.46038 · doi:10.1215/S0012-7094-75-04265-9 [2] Adams, D. R.; Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities, Indiana univ. Math. J. 53, No. 6, 1631-1666 (2004) · Zbl 1100.31009 · doi:10.1512/iumj.2004.53.2470 [3] Alvarez, J.: The distribution function in the Morrey space, Proc. amer. Math. soc. 83, 693-699 (1981) · Zbl 0478.46025 · doi:10.2307/2044236 [4] Arai, H.; Mizuhara, T.: Morrey spaces on spaces of homogeneous type and estimates for $\square b$ and the Cauchy-Szego projection, Math. nachr. 185, No. 1, 5-20 (1997) · Zbl 0876.42011 · doi:10.1002/mana.3211850102 [5] Bari, N. K.; Stechkin, S. B.: Best approximations and differential properties of two conjugate functions, Proc. Moscow math. Soc. 5, 483-522 (1956) [6] Bennett, C.; Sharpley, R.: Interpolation of operators, Pure appl. Math. 129 (1988) · Zbl 0647.46057 [7] Burenkov, V. I.; Guliyev, H.: Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces, Studia math. 163, No. 2, 157-176 (2004) · Zbl 1044.42015 · doi:10.4064/sm163-2-4 · doi:http://journals.impan.gov.pl/sm/Inf/163-2-4.html [8] Burenkov, V. I.; Guliev, V.; Guliyev, H.: Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces, J. comput. Appl. math. 208, No. 1, 280-301 (2007) · Zbl 1134.46014 · doi:10.1016/j.cam.2006.10.085 [9] Chiarenza, F.; Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function, Rend. math. 7, 273-279 (1987) · Zbl 0717.42023 [10] Di Fazio, G.; Ragusa, M. A.: Commutators and Morrey spaces, Boll. unione mat. Ital. 7, No. 5-A, 323-332 (1991) · Zbl 0761.42009 [11] Ding, Y.; Lu, S.: Boundedness of homogeneous fractional integrals on lp for n/a