### One-dimensional and Multidimensional Hardy Operators in Grand Lebesgue Spaces

#### Abstract

Grand Lebesgue spaces over sets of innite measure are dened with using an additional

characteristic a() called a grandizer. Conditions on the grandizer a(x) for the Hardy operators to

be bounded in the grand Lebesgue spaces Lp)

a (Rn) are found, and the lower and upper estimates

for a sharp constant in the one-dimensional and multidimensional Hardy inequalities are given in

dependence on the grandizer. For some special choice of the grandizer it is proved that this sharp

constant is equal to the sharp constant for the classical Lebesgue spaces.

#### Keywords

#### References

G. Di Fratta and A. Fiorenza. A direct approach to the duality of grand and small

Lebesgue spaces. Nonlinear Analysis: Theory, Methods and Applications, 70(7):2582-2592, 2009.

D. E. Edmunds, V. Kokilashvili, and A. Meskhi. Bounded and compact integral oper-

ators, volume 543 of Mathematics and its Applications. Kluwer Academic Publishers,

Dordrecht, 2002.

A. Fiorenza. Duality and re

exivity in grand Lebesgue spaces. Collect. Math.,

(2):131{148, 2000.

A. Fiorenza, B. Gupta, and P. Jain. The maximal theorem in weighted grand Lebesgue

spaces. Studia Math., 188(2):123{133, 2008.

A. Fiorenza and G. E. Karadzhov. Grand and small Lebesgue spaces and their

analogs. Journal for Analysis and its Applications, 23(4):657{681, 2004.

A. Fiorenza and J. M. Rakotoson. Petits espaces de Lebesgue et leurs applications.

C. R. Acad. Sci. Paris Ser. I, 333:1{4, 2001.

L. Greco, T. Iwaniec, and C. Sbordone. Inverting the p-harmonic operator.

Manuscripta Math., 92:249{258, 1997.

G. H. Hardy, J. E. Littlewood, and G. Polya. Inequalities. Cambridge University

Press: Cambridge, 1934. 324 pages.

T. Iwaniec and C. Sbordone. On the integrability of the Jacobian under minimal

hypotheses. Arch. Rational Mech. Anal., 119:129{143, 1992.

V. Kokilashvili. Boundedness criterion for the Cauchy singular integral operator in

weighted grand Lebesgue spaces and application to the Riemann problem. Proc. A.

Razmadze Math. Inst., 151:129{133, 2009.

V. Kokilashvili. Boundedness criteria for singular integrals in weighted Grand

Lebesgue spaces. J. Math. Sci., 170(1):20{33, 2010.

V. Kokilashvili. The Riemann boundary value problem analytic functions in the frame

of grand Lp spaces. Bull. Georgian Nat. Acad. Sci., 4(1):5{7, 2010.

V. Kokilashvili and A. Meskhi. A note on the boundedness of the Hilbert transform

in weighted grand Lebesgue spaces. Georgian Math. J., 16(3):547{551, 2009.

V. Kokilashvili, A. Meskhi, and L.-E. Persson. Weighted norm inequalities for integral

transforms with productkernels. Nova Science Publishers, New York, 2010.

V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko. Integral Operators in Non-

standard Function Spaces. Vol. I{II. Birkhaser, 2016. 1{1009 pages.

S. G. Krein, Yu. I. Petunin, and E. M. Semenov. Interpolation of linear operators,

volume 54 of Translations of Mathematical Monographs. American Mathematical

Society, Providence, R. I., 1982.

A. Kufner, L. Maligranda, and L.-E. Persson. The prehistory of the Hardy inequality.

Amer. Math. Monthly, 113:715{732, 2006.

A. Kufner, L. Maligranda, and L.-E. Persson. The Hardy Inequality - About its History

and Some Related Results. Pilsen, 2007.

A. Kufner and L.-E. Persson. Weighted inequalities of Hardy type. World Scientic

Publishing Co. Inc., River Edge, NJ, 2003.

A. Meskhi. Weighted criteria for the Hardy transform under the Bp condition in

grand Lebesgue spaces and some applications. J. Math. Sci., 178(6):622{636, 2011.

E.-L. Persson and S. G. Samko. A note on the best constants in some Hardy inequalities.

J. Math. Inequal., 9(2):437{447, 2015.

E.-L. Persson, G. E. Shambilova, and V. D. Stepanov. Hardy-type inequalities on the

weighted cones of quasi-concave functions. Banach J. Math. Anal., 9(2):21{34, 2015.

S. G. Samko. Hypersingular Integrals and their Applications. London-New-York:

Taylor & Francis, Series "Analytical Methods and Special Functions" vol. 5, 2002.

+ xvii pages.

S. G. Samko and S. M. Umarkhadzhiev. On Iwaniec-Sbordone spaces on sets which

may have innite measure. Azerb. J. Math., 1(1):67{84, 2011.

S. G. Samko and S. M. Umarkhadzhiev. On Iwaniec-Sbordone spaces on sets which

may have innite measure: addendum. Azerb. J. Math., 1(2):143{144, 2011.

S. G. Samko and S. M. Umarkhadzhiev. Riesz fractional integrals in grand Lebesgue

spaces. Fract. Calc. Appl. Anal., 19(3):608{624, 2016.

S. G. Samko and S. M. Umarkhadzhiev. On grand Lebesgue spaces on sets of

innite measure. Mathematische Nachrichten, 2016. http:// dx.doi.org/ 10.1002/

mana.201600136.

S. M. Umarkhadzhiev. Boundedness of linear operators in wieghted generalized grand

Lebesgue spaces. Vestnik of Chechen Academy of Sciences, 19(2):5{9, 2013. (in

Russian).

S. M. Umarkhadzhiev. Boundedness of the Riesz potential operator in weighted

grand Lebesgue spaces. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math.

J.], 16(2):62{68, 2014. (in Russian).

S. M. Umarkhadzhiev. Generalization of the notion of grand Lebesgue space. Russian

Mathematics (Iz. VUZ), 4:42{51, 2014.

S. M. Umarkhadzhiev. The boundedness of the Riesz potential operator from generalized

grand Lebesgue spaces to generalized grand Morrey spaces. In Operator theory,

operator algebras and applications. Selected papers based on the presentations at the

workshop WOAT 2012, Lisbon, Portugal, September 11{14, 2012, pages 363{373.

Basel: Birkhauser/Springer, 2014.

S. M. Umarkhadzhiev. Denseness of the Lizorkin space in grand Lebesgue space.

Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], 17(3):75{83, 2015.

(in Russian).

### Refbacks

- There are currently no refbacks.