New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

• Autorzy: Agnieszka Kałamajska , Katarzyna Pietruska-Pałuba
• Języki publikacji: EN

Abstrakty

EN

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

Słowa kluczowe

EN

Hardy inequalities; Orlicz-Sobolev spaces; Nondoubling measures

Kategorie tematyczne

• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• 26D10: Inequalities involving derivatives and differential and integral operators

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 6
• Strony: 2033-2050

Daty

wydano

2012-12-01

online

2012-10-12

Twórcy

autor

Agnieszka Kałamajska

• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland

autor

Katarzyna Pietruska-Pałuba

• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland

Bibliografia

• [1] Bloom S., Kerman R., Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator, Studia Math., 1994, 110(2), 149–167
• [2] Bobkov S.G., Götze F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 1999, 163(1), 1–28 http://dx.doi.org/10.1006/jfan.1998.3326[Crossref]
• [3] Brandolini B., Chiacchio F., Trombetti C., Hardy type inequalities and Gaussian measure, Commun. Pure Appl. Anal., 2007, 6(2), 411–428 http://dx.doi.org/10.3934/cpaa.2007.6.411[Crossref]
• [4] Brown R.C., Hinton D.B., Interpolation inequalities with power weights for functions of one variable, J. Math. Anal. Appl., 1993, 172(1), 233–242 http://dx.doi.org/10.1006/jmaa.1993.1020[Crossref]
• [5] Brown R.C., Hinton D.B., Weighted interpolation and Hardy inequalities with some spectral-theoretic applications, In: Proceedings of the Fourth International Colloquium on Differential Equations, Plovdiv, August 18–23, 1993, VSP, Utrecht, 1994, 55–70
• [6] Caffarelli L., Kohn R., Nirenberg L., First order interpolation inequalities with weights, Compositio Math., 1984, 53(3), 259–275
• [7] Chua S.-K., On weighted Sobolev interpolation inequalities, Proc. Amer. Math. Soc., 1994, 121(2), 441–449 http://dx.doi.org/10.1090/S0002-9939-1994-1221721-4[Crossref]
• [8] Chua S.-K., Sharp conditions for weighted Sobolev interpolation inequalities, Forum Math., 2005, 17(3), 461–478 http://dx.doi.org/10.1515/form.2005.17.3.461[Crossref]
• [9] Cianchi A., Some results in the theory of Orlicz spaces and applications to variational problems, In: Nonlinear Analysis, Function Spaces and Applications, 6, Prague, May 31–June 6, 1998, Academy of Sciences of the Czech Republic, Mathematical Institute, Prague, 1999, 50–92
• [10] Fernández-Martínez P., Signes T., Real interpolation with symmetric spaces and slowly varying functions, Q. J. Math., 2012, 63(1), 133–164 http://dx.doi.org/10.1093/qmath/haq009[Crossref]
• [11] Fiorenza A., Krbec M., Indices of Orlicz spaces and some applications, Comment. Math. Univ. Carolin., 1997, 38(3), 433–451
• [12] Gossez J.-P., Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Am. Math. Soc., 1974, 190, 163–205 http://dx.doi.org/10.1090/S0002-9947-1974-0342854-2[Crossref]
• [13] Gustavsson J., Peetre J., Interpolation of Orlicz spaces, Studia Math., 1977, 60(1), 33–59
• [14] Gutiérrez C.E., Wheeden R.L., Sobolev interpolation inequalities with weights, Trans. Amer. Math. Soc., 1991, 323(1), 263–281
• [15] Haroske D., Skrzypczak L., Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases, J. Funct. Spaces Appl., 2011, 9(2), 129–178 http://dx.doi.org/10.1155/2011/928962[WoS][Crossref]
• [16] Kałamajska A., Pietruska-Pałuba K., On a variant of the Hardy inequality between weighted Orlicz spaces, Studia Math., 2009, 193(1), 1–28 http://dx.doi.org/10.4064/sm193-1-1[Crossref]
• [17] Kałamajska A., Pietruska-Pałuba K., On a variant of the Gagliardo-Nirenberg inequality deduced from the Hardy inequality, Bull. Pol. Acad. Sci. Math., 2011, 59(2), 133–149 http://dx.doi.org/10.4064/ba59-2-4[Crossref]
• [18] Kałamajska A., Pietruska-Pałuba K., Weighted Hardy-type inequalities in Orlicz spaces, J. Inequal. Appl., 2012, 12(4), 745–766
• [19] Kokilashvili V., Krbec M., Weighted inequalities in Lorentz and Orlicz spaces, World Scientific, River Edge, 1991
• [20] Koskela P., Lehrbäck J., Weighted pointwise Hardy inequalities, J. Lond. Math. Soc., 2009, 79(3), 757–779 http://dx.doi.org/10.1112/jlms/jdp013[Crossref]
• [21] Krasnosel’kii M.A., Rutickii Ya.B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961
• [22] Kufner A., Weighted Sobolev Spaces, John Wiley & Sons, New York, 1985
• [23] Kufner A., Maligranda L., Persson L.-E., The Hardy Inequality, Vydavatelský Servis, Plzeň, 2007
• [24] Kufner A., Persson L.-E., Weighted Inequalities of Hardy Type, World Scientific, River Edge, 2003
• [25] Lai Q., Weighted modular inequalities for Hardy type operators, Proc. London Math. Soc., 1999, 79(3), 649–672 http://dx.doi.org/10.1112/S0024611599012010[Crossref]
• [26] Mazýa V.G., Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, 1985
• [27] Mitronovic D.S., Pečaric J.E., Fink A.M., Classical and New Inequalities in Analysis, Math. Appl. (East European Ser.), 61, Kluwer, Dordrecht, 1993
• [28] Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 1972, 165, 207–226 http://dx.doi.org/10.1090/S0002-9947-1972-0293384-6[Crossref]
• [29] Oinarov R., On weighted norm inequalities with three weights, J. London Math. Soc., 1993, 48(1), 103–116 http://dx.doi.org/10.1112/jlms/s2-48.1.103[Crossref]
• [30] Oleszkiewicz K., Pietruska-Pałuba K., Orlicz-space Hardy and Landau-Kolmogorov inequalities for Gaussian measures, Demonstratio Math., 2012, 45(2), 227–241
• [31] Opic B., Kufner A., Hardy-Type Inequalities, Pitman Res. Notes Math. Ser., 219, Longman Scientific & Technical, Harlow, 1990
• [32] Pilarczyk D., Asymptotic stability of singular solution to nonlinear heat equation, Discrete Contin. Dyn. Syst., 2009, 25(3), 991–1001 http://dx.doi.org/10.3934/dcds.2009.25.991[Crossref]
• [33] Rao M.M., Ren Z.D., Theory of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math., 146, Marcel Dekker, New York, 1991
• [34] Simonenko I.B., Interpolation and extrapolation of linear operators in Orlicz spaces, Mat. Sb., 1964, 63(105), 536–553 (in Russian)
• [35] Vazquez J.L., Zuazua E., The Hardy inequality and the asymptotic behaviour of the heat equation with an inversesquare potential, J. Funct. Anal., 2000, 173(1), 103–153 http://dx.doi.org/10.1006/jfan.1999.3556[Crossref]

Identyfikatory

DOI

10.2478/s11533-012-0116-5