One and two weight estimates for one-sided operators in $L^{p(\cdot)}$ spaces

V. Kokilashvili ; A. Meskhi ; M. Sarwar

Eurasian mathematical journal, Tome 1 (2010) no. 1, pp. 73-110.

Résumé

Various type weighted norm estimates for one-sided maximal functions and potentials are established in variable exponent Lebesgue spaces $L^{p(\cdot)}$. In particular, sufficient conditions (in some cases necessary and sufficient conditions) governing one and two weight inequalities for these operators are derived. Among other results generalizations of the Hardy–Littlewood, Fefferman–Stein and trace inequalities are given in $L^{p(\cdot)}$ spaces.

Bibliographie

[1] K. Andersen, E. Sawyer, “Weighted norm inequalities for the Riemann–Liouville and Weyl fractional operators”, Trans. Amer. Math. Soc., 308:2 (1988), 547–558 | DOI | MR | Zbl

[2] U. Ashraf, V. Kokilashvili, A. Meskhi, “Weight characterization of the trace inequality for the generalized Riemann–Liouville transform in $L^{p(x)}$ spaces”, Math. Ineq. Appl. (to appear)

[3] D. Cruz-Uribe, A. Fiorenza, C. J. Neugebauer, “The maximal function on variable $L^p$ spaces”, Ann. Acad. Sci. Fenn. Math., 28:1 (2003), 223–238 | MR | Zbl

[4] L. Diening, “Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$”, Math. Ineq. Appl., 7:2 (2004), 245–253 | MR | Zbl

[5] L. Diening, “Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(x)}$ and $W^{k,p(x)}$”, Math. Nachr., 268 (2004), 31–43 | DOI | MR | Zbl

[6] L. Diening, “Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces”, Bull. Sci. Math., 129:8 (2005), 657–700 | DOI | MR | Zbl

[7] L. Diening, P. Hästö, Muckenhoupt weights in variable exponent spaces, Preprint, 2008

[8] L. Diening, S. Samko, “Hardy inequality in variable exponent Lebesgue spaces”, Frac. Calc. Appl. Anal., 10:1 (2007), 1–18 | MR | Zbl

[9] D. E. Edmunds, V. Kokilashvili, A. Meskhi, Bounded and compact integral operators, Mathematics and its Applications, 543, Kluwer Academic Publishers, Dordrecht, 2002 | MR | Zbl

[10] D. E. Edmunds, V. Kokilashvili, A. Meskhi, “A trace inequality for generalized potentials in Lebesgue spaces with variable exponent”, J. Funct. Spaces Appl., 2:1 (2004), 55–69 | DOI | MR | Zbl

[11] D. E. Edmunds, V. Kokilashvili, A. Meskhi, “On the boundedness and compactness of the weighted Hardy operators in $L^{p(x)}$ spaces”, Georgian Math. J., 12:1 (2005), 27–44 | MR | Zbl

[12] D. E. Edmunds, V. Kokilashvili, A. Meskhi, “Two-weight estimates in $L^{p(x)}$ spaces with applications to Fourier series”, Houston J. Math., 35:2 (2009), 665–689 | MR | Zbl

[13] D. E. Edmunds, V. Kokilashvili, A. Meskhi, “One-sided operators in $L^{p(x)}$ spaces”, Math. Nachr., 281:11 (2008), 1525–1548 | DOI | MR | Zbl

[14] D. E. Edmunds, A. Meskhi, “Potential-type operators in $L^{p(x)}$ spaces”, Z. Anal. Anwend., 21 (2002), 681–690 | DOI | MR | Zbl

[15] D. E. Edmunds, V. Kokilashvili, A. Meskhi, “On the boundedness and compactness of the weighted Hardy operators in $L^{p(x)}$ spaces”, Georgian Math. J., 12:1 (2005), 27–44 | MR | Zbl

[16] I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, Weight theory for integral transforms on spaces of homogeneous type, Pitman Monographs and Surveys in Pure and Applied Mathematics, 92, Longman, Harlow, 1998 | MR | Zbl

[17] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge Univ. Press, 1934

[18] L. Hörmander, “$L^p$ estimates for (pluri-) subharmonic functions”, Math. Scand, 20 (1967), 65–78 | MR | Zbl

[19] M. I. A. Canestro, P. O. Salvador, “Weighted weak type inequalities with variable exponents for Hardy and maximal operators”, Proc. Japan Acad. (A), 82 (2006), 126–130 | DOI | MR | Zbl

[20] V. M. Kokilashvili, “On Hardy's inequalities in weighted spaces”, Soobsch. Akad. Nauk Gruz. SSR, 96 (1979), 37–40 (in Russian) | MR | Zbl

[21] V. Kokilashvili, “On a progress in the theory of integral operators in weighted Banach function spaces”, Function Spaces, Differential Operators and Nonlinear Analysis, Proceedings of the Conference held in Milovy (Bohemian-Moravian Uplands, May 28–June 2), Math. Inst. Acad. Sci. of Czech Republic, Prague, 2004

[22] V. Kokilashvili, A. Meskhi, “On a trace inequality for one-sided potentials and applications to the solvability of nonlinear integral equations”, Georgian Math. J., 8:3 (2001), 521–536 | MR | Zbl

[23] V. Kokilashvili, A. Meskhi, “On two-weight criteria for maximal function in $L^{p(x)}$ spaces defined on an interval”, Proc. A. Razmadze Math. Inst., 145, 2007, 100–102 | MR | Zbl

[24] V. Kokilasvili, A. Meskhi, “Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent”, Integ. Trans. Spec. Func., 18:9 (2007), 609–628 | DOI | MR

[25] V. Kokilasvili, A. Meskhi, “On the maximal and Fourier operators in weighted Lebesgue spaces with variable exponent”, Proc. A. Razmadze Math. Inst., 146, 2008, 120–123 | MR

[26] V. Kokilashvili, S. Samko, “On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent”, Z. Anal. Anwendungen, 22:4 (2003), 899–910 | DOI | MR

[27] V. Kokilashvili, S. Samko, “Singular integrals in weighted Lebesgue spaces with variable exponent”, Georgian Math. J., 10:1 (2003), 145–156 | MR | Zbl

[28] V. Kokilashvili, S. Samko, “Maximal and fractional operators in weighted $L^{p(x)}$ spaces”, Rev. Mat. Iberoamericana, 20:2 (2004), 493–515 | DOI | MR | Zbl

[29] V. Kokilashvili, S. Samko, “Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent”, Acta Mathematica Sinica, 2008 (to appear) | DOI | MR

[30] V. Kokilashvili, S. Samko, “The maximal operator in weighted variable spaces on metric measure spaces”, Proc. A. Razmadze Math. Inst., 144, 2007, 137–144 | MR | Zbl

[31] V. Kokilashvili, S. Samko, “Operators of harmonis analysis in weighted spaces with non-standard growth”, J. Math. Anal. Appl., 2008 (to appear) | MR

[32] T. S. Kopaliani, “Infimal convolution and Muckenhoupt $A_{p(\cdot)}$ condition in variable $L^p$ spaces”, Arch. Math., 89 (2007), 185–192 | DOI | MR | Zbl

[33] O. Kovácik, J. Rákosník, “On spaces $L^{p(x)}$ and $W^{k,p(x)}$”, Czechoslovak Math. J., 41:4 (1991), 592–618 | MR | Zbl

[34] M. Lorente, “A characterization of two-weight norm inequalities for one-sided operators of fractional type”, Canad. J. Math., 49 (1997), 839–848 | DOI | MR

[35] F. J. Martin-Reyes, P. Ortega Salvador, A. de la Torre, “Weighted inequalities for one-sided maximal functions”, Trans. Amer. Math. Soc., 319 (1990), 514–534 | DOI | MR

[36] F. J. Martin-Reyes, A. de la Torre, “Two-weight norm inequalities for fractional one-sided maximal operators”, Proc. Amer. Math. Soc., 117 (1993), 483–489 | DOI | MR | Zbl

[37] V. G. Maz'ya, Sobolev spaces, Springer Verlag, Berlin, Heidelberg, 1985 | MR | Zbl

[38] A. Meskhi, “Solution of some weight problems for Riemann–Liouville and Weyl operators”, Georgian Math. J., 5:6 (1998), 565–574 | DOI | MR | Zbl

[39] Sh. Ombrosi, “Weak weighted inequalities for a dyadic one-sided maximal functions in $\mathbb{R}^n$”, Proc. Amer. Math. Soc., 133 (2005), 1769–1775 | DOI | MR | Zbl

[40] D. V. Prokhorov, “On the boundedness of a class of integral operators”, J. London Math. Soc., 61:2 (2000), 617–628 | DOI | MR | Zbl

[41] E. T. Sawyer, “Weighted inequalities for one-sided Hardy–Littlewood maximal functions”, Trans. Amer. Math. Soc., 297 (1986), 53–61 | DOI | MR | Zbl

[42] S. Samko, “Convolution type operators in $L^{p(x)}$”, Integral Transf. and Spec. Funct., 7:1–2 (1998), 123–144 | DOI | MR | Zbl

[43] S. Samko, “Convolution type operators in $L^{p(x)}({\mathbb{R}}^n)$”, Integr. Transf. and Spec. Funct., 7:3–4 (1998), 261–284 | DOI | MR | Zbl

[44] S. Samko, “On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators”, Integral Transforms Spec. Funct., 16:5–6 (2005), 461–482 | DOI | MR | Zbl

[45] S. Samko, E. Shargorodsky, B. Vakulov, “Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators II”, J. Math. Anal. Appl., 325:1 (2007), 745–751 | DOI | MR | Zbl

[46] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives. Theory and applications, Gordon and Breach Sci. Publishers, London, New York, 1993 | MR | Zbl

[47] S. Samko, B. Vakulov, “Weighted Sobolev theorem with variable exponent”, J. Math. Anal. Appl., 310 (2005), 229–246 | DOI | MR | Zbl

[48] E. Sawyer, “Weighted inequalities for the one-sided Hardy–Littlewood maximal functions”, Trans. Amer. Math. Soc., 297 (1986), 53–61 | DOI | MR | Zbl

[49] E. T. Sawyer, “Weighted norm inequalities for fractional maximal operators”, Seminar on Harmonic Analysis (Montreal, Que., 1980), CMS Conf. Proc., 1, Amer. Math. Soc., Providence, R.I., 1981, 283–309 | MR

[50] E. T. Sawyer, R. L. Wheeden, “Carleson conditions for the Poisson integral”, Indiana Univ. Math. J., 40:2 (1991), 639–676 | DOI | MR | Zbl

[51] J. O. Strömberg, A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Math., 1381, Springer Verlag, Berlin, 1989 | MR | Zbl

[52] K. Tachizawa, “On weighted dyadic Carleson's inequalities”, J. Ineq. Appl., 6:4 (2001), 415–433 | DOI | MR | Zbl

[53] I. E. Verbitsky, “Weighted norm inequalities for maximal operators and Pisier's theorem on factorization through $L^{p\infty}$”, Integral Equations and Operator Theory, 15:1 (1992), 124–153 | DOI | MR | Zbl