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Keywords:
variable Lebesgue space; maximal operator; $\gamma $-rectangle; Besicovitch's covering theorem; weak-type inequality; strong-type inequality
variable Lebesgue space; maximal operator; $\gamma $-rectangle; Besicovitch's covering theorem; weak-type inequality; strong-type inequality
Summary:
The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_{p}(\mathbb {R}^d)$ (in the case $p >1$), but (in the case when $1/p(\cdot )$ is log-Hölder continuous and $p_{-} = \inf \{ p(x) \colon x \in \mathbb R^d \} > 1$) on the variable Lebesgue spaces $L_{p(\cdot )}(\mathbb {R}^d)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch's covering theorem for the so-called $\gamma $-rectangles. We introduce a general maximal operator $M_{s}^{\gamma ,\delta }$ and with the help of generalized $\Phi $-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function $1/p(\cdot )$ is log-Hölder continuous and $p_{-} > s$, where $1 \leq s \leq \infty $ is arbitrary (or $p_{-} \geq s$), then the maximal operator $M_{s}^{\gamma ,\delta }$ is bounded on the space $L_{p(\cdot )}(\mathbb {R}^d)$ (or the maximal operator is of weak-type $(p(\cdot ),p(\cdot ))$).
References:
[1] Almeida, A., Drihem, D.: Maximal, potential and singular type operators on Herz spaces with variable exponents. J. Math. Anal. Appl. 394 (2012), 781-795. DOI 10.1016/j.jmaa.2012.04.043 | MR 2927498 | Zbl 1250.42077
[2] Besicovitch, A. S.: A general form of the covering principle and relative differentiation of additive functions. Proc. Camb. Philos. Soc. 41 (1945), 103-110. DOI 10.1017/S0305004100022453 | MR 0012325 | Zbl 0063.00352
[3] Besicovitch, A. S.: A general form of the covering principle and relative differentiation of additive functions II. Proc. Camb. Philos. Soc. 42 (1946), 1-10. DOI 10.1017/S0305004100022660 | MR 0014414 | Zbl 0063.00353
[4] Cruz-Uribe, D., Diening, L., Fiorenza, A.: A new proof of the boundedness of maximal operators on variable Lebesgue spaces. Boll. Unione Mat. Ital. (9) 2 (2009), 151-173. MR 2493649 | Zbl 1207.42011
[5] Cruz-Uribe, D., Diening, L., Hästö, P.: The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal. 14 (2011), 361-374. DOI 10.2478/s13540-011-0023-7 | MR 2837636 | Zbl 1273.42018
[6] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis Birkhäuser/Springer, New York (2013). MR 3026953 | Zbl 1268.46002
[7] Cruz-Uribe, D., Fiorenza, A., Martell, J. M., Pérez, C.: The boundedness of classical operators on variable $L^p$ spaces. Ann. Acad. Sci. Fenn., Math. 31 (2006), 239-264. MR 2210118 | Zbl 1100.42012
[8] Cruz-Uribe, D., Fiorenza, A., Neugebauer, C. J.: The maximal function on variable $L^p$ spaces. Ann. Acad. Sci. Fenn., Math. 28 (2003), 223-238. MR 1976842 | Zbl 1037.42023
[9] Cruz-Uribe, D., Fiorenza, A., Neugebauer, C. J.: Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl. 394 (2012), 744-760. DOI 10.1016/j.jmaa.2012.04.044 | MR 2927495 | Zbl 1298.42021
[10] Diening, L., Harjulehto, P., Hästö, P., Mizuta, Y., Shimomura, T.: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math. 34 (2009), 503-522. MR 2553809 | Zbl 1180.42010
[11] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics 2017 Springer, Berlin (2011). MR 2790542 | Zbl 1222.46002
[12] Gát, G.: Pointwise convergence of cone-like restricted two-dimensional {$(C,1)$} means of trigonometric Fourier series. J. Approximation Theory 149 (2007), 74-102. DOI 10.1016/j.jat.2006.08.006 | MR 2371615 | Zbl 1135.42007
[13] Kopaliani, T.: Interpolation theorems for variable exponent Lebesgue spaces. J. Funct. Anal. 257 (2009), 3541-3551. DOI 10.1016/j.jfa.2009.06.009 | MR 2572260 | Zbl 1202.46024
[14] Kováčik, O., k, J. Rákosní: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. MR 1134951
[15] Marcinkiewicz, J., Zygmund, A.: On the summability of double Fourier series. Fundam. Math. 32 (1939), 122-132. DOI 10.4064/fm-32-1-122-132 | Zbl 0022.01804
[16] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics 44 Cambridge University Press, Cambridge (1999). MR 1333890 | Zbl 0911.28005
[17] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43 Princeton University Press, Princeton (1993). MR 1232192 | Zbl 0821.42001
[18] Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series Princeton University Press, Princeton (1971). MR 0304972 | Zbl 0232.42007
[19] Szarvas, K., Weisz, F.: Convergence of integral operators and applications. Period. Math. Hung. 73 (2016), 1-27.
[20] Weisz, F.: Summability of Multi-Dimensional Fourier Series and Hardy Spaces. Mathematics and Its Applications 541 Springer, Dordrecht (2002). MR 2009144 | Zbl 1306.42003
[21] Weisz, F.: Herz spaces and restricted summability of Fourier transforms and Fourier series. J. Math. Anal. Appl. 344 (2008), 42-54. DOI 10.1016/j.jmaa.2008.02.035 | MR 2416292 | Zbl 1254.42012
[22] Weisz, F.: Summability of multi-dimensional trigonometric Fourier series. Surv. Approx. Theory (electronic only) 7 (2012), 1-179. MR 2943169 | Zbl 1285.42010
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