| Title:
|
Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces (English) |
| Author:
|
Szarvas, Kristóf |
| Author:
|
Weisz, Ferenc |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
1079-1101 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $(p(\cdot ),p(\cdot ))$). (English) |
| Keyword:
|
variable Lebesgue space |
| Keyword:
|
maximal operator |
| Keyword:
|
$\gamma $-rectangle |
| Keyword:
|
Besicovitch's covering theorem |
| Keyword:
|
weak-type inequality |
| Keyword:
|
strong-type inequality |
| MSC:
|
42B25 |
| MSC:
|
42B35 |
| MSC:
|
52C17 |
| idZBL:
|
Zbl 06674863 |
| idMR:
|
MR3572924 |
| DOI:
|
10.1007/s10587-016-0311-9 |
| . |
| Date available:
|
2016-11-26T20:51:35Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145920 |
| . |
| Reference:
|
[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. Zbl 1250.42077, MR 2927498, 10.1016/j.jmaa.2012.04.043 |
| Reference:
|
[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. Zbl 0063.00352, MR 0012325, 10.1017/S0305004100022453 |
| Reference:
|
[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. Zbl 0063.00353, MR 0014414, 10.1017/S0305004100022660 |
| Reference:
|
[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. Zbl 1207.42011, MR 2493649 |
| Reference:
|
[5] Cruz-Uribe, D., Diening, L., Hästö, P.: The maximal operator on weighted variable Lebesgue spaces.Fract. Calc. Appl. Anal. 14 (2011), 361-374. Zbl 1273.42018, MR 2837636, 10.2478/s13540-011-0023-7 |
| Reference:
|
[6] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis.Applied and Numerical Harmonic Analysis Birkhäuser/Springer, New York (2013). Zbl 1268.46002, MR 3026953 |
| Reference:
|
[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. Zbl 1100.42012, MR 2210118 |
| Reference:
|
[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. Zbl 1037.42023, MR 1976842 |
| Reference:
|
[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. Zbl 1298.42021, MR 2927495, 10.1016/j.jmaa.2012.04.044 |
| Reference:
|
[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. Zbl 1180.42010, MR 2553809 |
| Reference:
|
[11] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev spaces with variable exponents.Lecture Notes in Mathematics 2017 Springer, Berlin (2011). Zbl 1222.46002, MR 2790542 |
| Reference:
|
[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. Zbl 1135.42007, MR 2371615, 10.1016/j.jat.2006.08.006 |
| Reference:
|
[13] Kopaliani, T.: Interpolation theorems for variable exponent Lebesgue spaces.J. Funct. Anal. 257 (2009), 3541-3551. Zbl 1202.46024, MR 2572260, 10.1016/j.jfa.2009.06.009 |
| Reference:
|
[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 |
| Reference:
|
[15] Marcinkiewicz, J., Zygmund, A.: On the summability of double Fourier series.Fundam. Math. 32 (1939), 122-132. Zbl 0022.01804, 10.4064/fm-32-1-122-132 |
| Reference:
|
[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). Zbl 0911.28005, MR 1333890 |
| Reference:
|
[17] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.Princeton Mathematical Series 43 Princeton University Press, Princeton (1993). Zbl 0821.42001, MR 1232192 |
| Reference:
|
[18] Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces.Princeton Mathematical Series Princeton University Press, Princeton (1971). Zbl 0232.42007, MR 0304972 |
| Reference:
|
[19] Szarvas, K., Weisz, F.: Convergence of integral operators and applications.Period. Math. Hung. 73 (2016), 1-27. |
| Reference:
|
[20] Weisz, F.: Summability of Multi-Dimensional Fourier Series and Hardy Spaces.Mathematics and Its Applications 541 Springer, Dordrecht (2002). Zbl 1306.42003, MR 2009144 |
| Reference:
|
[21] Weisz, F.: Herz spaces and restricted summability of Fourier transforms and Fourier series.J. Math. Anal. Appl. 344 (2008), 42-54. Zbl 1254.42012, MR 2416292, 10.1016/j.jmaa.2008.02.035 |
| Reference:
|
[22] Weisz, F.: Summability of multi-dimensional trigonometric Fourier series.Surv. Approx. Theory (electronic only) 7 (2012), 1-179. Zbl 1285.42010, MR 2943169 |
| . |
