# zbMATH — the first resource for mathematics

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces. (English) Zbl 1413.42040
Let $$\gamma:(0,\infty)\rightarrow(0,\infty)^d$$, $$\gamma=(\gamma_1,\dots,\gamma_d)$$, be a vector function such that:
1) $$\gamma_1(t)=t$$ $$(t>0)$$;
2) for each $$k$$ with $$2\leq k\leq d$$ the function $$\gamma_k$$ is increasing, continuous, $$\gamma_k(1)=1$$, $$\lim\limits_{t\rightarrow\infty}\gamma_k(t)=\infty$$, $$\lim\limits_{t\rightarrow 0}\gamma_k(t)=0$$ and there are constants $$c_{1,k},c_{2,k},\xi_k>1$$ for which $$c_{1,k}\gamma_k(t)\leq\gamma_k(\xi_kt)\leq c_{2,k}\gamma_k(t)$$ $$(t>0)$$.
Suppose $$\delta_1=1,$$ $$\delta_k\geq 1\;(2\leq k\leq d)$$ and $$\delta=(\delta_1,\dots,\delta_d)$$.
Denote the cone like set generated by $$\gamma$$ and $$\delta$$ as follows $\mathbb{R}^d_{\gamma,\delta}=\{(t_1,\dots,t_d)\in(0,\infty)^d:\delta_k^{-1}\gamma_k(t_k)\leq t_k\leq\delta_k\gamma_k(t_k)\;\;(1\leq k\leq d)\}.$ By $$\mathcal{I}^{\gamma,\delta}$$ denote the family of all $$d$$-dimensional intervals $$I=I_1\times\dots\times I_d$$ with $$(|I_1|,\dots,|I_d|)\in\mathbb{R}^d_{\gamma,\delta}$$ and having the centre at the origin.
The the cone-like maximal operator generated by $$\gamma$$ and $$\delta$$ is defined as follows $M^{\gamma,\delta}(f)(x)=\sup_{I\in\mathcal{I}^{\gamma,\delta}}\frac{1}{|x+I|}\int_{x+I}|f|\;\;\;\;\;(f\in L(\mathbb{R}^d),x\in\mathbb{R}^d).$ In the paper some results on the boundedness of classical Hardy-Littlewood maximal operator in variable Lebesgue spaces are extended on cone-like maximal operators. In particular, the following results are proved:
a) Let an exponent function $$p(\cdot):\mathbb{R}^d\rightarrow[1,\infty)$$ is such that $$1/p$$ is both locally log-continuous and log-continuous at infinity. Then the cone-like maximal operator $$M^{\gamma,\delta}$$ satisfies the following weak type estimation $\sup_{\lambda>0}\|\lambda\chi_{\{M^{\gamma,\delta}(f)>\lambda\}}\|_{p(\cdot)}\leq C \| f\|_{p(\cdot)}\;\;\;\;(f\in L^{p(\cdot)}(\mathbb{R}^d)).$ b) Let an exponent function $$p(\cdot):\mathbb{R}^d\rightarrow[1,\infty)$$ is such that $$1/p$$ is both locally log-continuous and log-continuous at infinity and $$\inf p>1$$. Then the cone-like maximal operator $$M^{\gamma,\delta}$$ satisfies the following strong type estimation $\| M^{\gamma,\delta}(f)\|_{p(\cdot)}\leq C\| f\|_{p(\cdot)}\;\;\;\;(f\in L^{p(\cdot)}(\mathbb{R}^d)).$ Note that a Besicovitch type covering result (see Theorem 3.1) proved in the paper is a particular case of the corresponding one proved by M. de Guzman in the work [Stud. Math. 34, 299–317 (1970; Zbl 0192.48804)] (see Corollary 1.7).

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry)
Full Text:
##### References:
 [1] A. Almeida, D. Drihem: Maximal, potential and singular type operators on Herz spaces with variable exponents. J. Math. Anal. Appl. 394 (2012), 781–795. · Zbl 1250.42077 [2] A. S. Besicovitch: A general form of the covering principle and relative differentiation of additive functions. Proc. Camb. Philos. Soc. 41 (1945), 103–110. [3] A. S. Besicovitch: 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 [4] D. Cruz-Uribe, L. Diening, A. Fiorenza: 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 [5] D. Cruz-Uribe, L. Diening, P. Hästö: The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal. 14 (2011), 361–374. · Zbl 1273.42018 [6] D. V. Cruz-Uribe, A. Fiorenza: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013. · Zbl 1268.46002 [7] D. Cruz-Uribe, A. Fiorenza, J. M. Martell, C. Pérez: The boundedness of classical operators on variable Lp spaces. Ann. Acad. Sci. Fenn., Math. 31 (2006), 239–264. · Zbl 1100.42012 [8] D. Cruz-Uribe, A. Fiorenza, C. J. Neugebauer: The maximal function on variable Lp spaces. Ann. Acad. Sci. Fenn., Math. 28 (2003), 223–238. · Zbl 1037.42023 [9] D. Cruz-Uribe, A. Fiorenza, C. J. Neugebauer: Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl. 394 (2012), 744–760. · Zbl 1298.42021 [10] L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta, T. Shimomura: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math. 34 (2009), 503–522. · Zbl 1180.42010 [11] L. Diening, P. Harjulehto, P. Hästö, M. Ružicka: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics 2017, Springer, Berlin, 2011. [12] G. Gát: 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 [13] T. Kopaliani: Interpolation theorems for variable exponent Lebesgue spaces. J. Funct. Anal. 257 (2009), 3541–3551. · Zbl 1202.46024 [14] O. Kovácik, J. [15] J. Marcinkiewicz, A. Zygmund: On the summability of double Fourier series. Fundam. Math. 32 (1939), 122–132. zbl · JFM 65.0266.01 [16] P. Mattila: 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 [17] E. M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43, Princeton University Press, Princeton, 1993. · Zbl 0821.42001 [18] E. M. Stein, G. Weiss: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, Princeton University Press, Princeton, 1971. · Zbl 0232.42007 [19] K. Szarvas, F. Weisz: Convergence of integral operators and applications. Period. Math. Hung. 73 (2016), 1–27. · Zbl 1389.03008 [20] F. Weisz: Summability of Multi-Dimensional Fourier Series and Hardy Spaces. Mathematics and Its Applications 541, Springer, Dordrecht, 2002. · Zbl 1306.42003 [21] F. Weisz: Herz spaces and restricted summability of Fourier transforms and Fourier series. J. Math. Anal. Appl. 344 (2008), 42–54. · Zbl 1254.42012 [22] F. Weisz: Summability of multi-dimensional trigonometric Fourier series. Surv. Approx. Theory (electronic only) 7 (2012), 1–179. · Zbl 1285.42010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.