Abstract
It is well known that the weak (1, 1) estimate does not hold for the strong maximal operator, but it still enjoys certain weak \(L\log L\)-type norm inequality. Let \(\Phi _n(t)=t(1+(\log ^+t)^{n-1})\) and the space \(L_{\Phi _n}({\mathbb {R}^{n}})\) be the set of all measurable functions on \({\mathbb {R}^{n}}\) such that
In this paper, we introduce a new weak norm space \(L_{\Phi _n}^{1,\infty }({\mathbb {R}^{n}})\), which is more larger than \(L^{1,\infty }({\mathbb {R}^{n}})\) space, and establish the corresponding limiting weak-type behavior of the strong maximal operator. Similar result has been extended to the multilinear strong maximal operator.
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Alfonseca, A., Soria, F., Vargas, A.: A remark on maximal operators along directions in \(\mathbb{R}^2\). Math. Res. Lett. 10(1), 41–49 (2003)
Bagby, R.J.: Maximal functions and rearrangements: some new proofs. Indiana Univ. Math. J. 32(6), 879–891 (1983)
Bagby, R.J., Kurtz, D.S.: \(L({\rm log} L)\) spaces and weights for the strong maximal function. J. Anal. Math. 44, 21–31 (1984/85)
Bourgain, J.: On the \(L^p\)-bounds for maximal functions associated to convex bodies in \({\mathbb{R}^{n}}\). Isr. J. Math. 54(3), 257–265 (1986)
Cao, M., Xue, Q., Yabuta, K.: On multilinear fractional strong maximal operator associated with rectangles and multiple weights. Rev. Mat. Iberoam. 33(2), 555–572 (2017)
Cao, M., Xue, Q., Yabuta, K.: Corrigendum to “On multilinear fractional strong maximal operator associated with rectangles and multiple weights” [Rev. Mat. Iberoam. 33 (2017), no. 2, 555–572]. Rev. Mat. Iberoam. 34(1), 475–479 (2018)
Cao, M., Xue, Q., Yabuta, K.: On the boundedness of multilinear fractional strong maximal operators with multiple weights. Pac. J. Math. 303(2), 491–518 (2019)
Christ, M.: The strong maximal function on a nilpotent group. Trans. Am. Math. Soc. 331(1), 1–13 (1992)
Córdoba, A., Fefferman, R.: A geometric proof of the strong maximal theorem. Ann. Math. (2) 102(1), 95–100 (1975)
Davis, B.: On the weak type \((1, 1)\) inequality for conjugate functions. Proc. Am. Math. Soc. 44, 307–311 (1974)
Ding, Y., Lai, X.: \(L^1\) -Dini conditions and limiting behavior of weak type estimates for singular integrals. Rev. Mat. Iberoam. 33(4), 1267–1284 (2017)
Ding, Y., Lai, X.: Weak type \( (1,1)\) behavior for the maximal operator with \(L^1\)-Dini kernel. Potential Anal. 47(2), 169–187 (2017)
Grafakos, L., Kinnunen, J.: Sharp inequalities for maximal functions associated with general measures. Proc. R. Soc. Edinb. Sect. A 128(4), 717–723 (1998)
Grafakos, L., Liu, L., Pérez, C., Torres, R.H.: The multilinear strong maximal function. J. Geom. Anal. 21(1), 118–149 (2011)
Guo, W., He, J., Wu, H.: Limiting weak-type behaviors for certain classical operators in harmonic analysis. Potential Anal. 54(2), 307–330 (2021)
Hardy, G.H., Littlewood, J.E.: A maximal theorem with function-theoretic applications. Acta Math. 54(1), 81–116 (1930)
Hou, X., Guo, W., Wu, H.: Vector-valued estimates on limiting weak-type behaviors of singular integrals and maximal operators. J. Math. Anal. Appl. 472(2), 1293–1312 (2019)
Hou, X., Wu, H.: On the limiting weak-type behaviors for maximal operators associated with power weighted measure. Can. Math. Bull. 62(2), 313–326 (2019)
Hou, X., Wu, H.: Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting. Front. Math. China 14(3), 535–550 (2019)
Hu, J., Huang, X.: A note on the limiting weak-type behavior for maximal operators. Proc. Am. Math. Soc. 136(5), 1599–1607 (2008)
Janakiraman, P.: Weak-type estimates for singular integrals and the Riesz transform. Indiana Univ. Math. J. 53(2), 533–555 (2004)
Janakiraman, P.: Limiting weak-type behavior for singular integral and maximal operators. Trans. Am. Math. Soc. 358(5), 1937–1952 (2006)
Jessen, B., Marcinkiewicz, J., Zygmund, A.: Note on the differentiability of multiple integrals. Fund. Math. 25, 217–234 (1935)
Katz, N.H.: A counterexample for maximal operators over a Cantor set of directions. Math. Res. Lett. 3(4), 527–536 (1996)
Katz, N.H.: Maximal operators over arbitrary sets of directions. Duke Math. J. 97, 67–79 (1999)
Liu, F., Xue, Q., Yabuta, K.: Regularity and continuity of the multilinear strong maximal operators. J. Math. Pures Appl. 138(9), 204–241 (2020)
Luque, T., Parissis, I.: The endpoint Fefferman–Stein inequality for the strong maximal function. J. Funct. Anal. 266(1), 199–212 (2014)
Melas, A.: The best constant for the centered Hardy-Littlewood maximal inequality. Ann. Math. 157(2), 647–688 (2003)
Mitsis, T.: The weighted weak type inequality for the strong maximal function. J. Fourier Anal. Appl. 12(6), 645–652 (2006)
Nagel, A., Stein, E.M., Wainger, S.: Differentiation in lacunary directions. Proc. Natl. Acad. Sci USA 75, 1060–1062 (1978)
Stein, E.M., Strömberg, J.-O.: Behavior of maximal functions in \({\mathbb{R}^{n}}\) for large \(n\). Ark. Mat. 21(2), 259–269 (1983)
Wiener, N.: The ergodic theorem. Duke Math. J. 5(1), 1–18 (1939)
Zhang, J., Saito, H., Xue, Q.: The Fefferman–Stein type inequalities for the multilinear strong maximal functions. Math. Inequal. Appl. 22(2), 539–552 (2019)
Zhang, J., Xue, Q.: Multilinear strong maximal operators on weighted mixed norm spaces. Publ. Math. Debr. 96(3–4), 347–361 (2020)
Acknowledgements
The authors want to express their sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable. The authors were supported partly by NSFC (Nos. 11671039, 11771358, 11871101) and 111 Project and the National Key Research and Development Program of China (Grant no. 2020YFA0712900).
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Communicated by Deguang Han.
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Qin, M., Wu, H. & Xue, Q. The limiting weak-type behaviors of the strong maximal operators. Banach J. Math. Anal. 15, 69 (2021). https://doi.org/10.1007/s43037-021-00154-6
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DOI: https://doi.org/10.1007/s43037-021-00154-6
Keywords
- Lower bound
- Best constant
- Limiting weak-type behavior
- Strong maximal operator
- Multilinear strong maximal operator