Abstract
Let $M_d$ be the centered HardyLittlewood maximal function associated to cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that $c_d \to \infty$ as $d\to \infty$, thus answering, for the case of cubes, a longstanding open question of E. M. Stein and J. O. Strömberg.

[A] J. M. Aldaz, "Remarks on the HardyLittlewood maximal function," Proc. Roy. Soc. Edinburgh Sect. A, vol. 128, iss. 1, pp. 19, 1998.
@article {A, MRKEY = {1606325},
AUTHOR = {Aldaz, J. M.},
TITLE = {Remarks on the {H}ardy{L}ittlewood maximal function},
JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A},
FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics},
VOLUME = {128},
YEAR = {1998},
NUMBER = {1},
PAGES = {19},
ISSN = {03082105},
CODEN = {PEAMDU},
MRCLASS = {42B25},
MRNUMBER = {1606325},
MRREVIEWER = {Javier Soria},
ZBLNUMBER = {0892.42010},
} 
[A2] J. M. Aldaz, "Dimension dependency of the weak type $(1,1)$ bounds for maximal functions associated to finite radial measures," Bull. Lond. Math. Soc., vol. 39, iss. 2, pp. 203208, 2007.
@article {A2, MRKEY = {2323449},
AUTHOR = {Aldaz, J. M.},
TITLE = {Dimension dependency of the weak type {$(1,1)$} bounds for maximal functions associated to finite radial measures},
JOURNAL = {Bull. Lond. Math. Soc.},
FJOURNAL = {Bulletin of the London Mathematical Society},
VOLUME = {39},
YEAR = {2007},
NUMBER = {2},
PAGES = {203208},
ISSN = {00246093},
MRCLASS = {42B25},
MRNUMBER = {2323449},
MRREVIEWER = {Juha K. Kinnunen},
DOI = {10.1112/blms/bdl027},
ZBLNUMBER = {1133.42026},
} 
[ACP] J. M. Aldaz, L. Colzani, and J. Pérez Lázaro, Optimal bounds on the modulus of continuity of the uncentered HardyLittlewood maximal function.
@misc{ACP,
author={Aldaz, J. M. and Colzani, L. and Pérez Lázaro, J.},
TITLE={Optimal bounds on the modulus of continuity of the uncentered {H}ardy{L}ittlewood maximal function},
NOTE={{\it {J}. {G}eom. {A}nal.},
to appear},
ARXIV={1009.1359v1},
} 
[AV] J. M. Aldaz and J. L. Varona, "Singular measures and convolution operators," Acta Math. Sin. $($Engl. Ser.$)$, vol. 23, iss. 3, pp. 487490, 2007.
@article {AV, MRKEY = {2292694},
AUTHOR = {Aldaz, J. M. and Varona, Juan L.},
TITLE = {Singular measures and convolution operators},
JOURNAL = {Acta Math. Sin. $($Engl. Ser.$)$},
FJOURNAL = {Acta Mathematica Sinica (English Series)},
VOLUME = {23},
YEAR = {2007},
NUMBER = {3},
PAGES = {487490},
ISSN = {14398516},
MRCLASS = {43A10 (42B25 47B38)},
MRNUMBER = {2292694},
MRREVIEWER = {Nicholay G. Tomin},
DOI = {10.1007/s1011400506826},
ZBLNUMBER = {1165.42304},
} 
[BH] D. A. Brannan and W. K. Hayman, "Research problems in complex analysis," Bull. London Math. Soc., vol. 21, iss. 1, pp. 135, 1989.
@article {BH, MRKEY = {0967787},
AUTHOR = {Brannan, D. A. and Hayman, W. K.},
TITLE = {Research problems in complex analysis},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical Society},
VOLUME = {21},
YEAR = {1989},
NUMBER = {1},
PAGES = {135},
ISSN = {00246093},
CODEN = {LMSBBT},
MRCLASS = {3002 (00A07 3102)},
MRNUMBER = {0967787},
DOI = {10.1112/blms/21.1.1},
ZBLNUMBER = {0695.30001},
} 
[Bou1] J. Bourgain, "On highdimensional maximal functions associated to convex bodies," Amer. J. Math., vol. 108, iss. 6, pp. 14671476, 1986.
@article {Bou1, MRKEY = {0868898},
AUTHOR = {Bourgain, J.},
TITLE = {On highdimensional maximal functions associated to convex bodies},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {108},
YEAR = {1986},
NUMBER = {6},
PAGES = {14671476},
ISSN = {00029327},
CODEN = {AJMAAN},
MRCLASS = {42B25 (52A40)},
MRNUMBER = {0868898},
DOI = {10.2307/2374532},
ZBLNUMBER = {0621.42015},
} 
[Bou2] J. Bourgain, "On the $L^p$bounds for maximal functions associated to convex bodies in ${\bf R}^n$," Israel J. Math., vol. 54, iss. 3, pp. 257265, 1986.
@article {Bou2, MRKEY = {0853451},
AUTHOR = {Bourgain, J.},
TITLE = {On the {$L\sp p$}bounds for maximal functions associated to convex bodies in {${\bf R}\sp n$}},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {54},
YEAR = {1986},
NUMBER = {3},
PAGES = {257265},
ISSN = {00212172},
CODEN = {ISJMAP},
MRCLASS = {42B25},
MRNUMBER = {0853451},
MRREVIEWER = {Roger L. Jones},
DOI = {10.1007/BF02764955},
ZBLNUMBER = {0616.42013},
} 
[Bou3] J. Bourgain, "On dimension free maximal inequalities for convex symmetric bodies in ${\bf R}^n$," in Geometrical Aspects of Functional Analysis (1985/86), New York: SpringerVerlag, 1987, vol. 1267, pp. 168176.
@incollection {Bou3, MRKEY = {0907693},
AUTHOR = {Bourgain, J.},
TITLE = {On dimension free maximal inequalities for convex symmetric bodies in {${\bf R}\sp n$}},
BOOKTITLE = {Geometrical Aspects of Functional Analysis (1985/86)},
SERIES = {Lecture Notes in Math.},
VOLUME = {1267},
PAGES = {168176},
PUBLISHER = {SpringerVerlag},
ADDRESS = {New York},
YEAR = {1987},
MRCLASS = {42B25 (42B15)},
MRNUMBER = {0907693},
MRREVIEWER = {Roger L. Jones},
DOI = {10.1007/BFb0078144},
ZBLNUMBER = {0622.52005},
} 
[C] A. Carbery, "An almostorthogonality principle with applications to maximal functions associated to convex bodies," Bull. Amer. Math. Soc., vol. 14, iss. 2, pp. 269273, 1986.
@article {C, MRKEY = {0828824},
AUTHOR = {Carbery, Anthony},
TITLE = {An almostorthogonality principle with applications to maximal functions associated to convex bodies},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {14},
YEAR = {1986},
NUMBER = {2},
PAGES = {269273},
ISSN = {02730979},
CODEN = {BAMOAD},
MRCLASS = {42B25 (47A30)},
MRNUMBER = {0828824},
MRREVIEWER = {Jos{é} L. Torrea Hern{á}ndez},
DOI = {10.1090/S027309791986154364},
ZBLNUMBER = {0588.42012},
} 
[CF] O. N. Capri and N. A. Fava, "Strong differentiability with respect to product measures," Studia Math., vol. 78, iss. 2, pp. 173178, 1984.
@article {CF, MRKEY = {0766713},
AUTHOR = {Capri, O. N. and Fava, N. A.},
TITLE = {Strong differentiability with respect to product measures},
JOURNAL = {Studia Math.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
VOLUME = {78},
YEAR = {1984},
NUMBER = {2},
PAGES = {173178},
ISSN = {00393223},
CODEN = {SMATAZ},
MRCLASS = {26B05},
MRNUMBER = {0766713},
MRREVIEWER = {William Eames},
ZBLNUMBER = {0495.28002},
} 
[FSSjU] L. Forzani, R. Scotto, P. Sjögren, and W. Urbina, "On the $L^p$ boundedness of the noncentered Gaussian HardyLittlewood maximal function," Proc. Amer. Math. Soc., vol. 130, iss. 1, pp. 7379, 2002.
@article {FSSjU, MRKEY = {1855622},
AUTHOR = {Forzani, Liliana and Scotto, Roberto and Sj{ö}gren, Peter and Urbina, Wilfredo},
TITLE = {On the {$L\sp p$} boundedness of the noncentered {G}aussian {H}ardy{L}ittlewood maximal function},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {130},
YEAR = {2002},
NUMBER = {1},
PAGES = {7379},
ISSN = {00029939},
CODEN = {PAMYAR},
MRCLASS = {42B25 (60H99)},
MRNUMBER = {1855622},
MRREVIEWER = {Oscar Blasco},
DOI = {10.1090/S0002993901061561},
ZBLNUMBER = {0991.42012},
} 
[Gu] M. de Guzmán, Differentiation of Integrals in $R^{n}$, New York: SpringerVerlag, 1975, vol. 481.
@book {Gu, MRKEY = {0457661},
AUTHOR = {de Guzm{á}n, Miguel},
TITLE = {Differentiation of Integrals in {$R^{n}$}},
SERIES = {Lecture Notes in Math.},
VOLUME={481},
PUBLISHER = {SpringerVerlag},
ADDRESS = {New York},
YEAR = {1975},
PAGES = {xii+266},
MRCLASS = {28A15},
MRNUMBER = {0457661},
MRREVIEWER = {Calixto P. Calderon},
ZBLNUMBER = {0327.26010},
DOI={10.1007/BFb0081051},
} 
[Me1] A. D. Melas, "On the centered HardyLittlewood maximal operator," Trans. Amer. Math. Soc., vol. 354, iss. 8, pp. 32633273, 2002.
@article {Me1, MRKEY = {1897399},
AUTHOR = {Melas, Antonios D.},
TITLE = {On the centered {H}ardy{L}ittlewood maximal operator},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {354},
YEAR = {2002},
NUMBER = {8},
PAGES = {32633273},
ISSN = {00029947},
CODEN = {TAMTAM},
MRCLASS = {42B25 (47G10)},
MRNUMBER = {1897399},
MRREVIEWER = {Cristian E. Guti{é}rrez},
DOI = {10.1090/S0002994702029008},
ZBLNUMBER = {1015.42015},
} 
[Me2] A. D. Melas, "The best constant for the centered HardyLittlewood maximal inequality," Ann. of Math., vol. 157, iss. 2, pp. 647688, 2003.
@article {Me2, MRKEY = {1973058},
AUTHOR = {Melas, Antonios D.},
TITLE = {The best constant for the centered {H}ardy{L}ittlewood maximal inequality},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {157},
YEAR = {2003},
NUMBER = {2},
PAGES = {647688},
ISSN = {0003486X},
CODEN = {ANMAAH},
MRCLASS = {42B25},
MRNUMBER = {1973058},
MRREVIEWER = {B. Muckenhoupt},
DOI = {10.4007/annals.2003.157.647},
ZBLNUMBER = {1055.42013},
} 
[MS] M. Trinidad Menarguez and F. Soria, "Weak type $(1,1)$ inequalities of maximal convolution operators," Rend. Circ. Mat. Palermo, vol. 41, iss. 3, pp. 342352, 1992.
@article {MS, MRKEY = {1230582},
AUTHOR = {Trinidad Menarguez, M. and Soria, Fernando},
TITLE = {Weak type {$(1,1)$} inequalities of maximal convolution operators},
JOURNAL = {Rend. Circ. Mat. Palermo},
FJOURNAL = {Rendiconti del Circolo Matematico di Palermo. Serie II},
VOLUME = {41},
YEAR = {1992},
NUMBER = {3},
PAGES = {342352},
ISSN = {0009725X},
CODEN = {RCMMAR},
MRCLASS = {42B25 (42B10 47G10)},
MRNUMBER = {1230582},
MRREVIEWER = {Anthony Carbery},
DOI = {10.1007/BF02848939},
ZBLNUMBER = {0770.42013},
} 
[Mu] D. Müller, "A geometric bound for maximal functions associated to convex bodies," Pacific J. Math., vol. 142, iss. 2, pp. 297312, 1990.
@article {Mu, MRKEY = {1042048},
AUTHOR = {M{ü}ller, Detlef},
TITLE = {A geometric bound for maximal functions associated to convex bodies},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {142},
YEAR = {1990},
NUMBER = {2},
PAGES = {297312},
ISSN = {00308730},
CODEN = {PJMAAI},
MRCLASS = {42B25},
MRNUMBER = {1042048},
ZBLNUMBER = {0728.42015},
MRREVIEWER = {Gerald B. Folland},
URL = {http://projecteuclid.org/getRecord?id=euclid.pjm/1102646348},
} 
[Sj] P. Sjögren, "A remark on the maximal function for measures in ${\bf R}^{n}$," Amer. J. Math., vol. 105, iss. 5, pp. 12311233, 1983.
@article {Sj, MRKEY = {0714775},
AUTHOR = {Sj{ö}gren, Peter},
TITLE = {A remark on the maximal function for measures in {${\bf R}\sp{n}$}},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {105},
YEAR = {1983},
NUMBER = {5},
PAGES = {12311233},
ISSN = {00029327},
CODEN = {AJMAAN},
MRCLASS = {28A15},
MRNUMBER = {0714775},
MRREVIEWER = {P. L. Walker},
DOI = {10.2307/2374340},
ZBLNUMBER = {0528.42007},
} 
[St1] E. M. Stein, "The development of square functions in the work of A. Zygmund," Bull. Amer. Math. Soc., vol. 7, iss. 2, pp. 359376, 1982.
@article {St1, MRKEY = {0663787},
AUTHOR = {Stein, Elias M.},
TITLE = {The development of square functions in the work of {A}. {Z}ygmund},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {7},
YEAR = {1982},
NUMBER = {2},
PAGES = {359376},
ISSN = {02730979},
CODEN = {BAMOAD},
MRCLASS = {4202 (01A60 4203 42B99)},
MRNUMBER = {0663787},
MRREVIEWER = {Roger L. Jones},
DOI = {10.1090/S027309791982150406},
ZBLNUMBER = {0526.01021},
} 
[St2] E. M. Stein, "Three variations on the theme of maximal functions," in Recent Progress in Fourier Analysis, Amsterdam: NorthHolland, 1985, vol. 111, pp. 229244.
@incollection {St2, MRKEY = {0848150},
AUTHOR = {Stein, Elias M.},
TITLE = {Three variations on the theme of maximal functions},
BOOKTITLE = {Recent Progress in {F}ourier Analysis},
VENUE={{E}l {E}scorial, 1983},
SERIES = {NorthHolland Math. Stud.},
VOLUME = {111},
PAGES = {229244},
PUBLISHER = {NorthHolland},
ADDRESS = {Amsterdam},
YEAR = {1985},
MRCLASS = {42B25},
MRNUMBER = {0848150},
MRREVIEWER = {Jos{é} L. Rubio de Francia},
DOI = {10.1016/S03040208(08)702883},
ZBLNUMBER = {0602.42021},
} 
[St3] E. M. Stein, Harmonic Analysis: RealVariable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton Univ. Press, 1993, vol. 43.
@book {St3, MRKEY = {1232192},
AUTHOR = {Stein, Elias M.},
TITLE = {Harmonic Analysis: RealVariable Methods, Orthogonality, and Oscillatory Integrals},
SERIES = {Princeton Math. Series},
VOLUME = {43},
NOTE = {with the assistance of {T}imothy {S}. {M}urphy},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1993},
PAGES = {xiv+695},
ISBN = {0691032165},
MRCLASS = {4202 (35Sxx 4302 47G30)},
MRNUMBER = {1232192},
MRREVIEWER = {Michael Cowling},
ZBLNUMBER = {0821.42001},
} 
[StSt] E. M. Stein and J. O. Strömberg, "Behavior of maximal functions in ${\bf R}^{n}$ for large $n$," Ark. Mat., vol. 21, iss. 2, pp. 259269, 1983.
@article {StSt, MRKEY = {0727348},
AUTHOR = {Stein, Elias M. and Str{ö}mberg, J.O.},
TITLE = {Behavior of maximal functions in {${\bf R}\sp{n}$} for large {$n$}},
JOURNAL = {Ark. Mat.},
FJOURNAL = {Arkiv för Matematik},
VOLUME = {21},
YEAR = {1983},
NUMBER = {2},
PAGES = {259269},
ISSN = {00042080},
CODEN = {AKMTAJ},
MRCLASS = {42B25},
MRNUMBER = {0727348},
MRREVIEWER = {Gerald B. Folland},
DOI = {10.1007/BF02384314},
ZBLNUMBER = {0537.42018},
}