Mathematical Research Letters

Volume 20 (2013)

Number 5

Lower bounds for the weak type (1, 1) estimate for the maximal function associated to cubes in high dimensions

Pages: 907 – 918



A. S. Iakovlev (Institutionen för matematik, Kungliga Tekniska Högskolan, Stockholm, Sweden)

J.-O. Strömberg (Institutionen för matematik, Kungliga Tekniska Högskolan, Stockholm, Sweden)


In this paper, we will provide the quantitative estimation for the dependence of a lower bound of the Hardy-Littlewood maximal function. This work was inspired by the paper of Stein and Strömberg where general properties of the maximal function were studied. In that work, the increase with the dimension $d$ of the constant $A_d$ that appears in the weak type (1, 1) inequality for the maximal function was proved however no estimation were given. In a recent paper, J.M. Aldaz showed that the lowest constant $A_d$ tends to infinity as the dimension $d \to \infty$. In this paper, we improve the result of J.M. Aldaz providing quantitative estimation of $A_d \geq Cd^{1/4}$, where $C$ is a constant independent of $d$.

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