Some remarks on dimension-free estimates for the discrete Hardy-Littlewood maximal functions (Q6161726)

In this article, dependencies of the optimal constants in strong and weak type bounds between maximal functions corresponding to the Hardy-Littlewood averaging operators over convex symmetric bodies acting on \(\mathbb{R}^d\) and \(\mathbb{Z}^d\) are studied. The manuscript contains ``A brief overview of the paper'' in the Introduction where the main results obtained are described in detail, several important observations are made and some conjectures in this context are open to discussion. Let \(G\) be a convex symmetric body in \(\mathbb{R}^d\), i.e. a bounded closed and symmetric convex subset of \(\mathbb{R}^d\) with nonempty interior. One of the most classical examples are the \(q\)-balls \(B^q\) in \(\mathbb{R}^d\) defined by \begin{align*} B^q = B^q (d) = \left\{ x \in \mathbb{R}^d : |x|_q = \left( \sum_{i = 1}^d |x_i|^q \right)^{1/q} \leq 1 \right\}, \quad 1 \leq q < \infty , \end{align*} and \begin{align*} B^{\infty} = B^{\infty} (d) = \left\{ x \in \mathbb{R}^d : |x|_\infty = \max_{i= 1, \ldots , d} |x_i| \leq 1 \right\}, \quad q = \infty. \end{align*} The authors associate with a convex symmetric body \(G\) the families of continuous \((M_t^G)_{t > 0}\) and discrete \((\mathcal{M}_t^G)_{t > 0}\) averaging operators given respectively by \begin{align*} M_t^G F (x) = \frac{1}{|G_t|} \int_{G_t} F(x-y) \, dy \qquad \text{and} \qquad \mathcal{M}_t^G f(x) = \frac{1}{|G_t \cap \mathbb{Z}^d|} \sum_{G_t \cap \mathbb{Z}^d} f(x-y) , \end{align*} for \(F \in L^1_{loc} (\mathbb{R}^d)\) and \(f \in \ell^{\infty} (\mathbb{Z}^d)\), where \( G_t = \{ y \in \mathbb{R}^d : t^{-1} y \in G \} \). Then they define the corresponding maximal functions by \begin{align*} M_{\ast}^G F (x) = \sup_{t > 0} |M_t^G F (x)| \qquad \text{and} \qquad \mathcal{M}_{\ast}^G f(x) = \sup_{t > 0} |\mathcal{M}_t^G f (x)| . \end{align*} Both maximal functions are of weak type \((1,1)\) and of strong type \((p,p)\) for any \(1 < p \leq \infty\), but none of these maximal functions is of strong type \((1,1)\). Finally, for \(1 < p \leq \infty\), denote by \( C(G,p) \) the smallest positive constant \(C\) for which the following strong type inequality holds \begin{align*} \| M_{\ast}^G F \|_{L^p (\mathbb{R}^d)} \leq C \| F \|_{L^p (\mathbb{R}^d)}, \qquad \text{for all } F \in L^p (\mathbb{R}^d). \end{align*} Similarly, \( C(G,1) \) will stand for the smallest positive constant \(C\) satisfying \begin{align*} \sup_{ \lambda > 0 } \lambda | \{ x \in \mathbb{R}^d : M_{\ast}^G F (x) > \lambda \} | \leq C \| F \|_{L^1 (\mathbb{R}^d)} , \qquad \text{for all } F \in L^1 (\mathbb{R}^d). \end{align*} Analogously, define \( \mathcal{C} (G,p) \), \(1 \leq p \leq \infty\), referring to \(\mathcal{M}_{\ast}^G\) in place of \(M_{\ast}^G\). The main results of this article are as follows. Theorem 1. For each \(1 \leq p \leq \infty\) the following estimate holds \begin{align*} C(G,p) \leq \mathcal{C} (G,p). \end{align*} Moreover, for the \(d\)-dimensional cube \(B^\infty \) one has \begin{align*} C(B^\infty ,1) = \mathcal{C} (B^\infty ,1) . \end{align*} \noindent Theorem 2. Let \(2 < q < \infty \), \( 1 < p < \infty \) and \(a > 0\). There exists a positive constant \(C( p,q,a )\) independent of the dimension \(d\) such that \begin{align*} \left\| \sup_{ N \geq a \, d } | \mathcal{M}_{N}^{B^q} f | \right\|_{ \ell^p (\mathbb{Z}^d) } \leq C( p,q,a ) \| f \|_{ \ell^p (\mathbb{Z}^d) } , \qquad \text{for all } f \in \ell^p (\mathbb{Z}^d) . \end{align*} \noindent Theorem 3. Denote the dyadic values by \(\mathbb{D} = \{ 2^n : n \in \mathbb{Z} \}\) and fix \( 2 \leq q < \infty \). Let \(C_1 , C_2 > 0\) and define \begin{align*} \mathbb{D}_{C_1, C_2} = \{ N \in \mathbb{D} : C_1 d^{1/q} \leq N \leq C_2 d \} . \end{align*} Then there exists a positive constant \(C_q\) independent of the dimension \(d\) such that for every \(f \in \ell^2 (\mathbb{Z}^d)\) one has \begin{align*} \Big\| \sup_{N \in \mathbb{D}_{C_1, C_2}} | \mathcal{M}_N^{B^q} f | \Big\|_{\ell^2 (\mathbb{Z}^d)} \leq C_q \| f \|_{\ell^2 (\mathbb{Z}^d)} . \end{align*} As noted, Theorem 1 gives a quantitative dependence between \(C(G,p)\) and \(\mathcal{C} (G,p)\). At the same time, inequality \(C(G,p) \leq \mathcal{C} (G,p)\) coincides with a well-known phenomenon in harmonic analysis, which states that it is harder to establish bounds for discrete operators than the bounds for their continuous counterparts. Systematic studies of dimension-free estimates have an extensive literature and are currently still a subject of study.