Sharp weak type estimates for a family of Zygmund bases

Abstract

Let B \mathcal {B} be the collection of rectangular parallelepipeds in R 3 \mathbb {R}^3 whose sides are parallel to the coordinate axes and such that B \mathcal {B} consists of parallelepipeds with side lengths of the form s , 2 j s , t s, 2^j s, t , where s , t > 0 s, t > 0 and j j lies in a nonempty subset S S of the integers. In this paper, we prove the following: If S S is a finite set, then the associated geometric maximal operator M B M_\mathcal {B} satisfies the weak type estimate | { x ∈ R 3 : M B f ( x ) > α } | ≤ C ∫ R 3 | f | α ( 1 + log + ⁡ | f | α ) \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha }\left (1 + \log ^+ \frac {|f|}{\alpha }\right )\; \end{equation*} but does not satisfy an estimate of the form | { x ∈ R 3 : M B f ( x ) > α } | ≤ C ∫ R 3 ϕ ( | f | α ) \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right ) \end{equation*} for any convex increasing function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) \phi : \mathbb [0, \infty ) \rightarrow [0, \infty ) satisfying the condition lim x → ∞ ϕ ( x ) x ( log ⁡ ( 1 + x ) ) = 0. \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))} = 0. \end{equation*} On the other hand, if S S is an infinite set, then the associated geometric maximal operator M B M_\mathcal {B} satisfies the weak type estimate | { x ∈ R 3 : M B f ( x ) > α } | ≤ C ∫ R 3 | f | α ( 1 + log + ⁡ | f | α ) 2 \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha } \left (1 + \log ^+ \frac {|f|}{\alpha }\right )^{2} \end{equation*} but does not satisfy an estimate of the form | { x ∈ R 3 : M B f ( x ) > α } | ≤ C ∫ R 3 ϕ ( | f | α ) \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right ) \end{equation*} for any convex increasing function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) \phi : \mathbb [0, \infty ) \rightarrow [0, \infty ) satisfying the condition lim x → ∞ ϕ ( x ) x ( log ⁡ ( 1 + x ) ) 2 = 0. \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))^2} = 0. \end{equation*}