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Sharp weak type estimates for a family of Córdoba bases

Abstract

Let \(\mathcal {B}\) be a collection of rectangular parallelepipeds in \(\mathbb {R}^3\) whose sides are parallel to the coordinate axes and such that \(\mathcal {B}\) consists of parallelepipeds with sidelengths of the form \(s, t, 2^N st\), where \(s, t > 0\) and N lies in a nonempty subset S of the natural numbers. In this paper, we prove the following: If S is a finite set, then the associated geometric maximal operator \(M_\mathcal {B}\) satisfies the weak type estimate

$$\begin{aligned} \left| \left\{ x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right\} \right| \le C \int _{\mathbb {R}^3} \frac{|f|}{\alpha }\left( 1 + \log ^+ \frac{|f|}{\alpha }\right) \; \end{aligned}$$

but does not satisfy an estimate of the form

$$\begin{aligned} \left| \left\{ x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right\} \right| \le C \int _{\mathbb {R}^3} \phi \left( \frac{|f|}{\alpha }\right) \end{aligned}$$

for any convex increasing function \(\phi : [0, \infty ) \rightarrow [0, \infty )\) satisfying the condition

$$\begin{aligned} \lim _{x \rightarrow \infty }\frac{\phi (x)}{x (\log (1 + x))} = 0\;. \end{aligned}$$

Alternatively, if S is an infinite set, then the associated geometric maximal operator \(M_\mathcal {B}\) satisfies the weak type estimate

$$\begin{aligned} \left| \left\{ x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right\} \right| \le C \int _{\mathbb {R}^3} \frac{|f|}{\alpha } \left( 1 + \log ^+ \frac{|f|}{\alpha }\right) ^{2} \end{aligned}$$

but does not satisfy an estimate of the form

$$\begin{aligned} \left| \left\{ x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right\} \right| \le C \int _{\mathbb {R}^3} \phi \left( \frac{|f|}{\alpha }\right) \end{aligned}$$

for any convex increasing function \(\phi : [0, \infty ) \rightarrow [0, \infty )\) satisfying the condition

$$\begin{aligned} \lim _{x \rightarrow \infty }\frac{\phi (x)}{x (\log (1 + x))^2} = 0. \end{aligned}$$

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References

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Correspondence to Paul Hagelstein.

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P. H. is partially supported by a grant from the Simons Foundation (#521719 to Paul Hagelstein).

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Hagelstein, P., Stokolos, A. Sharp weak type estimates for a family of Córdoba bases. Collect. Math. (2022). https://doi.org/10.1007/s13348-022-00366-5

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  • DOI: https://doi.org/10.1007/s13348-022-00366-5

Keywords

  • Maximal functions
  • Differentiation basis
  • Geometric maximal operator

Mathematics Subject Classification

  • Primary 42B25