Sharp weak type estimates for a family of Córdoba bases
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Hagelstein, Paul [1]
;
Stokolos, Alex
[2]
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[1] Baylor University
Baylor University
Estados Unidos
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[2] Georgia Southern University
Georgia Southern University
Estados Unidos
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- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 3, 2023, págs. 595-603
- Idioma: inglés
- DOI: 10.1007/s13348-022-00366-5
- Texto completo no disponible (Saber más ...)
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Resumen
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} .
