Maximal functions associated with flat plane curves with mitigating factors. (English) Zbl 1409.42016
Summary: We study the boundedness problem for maximal operators \({\mathbb {M}}^{\sigma }\) associated with flat plane curves with mitigating factors, defined by
\[
{\mathbb {M}}^{\sigma }f(x) \, := \, \sup _{1 \leq t \leq 2} \left| \int _{0}^{1} f(x-t\Gamma (s)) \, \kappa (s)^{\sigma } \, \mathrm{d}s\right| ,
\]
where \(\kappa (s)\) denotes the curvature of the curve \(\Gamma (s)=(s, g(s)+1)\), \(g(s) \in C^5[0,1]\) in \({\mathbb {R}}^2\). Let \(\triangle \) be the closed triangle with vertices \(P=(\frac{2}{5}, \frac{1}{5})\), \(Q=(\frac{1}{2}, \frac{1}{2})\), \(R=(0, 0)\). In this paper, we prove that for \( (\frac{1}{p}, \frac{1}{q}) \in \left( \Delta {\setminus } \{P, Q\} \right) \cap \{(\frac{1}{p}, \frac{1}{q}) :q > \max \{\sigma ^{-1},2\} \}\), there is a constant \(B\) such that \( \| {\mathbb {M}}^{\sigma }f\| _{L^q({\mathbb {R}}^2)} \leq \, B \, \| f\| _{L^p({\mathbb {R}}^2)}\).
MSC:
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B15 | Multipliers for harmonic analysis in several variables |
| 46T30 | Distributions and generalized functions on nonlinear spaces |
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