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Mapping properties of the elliptic maximal function. (English) Zbl 1037.42024

Let \(\mathcal E\) be the set of all ellipses in \(\mathbb R^2\) centered at the origin with axial lengths in \([\frac12, 2]\). The author defines the elliptic maximal function \(M\) by \[ Mf(x)=\sup_{E\in \mathcal E}| E| ^{-1}\int_E f(x+s)d\sigma(s) \] for a real-valued continuous function \(f\), where \(d\sigma\) is the arclength measure on \(E\) and \(| E| \) is the length of \(E\). The author notes that \(M\) is not bounded in \(L^p(\mathbb R^2)\) for \(p\leq 4\), and then as a step to consider the case \(p>4\), considers the related maximal functions \(M_\delta\), defined by \(M_\delta f(x)=\sup_{E\in \mathcal E}| E_\delta| ^{-1}\int_{E_\delta} f(x+u)du\), where \(E_\delta\) is the \(\delta\)-neighbourhood of the ellipse \(E\) and \(| E_\delta| \) is the two-dimensional Lebesgue measure of \(E_\delta\).
The main result is: \[ \| M_\delta f\| _{L^{{24}/{7},\infty}} \leq C\delta^{-1/3}| \log\delta| ^{5/4}\| f\| _{L^{2,1}}. \] From this, it follows that for \(\varepsilon>0\), \[ \| M_\delta f\| _{L^4}\leq C_\varepsilon \delta^{-(1/6+\varepsilon)}\| f\| _{L^{4}}. \] This, in turn, implies that \(M\) maps the Sobolev space \(W_{4, 1/6+\varepsilon}(\mathbb R^2)\) into \(L^4(\mathbb R^2)\) \((\varepsilon>0)\). These concern with Bourgain’s result: the circular maximal function is bounded on \(L^p(\mathbb R^2)\) if \(p>2\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:

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