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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