Endpoint estimates for the circular maximal function

Lee, Sanghyuk

doi:10.1090/S0002-9939-02-06781-3

Endpoint estimates for the circular maximal function
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by Sanghyuk Lee PDF

Proc. Amer. Math. Soc. 131 (2003), 1433-1442 Request permission

Abstract:

We consider the problem of endpoint estimates for the circular maximal function defined by \[ Mf(x)=\sup _{1<t<2}\left |\int _{S^1} f(x-ty)d\sigma (y)\right | \] where $d\sigma$ is the normalized surface area measure on $S^1$. Let $\Delta$ be the closed triangle with vertices $(0,0), (1/2, 1/2), (2/5,1/5)$. We prove that for $(1/p,1/q)\in \Delta \setminus \{(1/2,1/2), (2/5,1/5)\}$, there is a constant $C$ such that $\|Mf\|_{L^q(\mathbb R^2)}\le C\|f\|_{L^p(\mathbb R^2)}.$ Furthermore, $\|Mf\|_{L^{5,\infty }(\mathbb R^2)}\le C \|f\|_{L^{5/2,1}(\mathbb R^2)}$.

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