Maximal operators associated with some singular submanifolds

Abstract:

Let $\mathrm {U}$ be a bounded open subset of $\mathbb {R}^d$ and let $\Omega$ be a Lebesgue measurable subset of $\mathrm {U}$. Let $\gamma =(\gamma _1, \cdots , \gamma _n) : \mathrm {U}\setminus \Omega \rightarrow \mathbb {R}^n$ be a Lebesgue measurable function, and let $\mu$ be a Borel measure on $\mathbb {R}^{d+n}$ defined by \begin{equation*} \langle \mu , f \rangle =\int _{\mathbb {R}^d} f(y, \gamma (y)) \psi (y) \chi _{\mathrm {U}\setminus \Omega }(y)\; dy, \end{equation*} where $\psi$ is a smooth function supported in $\mathrm {U}$. In this paper we give some conditions under which the Fourier decay estimates $|\widehat {\mu }(\xi )| \le C (1+|\xi |)^{-\epsilon }$ hold for some $\epsilon >0$. As a corollary we obtain the $L^p$-boundedness properties of the maximal operators $\mathrm {M}_{S}$ associated with a certain class of possibly non-smooth $n$-dimensional submanifolds of $\mathbb {R}^{d+n}$, i.e., \[ \mathrm {M}_Sf(x)=\sup _{r>0} r^{-d}\int _{|y|<r} \big {|}f\big {(}x-(y,\gamma (y))\big {)}\big {|} \chi _{\mathbb {R}^d \setminus \Omega _{\text {sym}}} dy,\] where $\Omega _{\text {sym}}$ is a radially symmetric Lebesgue measurable subset of $\mathbb {R}^d$, $\gamma (y)=(\gamma _1(y), \cdots , \gamma _n(y))$, $\gamma _i(t y)=t^{a_i} \gamma _i(y)$ for each $t>0$ where $a_i \in \mathbb {R}$, and the function $\gamma _i : \mathbb {R}^d \setminus \Omega _{\text {sym}} \rightarrow \mathbb {R}$ satisfies some singularity conditions over a certain subset of $\mathbb {R}^d$. Also we investigate the endpoint $(parabolic\; H^1, L^{1,\infty })$ mapping properties of the maximal operators $\mathrm {M}_H$ associated with a certain class of possibly non-smooth hypersurfaces, i.e., \[ \mathrm {M}_Hf(x)=\sup _{r>0}\left |\int _{\mathbb {R}^d} f\big {(}x-(y,\gamma (y))\big {)} r^{-d} \psi (r^{-1}y) dy \right |,\] where the function $\gamma : \mathbb {R}^d \rightarrow \mathbb {R}$ satisfies some singularity conditions over a certain subset of $\mathbb {R}^d$ and $\gamma (t y)=t^m \gamma (y)$ for each $t>0$ where $m>0$.