Maximal operators associated with some singular submanifolds

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