Rough maximal functions supported by subvarieties on Triebel–Lizorkin spaces

Abstract

In this paper, we establish the boundedness of a class of maximal functions related to rough singular integrals supported by compound subvarieties on Triebel–Lizorkin spaces and Besov spaces. As applications, several corresponding estimates of maximal functions related to parametric Marcinkiewicz integrals are also presented.

Appendix

In this section we give the definitions of several rough kernels we used. Recall that the Hardy space $H^1(\mathrm{S}^{n-1})$ is the set of all $L^1(\mathrm{S}^{n-1})$ functions which satisfy $\Vert f\Vert _{H^1(\mathrm{S}^{n-1})}<\infty$, where

$$\begin{aligned} \Vert \varOmega \Vert _{H^1(\mathrm{S}^{n-1})}:=\int _{\mathrm{S}^{n-1}}\sup \limits _{0\le r<1}\left| \int _{\mathrm{S}^{n-1}}\varOmega (\theta )\frac{1-r^2}{|rw-\theta |^n}d\sigma (\theta )\right| d\sigma (w). \end{aligned}$$

The class $L(\log ^+L)^\alpha (\mathrm{S}^{n-1})$ (for $\alpha >0$) denotes the class of all measurable functions $\varOmega$ on $\mathrm{S}^{n-1}$ which satisfy

$$\begin{aligned} \Vert \varOmega \Vert _{L(\log ^{+}L)^\alpha (\mathrm{S}^{n-1})}:=\int _{\mathrm{S}^{n-1}}|\varOmega (\theta )|\log ^\alpha (|\varOmega (\theta )|+2)d\sigma (\theta )<\infty . \end{aligned}$$

Now we recall the definition of the block space $B_q^{(0,v)}(\mathrm{S}^{n-1})$. The block spaces in $\mathbf {R}^n$ originated from the work of Taibleson and Weiss on the convergence of the Fourier series in connection with developments of the real Hardy spaces. The block spaces on $\mathrm{S}^{n-1}$ was introduced by Jiang and Lu [19] in studying the homogeneous singular integral operators. A q-block on $\mathrm{S}^{n-1}$ is an $L^q(\mathrm{S}^{n-1})$ ($1<q\le \infty )$ function b which satisfies $\mathrm{supp}(b)=I$ and $\Vert b\Vert _q\le |I|^{1-1/q}$, where $|I|=\sigma (I)$, and $I=\{x\in \mathrm{S}^{n-1}:|x-x_0|<\alpha \}$ for some $\alpha \in (0,1]$ and $x_0\in \mathrm{S}^{n-1}$. The block $B_q^{(0,v)}(\mathrm{S}^{n-1})$ is defined by

$$\begin{aligned} B_q^{(0,v)}(\mathrm{S}^{n-1}):=\{\varOmega \in L^1(\mathrm{S}^{n-1}):\varOmega =\sum \limits _{\mu =1}^\infty \lambda _\mu b_\mu ,\ \ M_q^{(0,v)}(\{\lambda _\mu \})<\infty \}, \end{aligned}$$

where $v>-1$, $\lambda _\mu \in \mathbf {C}, b_\mu$ is a q-block supported on a cap $I_\mu$ on $\mathrm{S}^{n-1}$ and

$$\begin{aligned} M_q^{(0,v)}(\{\lambda _\mu \})=\sum \limits _{\mu =1}^\infty |\lambda _\mu |\big (1+\log ^{(v+1)}(|I_\mu |^{-1})\big ). \end{aligned}$$

The norm of $B_q^{(0,v)}(\mathrm{S}^{n-1})$ is given by

$$\begin{aligned} \Vert \varOmega \Vert _{B_q^{(0,v)}(\mathrm{S}^{n-1})}:=N_q^{(0,v)}(\varOmega )=\inf \{M_q^{(0,v)}(\{\lambda _\mu \})\}, \end{aligned}$$

where the infimum is taken over all q-block decompositions of $\varOmega$.

It is known that for any $0<\beta <1, L(\log ^+L)^{\beta }(\mathrm{S}^{n-1})$ and $H^1(\mathrm{S}^{n-1})$ do not contained each other. Moreover, the following proper inclusion relations are known:

$$\begin{aligned}&\displaystyle L^r(\mathrm{S}^{n-1})\subset L(\log ^+L)^{\beta _1}(\mathrm{S}^{n-1})\subset L(\log ^+L)^{\beta _2}(\mathrm{S}^{n-1})\ \mathrm{for}\ r>1\ \mathrm{and}\ 0<\beta _2<\beta _1;\nonumber \\&\displaystyle L(\log ^+L)^{\beta }(\mathrm{S}^{n-1})\subset H^1(\mathrm{S}^{n-1})\ \mathrm{for}\ \beta \ge 1;\nonumber \\&\displaystyle \bigcup \limits _{r>1}L^r(\mathrm{S}^{n-1})\subset B_q^{(0,v)}(\mathrm{S}^{n-1})\ \mathrm{if} \ q>1\ \mathrm{and}\ v>-1;\nonumber \\&\displaystyle B_q^{(0,v_2)}(\mathrm{S}^{n-1})\subset B_q^{(0,v_1)}(\mathrm{S}^{n-1})\ \mathrm{if} \ q>1\ \mathrm{and}\ v_2>v_1>-1;\nonumber \\&\displaystyle B_q^{(0,v)}(\mathrm{S}^{n-1})\subset H^1(\mathrm{S}^{n-1})+L(\log ^+L)^{1+v}(\mathrm{S}^{n-1})\ \mathrm{for}\ q>1\ \mathrm{and}\ v>-1. \end{aligned}$$

(4.5)