Regularity Properties of Bilinear Maximal Function and Its Fractional Variant

Abstract

We introduce and study the bilinear fractional maximal operator

$$\begin{aligned} {\mathfrak {M}}_\alpha (f,g)(x)=\sup \limits _{r>0}\frac{1}{|B(O,r)|^{1-\frac{\alpha }{n}}}\int _{B(O,r)}|f(x+y)g(x-y)|dy,\ \ \ \ \ x\in {\mathbb {R}}^n, \end{aligned}$$

where $O=(0,0,\ldots ,0)\in {\mathbb {R}}^n$ and $0\le \alpha <n$, which recovers the classical bilinear maximal operator ${\mathfrak {M}}$, corresponding to the case $\alpha =0$. When $\alpha =0$, we demonstrate that ${\mathfrak {M}}$ is bounded and continuous from the product Triebel–Lizorkin spaces $F_s^{p_1,q}({\mathbb {R}}^n)\times F_s^{p_2,q}({\mathbb {R}}^n)$ to $F_s^{p,q}({\mathbb {R}}^n)$, from the product fractional Sobolev spaces $W^{s,p_1}({\mathbb {R}}^n)\times W^{s,p_2}({\mathbb {R}}^n)$ to $W^{s,p}({\mathbb {R}}^n)$ and from the product Besov spaces $B_s^{p_1,q}({\mathbb {R}}^n)\times B_s^{p_2,q} ({\mathbb {R}}^n)$ to $B_s^{p,q}({\mathbb {R}}^n)$, provided that $0<s<1$, $1<p_1,p_2,p,q<\infty$ and $1/p=1/p_1+1/p_2$. When $0<\alpha <n$, we show that ${\mathfrak {M}}_\alpha$ is bounded and continuous from $W^{1,p_1}({\mathbb {R}}^n)\times W^{1,p_2}({\mathbb {R}}^n)$ to $W^{1,q}({\mathbb {R}}^n)$, provided that $1<p_1,p_2<\infty$, $1\le {p_1p_2}/{(p_1+p_2)}<\infty$, $0\le \alpha <n({1}/{p_1}+{1}/{p_2})$ and ${1}/{q}={1}/{p_1}+{1}/{p_2}-{\alpha }/{n}$. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to establish the q-quasicontinuity of ${\mathfrak {M}}_\alpha (f,g)$.