Limiting weak-type behaviors for fractional maximal operators and fractional integrals with rough kernel (Q2117372)

In this paper, the authors first investigated the limiting weak-type behaviors for the fractional maximal operators \(M_\Omega^\alpha\) and the fractional integrals \(T_{|\Omega|}^\alpha\) without any smoothness assumption on \(\Omega\), where \(\alpha\in(0,1)\), \(\Omega\) is a homogeneous function of degree zero and \[M_\Omega^\alpha=\sup_{r>0}\frac{1}{r^{n-\alpha}}\int_{B(x,y)}|\Omega(x-y)|f(y)dy ,\ T_\Omega^\alpha f(x)=\int_{\mathbb{R}^n}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}f(y)dy.\] To be more precise, by a reduction method, the authors proved that: Theorem 1 Let \(\alpha\in(0,n)\), \(1<r<\infty\). Suppose that \(\Omega\in L^1(\mathbb{S}^{n-1})\) is a homogeneous function of degree zero. The following statements are equivalent. (1) \(\Omega\in L^{\frac{n}{n-\alpha}}(\mathbb{S}^{n-1})\); (2) \(M_\Omega^\alpha\) is bounded from \(L^1\) to \(L^{\frac{n}{n-\alpha},\infty}\); (3) \(T_{|\Omega|}^\alpha\) is bounded from \(L^1\) to \(L^{\frac{n}{n-\alpha},\infty}\); (4) \(M_\Omega^\alpha\) is bounded from \(L^1(l^r)\) to \(L^{\frac{n}{n-\alpha},\infty}(l^r)\); (5) \(T_{|\Omega|}^\alpha\) is bounded from \(L^1(l^r)\) to \(L^{\frac{n}{n-\alpha},\infty}(l^r)\). Moreover, let \(f\in L^1(\mathbb{R}^n)\) and \(\{f_j\}_{j\in\mathbb{N}}\in L^1(l^r)\), if one of the above statements holds, then (a) \(\|\Omega\|_{L^{\frac{n}{n-\alpha}}(\mathbb{S}^{n-1})}\sim\Big\|\frac{\Omega(\cdot)}{|\cdot|^{n-\alpha}}\Big\|_{L^{\frac{n}{n-\alpha},\infty}}\sim\|M_{\Omega}^\alpha\|_{L^1\rightarrow L^{\frac{n}{n-\alpha},\infty}}\sim\|T_{|\Omega|}^\alpha\|_{L^1\rightarrow L^{\frac{n}{n-\alpha},\infty}}\sim\\ \|M_{\Omega}^\alpha\|_{L^1(l^r)\rightarrow L^{\frac{n}{n-\alpha},\infty}(l^r)}\sim\|T_{|\Omega|}^\alpha\|_{L^1(l^r)\rightarrow L^{\frac{n}{n-\alpha},\infty}(l^r)}\); (b) \(\lim_{t\rightarrow0^+}\Big\|M_{\Omega}^\alpha f_t(\cdot)-\frac{\Omega(\cdot)}{|\cdot|^{n-\alpha}}\|f\|_{L^1(\mathbb{R}^n)}\Big\|_{L^{\frac{n}{n-\alpha},\infty}(\mathbb{R}^n\backslash B(0,\rho))}=0\) for every \(\rho>0\); (c) \(\lim_{t\rightarrow0^+}\Big\|T_{|\Omega|}^\alpha f_t(\cdot)-\frac{\Omega(\cdot)}{|\cdot|^{n-\alpha}}\int_{\mathbb{R}^n}f(x)dx\Big\|_{L^{\frac{n}{n-\alpha},\infty}(\mathbb{R}^n\backslash B(0,\rho))}=0\) for every \(\rho>0\); (d) \(\lim_{t\rightarrow0^+}\Big\|\|\{M_{\Omega}^\alpha f_{j,t}(\cdot)-\frac{\Omega(\cdot)}{|\cdot|^{n-\alpha}}\|f\|_{L^1(\mathbb{R}^n)}\}_{j\in\mathbb{N}}\|_{l^r}\Big\|_{L^{\frac{n}{n-\alpha},\infty}(\mathbb{R}^n\backslash B(0,\rho))}=0\) for every \(\rho>0\). (2) \(\lim_{t\rightarrow0^+}\Big\|\|\{T_{|\Omega|}^\alpha f_{j,t}(\cdot)-\frac{\Omega(\cdot)}{|\cdot|^{n-\alpha}}\int_{\mathbb{R}^n}f_j(x)dx\}_{j\in\mathbb{N}}\|_{l^r}\Big\|_{L^{\frac{n}{n-\alpha},\infty}(\mathbb{R}^n\backslash B(0,\rho))}=0\) for every \(\rho>0\). Theorem 1 essentially improves and extends the results in [\textit{W. Guo} et al., Potential Anal. 54, No. 2, 307--330 (2021; Zbl 07303865)].