Boundedness estimate for certain Calderón–Zygmund type singular integrals on \(\textrm{BMO}\) spaces

Abstract

Let \(\beta \in (0,n)\). In this paper, we study the boundedness of the Calderón–Zygmund type singular integral

$$ T(f)(x):=\mathrm {p.v.}\int \limits _{\mathbb {R}^n}\frac{\Omega (y)}{|y|^{n-\beta }}f(x-y)\,dy $$

on the space \(\textrm{BMO}(\mathbb {R}^n)\). Precisely, let \(q\in (1,\infty )\) and \(\beta \in (0,\frac{(q-1)n}{q})\). We prove that, for any \(f\in \textrm{BMO}(\mathbb {R}^n)\cap L^{q'}(\mathbb {R}^n)\), \(Tf\in \textrm{BMO}(\mathbb {R}^n)\) and

$$ \Vert Tf\Vert _{\textrm{BMO}(\mathbb {R}^n)}\le C\left[ \Vert f\Vert _{\textrm{BMO}(\mathbb {R}^n)}+\frac{\beta ^{\frac{(q-1)n}{q}}}{\root q \of {n(q-1)-\beta q}}\Vert f\Vert _{L^{q'}(\mathbb {R}^n)}\right] , $$

where \(q'\in (1,\infty )\) is given by \(1/q+1/q'=1\) and C is a positive constant independent of \(\beta \) and f. This estimate can be seen as a further development for the corresponding results in the scale of Lebesgue spaces, established by Chen and Guo (J Funct Anal 281:Paper No. 109196, 2021), in the endpoint case.