Boundedness of Certain Calderón–Zygmund Type Singular Integrals on Morrey Spaces

Let \(\beta \in (0,n)\) $\beta \in (0,n)$ . In this paper, we study the boundedness of the singular integral \(\begin{aligned} T(f)(x):=\mathrm {p.v.}\int _{\mathbb {R}^n}\frac{\Omega (y)}{|y|^{n-\beta }}f(x-y)\,dy, \end{aligned}\) $$\begin{array}{c}\hfill T(f)(x):=\mathrm{p}.\mathrm{v}.{\int }_{{\mathbb{R}}^n}\frac{\mathrm{\Omega }(y)}{{|y|}^{n-\beta }}f(x-y)dy,\end{array}$$ which can be viewed as an extension of the classical Calderón–Zygmund singular integral, on Morrey spaces. Precisely, let \(q\in (1,\infty )\) $q\in (1,\infty )$ and \(\theta \in (0,n]\) $\theta \in (0,n]$ . We prove that, for any \(f\in \mathcal {M}^{\theta }_q(\mathbb {R}^n)\cap \mathcal {M}^{\theta }_1(\mathbb {R}^n)\) $f\in {\mathcal{M}}_q^{\theta }({\mathbb{R}}^n)\cap {\mathcal{M}}_1^{\theta }({\mathbb{R}}^n)$ , \(Tf\in \mathcal {M}^{\theta }_q(\mathbb {R}^n)\) $Tf\in {\mathcal{M}}_q^{\theta }({\mathbb{R}}^n)$ and \(\begin{aligned} \Vert Tf\Vert _{\mathcal {M}^{\theta }_q(\mathbb {R}^n)}\le C\left[ \Vert f\Vert _{\mathcal {M}^{\theta }_q(\mathbb {R}^n)} +\frac{\beta ^{\frac{(q-1)n}{q}}}{\root q \of {n(q-1)-\beta q}}\Vert f\Vert _{\mathcal {M}^{\theta }_1(\mathbb {R}^n)}\right] , \end{aligned}\) $$\begin{array}{c}\hfill {\Vert Tf\Vert }_{{\mathcal{M}}_q^{\theta }({\mathbb{R}}^n)}\le C\left[{\Vert f\Vert }_{{\mathcal{M}}_q^{\theta }({\mathbb{R}}^n)},+,\frac{{\beta }^{\frac{(q-1)n}{q}}}{\sqrt[q]{n(q-1)-\beta q}},{\Vert f\Vert }_{{\mathcal{M}}_1^{\theta }({\mathbb{R}}^n)}\right],\end{array}$$ where the constant C is independent of \(\beta \) $\beta$ and f, and \(\mathcal {M}^{\theta }_q(\mathbb {R}^n)\) ${\mathcal{M}}_q^{\theta }({\mathbb{R}}^n)$ denotes the Morrey space on \(\mathbb {R}^n\) ${\mathbb{R}}^n$ .