Intrinsic square functions on functions spaces including weighted Morrey spaces (Q2872182)

For \(1\leq q\leq\alpha\leq p\leq \infty\) and \(w\in A_q\), the space \((L_w^q, L^p)^\alpha(\mathbb R^n)\) consists of all measurable functions \(f\) such that \(\|f\|_{q_w,p,\alpha}<\infty\), where NEWLINE\[NEWLINE\|f\|_{q_w,p,\alpha} :=\sup_{r>0}\left\{\int_{\mathbb R^n} \left[w(B(y,r))^{\frac1\alpha-\frac1q-\frac1p} \|f\chi_{B(y,r)}\|_{L_w^q(\mathbb R^n)}\right]^p\,dy\right\}^{\frac1p}.NEWLINE\]NEWLINENEWLINENEWLINEFor \(\gamma\in(0,1]\), denote by \(\mathcal C_\gamma\) the family of all functions \(\varphi\) defined on \(\mathbb R^n\) with support in the closed unit ball \(B(0,1)\) such that \(\int_{\mathbb R^n}\varphi(x)\,dx=0\) and, for all \(x, x'\in\mathbb R^n\), \(|\varphi(x)-\varphi(x')|\leq|x-x'|^\gamma\). The intrinsic square function \(S_\gamma(f)\) of \(f\) is defined by setting, for all \(f\in L_{\mathop{\mathrm{loc}}}^1({\mathbb R}^n)\) and \(x\in\mathbb R^n\), NEWLINE\[NEWLINES_\gamma(f)(x)=\left\{ \int_{|x-y|<t}\left[\sup_{\varphi\in\mathcal C_\gamma} |f*\varphi_t(y)|\right]^2\frac{dy\,dt}{t^{n+1}}\right\}^{\frac12}.NEWLINE\]NEWLINENEWLINENEWLINEIn this paper, the author proves that the intrinsic square functions, including the Lusin area integral \(S_\gamma\) and the Littlewood-Paley \(g_\lambda^*\)-function as defined by Wilson, are bounded in \((L_w^q, L^p)^\alpha(\mathbb R^n)\). The author also shows that the corresponding commutators generated by these intrinsic square functions and BMO functions are also bounded in \((L_w^q, L^p)^\alpha(\mathbb R^n)\).