Boundedness of commutators of singular and potential operators in generalized grand Morrey spaces and some applications (Q2848714)

Let \((X,d,\mu)\) be a space of homogeneous type, where \(d\) is a quasi-metric on \(X\) and \(\mu\) is a positive measure on \(X\) satisfying {\parindent=6mm \begin{itemize}\item[-] the doubling condition; \item[-] \(\mu(B(x,r))< \infty\) for all \(x\in X\) and all \(r>0\); \item[-] \(\mu\big(\frac{B(x,R)}{B(x,r)}\big)>0\) for all \(x\in X\) and all \(0<r<R<d_X\). NEWLINENEWLINE\end{itemize}} Here \(B(x,r)\) and \(d_X\) denote the ball \(\{y\in X: d(x,y)<r\}\) and the diameter of \(X\).NEWLINENEWLINEFor \(1\leq p < \infty\) and \(0\leq \lambda <1\), the usual Morrey space \(L^{p,\lambda}(X,\mu)\) is the set of all measurable functions such that NEWLINE\[NEWLINE\|f\|_{L^{p,\lambda}(X,\mu)}:= \sup_{_{\substack{ x\in X\\ 0<r<d_X}}} \bigg(\frac 1{\mu(B(x,r))^\lambda} \int_{B(x,r)} |f(y)|^p d\mu(y)\bigg)^{1/p} < \infty.NEWLINE\]NEWLINE Also let \(\mu\) satisfy \(\mu(B(x,r))\lesssim r^\gamma\) with \(\gamma >0\). Let \(\varphi\) be a positive bounded function with \(\lim_{t\to 0^+}\varphi(t)=0\), and \(A\geq 0\) be a non-decreasing real-valued function with \(\lim_{x\to 0^+}A(x)=0\). The \textit{generalized grand Morrey space} \(L^{p),\lambda)}_{\varphi,A}(X,\mu)\) is the set of all measurable functions such that NEWLINE\[NEWLINE\|f\|_{L^{p),\lambda)}_{\varphi,A}(X,\mu)} := \sup_{0<\varepsilon <\min\{p-1,a\}} \varphi(\varepsilon)^{1/(p-\varepsilon)} \|f\|_{L^{p-\varepsilon,\lambda-A(\varepsilon)}(X,\mu)} < \infty,NEWLINE\]NEWLINE where \(a=\sup\{x>0: A(x)\leq \lambda\}\).NEWLINENEWLINEThe authors first showNEWLINENEWLINETheorem 1. Let \(1<p < \infty\) and \(0< \lambda <1\). If \(T\) is a Calderón-Zygmund operator and \(b\in \text{BMO}(X,\mu)\), then the commutator \([b,T]\) satisfies NEWLINE\[NEWLINE\big\|[b,T]f\big\|_{L^{p,\lambda}(X,\mu)}\lesssim \|b\|_{\text{BMO}(X,\mu)} \|f\|_{L^{p,\lambda}(X,\mu)}\qquad\text{for}\;f\in L^\infty_c(X).NEWLINE\]NEWLINENEWLINENEWLINEAs a corollary of Theorem 1 together with an ``extended reduction lemma'', the authors obtain their first main result as follows.NEWLINENEWLINETheorem 2. Let \(1<p < \infty\), \(0< \lambda <1\) and \(\theta>0\). If \(T\) is a Calderón-Zygmund operator and \(b\in \text{BMO}(X,\mu)\), then the commutator \([b,T]\) is bounded on \(L^{p),\lambda)}_{\theta,A}(X,\mu)\).NEWLINENEWLINEFor \(0<\alpha <1\), the potential operator \(I^\alpha\) is defined by NEWLINE\[NEWLINEI^\alpha f(x)= \int_X \frac{f(y)}{\mu(B(x,d(x,y)))^{1-\alpha}} d\mu(y).NEWLINE\]NEWLINE By studying the boundedness of \(M([b,I^\alpha])\) acting on generalized grand Morrey spaces, where \(b\in \text{BMO}(X,\mu)\) and \(M\) denotes the Hardy-Littlewood maximal operator, the authors obtain the boundedness of the commutator \([b,I^\alpha]\) acting on generalized grand Morrey spaces, which is their second main result.NEWLINENEWLINEAs applications, interior estimates for solutions of elliptic equations are also established in the framework of generalized grand Morrey spaces.