Boundedness of Lusin-area and functions on localized BMO spaces over doubling metric measure spaces
Let \((X,d)\) be a metric space endowed with a regular Borel measure \(\mu\) such that \(0<\mu(B(x,r))<\infty\) \((x\in X\), \(r>0)\). \((X,d,\mu)\) is called a doubling metric measure space if \(\mu(B(x,2r))\leq C_1\mu(B(x,r))\). A positive function \(\rho\) on \(X\) is called admissible if, for some \(C_0\), \(k_0>0\), \(\rho(x)^{-1}\leq C_0\rho(y)^{-1}(1+\rho(y)^{-1}d(x,y))^{k_0}\) \((x,y\in X)\). For an admissible function \(\rho\) and \(q\in [1,\infty)\), a function \(f\in L_{\text{loc}}^q(X)\) is said to be in \(\text{BMO}_\rho^q(X)\) if \[ \begin{aligned}\|f\|_{\text{BMO}_\rho^q(X)}= &\sup_{B=B(x,r);\,x\in X,\, r<\rho(x)}\biggl( \mu(B)^{-1}\int_B|f(y)-f_B|^q\,d\mu(y)\biggr)^{1/q}\\ +&\sup_{B=B(x,r);\,x\in X,\, r\geq\rho(x)}\biggl(\mu(B)^{-1}\int_B|f(y)|^q\,d\mu(y) \biggr)^{1/q}<\infty.\end{aligned} \] Similarly, a real valued function \(f\in L_{\text{loc}}^q(X)\) is said to be in \(\text{BLO}_\rho^q(X)\) if, in the definition of \(\text{BMO}_\rho^q(X)\), \(f_B\) is replaced by \(\text{ess\, inf}_{u\in B}f(u)\). In the case \((X,d,\mu)= (\mathbb R^n,|\cdot|,dx)\) and \(\rho\equiv1\), \(\text{BMO}_\rho^q(X)\) coincides with the Goldberg's local BMO space bmo. If \(\rho\) is an admissible function, \(\text{BMO}_\rho^q(X)\cong\text{BMO}_\rho^1(X)=:\text{BMO}_\rho(X)\). Now let \(\rho\) be an admissible function and \(\{Q_t\}_{t\geq0}\) be a family of operators bounded on \(L^2(X)\) with integral kernels \(\{Q_t(x,y)\}_{t\geq0}\) satisfying that there exist \(C\), \(\delta_1>0\), \(\delta_2\in (0,1)\), \(\gamma>0\) such that \(|Q_t(x,y)|\leq C(\mu(B(x,t))+\mu(B(x,d(x,y)))^{-1}(1+d(x,y)/t)^{-\gamma} (1+t/\rho(x))^{-\delta_1}\), and \(|\int_XQ_t(x,z)\,d\mu(z)|\leq C(1+\rho(x)/t)^{-\delta_2}\). Using this family of operators, the authors define the Littlewood-Paley \(g\)-function \(g(f)\), Lusin's area function \(S(f)\) and the \(g_\lambda^*\) function \(g_\lambda^*(f)\) by \[ \begin{aligned} g(f)(x)&= \bigg(\int_0^\infty |Q_t(f)(x)|^2\,dt/t\bigg)^{1/2},\\ S(f)(x)&= \biggl(\int_0^\infty \int_{d(x,y)<t} |Q_t(f)(y)|^2 \mu(B(y,t))^{-1}\,d\mu(y)\,dt/t\biggr)^{1/2},\\ g_\lambda^*(f)(x)&= \biggl(\iint_{X\times(0,\infty)} |Q_t(f)(y)|^2 (1+d(x,y)/t)^{-\lambda}\mu(B(y,t))^{-1}\,d\mu(y)\,dt/t\biggr)^{1/2}.\end{aligned} \] Their main result is the following: Let \(X\) be a doubling metric measure space satisfying one more condition, ``the \(\delta\)-annular decay property''. Let \(\rho\) be an admissible function. Assume that the Littlewood-Paley \(g\)-function is bounded on \(L^2(X)\). Then \(\|S(f)^2\|_{\text{BLO}_\rho(X)}\leq C \|f\|_{\text{BMO}_\rho(X)}^2\). If \(3n<\lambda <\infty\), the same result holds for the \(g_\lambda^*\) function without assuming ``the \(\delta\)-annular decay property''. Relating to their boundedness results, the authors give a nonnegative \(f\in\text{bmo}(\mathbb R)\) which is not in \(\text{blo}(\mathbb R)\).
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