The intrinsic square function characterizations of weighted Hardy spaces (Q384308)

For \(0<\alpha\leq 1\), let \(\mathcal C_\alpha\) be the family of \(\alpha\)-Lipschitz functions \(\varphi\) with support in the unit ball of \(\mathbb R^n\) and satisfying the cancellation condition \(\int_{\mathbb R^n}\varphi(x)dx=0\). For \(0<\alpha\leq 1\) and \(\varepsilon>0\), let \(\mathcal C_{\alpha,\varepsilon}\) be the family of \(\alpha\)-Lipschitz functions \(\varphi\) defined on \(\mathbb R^n\) such that NEWLINE\[NEWLINE|\varphi(x)|\leq (1+|x|)^{-n-\varepsilon}NEWLINE\]NEWLINE NEWLINE\[NEWLINE|\varphi(x)-\varphi(x')|\leq |x-x'|^{\alpha}\bigl((1+|x|)^{-n-\varepsilon} +(1+|x'|)^{-n-\varepsilon}\bigr)NEWLINE\]NEWLINE and \(\int_{\mathbb R^n}\varphi(x)dx=0\). Set NEWLINE\[NEWLINEA_\alpha(f)(x,t)=\sup_{\varphi\in \mathcal C_\alpha}|f*\varphi_t(x)|NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\tilde A_{\alpha,\varepsilon}(f)(x,t) =\sup_{\varphi\in \mathcal C_{\alpha,\varepsilon}}|f*\varphi_t(x)|,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\varphi_t(x) =t^{-n}\varphi(x/t).NEWLINE\]NEWLINE Define the Littlewood-Paley \(g\) function, Lusin's area function and \(g_{\lambda}^*\) function by NEWLINE\[NEWLINEg_\alpha(f)(x)=(\int_0^\infty A_\alpha(f)(x,t)^2t^{-1}\,dt)^{1/2},NEWLINE\]NEWLINE NEWLINE\[NEWLINES_\alpha(f)(x)=(\int_{|x-y|<t} A_\alpha(f)(y,t)^2t^{-n-1}\,dydt)^{1/2},NEWLINE\]NEWLINE and NEWLINE\[NEWLINEg_{\lambda,\alpha}^*(f)(x)=(\int_{\mathbb R_+^{n+1}} A_\alpha(f)(y,t)^2 t^{-n-1}\,dydt)^{1/2},NEWLINE\]NEWLINE respectively. They are called the intrinsic functions by M.~Wilson. The authors introduce \(g_{\alpha,\varepsilon}(f)\), \(S_{\alpha,\varepsilon}(f)\), \(g_{\lambda,(\alpha,\varepsilon)}^*(f)\) in terms of \(\tilde A_{\alpha,\varepsilon}(f)\) in place of \(A_\alpha(f)\). Using them, they characterize weighted Hardy spaces as follows.NEWLINENEWLINELet \(0<\alpha\leq 1\), \(n/(n+\alpha)<p<1\), \(w\in A_{p(1+\alpha/n)}\), \(\varepsilon>\alpha\) and \(\lambda>(3n+2\alpha)/n\). Then for a tempered distribution \(f\in ({\text{Lip}}(\alpha,1,0))^*\) the followings are equivalent.NEWLINENEWLINE(i) \(f\in H_w^{p}(\mathbb R^n)\).NEWLINENEWLINE(ii) \(g_\alpha(f)\in L_w^p(\mathbb R^n)\) or \(g_{\alpha,\varepsilon}(f)\in L_w^p(\mathbb R^n)\) and \(f\) vanishes weakly at infinity.NEWLINENEWLINE(iii) \(S_\alpha(f)\in L_w^p(\mathbb R^n)\) or \(S_{\alpha,\varepsilon}(f)\in L_w^p(\mathbb R^n)\) and \(f\) vanishes weakly at infinity.NEWLINENEWLINE(iv) \(g_{\lambda,\alpha}^*(f)\in L_w^p(\mathbb R^n)\) or \(g_{\lambda,(\alpha,\varepsilon)}^*(f)\in L_w^p(\mathbb R^n)\) and \(f\) vanishes weakly at infinity.NEWLINENEWLINEIn the above, NEWLINE\[NEWLINE{\text{ Lip}}(\alpha,1,0) =\{b\in L_{\text{loc}}^1(\mathbb R^n): \sup_{Q:{\text{ cube}}}|Q|^{-1-\alpha/n}\int_Q |b(y)-|Q|^{-1}\int_Q f(z)\,dz|\,dy<\infty\},NEWLINE\]NEWLINE and ``\(f\) vanishes weakly at infinity'' means that \(\lim_{t\to\infty}t*\varphi=0\) in the sense of distributions for any \(\varphi\in \mathcal S(\mathbb R^n)\).