Two-weight norm inequalities for maximal functions on homogeneous spaces and boundary estimates

Let D be an open subset of a homogeneous space(X,d,μ). Consider the maximal function ${\displaystyle M_{\phi }f\left(x\right)=\supset \frac{1}{\phi }\left(B\right){ʃ}_{B\cap \partial D}\left|f\right|d\nu }$, x∈ D, where the supremum is taken over all balls of the form B = B(a(x),r) with r > t(x) = d(x,∂D), a(x)∈ ∂D is such that d(a(x),x) < 3/2 t(x)${\displaystyle \text{and}\phi isano\cap egativesetfunctiondef\in edf\text{or}allB\text{or}elsetsofXsatiy\in gthequasi-mo\neg onicity\text{and}doubl\in g\propto erties.Wegivea\ne cessary\text{and}sufficientconditionontheweightsw\text{and}vf\text{or}theweighted\parallel \in \parallel equality\left(0.1\right)}$(ʃ_D [M_φ(f)]^q wdμ)^{1/q} ≤ c(ʃ_{∂D} |f|^p vdν)^{1/p}${\displaystyle \to holdwhen1<p<q<\infty ,}$σdν = v^{1-p'}dν${\displaystyle isadoubl\in gweight,\text{and}d\nu isadoubl\in gmeasure,\text{and}giveasufficientconditionf\text{or}\left(0.1\right)when1<p\le q<\infty withoutas\sum \in gt\hat{\sigma }isadoubl\in gweightbutwithanextraas\sum ptionon\phi .A\neg hercharacterizationf\text{or}\left(0.1\right)isalsoprovdf\text{or}1<p\le q<\infty \text{and}Dofthef\text{or}mY\times (0,\infty ),whereYisahomo\ge \ne ousspacewithgroupstructure.Theseres\underline{t}s\ge \ne ralizesomeknownthe\text{or}ems\in thecasewhen}$M_φ${\displaystyle isthe\frac{t}{i}onalmaximalfunction\in }$ℝ^{n+1}_+${\displaystyle ,t\hat{i}s,when}$M_φ f(x,t) = M_γ f(x,t) = sup_{r>t} 1/(ν(B(x,r))^{1-γ}) ʃ_{B(x,r)} |f|dν${\displaystyle ,where}$(x,t) ∈ ℝ^{n+1}_+${\displaystyle ,0<\gamma <1,\text{and}\nu isadoubl\in gmeasure\in }$ℝ^n!$!.