Nontangential approach regions on groups (Q2712735)

Let \(X\) be a topological group with a nonnegative Borel measure \(\mu\) and a quasidistance \(d(x,y)\), such that the balls \(B(x,r)=\{y\in X; d(x,y)<r\}\) are measurable sets and \(\{B(x,r)\}_{r>0}\) is a basis of open neighbourhoods for all \(x\in X\), and \(0<\mu(B(x,2r)\leq K\mu(B(x,r)\) for all \(x\in X\) and \(r>0\). Assume further \(A\leq \frac{\mu(B(x,r)}{\mu(B(x,y)}\leq A'\) for all \(x, y\in X\) and \(r>0\), and \(x\cdot B(y,r)=B(x\cdot y, r)\) \((x, y\in X, r>0)\), \(\mu(x\cdot E)=\mu(E)\) \((x\in X, E\subset X\) measurable), \(\mu(E)=\mu(E^{-1})\) \(E\subset X\) measurable. A family of measurable sets \(\Omega =\{\Omega_x\}_{x\in X}\) in \(X\times (0,\infty)\) is said to be a family of approach regions if \((x,0)\) is in the closure of \(\Omega_x\) with respect to the product topology. The maximal operator related to \(\Omega\) for a function \(f\) is NEWLINE\[NEWLINEM_\Omega f(x)=\sup_{(y,t)\in \Omega_x}\frac{1}{\mu(B(y,t))} \int_{B(y,t)}|f(z)|d\mu(z).NEWLINE\]NEWLINE A weight \(u\) on \((X,\mu,d)\) is said to be in \(A_p\), \(1\leq 1<\infty\), if the Hardy-Littlewood maximal function is weak-\(L^p(u(x)\mu(x))\) bounded. And a weight \(u\) on \((X,\mu,d)\) is said to be in \(A_p^\Omega\), \(1\leq 1<\infty\), if \(M_\Omega\) is weak-\(L^p(u(x)\mu(x))\) bounded. Set NEWLINE\[NEWLINEW(\Omega)=\{u\in L_{\text{loc}}^1(\mu),\;u\geq 0;\exists C,\;u(S(x,t))\leq C u(B(x,t)), \forall (x,t)\},NEWLINE\]NEWLINE where \(S(x,t)=\{y\in X; \Omega_y(t)\cap B(x,t)\neq \emptyset \}\) and \(\Omega_y(t)=\{z\in Z; (z,t)\in \Omega_y\}\). The main result is: If \(\Omega_x=\{(x\cdot y, t)\); \((y,t)\in \Omega_e\}\) for a given approach region \(\Omega_e\) of the identity element \(e\) of \(X\), then the following statements are equivalent. (a) \(\exists C>0\), \(\theta>0\) such that \(M_\Omega f(x)\leq CM_{\Gamma_\theta}f(x)\). (b) \(A_p^\Omega=A_p\) for all \(1\leq p<\infty\). (c) There is \(p\geq 1\) such that \(A_p^\Omega=A_p\). (d) There is \(0<\gamma\leq 1\) such that \(u(y)=d(e,y)^\gamma\in W(\Omega)\). (e) There is \(\theta>0\) such that \(\Omega_e\subset \Gamma_\theta(e)\). Here \(\Gamma_\theta(x)=\{(y,t)\in X\times(0,\infty)\); \(d(y,x)<\theta t\}\). Some applications to the cases \(X=\mathbb R^n\), \(\mathbb H^n\) (Heisenberg group), connected nilpotent Lie group with a left-invariant Riemannian metric, etc. are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].