In this article, we continue the work in Guan-Li and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.

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Let$E$ç$S$^{1}be a set with Minkowski dimension$d(E)1$. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions of$E$. The main result of this paper shows that these operators are bounded on$L$_{$rad$}^{$p$}(R^{2}) for$p>1+d(E)$and unbounded when$p<1+d(E)$. We also give some end-point results.

In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups$G$which possess certain “density” properties (see$Moskowitz$, M., On the density theorems of Borel and Furstenberg,$Ark. Mat.$$16$(1978), 11–27, and$Moskowitz$, M., Some results on automorphisms of bounded displacement and bounded cocycles,$Monatsh. Math.$$85$(1978), 323–336). For such groups we show that Γ always has finite index in its normalizer$N$_{$G$}(Γ). We then investigate analogous questions for the automorphism group Aut($G$) proving, under appropriate conditions, that Stab_{Aut($G$)}(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$ , of Aut($G$) has finite index in Stab_{Aut($G$)}(Γ). We test the limits of our results with various examples and counterexamples.