About Hardy-Littlewood Maximal Inequality And Clarkson Inequality

We have proved that the uncentered and centered Maximal function is a bounded operator from L^p(dμ) to L^p(dμ) when dμ, is the Lebesgue measure on Rⁿ.It is a classical result ,that ifμsatisfy a doubling condition:μ(B(x,2r))≤Cμ(B(x,r)),forall x∈Rⁿ,r> 0, both of these operators are of weak type(l,l) and they map L^p(Rⁿ,μ),p > 1 into itself.But not all measures can satisfy doubling condition,we remark that dμ(x) = |x|^αdx is a doubling measure whenα> -n ,but dμ(x) = e^c|x|dx, c≠0 is not. As a result,it is proved that the uncentered Maximal function is not a bounded operator from L^p(dμ) to L^p(dμ) when dμis dμ(x) = e^-c|x|dx,c > 0 on Rⁿ ,in fact,it is not even weak-type(p,p).Clarkson inequality is an inequality about p-degree of absolute value of a complex number.In this paper we also extended Clarkson inequality in two aspects. One hand is using Hilbert space to replace complex nunber space;In the other hand, we extended the parallelogram law to higher dimensional.The result extended its applied range in high degree.