A Fractal Plancherel Theorem

MOLTER, URSULA MARIA; ZUBERMAN, LEANDRO

REAL ANALYSIS EXCHANGE

Michigan State University Press

Lugar: Louisville; Año: 2009 vol. 34 p. 69 - 85

0147-1937

Given a function $h:[0,infty) oR$ we say that a measure $mu$on $R^n$ is uniformly $h$-dimensional if $mu(B_r(x))leq h(r)$for all $xinR^n$ and for all $r>0$. Given $fin L^2(mu)$,denote by $Fmu f$ its Fourier transform with respect to $mu$.When $h(x)=x^alpha$ Strichartz obtained a kind of PlancherelTheorem (cite {Str90a}). We extend this result to a larger classof functions $h$. We consider non-decreasing, continuous, doubling$(h(2x)<kappa h(x)$) functions that satisfies $h(0)=0$. We provethat 1)If $lim_{t o 0}t^n/h(t)=0$ (it means that $h$defines a dimension no greater than $n$), then$$sup_{xinR^n}sup_{rgeq 1} rac{1}{r^nh(r^{-1})}int_{B_r(x)}abs{Fmu f(xi)}^2dxileq C orm{f}_2^2.$$2)If $E$ is a quasi regular set (this means the lower density of$E$ with respect to the Hausdorff measure see eqref{density}) isbounded away from zero a positive lower bound,  and$mu=hau^h_{llcorner E}$ then,[liminf_{r oinfty}rac{1}{r^nh(r^{-1})}int_{B_r(y)}abs{Fmu f(xi)}^2dxigeq cint_E|f|^2dhau^h,]where $c$ doesn´t depend on $y$, $hau^h$ is the $h$-dimensionalHausdorff measure and $hau^h_{llcorner E}$ its restriction to$E$.This is an important generalization because there are sets wichare quasi regular for a function $h$ that are not quasi regular ifwe restrict the functions to $alpha$ powers, as we show insection 5.