×

A remark on a paper by C. Fefferman. (English) Zbl 0694.46029

The author proposes to give a simplified proof of the result by C. Fefferman concerning the imbedding \[ \int_{{\mathbb{R}}^ n}| u(x)|^ p V(x)dx\leq c\int_{{\mathbb{R}}^ n}| \nabla u(x)|^ p dx\quad for\quad all\quad u\in C^{\infty}_ 0({\mathbb{R}}^ n). \] Their main result is the following:
Theorem. Let \(1<p<n\), \(1<r<a/p\), \(V\in L^{r,n-pr}\). Then it follows \[ \int_{{\mathbb{R}}^ n} | u(x)|^ p V(x)dx\leq C\| V\|_{r,n-pr}\int_{{\mathbb{R}}^ n}| Vu(x)|^ p dx \] for all \(u\in C^{\infty}_ 0({\mathbb{R}}^ n)\). Here C depends on n and p only.
For the proof, they use the following lemmas.
Let \(A_ 1=\{V;\) \(MV(x)\leq CV(x)\) a.e. in \({\mathbb{R}}^ n\) for some constant \(C>0\}\), where MV is the usual Hardy-Littlewood maximal functions.
Lemma 1. Let \(V\in L^{r,n-pr}\), \(1<p<n\), \(1<r\leq n/p\) and let \(r_ 1:\) \(1<r_ 1<r\). Then \((MV^{r_ 1})^{1/r_ 1}\in A_ 1\cap L^{r,n- pr}.\)
Lemma 2. Let \(V\in L^{r,n-pr}\), \(1<p<n\), \(1<r<n/p\). Then \[ | \int_{{\mathbb{R}}^ n}\frac{V(y)}{| x-y|^{n-1}}dy| \leq c(n,r,p)[MV(x)]^{(p-1)/p}\| V\|^{1/p}_{r,n-pr}. \]
Reviewer: S.Koshi

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory