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On weighted estimates for the Kakeya maximal operator. (English) Zbl 0742.42009

The author defines an appropriate \(A_ p\) class for the Kakeya maximal function in \(R^ 2\). The eccentricity of a rectangle is the ratio of its longest side to its shortest side. The Kakeya maximal function is the maximal operator formed by using rectangles of eccentricity \(N\) but with arbitrary directions – call it \({\mathfrak M}_ N f\). It is known from work of Córdoba on \(L^ 2(R^ 2)\) that there is a logarithmic dependence of the estimates on \(N\). The author says that \(w\in{\mathfrak U}_ p^{\mathfrak K}\) if \[ \int_{R^ 2}({\mathfrak M}_ N f)^ p w dx\leq C(\log 2N)^ \alpha \int_{R^ 2}f^ p w dx. \tag{1} \] Appropriate unweighted \((w=1)\) estimates on \(L^ p(R^ n)\) for \(n\geq 3\) are still unknown.
The first major result of the author is a geometric characterization of \(A_ 2^{\mathfrak K}\) which is the convex cone of weights associated with the operator \(M_{<N}f=\sup_{1\leq r\leq N}M_{1,r}f\), where \(M_{1,r}\) is the operator formed by taking the sup over rectangles congruent to the rectangles \([0,1]\times[0,r]\). He proves that \(w\in A_ 2^{\mathfrak K}\) iff (1) \(w\) satisfies the usual \(A_ 2\) condition of Muckenhoupt and (2) there exist \(\beta\geq 0\), \(B\) such that for any \(N\geq 1\), and any rectangle \(R_ i\) with smallest side of length 1 and eccentricity \(N\) such that \(R_ i\cap Q_ i\neq\emptyset\), where \(Q_ i\) is a unit cube, and if we set \(I_ i=\{j\mid\;Q_ j\cap R_ i\neq\emptyset\}\), one has \[ \sum_ i w_ i\sum_{j\in I_ i\cap I_ k} w_ j^{-1}\leq BN^ 2(\log 2N)^ \beta, \] for every \(k\). He next shows \(w\in{\mathfrak U}_ 2^{\mathfrak K}\) iff \(w(r\cdot)\) is uniformly in \(A_ 2^{\mathfrak K}\) for every \(r>0\), where the constants \(B\) and \(\beta\) above are independent of \(r\). The author shows that \(A_ 2^{\mathfrak K}\) is related to \(A_ 1\) on lines, which is the condition that for every \(\sigma\in SO_ 2\), \(w_ \sigma^ x(y)=w(\sigma(x,y))\in A_ 1(R)\), uniformly for a.e. \(x\in R\) and every \(\sigma\). He shows that the second condition (2) of \(A_ 2^{\mathfrak K}\) is implied by \[ {1 \over | R|}\int_ R w\leq A(\log 2N)^ \alpha \min_{i\in I_ R} w_ i,\tag{3} \] where \(I_ R\) is defined as above, and \(R\) is any rectangle with sides 1 and \(N\). Condition (3) above then is quite interesting; the author notes that it is equivalent \((\alpha=0)\) to \(A_ 1\) on lines. He concludes by showing that for radial monotone weights, condition (3) is equivalent to condition (2).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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