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Two weighted estimates for oscillating kernels. I. (English) Zbl 0704.42013

The authors solve certain two weight problems for the kernels \(K_{a,b}(t)=(1+| t|^ n)^{-b}e^{i| t|^ a},\) \(a>1\) (a,1) where n (a positive integer) coincides with the dimension of the variable t i.e. \(t=(t_ 1,t_ 2,...,t_ n)\), \(| t| =(t^ 2_ 1+t^ 2_ 2+...+t^ 2_ n)^{1/2}\). Let \(\| t\| =\max_{1\leq j\leq n}\{| t_ j| \}\), \(Tf(x)=\int K(x-t)f(t)dt.\) The problem is to determine those weights w, v for which \(\| Tf\|_{q,w}\leq c(\| f\|)_{p,v}\) where \(\| g\|_{s,u}=(\int \| g\|^ su(t)dt)^{1/s},\) \(b\leq 1-a/2\). The case where \(a=2\) and \(b=0\) in (0,1) which is identical to the Fourier kernel is included here. Since the Fourier transform is included among the results in this paper some conditions depending on p, q, \(\epsilon\), \(\delta\) are given. A number of propositions and theorems, involving long formulae are given. Generally set \(1/s+1/s'=1\), \(1<s<\infty\). Proposition 2.3. Let Tf be a convolution operator, and assume \(w>0\) and \(v>0\). If \(\| Tf\|_{q,w}\leq C\| f\|_{p,v},\) then \(\| Tf\|_{p',v^{1-p'}}\leq c\| f\|_{q',w^{1-q'}}.\) In later sections, the authors analyse which conditions in earlier theorems are necessary. The case \(a=2\) and \(b=0\) is among the cases analysed here.
Reviewer: S.M.Shah

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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