Interpolation of operators in Lebesgue spaces with mixed norm and its applications to Fourier analysis
Let M be a measure space and \(M^ m\) be an m product space \(\prod^{m- 1}_{j=0}M_ j\) where \(M_ j\) are copies of M. Let \(L^ t(L^ s)=L^ t(M^ n;L^ s(M^ m))\) be a Lebesgue space with mixed norm \(\int_{M^ n}(\int_{M^ m}| f|^ s)^{t/s})^{1/t}\). Introducing two auxiliary holomorphic functionals \(W^ z\) and \(F^ z\) in \(| z| <1\), and subdividing the unit circle into several arcs, the author estimates these functionals by \(L^ t(L^ s)\)-norm for z in an arc, where the choice of permutation of the measure spaces \(\{M_ j\}\) depends on the arc. This idea is then used to prove certain interpolation theorems for linear operators in the foregoing Lebesgue spaces. Restricting the domain of \(F^ z\) to functions in \(\prod^{m+n- 1}_{j=0}L^ u(M_ j)\), where \((1/u)=((m/s)+(n/t))/(m+n),\) bounds on the functionals have been obtained. The interpolation theorem is used to conclude two results in Fourier analysis. It is proved that \(\| s^{\epsilon}(f)\|_ p\leq const \| f\|_ p\) if (i) \(\epsilon >0\), \(2d/(d+1)\leq p\leq 2\) and (ii) f defined on \({\mathbb{R}}^ d\) is of the form \(f(x)=f_ 0(x_ 0)f_ 1(x_ 1)...f_{d-1}(x_{d-1})\), where \(s^{\epsilon}(f)\) is the Riesz Bochner mean of order \(\epsilon\). As a second application of the interpolation theorem to Fourier Analysis it is shown that the best upper bound on the restriction on u given by \textit{P. A. Tomas} [Bull. Am. Math. Soc. 81, 477-478 (1975; Zbl 0298.42011)] to conclude a certain inequality in terms of Fourier transform can be improved for radial functions to arrive at the corresponding inequality.
scientific article; zbMATH DE number 4140440
scientific article