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Oscillatory integrals and spherical harmonics. (English) Zbl 0636.42018

Let \[ S_ L^{\delta}f=(A_ L^{\delta})^{-1}\sum^{L}_{\ell =0}A^{\delta}_{L-\ell}H_{\ell}f,\quad L=0,1,2,... \] be the L-th Cesàro mean of index \(\delta\) where Re \(\delta\) \(>-1\), \(H_{\ell}\) is the \(\ell\)-th harmonic projection and \(A_ L^{\delta}=\left( \begin{matrix} \ell +\delta \\ \ell \end{matrix} \right)\). The operator \(S_ L^{\delta}\) is investigated in the space \(L^ p(\Sigma^ n)\) for \(\delta >\delta_ p=\max (n3/1/p-1/2| -1/2,0)\), where \(\Sigma^ n\) is the unit sphere in \(R^{n+1}\). It is shown that \(S_ L^{\delta}\) is uniformly bounded for any \(1\leq p\leq \infty\) in case \(n=2\), and for any p satisfying \(| 1/2-1/p| \geq 1/(n+1)\) in case \(n\geq 3\). Estimates for the harmonic projections and the operator defined by \((T^{\delta}f\hat)(\xi)=(1-| \xi |^ 2)_+^{\alpha}\hat f(\xi)\) are also established extending a result previously obtained by L. Carleson and P. Sjölin [Studia Math. 44, 287-299 (1972; Zbl 0215.183)].
Reviewer: H.Tanabe

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Citations:

Zbl 0215.183
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