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Calderon’s reproducing formulas for the spherical mean \(L^2\)-multiplier operators. (English) Zbl 1418.43003

The spherical mean operator \(\mathscr{R}\) is defined by \[ \mathscr{R}(f)(r,x)=\displaystyle\int_{S^n}f(r\eta, x+r\xi)d\sigma_n(\eta, \xi), \] where \(S^n\) is the unit sphere of \(\mathbb{R}\times \mathbb{R}^n\) and \(d\sigma_n\) is the normalized surface measure on \(S^n\). The spherical mean \(L^2\)-multiplier operator is defined by \[ T_{m,\varepsilon}f=\mathscr{F}^{-1}(m_\varepsilon\circ \theta)\mathscr{F}(f) ),\, \varepsilon >0 \] where \(\mathscr{F}\) is the Fourier transform associated with the spherical mean operator, \(m_\varepsilon (r,x)=m(\varepsilon r, \varepsilon x)\) and \(\theta\) some bijective function. The author studies the multiplier \(T_{m,\varepsilon}\) and proves, among other results, the Calderón reproducing kernel formulas.

MSC:

43A32 Other transforms and operators of Fourier type
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Full Text: DOI

References:

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