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Oscillating multipliers on noncompact symmetric spaces. (English) Zbl 0696.43007

Let G/K be a rank one symmetric space of the noncompact type. We consider the convolution operator \(T_{\alpha,\beta}\) associated to the radial multiplier \[ m_{\alpha,\beta}(\lambda)=(\lambda^ 2+\rho^ 2)^{- \beta /2}\exp (i(\lambda^ 2+\rho^ 2)^{\alpha /2}),\quad Re \beta \geq 0,\quad \alpha >0. \] The main result we prove is the following
(I) If \(\alpha >1,\) \(T_{\alpha,\beta}\) is bounded on \(L^ p(G/K)\) if and only if \(p=2.\)
(II) If \(\alpha =1,\) \(T_{\alpha,\beta}\) is bounded on \(L^ p(G/K)\) if \(| 1/p-1/2| \leq Re \beta /(n-1).\)
(III) If \(\alpha <1,\) \(T_{\alpha,\beta}\) is bounded on \(L^ p(G/K)\) if \(| 1/p-1/2| <Re \beta /\alpha n.\)
Claim (I) is a consequence of the properties of the spherical Fourier transform on symmetric spaces of the noncompact type. On the other side (II) and (III) depend on the fact that if \(\alpha\leq 1\) the “part at infinity” of \(T_{\alpha,\beta}\) behaves nicely, while the “local part” is essentially Euclidean.
Reviewer: S.Giulini

MSC:

43A85 Harmonic analysis on homogeneous spaces
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
53C35 Differential geometry of symmetric spaces
26A33 Fractional derivatives and integrals
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