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Fourier multipliers from \(L^p\)-spaces to Morrey spaces on the unit circle. (English) Zbl 1303.42003
The Morrey spaces \(L^{p,\lambda }(\mathbb{T)}\) for \(1\leq p\leq \infty \) and \(0\leq \lambda \leq 1\) are defined by \[ L^{p,\lambda }(\mathbb{T)=}\left\{ f:\sup_{I\text{ interval }}\left( \frac{1 }{|I|^{\lambda }}\int_{I}|f|^{p}dx\right) ^{1/p}<\infty \right\} . \] These are Banach spaces that generalize the classical \(L^{p}\) spaces; indeed, \(L^{p,0}(\mathbb{T)}=L^{p}(\mathbb{T)}\).
The authors study Fourier multipliers on these spaces. In analogy with the classical theory, they prove that the spaces of multipliers mapping \(L^{p}\) to \(L^{p,\lambda }\) are distinct for suitable choices of \(p\) and \(\lambda \) . They also show that an absolutely continuous measure belongs to a Lipschitz class \(\mathrm{Lip}(\alpha )\), for some \(0<\alpha <1\), if and only if the measure acts by convolution from \(L^{p}\) to \(L^{p,\lambda }\) for some \( 1<p<\infty \) and \(0<\lambda <1\). The Peetre interpolation theorem for Morrey spaces is an important idea in their proofs.
MSC:
42A45 Multipliers in one variable harmonic analysis
42A55 Lacunary series of trigonometric and other functions; Riesz products
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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