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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.
##### Keywords:
Fourier multipliers; Morrey spaces
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##### References:
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