Abstract
This chapter focuses on sharp necessary conditions for radial Fourier multipliers. It is shown how a modification of the classical Hausdorff-Young inequality for Fourier transforms gives necessary criteria for radial Fourier multipliers in terms of smoothness conditions. These criteria are best possible within the framework of weak bounded variation function spaces, and are comparable with some sufficient conditions. The Interpolation techniques are very useful in multiplier theory; in particular the classical Hausdorff-Young inequality can be used to derive Fourier multiplier criteria.
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