Some new Lizorkin multiplier theorems for Fourier series and transforms

Abstract: This Licentiate Thesis is devoted to the study of Fourier series and Fourier transform multipliers and contains three papers (papers A - C) together with an introduction, which put these papers into a general frame.

In paper A a generalization of the Lizorkin theorem on Fourier multipliers is proved. The proof is based on using the so-called net spaces and interpolation theorems. An example is given of a Fourier multiplier which satisfies the assumptions of the generalized theorem but does not satisfy the assumptions of the Lizorkin theorem.In paper B we prove and discuss a generalization and sharpening of the Lizorkin theorem concerning Fourier multipliers between $L_p$ and $L_q$. Some multidimensional Lorentz spaces and an interpolation technique (of Sparr type) are used as crucial tools in the proofs. The obtained results are discussed in the light of other generalizations of the Lizorkin theorem and some open questions are raised.Paper C deals with the Fourier series multipliers in the case with a regular system. This system is rather general. For example, trigonometrical systems, the Walsh system and all multiplicative system with bounded elements are regular. A generalization and sharpening of the Lizorkin type theorem concerning Fourier series multipliers between $L_p$ and $L_q$ in this general case is proved and discussed.