Some new Fourier multiplier results of Lizorkin and Hörmander types

Abstract: This PhD Thesis is devoted to the study of Fourier series and Fourier transform multipliers and it contains four papers (papers A - D) together with an introduction, which put these papers into a general frame. This extensive introduction can also be seen as an independent description of this important area including the historical development and the most important results. 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 the spaces Lp and Lq. In particular, 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 Fourier series multipliers in the case with a strong regular system. This system is rather general. For example, trigonometrical systems, the Walsh system and all multiplicative system with bounded elements are strong regular. A generalization and sharpening of the Lizorkin type theorem concerning Fourier series multipliers between the spaces Lp and Lq in this general case is proved and discussed. The Hörmander multiplier theorem from 1960 was later on proved and applied by R. E. Edwards also to the case with (one–dimensional) Fourier series multipliers. In paper D we generalize this result to the case with two-dimensional Fourier series multipliers and with a general regular system. For the case when the Fourier multiplier sequence is generalized monotonous also a lower bound is proved so that, in fact, we get a characterization of all possible such sequences.