| Multiplier theorem plays a significant role in studying Fourier analysis, since many important operators are special cases of multipliers. These operators include Hilbert transform, Riesz potentials, Riesz transforms, partial sum of a Fourier series, cesaro means of a Fourier series, certain oscillatory singular integrals, etc. In 1964, K. deLeeuw proved a famous theorem which says that, under certain regulations, a multiplier T is bounded in the Lebesgue space {dollar}Lsp{lcub}p{rcub} (Rsp{lcub}n{rcub}){dollar} if and only if its associated multiplier T is bounded in {dollar}Lsp{lcub}p{rcub} (Tsp{lcub}n{rcub}).{dollar}; In this thesis, we extend deLeeuw's theorem on {dollar}Lsp{lcub}p{rcub}{dollar} to the Hardy spaces and the local Lorentz type Hardy spaces. We also study deLeeuw's theorem for the maximal operator T*, as well as the boundedness of weak type (p,p). Partial results of this thesis will appear in STUDIA MATH.; Theorem 14 and 16 are known results (see (11)), but we gave an alternating proof. Theorems 25, 26, 27, 28, 29 and 30 are all new results. |
