SINGULAR INTEGRALS AND REAL ANALYSIS

(1)具有Calderon-Zygmund核和粗核的奇异积分(I)具有Calderon-Zygmund核的振荡奇异积分的A_1加权弱(1，1)估计。已知A_p(p&gt；1)情形。我们克服了p=1的困难。(Ii)设Tf(X)=∫K(x，y)f(Y)dy。在T1=0的条件下，T是从Hardy空间H^p到局部Hardy空间h^p有界的。我们可以证明在更弱的条件下，也就是Lipscitz条件下，同样的结果也成立。(2)向量值奇异积分和Littlewood-Paley理论关于参量化Marcinkiewicz积分在L^p、Lipschitz和Campanato空间中有界性的新结果，它们对应于Littlewood-Paley g-函数、g^*_λ-函数和Lusin面积积分。利用这一结果和最近关于多线性Calderon-Zygmund奇异积分的结果，我们得到了关于多线性化Littlewood-Paley算子的一个有趣的结果。(3)分数次积分。我们推广了分数次积分的概念，并研究了几种函数空间，如L^p，BMO，Lipschitz，Campanato，Morrey空间，Orlicz空间的性质。(4)离散型奇异积分和Hardy空间。我们得到了离散Hardy空间的分子刻画，并利用它给出了一个分数次积分定理和Marcinkiewicz型乘子定理。给出了广义离散希尔伯特变换的形式，给出了广义离散希尔伯特变换的弱L^1有界性。(5)几乎处处收敛的结果。引入了新的函数空间--模函数空间，并研究了它的有效性

(1) Singular integrals with Calderon-Zygmund type kernels and with rough kernels(i) A_1 weighted weak (1,1) estimates for oscillatory singular integrals with Calderon-Zygmund kernels. A_p (p>1) case is known. We have overcomed the difficulties in the case p=1.(ii) Let Tf (x) = ∫K (x, y) f (y) dy. It is known that T is bounded from the Hardy space H^p to the local Hardy space h^p, under the condition T1 = 0. We could show that under more weak condition, Lipscitz condition, the same result holds. A counterexample was given in the critical index case.(2) Vector valued singular integrals and Littlewood-Paley theoryNew results on the boundedness of parametrized Marcinkiewicz integrals in L^p, Lipschitz, and Campanato spaces, which correspond to Littlewood-Paley's g-functions, g^*_λ-functions and Lusin's area integral. Using this and a recent result on multilinear Calderon- Zygmund singular integrals, we have an interesting result on multilinearized Littlewood-Paley operators.(3) Fractional integrals.We have generalized the notion of fractional integrals and studied those properties in several function spaces, such as L^p, BMO, Lipschitz, Campanato, and Morrey spaces, Orlicz spaces.(4) Singlular integrals and Hardy spaces of discrete type.We have molecular characterization of discrete Hardy space, and using this, we have given a theorem of fractional integrals and multiplier theorem of Marcinkiewicz type. Also, generalized discrete Hilbert transform were formulated and its weak l^1 boundedness were given.(5) Almost everywhere convergence results.New function space, modular function space, were introduced, and its effectiveness were studied