SINGULAR INTEGRALS AND REAL ANALYSIS

(1)带有Calderon-Zygmund型核和粗糙核的奇异积分(i)带有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>1）情况。我们已经克服了p=1情况下的困难。（ii）设Tf (x) =∫K (x, y) f (y) dy，已知当T1 = 0时，T有界，从Hardy空间H^p到局部Hardy空间H^p。我们可以证明在更弱的条件下，Lipscitz条件下，同样的结果成立。给出了临界指标情况下的一个反例。(2)向量值奇异积分与Littlewood-Paley理论。在L^p、Lipschitz和Campanato空间中，得到了参数化Marcinkiewicz积分的有界性的新结果，这些积分对应于Littlewood-Paley的g函数、g^*_λ-函数和Lusin的面积积分。利用这个结果和最近关于多线性Calderon- Zygmund奇异积分的一个结果，我们得到了关于多线性化Littlewood-Paley算子的一个有趣的结果。(3)分数积分。推广了分数阶积分的概念，研究了分数阶积分在L^p、BMO、Lipschitz、Campanato以及Morrey空间、Orlicz空间等函数空间中的性质。(4)离散型的奇异积分与Hardy空间。给出了离散Hardy空间的分子表征，并由此给出了分数阶积分定理和Marcinkiewicz型乘数定理。建立了广义离散希尔伯特变换，并给出了其弱l^1有界性。(5)几乎处处都有收敛结果。引入了一种新的函数空间——模函数空间，并对其有效性进行了研究

(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