Bh ~ P-and Some Properties Of The Bl ~ P Space

L~p space and H~p space are two important spaces constructed with func-tions which contain their own specific characteristics. L~p, is very familiar to usbecause of its great contribution to differential equations, probability theory, and functional theory, while H~p space theory is the basic principle for func-tional theory, approximating theory and harmonic analysis. It should be noted, however, that the H~p space and the L~p space have a close relationship with eachother, since H~p is isomorphic to L~p when p＞1. As a result, we usually focuson the case p＜1 when we study the space H~p.One can easily recognize functional analysis' direct influence on the de-velopment of the H~p theory. It leads us to wonder whether we could relateit with the Banach space, which is one of the most significant spaces in func-tional analysis. To solve this problem, H~p space and L~p space were redefined, and the range of the value of functions in the two spaces were extended ontothe Banach space. This paper shall denote the new spaces generated by theseBanach-valued functions as BH~p and BL~p respectively. But because the valueof the functions in these spaces is on the Banach space, we have to improvethe methods we used in studying the L~p and H~p, which makes the work morechallengious. Some special methods, such as the linear functional theory, havebeen applied to study the new spaces.Based on the classic L~p theory and H~p theory, the corresponding theoremsare proved in the BH~p and BL~p spaces. After discussing about the Piossonintegral and non-thangential maximal functions in BL~p, by some properties ofcomplex functions and some extended inequalities, we obtained the Hardy'sconvexity theorem and the factorization theorem in BH~p.