Kakeya BMO Space In The Plane

Based on the Kakeya maximal function in the plane quoted, this paper defines a BMO_δ space.Firstly, we introduce an equivalent definition of the BMO_δ norm.Secondly, we discuss the relation between the BMO_δ space and L^∞:1. L^∞ is contained in the BMO_δ space but is not equal to it; 2. the BMO_δ norm of the Kakeya maximal function can be controlled by the L^∞ norm of the original function; 3. similarly, the BMO_δ norm of the Hardy-Littlewood maximal function can be controlled by the L^∞ norm of the original function.Meanwhile, we prove the complete and lattice space properties of the BMO_δ space. Compared with the BMO space, if a function f belongs to the BMO_δ space, then so does |f|.Moreover, we obtain the following results: 1. the (p, ∞)_δ atom space is contained in the (p, 2)_δ atom space and the H_δ^1,∞,0 space is contained in the H_δ^1,2,0 space; 2. the BMO_δ space is contained in the dual space of the H_δ¹ space; 3. the H_δ^1,∞,0 space is equal to the H_δ¹ space and the dual space of the H_δ^1,2,0 space is contained in the BMO_δ space. We explain why we cannot obtain the ideal results that the BMO_δ space is the dual space of the H_δ¹ space and the inequality is similar to John-Nirenberg inequality.Lastly, by comparing the BMO_δ space with the BMO space, we find that the BMO space is contained in the BMO_δ space but is not equal to it. Additionally, if f belongs to the BMO space, then so does its Hardy-Littlewood maximal function Mf.