Generalizations of $M$-spaces, I

Spaces and Non-commutative Generalizations I*

We give an elementary proof that the H p spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between H 1 and H ∞. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hilbert transform and the classical factorisation of a function in H p as a product of two functions in H q and H r with 1/q + 1/r ...

M-FUZZIFYING INTERVAL SPACES

In this paper,  we introduce  the notion of $M$-fuzzifying interval spaces, and discuss the relationship between $M$-fuzzifying interval spaces and $M$-fuzzifying convex structures.It is proved that  the category  {bf MYCSA2}  can be embedded in  the category  {bf  MYIS}  as a reflective subcategory, where  {bf MYCSA2} and   {bf  MYIS} denote  the category of $M$-fuzzifying convex structures of...

Balls in generalizations of metric spaces

This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let (X ,pb) be a partial b-metric space in the sense of Mustafa et al. For the family of all pb-open balls in (X ,pb), this paper proves that there are x, y ∈ B ∈ such that B′ B for all B′ ∈ , where B and B′ are with centers x and y, respectively. This result shows that is not a base of any topology on X...

I-28: Anti M

Spaces and Non-commutative Generalizations Ii*

We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the ¯ ∂-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution of the ¯ ∂-equation, which satisfies simultaneously a good L ∞ estimate and a good L 1 estimate. Th...