Martingale Hardy spaces and BMO spaces generated by an operator T are investigated. An atomic decomposition of the space $H_p^T$ is given if the operator T is predictable. We generalize the John-Nirenberg theorem, namely, we prove that the $BM{O}_q$ spaces generated by an operator T are all equivalent. The sharp operator is also considered and it is verified that the $L_p$ norm of the sharp operator is equivalent to the $H_p^T$ norm. The interpolation spaces between the Hardy and BMO spaces are identified by the real method....

We study the measure of non-compactness of operators between abstract real interpolation spaces. We prove an estimate of this measure, depending on the fundamental function of the space. An application to the spectral theory of linear operators is presented.

Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T: A₀ → B₀ or T: A₁ → B₁ is weakly compact, then so is $T:A_{\left[\theta \right]}\to B_{\left[\theta \right]}$ for all 0 < θ < 1, where $A_{\left[\theta \right]}$ and $B_{\left[\theta \right]}$ are interpolation spaces with respect to the pairs (A₀,A₁) and (B₀,B₁). Some formulae for this measure and relations to other quantities measuring weak noncompactness are established.

An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that b ⪯ a if and only if there is a doubly stochastic matrix A such that b = Aa. We prove that under monotonicity assumptions on the vectors a and b the matrix A may be chosen monotone. This result is then applied to show that $(\widetilde{L^p},L^{\infty })$ is a Calderón couple for 1 ≤ p < ∞, where $\widetilde{L^p}$ is the Köthe dual of the Cesàro space $Ce{s}_{p^{\text{'}}}$ (or equivalently the down space $L_{\downarrow }^{p^{\text{'}}}$). In particular, $(\widetilde{L¹},L^{\infty })$ is a Calderón couple, which gives a...

This paper is an informal presentation of material from [28]–[34]. The monotone envelopes of a function, including the level function, are introduced and their properties are studied. Applications to norm inequalities are given. The down space of a Banach function space is defined and connections are made between monotone envelopes and the norms of the down space and its dual. The connection is shown to be particularly close in the case of universally rearrangement invariant spaces. Next, two equivalent...