Sharp logarithmic estimates for positive dyadic shifts

The paper contains the study of sharp logarithmic estimates for positive dyadic shifts A given on probability spaces (X,μ) equipped with a tree-like structure. For any K>0 we determine the smallest constant L=L(K) such that∫E|Af|dμ≤K∫RΨ(|f|)dμ+L(K)⋅μ(E), where Ψ(t)=(t+1)log⁡(t+1)−t, E is an arbitrary measurable subset of X and f is an integrable function on X. The proof exploits Bellman function method: we extract the above estimate from the existence of an appropriate special function, enjoying certain size and concavity-type conditions. As a corollary, a dual exponential bound is obtained.