Research proof index

Alternative form of Submartingale.ae_bddAbove.

For a submartingale indexed by a countable type, the predictable part is monotone a.e.

{τₙ(c) < 1} = {c < Aⁿ₁}.

A family of random variables is uniformly integrable iff its L¹ tails above c tend to zero uniformly in the index.

An analogue of uniformIntegrable_iff, but the nonstrict inequality in C ≤ ‖X i x‖₊ is replaced by a strict inequality.

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. In the project’s cancellative metric-space setting, assume the associated nontangential operators satisfy the required uniform $L^2$ bound. If $F$ and $G$ are measurable and $f$ is measurable with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, then the Carleson operator obeys the restricted estimate

$\int_G^+ \mathcal{C}_K f(x)\,dx \le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$