Research proof index

]]σ,τ]] = ⋃ᵢ (i, i+1] × {σ ≤ i < τ} as subsets of ℕ × Ω , a purely arithmetic identity on ℕ∞, valid for any σ, τ.

]]σ,τ]] is an elementary predictable set for bounded stopping times on ℕ (blueprint lem:elementaryPredictableSet_stochasticInterval). With τ ≤ n, ]]σ,τ]] = ⋃_{i < n} (i, i+1] × {σ ≤ i < τ} is a finite disjoint union of predictable rectangles, which is exactly the data of an ElementaryPredictableSet.

Along a strictly decreasing sequence u → x from the right, the regularized values r (u k) tend to the right limit of h at x along T' ⊇ T.

Along a strictly increasing sequence u → x from the left, the regularized values r (u k) tend to the left limit of h at x along T' ⊇ T.

The right-limit regularization inherits left limits of h along T.

Let $1<q\le2$ and let $q'$ be its Hölder conjugate. For measurable sets $F,G\subseteq\mathbb{R}$ and measurable $f$ with $\lVert f(x)\rVert\le\mathbf{1}_F(x)$, the real-line Carleson operator satisfies

$\int_G^+ T f(x)\,dx \le C(q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$