Research proof index

For Antitone functions, the InfRegularized is the supremum of values.

Under the support condition σ.M.ker ≤ ρ.M.ker (i.e., supp(ρ) ⊆ supp(σ)), conjugation by σ^γ and σ^{-γ} preserves the inner product: ⟪ρ.M, H⟫ = ⟪σ^γ ρ σ^γ, σ^{-γ} H σ^{-γ}⟫. This holds because the kernel condition ensures ρ is supported on supp(σ), where σ^γ σ^{-γ} acts as the identity.

Sub-lemma for Step 1b: the conj of H_hat by σ^{−γ} simplifies to (ρ.M.conj (σ^γ).mat)^{α−1}. This uses σ^{−γ} · σ^γ = identity (on support) to cancel the outer σ^γ factors. -/ theorem H_hat_conj_sigma (hα : 1 < α) (ρ σ : MState d) : let γ := (1 - α) / (2 * α) (H_hat α ρ σ).conj (σ.M ^ (-γ)).mat = (ρ.M.conj (σ.M ^ γ).mat) ^ (α - 1) := by intro γ have hγ :...

In the function-distance space used for the real-line Carleson argument, every ball of radius $2R'$ can be covered by at most three balls of radius $R'$.

The function y ↦ Ks s x y is integrable.

An integration-by-parts identity for real- or complex-valued $f$. If $a\le b$, $f$ is differentiable on $[a,b]$, and $f'$ is integrable there, then for every constant $c$, $\int_a^b(t+c)f'(t)\,dt=(b+c)f(b)-(a+c)f(a)-\int_a^b f(t)\,dt.$