Research proof index

The trace functional is invariant under 1_A ⊗ V conjugation. -/ theorem sandwichedTraceFunctional_conj_tensorUnitary (ρ σ : MState (dA × dB)) (V : Matrix.unitaryGroup dB ℂ) : Q̃_ α(ρ.conjTensorUnitary V‖σ.conjTensorUnitary V) = Q̃_ α(ρ‖σ) := by exact sandwichedTraceFunctional_conj_unitary_MState _ ρ σ section twirling variable {dA dB : Type*} variable [...

The trace functional is invariant under 1_A ⊗ V conjugation. -/ theorem sandwichedTraceFunctional_conj_tensorUnitary (ρ σ : MState (dA × dB)) (V : Matrix.unitaryGroup dB ℂ) : Q̃_ α(ρ.conjTensorUnitary V‖σ.conjTensorUnitary V) = Q̃_ α(ρ‖σ) := by exact sandwichedTraceFunctional_conj_unitary_MState _ ρ σ section twirling variable {dA dB : Type*} variable [...

Let $1<q\le2$ and let $q'$ be its Hölder conjugate. Assume $a\ge4$ and that the truncated Calderón-Zygmund operators $T_r$ satisfy the required uniform strong $L^2$ estimate for every $r>0$. If $F$ and $G$ are measurable and $f$ is measurable with $\lVert f(x)\rVert\le\mathbf{1}_F(x)$, then the two-sided metric Carleson operator satisfies

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

Assume the phase space is countable, $a\ge4$, $1<q<2$, and every truncated Calderón-Zygmund operator $T_r$ has the required uniform strong $L^2$ bound. Then the two-sided metric Carleson operator is bounded from Lorentz $L^{q,1}$ to weak Lorentz $L^{q,\infty}$, with the explicit project constant $4C(a,q)/q$.

A lemma invalidating the step in https://arxiv.org/pdf/hep-ph/0605184 leading to equation (4.4).

A function related to the quartic term of the potential, used in the stableness condition and equivalent to the term J4 in https://arxiv.org/abs/hep-ph/0605184. -/ noncomputable def quarticTermReduced (P : PotentialParameters) (k : EuclideanSpace ℝ (Fin 3)) : ℝ := P.η (Sum.inl 0) (Sum.inl 0) + 2 * ∑ b, k b * P.η (Sum.inl 0) (Sum.inr b) + ∑ a, ∑ b, k a *...