Research proof index

H₂, H₃, H₄ are modular forms of weight 2 and level Γ(2) -/ noncomputable def H₂_SIF : SlashInvariantForm (Γ 2) 2 where toFun := H₂ slash_action_eq' := slashaction_generators_Γ2 H₂ (2 : ℤ) H₂_α_action H₂_β_action H₂_negI_action noncomputable def H₃_SIF : SlashInvariantForm (Γ 2) 2 where toFun := H₃ slash_action_eq' := slashaction_generators_Γ2 H₃ (2 : ℤ) H...

lim_{t→∞} L₁,₀(it)/(F(it)G(it)) = 1/2.

For $x\in[0,2\pi]$ and an interval-integrable function $g$, the norm of the localized Dirichlet-kernel integral over $[x-\pi,x+\pi]$, with cutoff $\max(1-|x-y|,0)$, is bounded by the sum of the real Carleson operators applied to $g$ and to its complex conjugate.

For every bounded, finitely supported function $f$, almost every point $x$ admits a sequence of balls $B(c_i,r_i)$ that all contain $x$, whose radii tend to $0$ from above, and whose averages converge to $f(x)$:

$\dashint_{B(c_i,r_i)} f(y)\,dy\longrightarrow f(x).$

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. If every phase-linearized nontangential operator has the required uniform $L^2$ bound, then for measurable $F,G$ and measurable $f$ with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, the linearized Carleson operator satisfies

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

Let $\mathfrak A$ be an antichain contained in the tile class $\mathfrak C(k,n)$, and let $f$ be measurable with $\lVert f(x)\rVert\le\mathbf 1_F(x)$. On $G\setminus G'$, the Carleson sum over the selected positive tiles in $\mathfrak A$ has an $L^1$ bound with exponential decay in the layer index $n$:

$\int_{G\setminus G'}^+\!\lVert\operatorname{CarlesonSum}f(x)\rVert\,dx\le C(a,q)\,\mu(G)^{1-1/q}\mu(F)^{1/q}\,2^{-((q-1)/(8a^4))n}.$