1.10. Two-sided Metric Space Carleson🔗

We prove a variant of Theorem 1.1.1.1 for a two-sided Calderon--Zygmund kernel on the doubling metric measure space (X,\rho,\mu,a). This means a one-sided Calderon--Zygmund kernel K which additionally satisfies, for all x,x',y\in X with x\ne y and 2\rho(x,x')\le \rho(x,y), |K(x,y)-K(x',y)| \le \left(\frac{\rho(x,x')}{\rho(x,y)}\right)^{\frac1a} \frac{2^{a^3}}{V(x,y)}. By the additional regularity, we can weaken the assumption nontanbound to a family of operators that is easier to work with in applications. Namely, for r>0, x\in X, and a bounded measurable function f:X\to\mathbb{C} supported on a set of finite measure, define T_r f(x) := \int_{r\le\rho(x,y)} K(x,y)f(y)\,d\mu(y) = \int_{X\setminus B(x,r)} K(x,y)f(y)\,d\mu(y).

Theorem1.10.1

Statement uses 2

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Theorem 1.1.1.1«metric-space-Carleson»statement
«nontangential-from-simple»«nontangential-from-simple»statement

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Theorem 1.1.1.1

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For all integers a\ge 4 and real numbers 1<q\le 2, the following holds. Let (X,\rho,\mu,a) be a doubling metric measure space. Let \Mf be a cancellative compatible collection of functions and let K be a two-sided Calderon--Zygmund kernel on (X,\rho,\mu,a). Assume that for every bounded measurable function g on X supported on a set of finite measure and all r>0 we have \|T_r g\|_2 \le 2^{a^3}\|g\|_2. Then for all Borel sets F and G in X and all Borel functions f:X\to\mathbb{C} with |f|\le\mathbf{1}_F, we have, with T defined in def-main-op, \left|\int_G T f\,d\mu\right| \le \frac{2^{474a^3}}{(q-1)^6} \mu(G)^{1-\frac1q}\mu(F)^{\frac1q}.

Lean code for Theorem1.10.1●1 theorem

theoremdefined in Carleson/TwoSidedCarleson/MainTheorem.lean

complete

theorem two_sided_metric_carleson.{u_1} {X : Type u_1} {a : ℕ} [MetricSpace X]
  [MeasureTheory.DoublingMeasure X ↑(defaultA a)] {q q' : NNReal}
  {F G : Set X} {K : X → X → ℂ} [IsTwoSidedKernel a K]
  [CompatibleFunctions ℝ X (defaultA a)] [IsCancellative X (defaultτ a)]
  (ha : 4 ≤ a) (hq : q ∈ Set.Ioc 1 2) (hqq' : q.HolderConjugate q')
  (hF : MeasurableSet F) (hG : MeasurableSet G)
  (hT :
    ∀ r > 0,
      MeasureTheory.HasBoundedStrongType (czOperator K r) 2 2
        MeasureTheory.volume MeasureTheory.volume ↑(C_Ts a))
  {f : X → ℂ} (hmf : Measurable f)
  (hf : ∀ (x : X), ‖f x‖ ≤ F.indicator 1 x) :
  ∫⁻ (x : X) in G, carlesonOperator K f x ≤
    ↑(C10_0_1 a q) * MeasureTheory.volume G ^ (↑q')⁻¹ *
      MeasureTheory.volume F ^ (↑q)⁻¹

theorem two_sided_metric_carleson.{u_1}
  {X : Type u_1} {a : ℕ} [MetricSpace X]
  [MeasureTheory.DoublingMeasure X
      ↑(defaultA a)]
  {q q' : NNReal} {F G : Set X}
  {K : X → X → ℂ} [IsTwoSidedKernel a K]
  [CompatibleFunctions ℝ X (defaultA a)]
  [IsCancellative X (defaultτ a)]
  (ha : 4 ≤ a) (hq : q ∈ Set.Ioc 1 2)
  (hqq' : q.HolderConjugate q')
  (hF : MeasurableSet F)
  (hG : MeasurableSet G)
  (hT :
    ∀ r > 0,
      MeasureTheory.HasBoundedStrongType
        (czOperator K r) 2 2
        MeasureTheory.volume
        MeasureTheory.volume ↑(C_Ts a))
  {f : X → ℂ} (hmf : Measurable f)
  (hf :
    ∀ (x : X), ‖f x‖ ≤ F.indicator 1 x) :
  ∫⁻ (x : X) in G,
      carlesonOperator K f x ≤
    ↑(C10_0_1 a q) *
        MeasureTheory.volume G ^ (↑q')⁻¹ *
      MeasureTheory.volume F ^ (↑q)⁻¹