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The grand maximal function is pointwise dominated by a tangential maximal function

Statement

Assume Countable Choice. Let $n \geq 1$ , $T > 0$ and let $φ \in S (R^{n})$ with $\int φ \neq 0$ . Then there are $N = N (n, φ, T)$ and $C = C (n, φ, T) < \infty$ such that for every $f \in S^{'} (R^{n})$ and every $x \in R^{n}$ , $M_{N} f (x) \leq C M_{φ, T} f (x),$ where $M_{N}$ is the grand maximal function of Grand maximal test class of order N and the grand maximal function and $M_{φ, T}$ is the tangential maximal function $M_{φ, T} f (x) = \sup_{t > 0} \sup_{y} ∣ (f * φ_{t}) (x - y) ∣ (1 + ∣ y ∣ / t)^{- T}$ . Consequently $∥ M_{N} f ∥_{L^{p}} \leq C ∥ M_{φ, T} f ∥_{L^{p}}$ for every $0 < p \leq \infty$ , both sides extended values.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $T > 0$ , $φ \in S$ with $\int φ \neq 0$ , $f \in S^{'}$ .

[F1]

Deconvolution: for every $ψ \in S$ and every choice of positive integer parameters $L^{'}, N^{'}$ there are $C_{d e c}, M_{d e c}, s_{0}$ and $η^{j} \in S$ with $ψ = \sum_{j \geq 0} η^{j} * φ_{s_{0} 2^{- j}}$ in $S$ and $∥ η^{j} ∥_{S_{N^{'}}} \leq C_{d e c} 2^{- j n L^{'}} ∥ ψ ∥_{S_{M_{d e c}}}$ (Deconvolution of a Schwartz function along the dyadic dilates of a fixed kernel); here $S_{M}$ is the norm $h \mapsto \sup_{w} (1 + ∣ w ∣)^{M} \max_{∣ α ∣ \leq M} ∣ \partial^{α} h (w) ∣$ . The supplier's weight $\max (1, ∣ w ∣)^{M}$ and this weight satisfy $\max (1, ∣ w ∣)^{M} \leq (1 + ∣ w ∣)^{M} \leq 2^{M} \max (1, ∣ w ∣)^{M}$ ; absorb $2^{N^{'}}$ in $C_{d e c}$ .

[F2]

If $Ψ \in S$ and $t > 0$ , then $(f * Ψ_{t}) (x) = \int_{R^{n}} (f * φ_{s t}) (x - w) η_{t} (w) d w$ whenever $Ψ = η * φ_{s}$ in the sense of the decomposition of [F1]: apply Schwartz parameter pairing and integral interchange to $H (w) (z) = η_{t} (w) φ_{s t} (x - w - z)$ . Each Schwartz seminorm of this family is bounded by $C (1 + ∣ w ∣)^{m} ∣ η_{t} (w) ∣$ , an integrable function because $η_{t}$ is Schwartz. The interchange proves the identity. Applying continuity of $f$ to the reflected translate of each partial sum in [F1] also justifies passage to the series; the subsequent nonnegative estimates apply to finite sums first, then to their limit.

[F3]

The translated kernel $ψ^{v} (w) = ψ (w + v)$ satisfies $S_{M} (ψ^{v}) \leq 2^{M} S_{M} (ψ)$ for $∣ v ∣ \leq 1$ , and if $P_{N} (ψ) \leq 1$ then $S_{M} (ψ) \leq 1$ for $M \leq N$ (Grand maximal test class of order N and the grand maximal function, Schwartz space and its seminorms).

Proof technique: insert the dyadic deconvolution identity and sum the rapidly decaying coefficients.

Proof

technique · direct

1.1F1F2F3algebra

Single-kernel estimate. Choose integers $L^{'} > T$ and $N^{'} > T + n$ (for example, $L^{'} = ⌊ T ⌋ + 1$ and $N^{'} = ⌊ T + n ⌋ + 1$ ), and apply [F1] with these parameters, obtaining $C_{d e c}, M_{d e c}, s_{0}$ . By the smaller-scale clause of [F1], decrease $s_{0}$ if necessary so that $s_{0} \leq 1$ . Take an integer $N \geq \max (M_{d e c}, N^{'})$ . Fix $ψ \in F_{N}$ , $t > 0$ and $x$ ; write the deconvolution of [F1] and set $s = s_{0} 2^{- j}$ . If $M_{φ, T} f (x) = + \infty$ , the desired estimate is immediate. Otherwise, by [F2], $∣ (f * ψ_{t}) (x) ∣ \leq \sum_{j \geq 0} \int_{R^{n}} ∣ (f * φ_{s t}) (x - w) ∣ ∣ η_{t}^{j} (w) ∣ d w \leq M_{φ, T} f (x) \sum_{j \geq 0} \int_{R^{n}} (1 + \frac{∣ w ∣}{s t})^{T} ∣ η_{t}^{j} (w) ∣ d w,$ where we used the definition of $M_{φ, T} f (x)$ with the displacement $w$ at scale $s t$ . Substituting $w = t u$ and using $η_{t}^{j} (t u) = t^{- n} η^{j} (u)$ gives $\int_{R^{n}} (1 + \frac{∣ w ∣}{s t})^{T} ∣ η_{t}^{j} (w) ∣ d w = \int_{R^{n}} (1 + \frac{∣ u ∣}{s})^{T} ∣ η^{j} (u) ∣ d u \leq s^{- T} \int_{R^{n}} (1 + ∣ u ∣)^{T} ∣ η^{j} (u) ∣ d u \leq C_{T, N^{'}} s^{- T} ∥ η^{j} ∥_{S_{N^{'}}} \leq C^{'} 2^{j T} 2^{- j n L^{'}} ∥ ψ ∥_{S_{M_{d e c}}},$ since $s \leq 1$ , $N^{'} > T + n$ makes $(1 + ∣ u ∣)^{T - N^{'}}$ integrable, and [F1] gives $∥ η^{j} ∥_{S_{N^{'}}} \leq C 2^{- j n L^{'}} ∥ ψ ∥_{S_{M_{d e c}}}$ . Also $∥ ψ ∥_{S_{M_{d e c}}} \leq 1$ by the choice of $N$ . The series converges because $n L^{'} > T$ , so $∣ (f * ψ_{t}) (x) ∣ \leq C^{''} M_{φ, T} f (x)$ with $C^{''}$ independent of $ψ, t, x$ .

2.1step 1.1F3givenalgebra

Aperture. For $t > 0$ and $y$ with $∣ x - y ∣ \leq t$ write $v = (y - x) / t$ , so $∣ v ∣ \leq 1$ , and let $ψ^{v} (w) = ψ (w + v)$ ; then $(f * ψ_{t}) (y) = (f * ψ_{t}^{v}) (x)$ : both equal $t^{- n} ⟨ f_{w}, ψ ((y - w) / t) ⟩$ and $t^{- n} ⟨ f_{w}, ψ ((x - w) / t + v) ⟩ = t^{- n} ⟨ f_{w}, ψ ((y - w) / t) ⟩$ . By [F3], $S_{M_{d e c}} (ψ^{v}) \leq 2^{M_{d e c}} S_{M_{d e c}} (ψ) \leq 2^{M_{d e c}}$ , so the kernel $Ψ = ψ^{v} / 2^{M_{d e c}}$ satisfies $∥ Ψ ∥_{S_{M_{d e c}}} \leq 1$ ; the argument of step 1.1 uses only this bound on the kernel and the deconvolution of [F1] with the kernel $φ$ , so it applies verbatim to $Ψ$ and gives $∣ (f * Ψ_{t}) (x) ∣ \leq C^{''} M_{φ, T} f (x)$ . Since $(ψ^{v} / 2^{M_{d e c}})_{t} = ψ_{t}^{v} / 2^{M_{d e c}}$ and $(f * ψ_{t}) (y) = (f * ψ_{t}^{v}) (x)$ , multiplying by $2^{M_{d e c}}$ gives $∣ (f * ψ_{t}) (y) ∣ \leq C^{''} 2^{M_{d e c}} M_{φ, T} f (x)$ . Taking the supremum over $ψ \in F_{N}$ , $t > 0$ and $∣ x - y ∣ \leq t$ gives $M_{N} f (x) \leq C M_{φ, T} f (x)$ . The $L^{p}$ statement follows by monotonicity of the integral; for $p = \infty$ it is the pointwise bound.

3.1step 1.1step 2.1∎

Conclusion. Step 1.1 controls a single test kernel by the tangential maximal function with a rapidly convergent deconvolution expansion, and step 2.1 removes the aperture restriction by translating the kernel; the class $F_{N}$ is then dominated pointwise. This proves the lemma.

Depends on

Used by

Maximal-function characterisations of real Hardy spaces Theorem

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David Cruz-Uribe SFO, Li-An Daniel Wang, Variable Hardy Spaces, arXiv:1211.6505 (2012) (standard reference, not scraped)
Marcin Bownik, Anisotropic Hardy Spaces and Wavelets, Memoirs of the American Mathematical Society 164 (2003), no. 781 (standard reference, not scraped)