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Unweighted local good-lambda estimate for maximal truncations

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 1$ , $0 < δ \leq 1$ , and let $k$ (pointwise size $A_{1}$ , standard $δ$ -Hölder $A_{2}^{'}$ , cancellation $A_{3}$ ), a principal-value distribution $W$ for $k$ , and the associated $L^{2}$ -bounded convolution operator $T$ with norm $B$ be as in the published maximal-truncation theorem, with truncations $T_{ε}$ , $T^{(ε, N)}$ and maximal truncations $T^{*}, T^{* *}$ (Maximal truncated singular integrals, Standard (Hölder) Calderón–Zygmund kernels, Calderón–Zygmund kernels and their associated operators). Let $f \in L_{l o c}^{1} (R^{n})$ be such that $\int_{∣ x - y ∣ \geq ε} ∣ f (y) ∣ ∣ x - y ∣^{- n} d y < \infty$ for every $x$ and $ε > 0$ , so that $T_{ε} f$ , $T^{(ε, N)} f$ , $T^{*} f$ and $T^{* *} f$ are defined at every point, and let $λ > 0$ be such that $Ω_{λ} = {T^{* *} f > λ}$ is a proper open set. Then there are constants $γ_{0} = c_{0} (n, δ) (A_{1} + A_{2}^{'} + A_{3})^{- 1}$ and $C_{n}$ , depending only on $n$ and $δ$ , such that for every $0 < γ < γ_{0}$ , $∣ {T^{* *} f > 2 λ} \cap {M f \leq γ λ} ∣ \leq C_{n} γ (A_{1} + A_{2}^{'} + A_{3} + B) ∣ {T^{* *} f > λ} ∣,$ The same inequality holds with $T^{*}$ throughout whenever ${T^{*} f > λ}$ is a proper open set. More precisely, for either $U = T^{* *}$ or $U = T^{*}$ under its stipulated level-set hypothesis, each Whitney cube $Q_{j}$ used in the proof satisfies $∣ Q_{j} \cap {U f > 2 λ} \cap {M f \leq γ λ} ∣ \leq C_{n} γ (A_{1} + A_{2}^{'} + A_{3} + B) ∣ Q_{j} ∣ .$ If $A_{1} + A_{2}^{'} + A_{3} = 0$ , interpret $γ_{0} = + \infty$ ; then the kernel and both maximal truncations vanish.

Facts & Assumptions

Given: Countable Choice; $n \geq 1$ , $0 < δ \leq 1$ , constants $A_{1}, A_{2}^{'}, A_{3}, B$ ; the kernel $k$ , distribution $W$ , operator $T$ and function $f$ of the Statement; $λ > 0$ with $Ω_{λ}$ a proper open set; $0 < γ < γ_{0}$ with $γ_{0}$ fixed in step 3.1; the centred maximal function $M$ (The centered and uncentered Hardy-Littlewood maximal functions).

[F1]

$∣ k (y) ∣ \leq A_{1} ∣ y ∣^{- n}$ for $y \neq 0$ and $∣ k (x - y) - k (x) ∣ \leq A_{2}^{'} ∣ y ∣^{δ} ∣ x ∣^{- n - δ}$ whenever $∣ x ∣ \geq 2 ∣ y ∣ > 0$ (Maximal truncated singular integrals, Standard (Hölder) Calderón–Zygmund kernels), and for $h \in L^{1} (R^{n})$ and $μ > 0$ one has $∣ {T^{* *} h > μ} ∣ \leq C_{n, δ} (A_{1} + A_{2}^{'} + A_{3} + B) μ^{- 1} ∥ h ∥_{1}$ , with $T^{*} h \leq T^{* *} h \leq 2 T^{*} h$ pointwise (Maximal truncations: weak (1,1) and strong Lp bounds).

[F2]

For a nonempty open proper $Ω \subseteq R^{n}$ the Whitney family of Maximal dyadic cubes covering a proper open set consists of pairwise disjoint dyadic cubes $Q_{j}$ , at most countable with $⋃_{j} Q_{j} = Ω$ , and each $Q_{j}$ comes with $y_{j} \notin Ω$ satisfying $∣ x - y_{j} ∣ \leq 6 n \sqrt{n} ℓ (Q_{j})$ for all $x \in Q_{j}$ .

[F3]

For $g \in L_{l o c}^{1} (R^{n})$ , $z \in R^{n}$ , $r > 0$ and $δ > 0$ , $\int_{∣ t - z ∣ \geq r} ∣ g (t) ∣ ∣ t - z ∣^{- n - δ} d t \leq C_{n, δ} r^{- δ} M g (z)$ (Dyadic annulus far-field estimates for the maximal function).

[F4]

For every cube $Q$ and $z \in Q$ one has $∣ Q ∣^{- 1} \int_{Q} ∣ g ∣ \leq C_{n} M g (z)$ ; in particular if $M g (z) \leq γ λ$ then $\int_{Q} ∣ g ∣ \leq C_{n} ∣ Q ∣ γ λ$ (Ball and cube maximal functions are pointwise comparable).

[F5]

Lebesgue measure (and any measure) is countably additive on pairwise disjoint measurable sets, so for pairwise disjoint measurable sets $A_{j}$ one has $∣ ⋃_{j} A_{j} ∣ = \sum_{j} ∣ A_{j} ∣$ (Countable additivity and continuity of finitely additive set functions).

Proof

technique · direct

1.1F2givenchoosealgebra

Whitney geometry. Treat either $U = T^{* *}$ or $U = T^{*}$ , with its own proper open set $Ω_{λ} = {U f > λ}$ ; if this set is empty the assertion is immediate. Apply [F2] to obtain disjoint cubes $Q_{j}$ covering it and points $y_{j} \notin Ω_{λ}$ with $∣ x - y_{j} ∣ \leq d_{n} ℓ (Q_{j})$ for $x \in Q_{j}$ , where $d_{n} = 6 n \sqrt{n}$ . Put $L_{j} = ℓ (Q_{j})$ , $Λ = 24 n \sqrt{n}$ and $V_{j} = Λ^{2} Q_{j}$ . Choose $z_{j} \in Q_{j}$ with $M f (z_{j}) \leq γ λ$ whenever such a point exists. For $t \notin V_{j}$ and $x, z_{j} \in Q_{j}$ , $∣ t - y_{j} ∣ \geq (Λ^{2} / 2 - d_{n} - 1 / 2) L_{j} = : b_{n} L_{j}$ , with $b_{n} > 2 d_{n}$ ; the distances $∣ t - x ∣, ∣ t - y_{j} ∣, ∣ t - z_{j} ∣$ are mutually comparable. Also $V_{j} \subseteq B (y_{j}, R_{j})$ with $R_{j} \leq C_{n} L_{j}$ . These follow from coordinate bounds and the triangle inequality.

2.1F1F4step 1.1givenalgebra

Local part. Put $u_{j} = f 1_{V_{j}}$ and $g_{j} = f - u_{j}$ . By [F4], $∥ u_{j} ∥_{1} \leq C_{n} ∣ V_{j} ∣ M f (z_{j}) \leq C_{n} γ λ ∣ Q_{j} ∣$ . The weak $(1, 1)$ estimate [F1] for either maximal truncation therefore gives $∣ {U u_{j} > λ / 2} ∣ \leq C_{n, δ} (A_{1} + A_{2}^{'} + A_{3} + B) γ ∣ Q_{j} ∣$ . Cubes with no $z_{j}$ contribute nothing to the target set.

2.2F1F3step 1.1givenalgebra

Uniform difference of far truncations. Fix $x \in Q_{j}$ and $y = y_{j}$ . On the common part of the cutoff domains, Hölder smoothness bounds the integral of the kernel difference by $A_{2}^{'} ∣ x - y ∣^{δ} \int_{V_{j}^{c}} ∣ f (t) ∣ ∣ t - y ∣^{- n - δ} d t \leq C_{n, δ} A_{2}^{'} M f (z_{j})$ : compare $∣ t - y ∣$ with $∣ t - z_{j} ∣$ , use $∣ t - z_{j} ∣ \geq c_{n} L_{j}$ , and apply [F3] at $z_{j}$ . For each cutoff radius $a$ (the lower radius $ε$ and, for double truncations, also the upper radius $N$ ), a mismatch lies where one of $∣ t - x ∣, ∣ t - y ∣$ is at most $a$ and the other is greater than $a$ . Their difference is at most $∣ x - y ∣$ . If the mismatch meets $V_{j}^{c}$ , the geometry implies $a \geq c_{n} L_{j}$ , both distances are comparable to $a$ , and $∣ t - z_{j} ∣ \leq C_{n} a$ . Thus the kernel contributing on either mismatch is at most $C_{n} A_{1} a^{- n}$ and its integral is bounded by $C_{n} A_{1} a^{- n} \int_{B (z_{j}, C_{n} a)} ∣ f ∣ \leq C_{n} A_{1} M f (z_{j})$ . There are at most four mismatch pieces. It follows uniformly in all admissible cutoffs that the two far truncations at $x$ and $y_{j}$ differ by at most $C_{n, δ} (A_{1} + A_{2}^{'}) M f (z_{j})$ . For $T^{*}$ use only the lower cutoff.

2.3F1F4step 1.1givenalgebra

Far truncations at the boundary point. Choose $R_{j}$ with $V_{j} \subseteq B (y_{j}, R_{j})$ and $R_{j} \leq C_{n} L_{j}$ . If $ε \geq R_{j}$ , the far truncation equals the corresponding truncation of $f$ , bounded by $U f (y_{j}) \leq λ$ . If $ε < R_{j} < N$ , split the far integral at $R_{j}$ ; the part beyond $R_{j}$ is the corresponding truncation of $f$ and is at most $λ$ , while the part below $R_{j}$ is bounded by $A_{1} (b_{n} L_{j})^{- n} \int_{B (z_{j}, C_{n} L_{j})} ∣ f ∣ \leq C_{n} A_{1} M f (z_{j})$ because $g_{j}$ vanishes within distance $b_{n} L_{j}$ of $y_{j}$ . If $N \leq R_{j}$ , only this latter bound is needed. For single truncations the same proof uses $N = \infty$ . Consequently $U g_{j} (y_{j}) \leq λ + C_{n} A_{1} M f (z_{j})$ .

3.1F5step 2.1step 2.2step 2.3givenalgebra∎

Combining steps 2.2 and 2.3 gives $U g_{j} (x) \leq λ + C_{n, δ} (A_{1} + A_{2}^{'}) γ λ$ on $Q_{j}$ . Choose the dimensional $c_{0}$ in $γ_{0} = c_{0} / (A_{1} + A_{2}^{'} + A_{3})$ so that this is at most $3 λ / 2$ for $γ < γ_{0}$ . Subadditivity then implies $Q_{j} \cap {U f > 2 λ} \cap {M f \leq γ λ} \subseteq {U u_{j} > λ / 2}$ . Apply step 2.1 and sum over the disjoint Whitney cubes using [F5] to obtain $∣ {U f > 2 λ} \cap {M f \leq γ λ} ∣ \leq C_{n, δ} γ (A_{1} + A_{2}^{'} + A_{3} + B) ∣ {U f > λ} ∣$ . The proof applies separately to both $U$ , establishing the two asserted estimates.

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Sources

Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (Springer GTM 249, 2014) (standard reference, not scraped)
Juha Kinnunen, Harmonic Analysis (Aalto University lecture notes) (standard reference, not scraped)