Sources for the mixed-norm maximal estimate

H's Lemma 7.1, expanded in the mixed-norm proof in the main text, cites Theorem 3.9 of Cruz-Uribe, Martell, and Pérez's monograph Weights, Extrapolation and the Theory of Rubio de Francia (2011). The original book pages for that theorem have not been directly verified here, so no specific book page is presented as verified. The primary-source versions below suffice on their own to support the proof of Lemma 7.1.

1. Weighted extrapolation for arbitrary nonnegative pairs

H, p. 52 · Return to the mixed-norm proof

Cruz-Uribe, Martell, and Pérez, Extensions of Rubio de Francia's extrapolation theorem, Collectanea Mathematica Extra (2006), 195–231, Theorem 3.1(a), published p. 206. In the authors' manuscript, the corresponding theorem is on p. 9, with conventions beginning on p. 8.

Let $\mathcal F$ be a family of pairs $(U,V)$ of nonnegative measurable functions. Suppose that for some $1\le a_0<\infty$ and every $v\in A_{a_0}$ ,

\int U^{a_0}v\le C\int V^{a_0}v,

Then for every $1<a<\infty$ and every $v\in A_a$ , the corresponding estimate with power $a$ holds. The constant must be uniform over the family of pairs, with weight dependence controlled by the $A_p$ characteristic. The conventions preceding the theorem on p. 206 interpret both hypothesis and conclusion as applying to pairs whose respective left-hand sides are finite. No operator relation between $U$ and $V$ is required.

Published paper · Authors' manuscript

Cruz-Uribe's Extrapolation and Factorization (2017), Theorem 5.1 on p. 20 and its proof on pp. 20–22, gives the same formulation for pairs. Pages 19–20 explain the finite-left-hand-side and constant conventions. Equation (7.1) on p. 28 further states that if a nondecreasing function $N_{a_0}([v]_{A_{a_0}})$ controls the initial norm constant, then the target constant is controlled by

c_1N_{a_0}\!\left(c_2[v]_{A_a}^{\,1+(a_0-1)/(a-1)}\right)

where $c_1,c_2$ depend only on the dimension and the two exponents. We do not need the optimal power of the weight characteristic.

Author's text, Theorem 5.1 · Quantitative formula (7.1)

2. Finite left-hand sides and the scalar maximal estimate

H, p. 52 · H, p. 53

Theorem 2.3 on pp. 6–7 of Cruz-Uribe's notes gives the following bound for the usual uncentered Hardy–Littlewood maximal operator: if $1<a<\infty$ and $v\in A_a(\mathbb R^n)$ , then

\|Mf\|_{L^a(v)} \le c_{n,a}[v]_{A_a}^{1/(a-1)}\|f\|_{L^a(v)}.

The strong-type formula is on p. 7. The notes give the detailed proof only in the dyadic case and provide references for the general case.

Author's text, Theorem 2.3

H first restricts the $y$ variable to a bounded interval $I$ and takes $g$ bounded and supported there. Each inner norm is controlled by a constant multiple of $M_y\mathbf1_I$ . The scalar weighted maximal estimate makes all these left-hand sides finite for every required $a>1$ and $v\in A_a$ . Then take

g_n=\min(g,n)\mathbf1_{[-n,n]}(y)

and apply monotone convergence at each level, with $M_yg_n\uparrow M_yg$ as well. Thus the finite-left-hand-side formulation is sufficient.

Martell and Prisuelos-Arribas, Weighted Hardy spaces associated with elliptic operators. Part I, arXiv:1406.6285v2, Lemma 3.3 on pp. 10–11, also gives extrapolation for pairs and discusses truncation. Page 11 specifically explains the finite-left-hand-side convention in Theorem 3.9 of the 2011 monograph. This is an author's explanation of that convention, not a substitute for checking the original book pages.

Authors' text, Lemma 3.3 and discussion

3. Matching the norm order and constants in H

H, p. 52 · H, p. 53 · H, p. 56 · H, p. 57

Lemma 7.1 proceeds from the scalar weighted $L_y^t$ estimate by integrating in $w$ , extrapolating the resulting pairs to $L_y^s$ , and then integrating in $\xi$ . Finally it extrapolates the new pairs to $L_y^r$ . The final norm always has the order

L_y^r(v;L_\xi^s(L_w^t)).

No norms with different exponents are interchanged, and no ordering by size of $r,s,t$ is required. Both integrations occur at matching powers. Together with quantitative extrapolation, this makes the constant independent of the numbers of atoms in the two finite measure spaces.

In Lemma 7.2, with $2<q<3$ fixed, the actual substitution is

r=p_j/q,\quad s=4/q,\quad t=h_j/q.

H chooses $p_j,4,h_j$ all strictly greater than $q$ , so the three extrapolation exponents exceed one. For fixed $q$ , the resulting constant may depend on $q$ , but not on $K,N$ or the number of random signs. As $q\downarrow2$ , some exponents tend to one and the constant may diverge. This does not give the endpoint estimate at $q=2$ .

4. Explicit nonsharp weight-characteristic bounds for the two extrapolations

H, p. 52 · H, p. 53 · Where both extrapolations enter

To ensure that a separate constant for each weight does not conceal a lack of uniformity, take the initial bound to be a finite nondecreasing function. At the scalar starting point,

B_t(A)\le c_t A^{1/(t-1)},\quad A\ge1.

Using formula (7.1) cited above to extrapolate from $t$ to $s$ , we may take

D_{s,t}(A)\le c_{s,t}A^{1/(t-1)+1/(s-1)}.

After integrating at the matching exponent and extrapolating from $s$ to $r$ , we may take

C_{r,s,t}(A) \le c_{r,s,t} A^{[1/(t-1)+1/(s-1)][1+(s-1)/(r-1)]}.

These powers are not optimal and are not used as later powers of $K$ . The actual application is unweighted, with $A=1$ . Once $q$ is fixed, all three exponents are fixed and the entire cost enters $C_q$ . These nonsharp bounds establish only that integration and extrapolation introduce no hidden dependence on the number of atoms, $K$ , or $N$ . They do not provide uniform control as $q\downarrow2$ .