This is another short post meant to record some calculations for later things, and store a few notes-to-self.

A result along these lines is used in the construction of the cube system in Zorin-Kranich’s paper on polynomial Carleson:

Proposition: Say a subfamily \(\mathcal{F} \subseteq \mathcal{D}\) of generalized dyadic cubes satisfies the \(\tilde{C}\)-Carleson packing condition (or is \(\tilde{C}\)-Carleson, for short) if there exists a \(\tilde{C} > 0\) such that for all \(Q_0 \in \mathcal{D}\), we have $$\sum_{\substack{Q \in \mathcal{F} \\ Q \subseteq Q_0}} \lambda^d(Q) \leq \tilde{C} \lambda^d(Q_0).$$ If \(\bigcup_{Q \in \mathcal{F}} Q \subseteq Q’\) for some generalized dyadic cube \(Q’ \in \mathcal{D}\), there exists an absolute \(0 < c_{d, \mathcal{D}} < 1\) such that for all \(\lambda \geq 0\), $$\lambda^d \bigg( \bigg\{ \sum_{Q \in \mathcal{F}} \chi_Q > \lambda \tilde{C} \bigg\} \bigg) \leq 2 \exp(- c \lambda) \lambda^d(Q’).$$

I’ll use the strategy of Hart Smith’s proof of the John-Nirenberg inequality. Namely, we let \(C_d > 1\) be some dimensional constant to be chosen later, and take a Calderón-Zygmund decomposition of \(f = \sum_{Q \in \mathcal{F}} \chi_Q\) at height \(C_d \tilde{C}\); then we have a decomposition of $$f = g + b = \bigg( f (1 \ – \, \chi_{\bigcup_i Q_i^{\text{C-Z}}}) + \sum_i \chi_{Q_i^{\text{C-Z}}} (f)^{Q_i^{\text{C-Z}}} \bigg) + \sum_i \chi_{Q_i^{\text{C-Z}}} (f \ – \, (f)^{Q_i^{\text{C-Z}}}),$$ and we have (for each \(i\)) $$C_d \tilde{C} < \frac{1}{\lambda^d(Q_i^{\text{C-Z}})} \int_{Q_i^{\text{C-Z}}} |f| \, d\lambda^d \leq D^d C_d \tilde{C},$$ where \(D \in \mathbb{N} \cap (1, \infty)\) is the generational ratio for the generalized dyadic system \(\mathcal{D}\). We note that because \(C_d > 1\) and each \(Q \in \mathcal{F}\) is contained in \(Q’\), our Calderón-Zygmund cubes are necessarily dyadic sub-cubes of \(Q’\).

In particular, it follows that $$\sum_i \lambda^d(Q_i^{\text{C-Z}}) \leq \frac{1}{C_d \tilde{C}} \sum_i \int_{Q_i^{\text{C-Z}}} |f| \, d\lambda^d \leq \frac{1}{C_d} \lambda^d(Q’).$$

We consider the effect of subtracting-off the average of \(f\) on the cube \(Q_i^{\text{C-Z}}\); because (generalized) dyadic cubes are either disjoint or nested, when looking at each cube, we have the complete expression $$\sum_{\substack{Q \in \mathcal{F} \\ Q \subseteq Q_i^{\text{C-Z}}}} \chi_Q \ – \, \frac{1}{\lambda^d(Q_i^{\text{C-Z}})} \int_{Q_i^{\text{C-Z}}} \sum_{\substack{Q \in \mathcal{F} \\ Q \subseteq Q_i^{\text{C-Z}}}} \chi_Q \, d\lambda^d \\ + \sum_{\substack{Q \in \mathcal{F} \\ Q \supset Q_i^{\text{C-Z}}}} \chi_Q \ – \, \frac{1}{\lambda^d(Q_i^{\text{C-Z}})} \sum_{\substack{Q \in \mathcal{F} \\ Q \supset Q_i^{\text{C-Z}}}} \lambda^d(Q \cap Q_i^{\text{C-Z}}).$$

On the cube \(Q_i^{\text{C-Z}}\), the second line cancels out to zero. So we need only consider the first line, which is equal to the difference of a variable part and a positive constant; and this last item is \(\leq \tilde{C}\) (by the Carleson packing condition).

We shift this constant contribution to the good part; then we have a remnant which is exactly in the form for us to apply this decomposition process again, at height \(C_d \tilde{C}\). Inductively proceeding, we do this as many times as necessary to reach the height of (the nearest integer below) \(\lambda\). The measure of the exceptional set falls off exponentially as we iterate.

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We also have the following result, to complement our upcoming discussion about spherical maximal functions:

Lemma: Suppose \(\sigma \geq 0\) is a compactly-supported Borel measure, and that for every \(f \in \mathcal{S}(\mathbb{R}^d)\), we have the uniform a priori bound for the maximal operator $$(T_* f)(x) = \sup_{t > 0} \left| \int_{\mathbb{R}^d} f(x \ – \, t y) \, d\sigma \right|$$ of $$\|T_* f\|_{L^p(\mathbb{R}^d)} \lesssim \|f\|_{L^p(\mathbb{R}^d)}$$ for some \(1 \leq p < \infty\). Then \(T_* f\) is well-defined and measurable for any \(f \in L^p(\mathbb{R}^d)\), and obeys the same bound.

We first show this for real-valued functions \(f\) obeying \(f = 0\) \(\lambda^d\)-a.e. on \(\mathbb{R}^d\). Specifically, we show that for such \(f\), \(T_* f = 0\) almost-everywhere also, and hence is Lebesgue measurable.

First, we let \(0 \leq f \leq \chi_U\), for an open set \(U \subseteq \mathbb{R}^d\) of finite-measure; then, taking a sequence in \(C^{\infty}_c(\mathbb{R}^d)\) which increases to \(\chi_U\) pointwise everywhere, we get that \(T_* (\chi_U) = \lim_{n \to \infty} T_* (f_n)\) pointwise everywhere (by interchanging limit and supremum). It then follows that \(T_* (\chi_U)\) is Borel measurable (indeed, Baire class one).

Moreover, the same logic shows that if \(N \in \mathbb{N}\), \((U_n)_{n = 1}^{N} \subseteq \tau_{\mathbb{R}^d}\) is a collection of open subsets of finite \(\lambda^d\)-measure, and \((a_n)_{n = 1}^{N}\) is a sequence of nonnegative real numbers, then $$T_* \bigg( \sum_{n = 1}^{N} a_n \chi_{U_n} \bigg)$$ is Borel measurable and obeys the expected \(L^p\) bound.

We then see that if \(E \subseteq \mathbb{R}^d\) has Lebesgue measure zero, \(T_* (\chi_E) = 0\) a.e., from monotonicity and taking a downward limit of open sets to produce a \(G_{\delta}\) containing \(E\). By using subadditivity and decomposing the range, we see that \(T_*\) is well-defined and also vanishes when applied to any real-valued function equal to zero almost-everywhere.

Next, for positive simple functions: we note that $$\bigg| T_* \bigg( \sum_{n = 1}^{N} a_n \chi_{E_n} \bigg) \ – \, T_* \bigg( \sum_{n = 1}^{N} a_n \chi_{U_{n, k}} \bigg) \bigg| \leq T_* \bigg( \sum_{n = 1}^{N} a_n \chi_{U_{n, k} \setminus E_n} \bigg) \leq \sum_{n = 1}^{N} a_n T_* (\chi_{U_{n, k} \setminus K_{n, k}}),$$ where all of this occurs without caring about measurability (here we use the outer and inner regularity of \(\lambda^d\)). Now, we see that each element of the sum, right-most above, is measurable; and so we have a situation where $$|f \ – \, g_k| \leq h_k \to 0$$ in \(L^p(\mathbb{R}^d)\), where \(g_k\) and \(h_k\) are Borel measurable and elements of \(L^p(\mathbb{R}^d)\), and \(f\) is to be determined. It follows that \((g_k)\) is a Cauchy sequence in \(L^p(\mathbb{R}^d)\), and by passing to pointwise subsequences that \(g_{k_r} \to f\) pointwise a.e.; hence \(f = T_* (\sum_n a_n \chi_{E_n})\) is Lebesgue measurable.

Then, by monotonicity, taking a sequence of simple functions \(0 \leq f_n \leq f\) for any nonnegative Lebesgue-measurable function \(f \in L^p(\mathbb{R}^d)\), we get that \(T_* f\) is measurable. This handles the positive case.

For the remaining case, we take \(f \in L^p(\mathbb{R}^d)\); then $$\left\| \sup_{t > 0} \int_{\mathbb{R}^d} |f(x \ – \, t y)| \, d\sigma \right\|_{L^p_x(\mathbb{R}^d)} < \infty,$$ where the object inside the \(L^p\) norm is measurable; this is by the above result. As a consequence, for \(\lambda^d\)-a.e. \(x \in \mathbb{R}^d\), we have \(y \mapsto f(x \ – \, t y)\) being in \(L^1(\mathbb{R}^d, \sigma)\) for every \(t > 0\) (and uniformly, even). Consequently, for a.e. \(x\), \((T_* f)(x)\) is well-defined, with the (family of) integral(s) inside the supremum making sense, even for signed \(f\).

Moreover, we get that $$|T_*(f) \ – \, T_*(f_n)| \leq T_*(f_n \ – \, f) \leq T_* ( |f_n \ – \, f|),$$ where this holds as a pure pointwise (a.e.) statement at first, and we do not care about measurability; next, we recognize that the dominating function is measurable and tending to zero in \(L^p(\mathbb{R}^d)\), using our newly-minted result for positive functions, and so we get measurability once more from the same comparison argument; thus \(T_*(f)\) is a Lebesgue measurable function.

Up until now, we have chosen to work with a single fixed function \(f : \mathbb{R}^d \to \mathbb{C}\); but with pointwise changes up to sets of measure zero, we recall our above remarks on functions vanishing \(\lambda^d\)-a.e.; using sublinearity (the triangle inequality, really), this gives the well-definedness of \(T_*\) on \(L^p(\mathbb{R}^d)\) elements, which we recall are merely equivalence classes.