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Maximal-function characterisations of real Hardy spaces

Statement

Assume Countable Choice. Let $n \geq 1$ , $0 < p < \infty$ and let $φ \in S (R^{n})$ with $\int φ \neq 0$ . Then there is $N_{0} (n, p, φ) < \infty$ , depending only on $n$ , $p$ and the fixed kernel $φ$ , such that for every $N \geq N_{0} (n, p, φ)$ and every $f \in S^{'} (R^{n})$ the following assertions are equivalent:

$M_{φ}^{0} f \in L^{p} (R^{n})$ ;
$M_{φ}^{*, a} f \in L^{p} (R^{n})$ for some $a \geq 1$ (equivalently, for every $a \geq 1$ );
$M_{N} f \in L^{p} (R^{n})$ .

Here $M_{φ}^{0} f$ , $M_{φ}^{*, a} f$ are the maximal functions of Radial and nontangential maximal functions of a tempered distribution and $M_{N}$ is the grand maximal function of Grand maximal test class of order N and the grand maximal function. Moreover the extended quantities $∥ M_{φ}^{0} f ∥_{L^{p}}$ , $∥ M_{φ}^{*, a} f ∥_{L^{p}}$ and $∥ M_{N} f ∥_{L^{p}}$ are finite exactly on the common set of $f$ satisfying 1-3, and on that set they are equivalent: $∥ M_{φ}^{0} f ∥_{L^{p}} \leq ∥ M_{φ}^{*, a} f ∥_{L^{p}} \leq (1 + a)^{N} P_{N} (φ) ∥ M_{N} f ∥_{L^{p}}, ∥ M_{N} f ∥_{L^{p}} \leq C ∥ M_{φ}^{0} f ∥_{L^{p}},$ with $C = C (n, p, N, φ) < \infty$ depending only on $n, p, N$ and finitely many Schwartz seminorms of $φ$ together with quantitative nonvanishing data for $\hat{φ}$ near zero (as used in the deconvolution lemma). Consequently the space $H^{p} (R^{n})$ of The real Hardy space $H^{p}$ defined by a radial maximal function does not depend on the choice of admissible $φ$ , and for each admissible $φ$ the grand maximal function may be used to define the same space with an equivalent quasi-norm for every order $N \geq N_{0} (n, p, φ)$ ; for two admissible kernels $φ, ψ$ the two radial definitions agree because both are equivalent to $M_{N}$ for every $N \geq \max (N_{0} (n, p, φ), N_{0} (n, p, ψ))$ . The recorded admissible thresholds of the sources are $N \geq ⌊ n / p ⌋ + 1$ for the nontangential class $F_{N}$ with derivatives through $N + 1$ [DKKP], $N > 1 + n / p$ for the radial class $B_{N}$ with derivatives through $N$ [MSV, section 1, p. 16, for $0 < p \leq 1$ ], and $N > n / p + n + 1$ [CUW], stated there for the grand maximal functions normalised by the test classes of those papers; the proof below uses an unspecified finite $N_{0} (n, p, φ)$ that is at least as large as the order thresholds consumed by the finitely many comparison estimates for the fixed kernel $φ$ (the deconvolution constants of the comparison lemmas depend on the kernel, so the order threshold asserted here depends on $φ$ as well as on $n$ and $p$ ), and the existence of such a finite threshold is what is asserted.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $0 < p < \infty$ , $φ \in S$ with $\int φ \neq 0$ , $f \in S^{'}$ , and a fixed order $N$ .

[F1]

Pointwise domination by the grand maximal function: $M_{φ}^{0} f \leq M_{φ}^{*, 1} f \leq 2^{N} P_{N} (φ) M_{N} f$ and $M_{φ}^{*, a} f \leq (1 + a)^{N} P_{N} (φ) M_{N} f$ for every $a \geq 1$ , provided $N$ is the order of the grand maximal function (The grand maximal function dominates every admissible radial and nontangential maximal function).

[F2]

Tangential comparison: for $0 < q < p$ and $T = n / q$ , $∥ M_{φ, T} f ∥_{L^{p}} \leq C_{n, p, q} ∥ M_{φ}^{*, 1} f ∥_{L^{p}}$ (The tangential maximal function is controlled by the aperture-one nontangential maximal function in $L^{p}$ ).

[F3]

Grand dominated by tangential: for every $T > 0$ there are $N_{2} (n, φ, T)$ and $C_{0}$ with $M_{N} f \leq C_{0} M_{φ, T} f$ whenever $N \geq N_{2}$ (The grand maximal function is pointwise dominated by a tangential maximal function).

[F4]

Good-set estimates: assertion 5 of Truncated maximal functions: finiteness, comparison estimates and the good-set bound uses the extended centered average $\tilde{M}$ of that item for general nonnegative Borel inputs. When $M_{φ}^{0} f \in L^{p}$ and $q_{0} = p / 2$ , the input $(M_{φ}^{0} f)^{q_{0}}$ belongs to $L^{2} \subset L_{l o c}^{1}$ , so $\tilde{M} = M$ and the standard maximal operator can be used. Assertions 1-4 of the same lemma provide the truncated functions $M_{φ, j}^{ϵ, L}$ and their finiteness, grand/tangential comparison, tangential/aperture-one comparison and good-set bound.

[F5]

Hardy-Littlewood boundedness: $∥ M g ∥_{L^{r}} \leq C_{n, r} ∥ g ∥_{L^{r}}$ for $1 < r < \infty$ (The centered maximal operator is bounded on $L^{p} (R^{n})$ for $1 < p < \infty$ ).

[F6]

The maximal functions are Borel measurable, so all $L^{p}$ expressions are meaningful with values in $[0, \infty]$ (Measurability and lower semicontinuity of the smooth maximal functions).

[F7]

$L^{2} (R^{n}) \subset L_{l o c}^{1} (R^{n})$ : on each compact $K$ , Holder gives $\int_{K} ∣ g ∣ \leq λ (K)^{1 / 2} ∥ g ∥_{2}$ , and compact sets have finite measure because they are bounded (Holder's inequality for integrals, including the endpoint cases, Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F8]

If nonnegative measurable functions $h_{m}$ increase pointwise to $h$ , then their integrals increase to $\int h$ by the monotone convergence theorem (Monotone convergence for the integral).

Proof technique: a priori good-set argument, then removal of the a priori finiteness by truncation, then the pointwise domination for the converse.

Proof

technique · direct

1.1F1F2F3F4F5F6F7algebra

A priori estimate. Assume $M_{φ}^{*, 1} f \in L^{p}$ and let $q_{0} = p / 2$ , $T = 2 n / p$ , so that $T = n / q_{0}$ and $q_{0} < p$ . Let $N_{2} = N_{2} (n, φ, T)$ and $C_{0} = C_{0} (n, φ, T)$ be the constants of [F3], let $C_{T} = C_{n, p, q_{0}}$ be the norm constant from [F2], and let $C_{H} = C_{n, 2}^{1 / q_{0}}$ be the powered Hardy-Littlewood constant from [F5] at $r = 2$ . Put $C_{1} = C_{0} C_{T}$ and $λ = 2^{1 / p} C_{1}$ , so that $∥ M_{N} f ∥_{L^{p}} \leq C_{1} ∥ M_{φ}^{*, 1} f ∥_{L^{p}}$ by [F2], [F3] for every $N \geq N_{2}$ . Let $N_{3} = N_{3} (n, φ, q_{0}, λ)$ be the threshold of [F4, assertion 5]; fix $N \geq \max (N_{2}, N_{3})$ and let $C_{4} = C_{4} (n, φ, q_{0}, λ, N)$ be the constant of [F4, assertion 5] at this order. Set $F = {x : M_{N} f (x) \leq λ M_{φ}^{*, 1} f (x)}$ . On $F^{c}$ one has $M_{φ}^{*, 1} f \leq λ^{- 1} M_{N} f$ pointwise, hence $∥ M_{φ}^{*, 1} f χ_{F^{c}} ∥_{L^{p}}^{p} \leq λ^{- p} ∥ M_{N} f ∥_{L^{p}}^{p} \leq (C_{1} / λ)^{p} ∥ M_{φ}^{*, 1} f ∥_{L^{p}}^{p} = \frac{1}{2} ∥ M_{φ}^{*, 1} f ∥_{L^{p}}^{p}$ . On $F$ , [F4, assertion 5] gives $M_{φ}^{*, 1} f \leq C_{4} \tilde{M} ((M_{φ}^{0} f)^{q_{0}})^{1 / q_{0}}$ at every point at which $M_{φ}^{*, 1} f$ is finite, hence almost everywhere. Since [F1] gives $M_{φ}^{0} f \leq M_{φ}^{*, 1} f$ , we have $(M_{φ}^{0} f)^{q_{0}} \in L^{2}$ ; [F7] gives its local integrability and hence $\tilde{M} = M$ on this input. Integrating, $∥ M_{φ}^{*, 1} f χ_{F} ∥_{L^{p}} \leq C_{4} ∥ M ((M_{φ}^{0} f)^{q_{0}})^{1 / q_{0}} ∥_{L^{p}} = C_{4} ∥ M ((M_{φ}^{0} f)^{q_{0}}) ∥_{L^{p / q_{0}}}^{1 / q_{0}} \leq C_{4} C_{H} ∥ (M_{φ}^{0} f)^{q_{0}} ∥_{L^{p / q_{0}}}^{1 / q_{0}} = C_{4} C_{H} ∥ M_{φ}^{0} f ∥_{L^{p}}$ by [F5] with $r = p / q_{0} = 2 > 1$ . Combining the two pieces, $∥ M_{φ}^{*, 1} f ∥_{L^{p}}^{p} \leq (C_{4} C_{H})^{p} ∥ M_{φ}^{0} f ∥_{L^{p}}^{p} + \frac{1}{2} ∥ M_{φ}^{*, 1} f ∥_{L^{p}}^{p}$ , so $∥ M_{φ}^{*, 1} f ∥_{L^{p}} \leq 2^{1 / p} C_{4} C_{H} ∥ M_{φ}^{0} f ∥_{L^{p}}$ ; this is the a priori estimate, with a constant independent of $f$ at each fixed order $N \geq \max (N_{2}, N_{3})$ .

2.1F4F5F6F7F8step 1.1algebra

Finiteness of $M_{φ}^{*, 1} f$ when $M_{φ}^{0} f \in L^{p}$ . Let $f$ be arbitrary with $M_{φ}^{0} f \in L^{p}$ . Choose $L \geq L_{0} (f, n, φ, p)$ as in assertion 1 of [F4] for the fixed exponent $p$ of the theorem and then $N^{'} = N^{'} (n, φ, T, L)$ as in assertions 2-4 of [F4], where $T = 2 n / p$ and $q_{0} = p / 2$ are as above (note that $N^{'}$ may be much larger than the theorem's fixed $N$ ; it is used only to prove finiteness). With $0 < ϵ \leq 1 / 2$ set $M_{1} = M_{φ, 1}^{ϵ, L} f$ , $M_{0} = M_{φ, 0}^{ϵ, L} f$ , $M_{T} = M_{φ, T}^{ϵ, L} f$ , $M_{G} = M_{N^{'}}^{ϵ, L} f$ and $F_{ϵ} = {x : M_{G} (x) < λ M_{1} (x)}$ with $λ = 2^{1 / p} C_{1}^{'}$ , $C_{1}^{'} = C_{1}^{'} (n, φ, T, L)$ the product of the constants in assertions 2 and 3 of [F4]. Assertion 1 of [F4] gives $∥ M_{1} ∥_{L^{p}} < \infty$ . On $F_{ϵ}^{c}$ one has $M_{1} \leq λ^{- 1} M_{G}$ , so $∥ M_{1} χ_{F_{ϵ}^{c}} ∥_{L^{p}}^{p} \leq λ^{- p} ∥ M_{G} ∥_{L^{p}}^{p} \leq (C_{1}^{'} / λ)^{p} ∥ M_{1} ∥_{L^{p}}^{p} = \frac{1}{2} ∥ M_{1} ∥_{L^{p}}^{p}$ . On $F_{ϵ}$ , assertion 4 of [F4] gives $M_{1} \leq C_{3} \tilde{M} ((M_{φ}^{0} f)^{q_{0}})^{1 / q_{0}}$ with $C_{3} = C_{3} (n, φ, q_{0}, T, L, λ, N^{'})$ . Since $(M_{φ}^{0} f)^{q_{0}} \in L^{2}$ , [F7] gives $\tilde{M} = M$ on this input; integrating as in step 1.1 and using [F5] gives $∥ M_{1} χ_{F_{ϵ}} ∥_{L^{p}} \leq C^{''} ∥ M_{φ}^{0} f ∥_{L^{p}}$ with $C^{''}$ independent of $ϵ$ (but depending on the fixed $L$ and hence on $f$ ). Hence $∥ M_{φ, 1}^{ϵ, L} f ∥_{L^{p}} \leq 2^{1 / p} C^{''} ∥ M_{φ}^{0} f ∥_{L^{p}}$ for every $0 < ϵ \leq 1 / 2$ . Take $ϵ_{m} = 1 / (m + 2)$ for $m \geq 0$ . Then the weights $t^{L} / (t + ϵ_{m} + ϵ_{m} ∣ y ∣)^{L}$ increase pointwise to $1$ and the ranges $0 < t < 1 / ϵ_{m}$ increase to $(0, \infty)$ . For each fixed witness $(t, y)$ , eventually $t < 1 / ϵ_{m}$ and its weight tends to $1$ , so $M_{φ, 1}^{ϵ_{m}, L} f ↗ M_{φ}^{*, 1} f$ pointwise. By [F8], monotone convergence gives $∥ M_{φ}^{*, 1} f ∥_{L^{p}} \leq 2^{1 / p} C^{''} ∥ M_{φ}^{0} f ∥_{L^{p}} < \infty$ .

3.1step 1.1step 2.1F2F3F5algebra

The implication 1 $\Rightarrow$ 3 and the norm bound. Let $f$ satisfy 1. By step 2.1, $M_{φ}^{*, 1} f \in L^{p}$ , so the a priori estimate of step 1.1 applies at every order $N \geq N_{0} (n, p, φ) : = \max (N_{2} (n, φ, 2 n / p), N_{3} (n, φ, p / 2, 2^{1 / p} C_{1}))$ and gives $∥ M_{φ}^{*, 1} f ∥_{L^{p}} \leq C ∥ M_{φ}^{0} f ∥_{L^{p}}$ with $C = 2^{1 / p} C_{4} (n, φ, p / 2, 2^{1 / p} C_{1}, N) C_{H}$ independent of $f$ , where $C_{H} = C_{n, 2}^{1 / q_{0}}$ is the constant from [F5] at $r = 2$ ; then [F2], [F3] give $∥ M_{N} f ∥_{L^{p}} \leq C_{1} ∥ M_{φ}^{*, 1} f ∥_{L^{p}} \leq C_{1} C ∥ M_{φ}^{0} f ∥_{L^{p}}$ for the same orders $N \geq N_{0} (n, p, φ)$ , since the estimates of steps 1.1 and 2.1 hold with the stated constants for every such $N$ . Thus 1 implies 3 for every $N \geq N_{0} (n, p, φ)$ , with the stated norm bound.

4.1step 3.1F1F6∎

The remaining implications. If 3 holds, then [F1] gives $M_{φ}^{0} f \leq 2^{N} P_{N} (φ) M_{N} f \in L^{p}$ and $M_{φ}^{*, a} f \leq (1 + a)^{N} P_{N} (φ) M_{N} f \in L^{p}$ for every $a \geq 1$ , so 3 implies 1 and 2 for every aperture, with the displayed bounds (the first inequality $∥ M^{0} ∥ \leq ∥ M^{*, a} ∥$ is pointwise since $M_{φ}^{0} f \leq M_{φ}^{*, a} f$ ). If 2 holds for some $a$ , then $M_{φ}^{0} f \leq M_{φ}^{*, a} f \in L^{p}$ pointwise, so 2 implies 1. Hence all three assertions are equivalent, the quantities are finite exactly on the common set, and the displayed equivalence of extended norms holds with constants depending only on $n, p, N$ and the kernel data used in the deconvolution comparison and in $P_{N} (φ)$ . The last sentence about the kernel-independence of $H^{p}$ follows by applying the equivalence to two admissible kernels $φ, ψ$ and a common order $N \geq \max (N_{0} (n, p, φ), N_{0} (n, p, ψ))$ . This proves the theorem.

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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)
Shai Dekel, Gerard Kerkyacharian, George Kyriazis, Pencho Petrushev, A New Proof of the Atomic Decomposition of Hardy Spaces, Constructive Theory of Functions (Sozopol 2016), pp. 59-73 (standard reference, not scraped)
Stefano Meda, Peter Sjogren, Maria Vallarino, Atomic decompositions and operators on Hardy spaces, Revista de la Union Matematica Argentina 50 (2009), no. 2, 15-22 (standard reference, not scraped)
Li-An Daniel Wang, Multiplier Theorems on Anisotropic Hardy Spaces (PhD dissertation, University of Oregon, 2012) (standard reference, not scraped)