Endpoint completion of the Coifman–Meyer–Stein tent spaces: the $p=\infty$ behaviour, the abandoned $T_p^\infty$ scale, and sharp borderline equivalences

The 1985 Coifman–Meyer–Stein paper introduced the tent spaces $T_q^p$ on the upper half-space — function spaces measured by conical square-function functionals — together with their duality, atomic decompositions, factorization and change-of-aperture results, and it explicitly set aside several endpoint and definitional issues: the behaviour at $p=\infty$, where the aperture equivalence fails and a Carleson-type reformulation is required; the scale $T_p^\infty$ with the roles of $p$ and $\infty$ interchanged, which the authors abandoned as ill-defined and whose duals require measures; atomic decompositions at endpoint indices; and the sharp borderline ranges of the norm equivalences such as the comparison between the conical and Carleson functionals. Later literature has built a substantial endpoint theory, largely within weighted and Whitney-averaged generalizations of the classical spaces (sharp change of angle by Auscher, Carleson duality preduals by Hytönen–Rosén, the unified Whitney scale of Huang, Amenta, Auscher–Bechtel–Haardt and Haardt, and the tent-type John–Nirenberg scale of Shaabani). The original objective nevertheless retains a precise open core: canonical definitions and measure-valued duals of the pointwise $T_p^\infty$-type scale for $1<p<\infty$ outside the dyadic model, canonical endpoint atoms with controlled sharp constants, and the complete sharp borderline description of the classical two-parameter scale.

Background

Square functions are the workhorse of modern harmonic analysis: many operators are controlled by averaging, over regions of the upper half-space $\mathbb{R}^{n+1}_{+}=\mathbb{R}^{n}\times(0,\infty)$, the functions that encode them. For a cone of aperture $\alpha>0$ with vertex $x\in\mathbb{R}^{n}$, $\Gamma_{\alpha}(x)=\{(y,t):|y-x|<\alpha t\}$, and a measurable $f$ on the half-space, the conical functional is $A_q^{(\alpha)}(f)(x)=(\iint_{\Gamma_{\alpha}(x)}|f(y,t)|^q\,dy\,dt/t)^{1/q}$ (up to a harmless normalization), and the tent space $T_q^p$ consists of those $f$ with $A_q(f)\in L^p(\mathbb{R}^{n})$. Dually, for a Euclidean ball $B$, the tent $T(B)$ is the cone-like region $\{(y,t):\mathrm{dist}(y,B^{c})\ge t,\,0<t\le r_B\}$, and the Carleson functional $C_q(f)(x)=\sup_{B\ni x}\big(|B|^{-1}\iint_{T(B)}|f|^q\,dy\,dt/t\big)^{1/q}$ measures a Carleson-measure type control (a Carleson measure is a measure on the half-space whose mass over each tent is bounded by the volume of the ball below it). Coifman, Meyer and Stein (1985) developed the functional-analytic theory of these spaces — duality $(T_q^p)^*\simeq T_{q'}^{p'}$ for $1<p,q<\infty$ and $(T_q^1)^*\simeq T_{q'}^\infty$ (defined through $C_{q'}$) for $q>1$, atomic decompositions of $T_q^1$ into atoms supported in single tents, factorization, and the independence of the spaces of the aperture $\alpha$ for $p<\infty$ — and used them to reorganize the theory of Hardy spaces and of BMO (the space of locally integrable functions of bounded mean oscillation), the Cauchy integral on Lipschitz curves, and multilinear singular integrals. The paper deliberately left the endpoint and definitional structure incomplete. First, the change of aperture degenerates at $p=\infty$: enlarging the cone increases the conical supremum, so $A^{(\alpha)}(f)\in L^\infty$ need not imply $A^{(\beta)}(f)\in L^\infty$ for $\beta>\alpha$, and the $p=\infty$ theory must run through the Carleson functional rather than the conical one. Second, the scale $T_p^\infty$ — the member of the family with cone exponent $p$ and base exponent $\infty$, i.e. the tent-space analogue of the space of functions with a bounded $p$-square function (those $f$ for which the conical $p$-functional $A_p(f)$ is essentially bounded over the base variable) — was judged ill-defined and abandoned: the passage by Shaabani (2025) records that in the 1985 paper this scale 'was considered to be ill-defined and therefore was almost excluded' (remark in Section 3) and that 'their definition was modified, and the scale of $T_p^\infty$ spaces was abandoned' (remark (a) of Section 6), with the dual description of such spaces forced to admit measures in the target. Third, atomic decompositions at endpoint indices require care (atoms obtained from the classical decompositions need not have the regularity — e.g. continuity — of the ambient class, and sharp constants for endpoint atoms were not supplied). Fourth, comparisons between the conical and Carleson norms such as $\|A_q f\|_{L^p}\lesssim\|C_q f\|_{L^p}$ were established only in restricted, sharp ranges of the indices $(p,q)$, leaving the borderline behaviour unsettled. A large subsequent literature has built endpoint theories around generalized tent spaces: weighted tent spaces with local $L^r$-Whitney averages $\mathcal{W}_r$ (averages of $|f|^r$ over Whitney boxes $B(y,t/2)\times(t/2,t)$, used to replace pointwise values by local means) introduced by Hofmann–Mayboroda–McIntosh for elliptic boundary value problems and unified by Huang (2016); the three-parameter scales $T^{p,q,r}_\beta$ and $Z^{p,q,r}_\beta$ of Auscher–Bechtel–Haardt and Haardt; the Carleson-duality preduals of Hytönen–Rosén; the endpoint tent spaces $T^1$ and $T^\infty$ in the scalar and vector-valued settings; and the tent-type John–Nirenberg scale of Shaabani, which connects the abandoned $T_p^\infty$ scale to the $JN_p$ spaces (the John–Nirenberg spaces of functions whose mean oscillation over disjoint cubes is uniformly $L^p$-summable, sitting between BMO and $L^p$). Concretely, Shaabani fixes a symmetric convex body $K\subset\mathbb{R}^n$ (a compact convex set with nonempty interior, symmetric about the origin), identifies each point $(x,t)$ of the half-space with the $K$-ball $B(x,t)\subset\mathbb{R}^n$ (the closed ball, for the norm $\|\cdot\|_K$ with unit ball $K$, centred at $x$ with radius $t$) and writes $f_B=f(x,t)$; his space $S^p(K)$ consists of the continuous functions on the half-space with $\|f\|_{S^p(K)}=(\sup_{\mathcal C}\sum_{B\in\mathcal C}|f_B|^p)^{1/p}<\infty$, the supremum over finite pairwise-disjoint collections $\mathcal C$ of $K$-balls, and $S^p_0(K)$ is the closure in $S^p(K)$ of the compactly supported continuous functions. These developments supply the modern context in which the original endpoint questions are to be completed.

Problem Statement

Complete the functional-analytic theory of the Coifman–Meyer–Stein tent spaces at the endpoint and definitional boundary left open in the 1985 development. The endpoint theory of the tent-space scale forms one coherent objective; of the results below, item (1) is background that later work has substantially settled, while the required results are (2)–(4): (1) the $p=\infty$ endpoint: identify the appropriate definition(s) of tent-type spaces at $p=\infty$ that capture non-tangential boundedness (the conical aperture equivalence fails at $p=\infty$ since enlarging the cone can destroy $L^\infty$ membership, so the $p=\infty$ theory must be organized around Carleson-type functionals), together with sharp quantitative control of the norm equivalence under change of aperture; (2) the $T_p^\infty$ scale (the roles of $p$ and $\infty$ interchanged: the member of the family with cone exponent $p$ and base exponent $\infty$ — an essential supremum over the base variable of the conical $p$-functional, i.e. the tent-space analogue of the space of functions with a bounded $p$-square function), abandoned in the 1985 paper as ill-defined: produce canonical definitions and complete dual/predual descriptions, allowing the dual spaces to be spaces of measures where the function-valued dual is inadequate — the concrete currently open case being the dual of the $S^p_0$-type continuous tent scale for $1<p<\infty$ in the non-dyadic setting (resolved only for $p=1$ and in dyadic models, in which the continuous geometry is replaced by the grid of dyadic cubes), so that supplying the canonical definitions together with the dual/predual description in this non-dyadic $1<p<\infty$ case resolves item (2) — the cases already settled ($p=1$ and the dyadic model for all $1\le p<\infty$) need no re-derivation; (3) atomic decompositions at the endpoint indices (in particular for the $\infty$-in-cone spaces and at $p=1$), with atoms belonging to the ambient class (e.g. continuous atoms) and with controlled sharp constants; and (4) the sharp borderline description of the norm equivalences of the classical two-parameter scale, namely the comparison between the conical functional $A_q$ and the Carleson functional $C_q$ in $L^p$ — this comparison is the sole required target of item (4) — whose known range is restricted and sharp. The endpoint theory to be completed consists of definitions, dualities, atomic decompositions and sharp endpoint estimates that are consistent across the whole $T_q^p$ scale and with the surrounding Littlewood–Paley, BMO/John–Nirenberg and elliptic boundary-value-problem applications; the currently unresolved part of the question is the pointwise (non-Whitney, non-dyadic) endpoint theory described in (2)–(4), since the $p=\infty$ aperture and weighted/Whitney-averaged endpoint theories have been substantially settled in later work.

References

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