Endpoint estimates for the biest operator, the boundedness regime of reparametrized simplex multipliers, and maximal higher-order ordering operators

The biest is the trilinear Fourier multiplier operator $T(f_1,f_2,f_3)(x)=\int_{\xi_1<\xi_2<\xi_3}\hat f_1(\xi_1)\hat f_2(\xi_2)\hat f_3(\xi_3)e^{2\pi ix(\xi_1+\xi_2+\xi_3)}\,d\xi_1d\xi_2d\xi_3$, whose symbol is the characteristic function of a frequency simplex; it emerged from the study of eigenfunction expansions of one-dimensional Schrodinger operators. The 2004 paper 'L^p estimates for the biest I. The Walsh case' proved Holder-type bounds for the dyadic (Walsh) model on an open exponent polytope $D$ and explicitly deferred three directions: the Fourier model, endpoint behavior on the boundary of $D$, and higher-order ('triest'/'multiest') analogues; the classification of reparametrized simplex multipliers was subsequently posed as an open question by the later literature (the 2003 Christ-Kiselev counterexample paper, the multiest paper, and Kesler's papers). The Fourier case was resolved by the 2004 sequel (and extended to arbitrary product symbols by Jung in 2017); the reparametrized-multiplier question is largely delineated through degeneracy classes (degenerate tuples admit no $L^p$ estimates, non-degenerate tuples a wide range, semi-degenerate tuples a sharp intermediate range); and the $n$-linear simplex operators are bounded from $L^2\times\cdots\times L^2$ into $L^{2/n}$ for every $n$. The surviving open core comprises: (a) endpoint and weak-type behavior of the biest exactly on the boundary of $D$, in both the Walsh and Fourier models; (b) completion of the parameter delineation to all tuples and all degrees with sharp exponents; and (c) $L^2$/$L^p$ estimates for the maximal Carleson-type ordering operators $M^n$ with $n\ge3$, announced as forthcoming by the authors in 2007, never published, and identified by them as the obstruction to pointwise convergence of eigenfunction expansions for general upper-triangular systems.

Background

A multilinear Fourier multiplier operator acts on several input functions by multiplying their Fourier transforms by a fixed function of the frequency variables (the symbol) and transforming back. The emblematic case is the bilinear Hilbert transform $H(f,g)(x)=\mathrm{p.v.}\int_{\mathbb{R}}f(x+y)g(x-y)\,dy/y$, whose symbol is singular along a diagonal in frequency space; Lacey and Thiele proved that it maps $L^p\times L^q$ boundedly into $L^r$ whenever $1<p,q\le\infty$ and $1/p+1/q=1/r$ with $2/3<r<\infty$, resolving a 1960s conjecture of A. P. Calderon. The present problem concerns the next orders of this theory.

The biest operator of Muscalu, Tao and Thiele is the trilinear Fourier multiplier

$$T(f_1,f_2,f_3)(x)=\int_{\xi_1<\xi_2<\xi_3}\hat f_1(\xi_1)\hat f_2(\xi_2)\hat f_3(\xi_3)e^{2\pi ix(\xi_1+\xi_2+\xi_3)}\,d\xi_1d\xi_2d\xi_3,$$

whose symbol is the characteristic function of the frequency simplex $\xi_1<\xi_2<\xi_3$; the name (a deliberate misspelling of 'beast') signals that it was the first genuinely trilinear time-frequency operator to be estimated. It is not an isolated curiosity: as the source paper explains, multilinear singular integrals of exactly this type occur in the WKB-type eigenfunction expansions of one-dimensional Schrodinger operators (WKB, after Wentzel, Kramers and Brillouin: the high-frequency asymptotic expansion of the eigenfunctions) and, more generally, of AKNS systems (the integrable-systems framework of Ablowitz, Kaup, Newell and Segur that contains Schrodinger and Dirac operators as special cases) - iterating the expansion produces a multilinear series whose terms are multipliers supported on ordering regions of the frequencies. The bilinear member of this family is essentially the bilinear Hilbert transform, and the trilinear member is the biest. In the Schrodinger application the inputs are $L^2$ functions, so the desired bound is $L^2\times L^2\times L^2\to L^{2/3}$ (note $2/3<1$: the target is quasi-Banach).

The source paper proved its theorem in the Walsh model. The Walsh model is a dyadic toy version of Fourier analysis in which the characters are the Walsh functions, $\{0,1\}$-valued analogues of the exponentials built by repeatedly dilating and reflecting $\chi_{[0,1)}$, so that wave packets are exactly localized on quartiles - time-frequency rectangles $I\times\omega$ that are products of dyadic intervals with area $|I|\,|\omega|=4$. Exact localization isolates the combinatorial core of the problem from kernel regularity, and it is standard to prove the Walsh theorem first and transfer to the Fourier setting afterwards. The Walsh biest is decomposed into two pieces (one coupling the middle input with the first, the other with the third), each analyzed by organizing quartiles into trees and controlling them through size and energy functionals, and the $L^p$ bounds are then deduced by multilinear interpolation of restricted weak-type estimates, i.e. bounds tested against indicator functions of finite-measure sets, which is how quasi-Banach targets are handled.

The shape of the result is best expressed in exponent coordinates. Writing the target exponent as $p_4'$ (so that $1/p_1+1/p_2+1/p_3=1/p_4'$, the same homogeneity as the pointwise product of three functions) and setting $\alpha=(1/p_1,1/p_2,1/p_3,1-1/p_4')$, the tuples for which Theorem 1.8 of the source paper asserts the bounds form an open polytope $D=D'\cap D''$ inside the hyperplane $\alpha_1+\alpha_2+\alpha_3+\alpha_4=1$, where $D'$ and $D''$ are the interiors of the convex hulls of the twelve explicit vertices

$$\begin{array}{llll} A_1=(1,1/2,1,-3/2) & A_2=(1/2,1,1,-3/2) & A_3=(1/2,1,-3/2,1) & A_4=(1,1/2,-3/2,1)\\ A_5=(1,-1/2,0,1/2) & A_6=(1,-1/2,1/2,0) & A_7=(1/2,-1/2,0,1) & A_8=(1/2,-1/2,1,0)\\ A_9=(-1/2,1,0,1/2) & A_{10}=(-1/2,1,1/2,0) & A_{11}=(-1/2,1/2,1,0) & A_{12}=(-1/2,1/2,0,1) \end{array}$$

and of their index-swapped counterparts $B_1,\dots,B_{12}$, each $B_j$ being obtained from $A_j$ by permuting the indices 1 and 3 (for instance $B_2=(1,1,1/2,-3/2)$). The theorem asserts that the Walsh biest maps $L^{p_1}\times L^{p_2}\times L^{p_3}$ into $L^{p_4'}$ for every tuple in $D$; the center point $(1/2,1/2,1/2,-1/2)$ is the $L^2\times L^2\times L^2\to L^{2/3}$ bound. Nothing is asserted on the boundary $\partial D$: the vertices themselves, such as the $L^1\times L^2\times L^1\to L^{2/5}$ configuration corresponding to $A_1$, are precisely the endpoint questions.

Two extensions of the operator frame the open problem. First, for a parameter tuple $\beta=(\beta_1,\beta_2,\beta_3)$ with nonzero entries one can reparametrize the ordering, defining (following the later literature)

$$C^\beta(f_1,f_2,f_3)(x)=\int_{\xi_1/\beta_1<\xi_2/\beta_2<\xi_3/\beta_3}\hat f_1(\xi_1)\hat f_2(\xi_2)\hat f_3(\xi_3)e^{2\pi ix(\xi_1+\xi_2+\xi_3)}\,d\xi_1d\xi_2d\xi_3,$$

so that the biest is essentially $C^{(1,1,1)}$; the companion papers and the subsequent literature raise the question of which $\beta$ admit analogous estimates. The answer is governed by the degeneracy of the partial sums of $\beta$: $\beta$ is called degenerate if some adjacent partial sum $\beta_i+\beta_{i+1}$ vanishes, semi-degenerate if some non-adjacent partial sum vanishes (for three inputs, the full sum $\beta_1+\beta_2+\beta_3$) while no adjacent one does, and non-degenerate if no partial sum $\sum_{k=i}^j\beta_k$ vanishes. Second, there is the higher-order ('triest' or 'multiest') hierarchy: the $n$-linear operator $T^n$ with symbol $\chi_{\{\xi_1<\dots<\xi_n\}}$, and its maximal Carleson-type variants

$$M^n(f_1,\dots,f_n)(x)=\sup_N\Big|\int_{\xi_1<\dots<\xi_n<N}\prod_{j=1}^n\hat f_j(\xi_j)\,e^{2\pi ix(\xi_1+\dots+\xi_n)}\,d\xi_1\dots d\xi_n\Big|,$$

in which the upper cutoff of the ordering region is replaced by a supremum over all cutoffs - the ordering-analogue of the Carleson-Hunt theorem on almost-everywhere convergence of Fourier series. For a parameter tuple $\beta=(\beta_1,\dots,\beta_n)$ with nonzero entries, the reparametrized maximal variants $M^n_\beta$ are defined by the same supremum with the ordering constraint $\xi_1/\beta_1<\dots<\xi_n/\beta_n<N$, with $\beta$ non-degenerate in the sense above (no partial sum $\sum_{k=i}^{j}\beta_k$ vanishes), as in the multiest paper. The maximal operators are the ones needed for pointwise (not merely $L^2$) convergence of the eigenfunction expansions of general triangular systems. By contrast, the trilinear Hilbert transform - the analogue in which the ordering constraint is replaced by a condition of the form $\xi_1+2\xi_2+3\xi_3>0$ - is described in the source paper as just barely beyond the reach of known multilinear techniques and remains open.

Problem Statement

Determine the remaining mapping properties of the biest operator and its relatives - the problem left open by 'L^p estimates for the biest I. The Walsh case' (Muscalu, Tao and Thiele, 2004), which deferred three directions: the Fourier model, endpoint behavior on the boundary of $D$, and the higher-order ('triest'/'multiest') analogues. One direction is now closed: the Fourier (non-dyadic) model of the biest itself is bounded on the same polytope $D$ (the 2004 sequel 'L^p estimates for the biest II. The Fourier case'), and its symbol class was later extended to arbitrary sufficiently regular product symbols (Jung, 2017). The classification question in part (2) below was posed as an open question by the subsequent literature (the 2003 Christ-Kiselev counterexample paper, the multiest paper arXiv:0712.2420, and Kesler's papers). The open problem consists of three linked parts.

(1) Endpoints of the biest. For the Walsh and the Fourier biest, determine the behavior exactly on the boundary $\partial D$ of the open exponent polytope $D=D'\cap D''$ of tuples $\alpha=(1/p_1,1/p_2,1/p_3,1-1/p_4')$ in the hyperplane $\alpha_1+\alpha_2+\alpha_3+\alpha_4=1$ - the polytope, constructed in the Background above from the twelve explicit vertices $A_1,\dots,A_{12}$ and their index-swapped counterparts $B_1,\dots,B_{12}$, on which Theorem 1.8 of the source paper (and, for the Fourier model, Theorem 1.2 of the sequel) asserts the bounds under $1<p_1,p_2,p_3\le\infty$ and $0<p_4'<\infty$: for which boundary tuples of $D$ in the closure of the admissible exponent class in which the sources pose their interpolation framework (Definition 2.1 of the source paper / Definition 3.1 of the sequel: $\alpha_i<1$, $\alpha_1+\alpha_2+\alpha_3+\alpha_4=1$, at most one negative $\alpha_i$) does a strong-type or weak-type estimate $L^{p_1}\times L^{p_2}\times L^{p_3}\to L^{p_4'}$ (or a refinement in Lorentz spaces $L^{p,r}$, which refine the $L^p$ scale by recording weak-type behavior) hold - including the vertex configurations such as $\alpha=(1,1/2,1,-3/2)$, i.e. $L^1\times L^2\times L^1\to L^{2/5}$, and the facets - and at which boundary points does boundedness fail? The boundary of $D$ also contains tuples with two negative $\alpha_i$, which lie outside the sources' framework (the estimate there is not defined) and are excluded from this question.

(2) The reparametrized family. For the operators $C^\beta$ defined by the ordering constraint $\xi_1/\beta_1<\xi_2/\beta_2<\xi_3/\beta_3$ with $\beta$ having nonzero entries, and for their higher-degree analogues, complete the delineation of the boundedness regime: characterize exactly the tuples $\beta$, in every degree $n$, for which Holder-type estimates hold, with the sharp range of exponents in each case. The known partial delineation - non-degenerate tuples satisfy a wide range of $L^p$ estimates; degenerate tuples admit none; representative semi-degenerate tuples such as $(1,1,-2)$ are bounded into $L^p$ exactly for $1/2<p<\infty$ - is to be completed to all degeneracy patterns, including every configuration of vanishing partial sums in degrees $n\ge4$.

(3) Maximal ordering operators. Prove or disprove $L^2$ (and $L^p$) estimates for the maximal Carleson-type variants $M^n$ and $M^n_\beta$ for $n\ge3$ - the supremum over frequency cutoffs of the ordering multipliers, with non-degenerate $\beta$ - whose boundedness was announced as forthcoming by the authors in the 2007 multiest paper and identified there as the exact reduction target for almost-everywhere convergence of the eigenfunction (WKB) expansions of general upper-triangular systems.

References

1. Camil Muscalu, Terence Tao, and Christoph Thiele, 'L^p estimates for the biest I. The Walsh case', Mathematische Annalen 329(3):401-426, 2004. DOI: 10.1007/s00208-004-0518-1; arXiv:math/0102083; https://arxiv.org/abs/math/0102083
2. Camil Muscalu, Terence Tao, and Christoph Thiele, 'L^p estimates for the biest II. The Fourier case', Mathematische Annalen 329(3):427-461, 2004. DOI: 10.1007/s00208-003-0508-8; arXiv:math/0102084; https://arxiv.org/abs/math/0102084
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