Sharp $L^p$ bounds for Hörmander-type oscillatory integral operators in the restriction range $q > 2d/(d-1)$: the constant-coefficient model case of arbitrary signature

Lars Hörmander's 1973 paper conjectured that oscillatory integral operators $T_N f(x)=\int e^{iN\varphi(x,y)}a(x,y)f(y)\,dy$ whose phase has mixed Hessian of maximal rank $d-1$ and whose level hypersurfaces have nonvanishing Gaussian curvature satisfy the sharp estimate $\|T_N f\|_{L^q}\le C N^{-d/q}(1/2-1/(2d)-1/q)^{1/(2d)-1/2}\|f\|_{L^r}$ throughout the range $1/q<1/2-1/(2d)$, and proved this in the plane $d=2$ with the sharp constant. The conjecture was later refuted in its full variable-coefficient generality: Jean Bourgain constructed counterexamples in every odd dimension $d\ge 3$, and Bourgain–Guth, together with sharpness examples going back to Wisewell, settled the even-dimensional range; the sharp signature-dependent ranges for the general variable-coefficient class are now known (Stein; Bourgain–Guth; Guth–Hickman–Iliopoulou; Hickman–Iliopoulou). What remains open is the constant-coefficient model case $Sf(x)=\int e^{i\langle x',y\rangle+ix_d\langle Ay,y\rangle/2}f(y)\,dy$ with $A$ an arbitrary real symmetric nonsingular matrix, i.e. the Fourier extension operator for a surface of nonvanishing Gaussian curvature of arbitrary signature: whether $\|Sf\|_{L^q}\le C(1/2-1/(2d)-1/q)^{1/(2d)-1/2}\|f\|_{L^r}$ holds whenever $1/q<1/2-1/(2d)$ and the necessary condition $(d+1)/((d-1)q)+1/r\le 1$ holds, for $d\ge 3$, together with the endpoint and sharp-constant behavior. In the elliptic (definite $A$) case this is the classical Fourier restriction conjecture, open since the 1970s; the best known three-dimensional range is $p>22/7$ versus the conjectured $p>3$.

Background

An oscillatory integral operator is a map of the form $T_N f(x)=\int e^{iN\varphi(x,y)}a(x,y)f(y)\,dy$, where $\varphi$ is a smooth real-valued phase function of variables $x\in\mathbb{R}^d$, $y\in\mathbb{R}^{d-1}$, $a$ is a smooth compactly supported amplitude (cutoff), and $N\to\infty$ is a large frequency parameter. Hörmander's 1973 paper introduced, under the so-called Carleson–Sjölin hypotheses, the class now called Hörmander-type operators: the mixed Hessian $\partial^2\varphi/\partial x\partial y$ is required to have maximal rank $d-1$, and for each fixed $x$ the level hypersurface $\{\partial_x\varphi(x,y):y\in\operatorname{supp}a(x,\cdot)\}\subset\mathbb{R}^d$ is required to have nonvanishing Gaussian curvature, meaning that its Gauss map $y\mapsto\partial_x\varphi(x,y)/|\partial_x\varphi(x,y)|$ has a nonsingular Jacobian (the hypersurface is nowhere flat). The prototypical example is the model operator with quadratic phase $Sf(x)=\int e^{i\langle x',y\rangle+ix_d\langle Ay,y\rangle/2}f(y)\,dy,$ where $x=(x',x_d)$, $y\in\mathbb{R}^{d-1}$ and $A$ is a real symmetric nonsingular $(d-1)\times(d-1)$ matrix; this is exactly the Fourier extension operator $f\mapsto \widehat{f\,d\sigma}$ for the graph of the quadratic form $y\mapsto\langle Ay,y\rangle/2$, where $d\sigma$ denotes the induced surface measure, and its adjoint is the Fourier restriction operator $g\mapsto\hat g|_S$ restricting a Fourier transform to that surface. The signature of $A$ (the number of positive minus negative eigenvalues, up to sign) records the curvature type of the surface: definite $A$ gives an elliptic surface such as a paraboloid or sphere, while indefinite $A$ gives a hyperbolic surface such as a saddle. Stationary phase (the asymptotic evaluation of $T_N$ on localized inputs) yields the lower bounds for $\|T_N f\|_q$ that any upper bound must match, and it shows that the conjecturally sharp estimate is $\|T_N f\|_{L^q}\le C N^{-d/q}\bigl(1/2-1/(2d)-1/q\bigr)^{1/(2d)-1/2}\|f\|_{L^r},$ valid in the range $1/q<1/2-1/(2d)$ (equivalently $q>2d/(d-1)$) under the necessary condition $(d+1)/((d-1)q)+1/r\le 1$; on the diagonal $r=q$ both conditions reduce to $q>2d/(d-1)$. At $d=2$ the prefactor becomes $(1/4-1/q)^{-1/4}\sim(q/(q-4))^{1/4}$, which is the sharp constant in Hörmander's proof of the planar case, extending the Carleson–Sjölin theorem on oscillatory integrals in the plane. The planar theory was the engine of the classical results on Bochner–Riesz means (radial Fourier multipliers of the form $(1-|\xi|^2)^{\delta}_+$), because summability of such means reduces, via the Carleson–Sjölin reduction, to estimates for precisely these oscillatory integrals. In the model case with definite $A$ the conjecture specializes to the Fourier restriction conjecture posed by Elias Stein: the extension operator for any surface of nonvanishing Gaussian curvature should map $L^p$ of the surface to $L^p(\mathbb{R}^d)$ for all $p>2d/(d-1)$; in its general arbitrary-signature form it is quoted from Stein by Hickman–Iliopoulou (2020) as holding for any such surface regardless of signature. Stein's conjecture, and its extension to arbitrary signature in the model case, is one of the central open problems of Euclidean harmonic analysis; it is closely tied to the Kakeya conjecture on the dimension of sets containing a line segment in every direction (restriction-type estimates imply Kakeya-type bounds, though not conversely), to Bochner–Riesz summability, and to local smoothing for the wave and Schrödinger equations.

Problem Statement

Let $d\ge 3$ and let $A$ be a real symmetric nonsingular $(d-1)\times(d-1)$ matrix, and consider the model oscillatory integral (Fourier extension) operator $Sf(x)=\int e^{i\langle x',y\rangle+ix_d\langle Ay,y\rangle/2}f(y)\,dy,\qquad x=(x',x_d)\in\mathbb{R}^d,\; y\in\mathbb{R}^{d-1},$ and, more generally, the extension operator for any smooth compactly supported piece of a surface with nonvanishing Gaussian curvature (of arbitrary signature). (1) Do the necessary conditions $1/q<1/2-1/(2d)$ (equivalently $q>2d/(d-1)$; on the diagonal $r=q$ the range condition and the necessary condition coincide, so this is the binding condition) and $(d+1)/((d-1)q)+1/r\le 1$ imply the conjectured sharp estimate $\|Sf\|_{L^q(\mathbb{R}^d)}\le C\bigl(1/2-1/(2d)-1/q\bigr)^{1/(2d)-1/2}\|f\|_{L^r(\mathbb{R}^{d-1})}$ for all such surfaces and all admissible $(r,q)$, for every $d\ge 3$, with $C$ depending on $d$, $r$, $q$, the surface, and the amplitude cutoff, but not on $f$? In the diagonal case $r=q$ this asks whether $\|Sf\|_{L^p}\le C\|f\|_{L^p}$ for all $p>2d/(d-1)$ and every signature of $A$; for definite $A$ this is the Fourier restriction conjecture and for indefinite $A$ its hyperbolic counterpart. (2) Does the corresponding large-parameter estimate $\|T_N f\|_{L^q}\le CN^{-d/q}(1/2-1/(2d)-1/q)^{1/(2d)-1/2}\|f\|_{L^r}$ hold in the same range for the model quadratic phase with the sharp behavior of the constant (no $\varepsilon$-loss in the frequency parameter), in particular at the endpoint of the second condition, i.e. $(d+1)/((d-1)q)+1/r=1$ with the strict range condition $1/q<1/2-1/(2d)$ still in force? On the diagonal $r=q$ the two conditions coincide at $q=2d/(d-1)$, so the strict range condition excludes the endpoint itself and the endpoint question there is understood as the limiting sharp-constant behavior as $q$ decreases to $2d/(d-1)$, where the prefactor $(1/2-1/(2d)-1/q)^{1/(2d)-1/2}$ degenerates. The same question for the genuinely variable-coefficient phases - those satisfying the Carleson–Sjölin hypotheses (rank $d-1$ mixed Hessian and nonvanishing Gaussian curvature of the level hypersurfaces) with the level hypersurfaces genuinely depending on $x$, in contrast to the constant-coefficient phases of (1) whose level hypersurfaces are independent of $x$ - is now known to have a negative answer for $d\ge 3$: Bourgain's odd-dimensional counterexamples and the even-dimensional lower bounds of Bourgain, Wisewell and Bourgain–Guth show the range $1/q<1/2-1/(2d)$ fails for that $x$-dependent class, whose sharp signature-dependent ranges are now known. The open problem is therefore the constant-coefficient model case above - the extension operator for the model quadratic phase and, more generally, for any smooth compactly supported piece of a surface with nonvanishing Gaussian curvature, of arbitrary signature - together with its endpoint and sharp-constant refinements, for $d\ge 3$. Parts (1) and (2) are not independent: for the model quadratic phase, (2) is the rescaled form of (1) - the large-parameter operator is the model extension operator at scale $N$, $T_N f(x)=S f(Nx)$, so $\|T_N f\|_{L^q}=N^{-d/q}\|S f\|_{L^q}$ up to the bounded amplitude cutoff - so the endpoint and sharp-constant content of (2) is already contained in the non-strict condition $(d+1)/((d-1)q)+1/r\le 1$ and the displayed constant of (1). For the model quadratic phase, (2) is therefore contained in (1). The diagonal case $r=q$, the model case alone, and a single signature of $A$ are strictly weaker statements than (1), which requires every $d\ge 3$, both the model quadratic phase and the general-surface case, and every admissible $(r,q)$.

References

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