Completing the sharp $L^2$ theory of Fourier integral operators with complex phase beyond the global injectivity hypothesis of Hörmander (1983)

For a Fourier integral operator $A$ of order $0$ with complex phase, associated with a positive complex canonical relation $C$, Hörmander's 1983 memoir determined exactly when the whole operator class is $L^2$-continuous — namely, when the real tangent planes of $C$ project injectively at every real point — and, under global injectivity of the projections of the real part $C_{\mathbb{R}}$, proved that $A^*A$ is a pseudo-differential operator and that the sharp estimate $\|Au\|\le(M+\varepsilon)\|u\|+C_{\varepsilon,K}\|u\|_{H^{(-1)}}$ is equivalent to the bound $M$ on the supremum of the invariant symbol half-density norm over the real points of $C$. The theory outside the global injectivity hypothesis is unfinished. This problem asks for its completion: when $C_{\mathbb{R}}$ fails to project injectively — several distinct real points of $C$ over the same base covector, a situation the memoir itself flags — prove or refute the equivalence between the sharp estimate and the symbol-norm bound. Necessity of the symbol-norm bound is known in full generality; sufficiency is open, and the later literature (1983–2026) develops $L^2$ and norm results only under graph-type or linear hypotheses.

Background

Let $X$ and $Y$ be smooth ($C^\infty$) paracompact manifolds, and let $T^*(X)$, $T^*(Y)$ be their cotangent bundles, with the zero section removed where indicated by $\setminus 0$. A Fourier integral operator (FIO) is an operator $A:C_0^\infty(Y)\to\mathcal{D}'(X)$ whose distribution kernel is locally an oscillatory integral $\int e^{i\varphi(x,y,\theta)}\,a(x,y,\theta)\,d\theta$ with amplitude $a$ and phase function $\varphi$; the kernel belongs to the class $I^m$ when the amplitude is of order $m$ in the dual (phase) variables $\theta$, and $m=0$ is the $L^2$-critical order. The canonical relation of $A$ is the image of the differentials of $\varphi$ under integration over the phase variables — a conic (invariant under positive dilation of the covariables) Lagrangian submanifold $C\subset(T^*(X)\setminus 0)\times(T^*(Y)\setminus 0)$ for the difference of the two canonical symplectic forms. Operators with complex phase were introduced by Duistermaat–Sjöstrand (1973) and developed by Melin–Sjöstrand (1975, 1976) through the almost analytic continuation calculus: the phase function takes complex values, the kernel is defined via almost analytic extensions, and $C$ becomes a complex conic Lagrangian. It is called a positive complex canonical relation when the Hermitian form $\frac{1}{i}(\sigma_X-\sigma_Y)$ — the difference of the symplectic forms of the two factors, divided by $i$ — is non-negative on the complex tangent spaces of $C$. Positivity produces exponential decay of the kernel away from the real set and is the natural condition arising in parametrices of differential operators of real principal type and in semiclassical propagator approximations. The real part $C_{\mathbb{R}}$ of $C$ is its set of points fixed by complex conjugation, a conic Lagrangian in the real product $(T^*(X)\setminus 0)\times(T^*(Y)\setminus 0)$; at a real point $\gamma\in C_{\mathbb{R}}$ the real tangent plane $T_\gamma(C)_{\mathbb{R}}$ is the set of real vectors in the tangent space of $C$ at $\gamma$. The two natural projections of $T_\gamma(C)_{\mathbb{R}}$, to the tangent spaces of $T^*(X)$ and of $T^*(Y)$, may or may not be injective; this injectivity is the microlocal counterpart of the local graph condition under which real-phase FIO theory is usually developed.

The source memoir (Hörmander, Arkiv för Matematik 21 (1983), 283–307) works with properly supported operators $A\in I^0(Y\times X,C')$ acting on half-densities (densities to the power $1/2$, written $\Omega^{1/2}$, so that $L^2$ is defined without choosing a volume measure); here $C'$ denotes the image of $C$ under complex conjugation together with interchange of the two factors, the convention under which a kernel in $I^0(Y\times X,C')$ defines an operator from functions on $Y$ to functions on $X$ with canonical relation $C$. It proves four groups of results. (a) The exact class-level dichotomy: every such $A$ is bounded on $L^2$ if and only if, at every real point of $C$, the two projections of the real tangent plane are injective (Theorems 2.2 and 3.5). (b) If, in addition to the tangent-plane injectivity of (a), the projection $C_{\mathbb{R}}\to T^*(X)\setminus 0$ is injective, then $A^*A$ is a pseudo-differential operator of order $0$ and type $(1/2,1/2)$ (Theorem 3.6); here “type $(1/2,1/2)$” refers to the anisotropic symbol metric in which the symbol estimates hold, as in the Weyl calculus of pseudo-differential operators. (c) For positive linear canonical relations with injective real part, an exact normal form (Theorem 4.1): after linear changes of the variables, the model operators are tensor products of explicitly given quadratic Fourier multipliers $R(z)$, of the operators $E(a)$ generated by $e^{iax^2D_x}$-type flows, and of the model relations $\xi=ix$ and $\eta=-iy$; on the real part there is an invariant (construction-independent) half density $d_C$ (Definition 4.2), the norm of a model operator equals exactly the norm of its symbol half density (Theorem 4.3), and $d_C$ depends continuously on the relation (Theorem 4.4). (d) The sharp norm estimate (Theorem 5.1): under hypothesis (i) — both global projections $C_{\mathbb{R}}\to T^*(X)\setminus 0$ and $C_{\mathbb{R}}\to T^*(Y)\setminus 0$ injective — and hypothesis (ii) — the tangent-plane injectivity of (a) — one has, for every compact $K\subset Y$ and $\varepsilon>0$, that the estimate

$$\|Au\|_{L^2(X;\Omega^{1/2})}\le(M+\varepsilon)\|u\|_{L^2}+C_{\varepsilon,K}\|u\|_{H^{(-1)}},\qquad u\in C_0^\infty(K;\Omega^{1/2}(Y))$$

holds if and only if $M$ bounds the supremum, over the real points of $C$ lying above $K$, of the norm of the invariant symbol half density of $A$ (the principal symbol measured against $d_C$); here $H^{(-1)}$ is the Sobolev space of order $-1$. The necessity of the symbol-norm bound is proved without hypothesis (i); the sufficiency direction, and the pseudo-differential reduction (b), are not available without it. The memoir states the obstruction explicitly: over a real $(x,\xi)\in T^*(X)\setminus 0$ there may exist several distinct $(y,\eta)\in T^*(Y)\setminus 0$ with $(x,\xi,y,\eta)\in C$, although the local results on the composition $C^{-1}\circ C$ prove that the set of such points is discrete; thus the local pieces of $A^*A$ need not assemble into a pseudo-differential operator, and the norm may a priori involve interaction between the several real branches.

Problem Statement

Let $X$ and $Y$ be smooth manifolds, let $\Omega^{1/2}$ denote half-densities, and let $C\subset(T^*(X)\setminus 0)\times(T^*(Y)\setminus 0)$ be a conic closed positive complex canonical relation in the sense of Melin–Sjöstrand (positive with respect to $\frac{1}{i}(\sigma_X-\sigma_Y)$), with real part $C_{\mathbb{R}}$ and real tangent planes $T_\gamma(C)_{\mathbb{R}}$. The objective is to complete the sharp $L^2$ theory of Hörmander (1983) for the classes $I^0(Y\times X,C')$ outside its global injectivity hypothesis.

Assume the tangent-plane injectivity at every real point of $C$ (so that, by Theorems 2.2 and 3.5 of the source, every properly supported $A\in I^0$ with relation $C$ is $L^2$-continuous), but assume the projection $C_{\mathbb{R}}\to T^*(X)\setminus 0$ (or $C_{\mathbb{R}}\to T^*(Y)\setminus 0$) is not injective: there exist distinct real points $(x,\xi,y_1,\eta_1)$ and $(x,\xi,y_2,\eta_2)$ of $C$ over the same base covector, a set known from the source to be discrete. Prove, or disprove by an explicit example, that for every properly supported $A\in I^0(Y\times X,C')$, every compact $K\subset Y$, and every $\varepsilon>0$, the estimate

$$\|Au\|_{L^2(X;\Omega^{1/2})}\le(M+\varepsilon)\|u\|_{L^2}+C_{\varepsilon,K}\|u\|_{H^{(-1)}},\qquad u\in C_0^\infty(K;\Omega^{1/2}(Y))$$

is equivalent to $M$ bounding the supremum, over the real points of $C$ above $K$, of the invariant symbol half-density norm of $A$. Necessity is already known in this generality; only sufficiency is at issue — the direction proved in the source under global injectivity through the pseudo-differential reduction of $A^*A$ (Theorem 3.6), which is not available without it. If the equivalence fails, a disproof consists of an explicit counterexample: a positive complex canonical relation with several real branches over a common base covector, an order-$0$ symbol whose invariant symbol half-density norm stays bounded, and, for some $\varepsilon>0$ and every constant $C_{\varepsilon,K}$, a sequence of test half-densities along which the estimate fails. Since tangent-plane injectivity keeps every such operator $L^2$-continuous, failure means breakdown of the $(M+\varepsilon)$-estimate along high-frequency test half-densities, not unboundedness of the operator norm.

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