A general paradifferential/paracomposition framework extending Hörmander's Nash-Moser method to composition operators

Hörmander's 1990 chapter 'The Nash-Moser Theorem and Paradifferential Operators' showed that for nonlinear equations whose linearization admits a right inverse within (para-)differential symbolic calculus, the loss of derivatives can be handled by paradifferential analysis itself, so that the Nash-Moser iteration can be replaced by elementary nonlinear functional analysis (the 'para-inverse' equation). The method does not cover equations whose nonlinearity involves composition of the unknown - free-boundary water-wave problems, unknown diffeomorphisms, small-divisor conjugacy problems - for which Alinhac's paracomposition calculus is the natural tool; the chapter cites it (reference [1]) but does not carry out the extension. The open problem is to develop the general paradifferential/paracomposition framework: a symbolic calculus for low-regularity changes of variables with tame estimates and conjugation rules, plus a general implicit-function-type theorem whose hypotheses are stated inside that calculus, covering the composition-involving range of applications of the Nash-Moser theorem. The problem remains open: the Alinhac-Gérard treatise (1991/2007) contains no paracomposition; water waves, periodic Schrödinger solutions, and Landau damping have been treated by paradifferential methods only case-by-case; Said (2023) is still generalizing the basic paracomposition calculus; and Alazard-Shao (2023-2026) replace Nash-Moser/KAM schemes by paradifferential and paracomposition arguments only for model problems, explicitly presenting the general framework as a program for future work.

Background

Many nonlinear problems of analysis - from the isometric embedding of Riemannian manifolds to small-divisor problems and free-boundary problems for water waves - share a structural feature: the equation $F(u)=0$ can be linearized, but the linearized equation $F'(u)v=f$ is solvable only at the price of a loss of derivatives, meaning that solving for $v$ costs more Sobolev regularity than $f$ possesses (the Sobolev space $H^s$ consists of the functions whose derivatives up to order $s$ are square-integrable). No contraction argument then survives in a single Banach space and the classical implicit function theorem fails. The Nash-Moser theorem, introduced for the isometric embedding problem and codified in Hamilton's 1982 survey, restores solvability by Newton iteration combined with smoothing operators (operators $S_\theta$, $0<\theta<1$, mapping a rough function to a smooth approximation converging to it as $\theta\to1$) inside a scale of Banach spaces, with tame estimates in which each derivative of an output is controlled by one higher derivative of the input.

Bony's paradifferential calculus (1981) provides the linearization tool adapted to limited regularity. Based on the Littlewood-Paley decomposition of a function into dyadic frequency blocks $\Delta_j u$ (each block selected by a smooth cutoff in the frequency variable; these cutoffs are the ones referred to below as the frequency cutoffs), it splits products and compositions into paraproducts (a low-frequency factor acting as coefficient on high-frequency factors), and paralinearization writes a nonlinear function of $u$ as a linear paradifferential operator $T_{F'(u)}u$, essentially a pseudo-differential operator (a Fourier multiplier-type operator defined by a symbol $a(x,\xi)$) whose symbol retains only the low frequencies of $F'(u)$, acting on $u$, plus a smoother remainder. Alinhac's paracomposition (1986) is the analogue of the pull-back $f\mapsto f\circ\chi$ by a low-regularity diffeomorphism $\chi$: it decomposes $f\circ\chi$ into a paradifferential part plus a remainder, plays the role that Fourier integral operators (operators such as pull-backs by changes of variables, which transport singularities along canonical transformations) play within pseudo-differential calculus, and comes with rules for conjugating a paradifferential operator by a change of variables; it underlies Alinhac's 'good unknown', which removes the spurious loss of regularity caused by coordinate changes in free-boundary problems.

Hörmander, in a 1985 note and in his 1990 chapter, reformulated this circle of ideas: the dyadic decompositions implicit in Nash-Moser smoothing and in paradifferential calculus are the same mechanism, and for equations whose linearization admits a right inverse given by (para-)differential symbolic calculus, one can solve $F(u)=0$ by elementary nonlinear functional analysis - a fixed-point formulation Hörmander called the para-inverse equation - rather than by a Nash-Moser iteration. The chapter proves an abstract implicit function theorem with sharp loss of regularity, introduces the para-inverse, and includes a paradifferential proof of the isometric embedding theorem. The limitation, as later experts summarize it, is that the framework requires the linearized equation to be invertible within (para-)differential calculus, with exactly invertible linearized operators, and it was not extended to problems whose nonlinearity involves composition of the unknown - the situation in which the unknown is, or determines, a diffeomorphism. That is precisely the setting of free-boundary water-wave problems (flattening the moving domain requires composition with a diffeomorphism determined by the unknown, and the Dirichlet-to-Neumann operator - the operator mapping the boundary trace of a harmonic function to its normal derivative, which governs the water-wave dynamics - depends nonlinearly on this domain); of conjugacy problems of KAM type (Kolmogorov-Arnold-Moser theory constructs quasiperiodic solutions by conjugating a system to a constant-coefficient model; the frequencies $\omega$ are Diophantine, i.e. $|\omega\cdot k|\ge c|k|^{-N}$ for all integer vectors $k$, so the inverse of $\omega\cdot\partial$ loses $N$ derivatives); and of equations posed on groups of diffeomorphisms, where composition itself loses derivatives. The candidate tool for the extension is Alinhac's paracomposition - reference [1] of the chapter - but the extension is not carried out there.

Problem Statement

Develop a general paradifferential/paracomposition framework that extends Hörmander's paradifferential approach to Nash-Moser-type problems involving composition, with one overarching objective: that the loss of derivatives in nonlinear equations whose nonlinearity involves composition of the unknown be absorbed by the symbolic calculus itself, rather than by Nash-Moser smoothing iterations. Concretely, the required framework consists of jointly linked components: (i) a symbolic calculus for paracomposition operators - definitions valid for low-regularity changes of variables, boundedness and tame estimates in Sobolev-type scales with sharp tracking of the dependence on the diffeomorphism, composition and conjugation rules compatible with the paradifferential calculus, and control of the degradation of the frequency cutoffs under the successive compositions generated by an iteration; and (ii) a general implicit-function-type theorem whose hypotheses are stated inside this calculus (invertibility of the paralinearized operator and of the associated change-of-variable data), and which is established without Nash-Moser smoothing operators, stated at a level of generality that does not assume properties specific to a single application class. The framework is intended to cover the canonical composition-heavy classes of applications of the Nash-Moser theorem - free-boundary/water-wave-type problems and small-divisor conjugacy/KAM-type problems in which the unknown is or contains a diffeomorphism. The target is the general framework, not a collection of independent problem-specific constructions; a framework covering only renamed special cases would not answer the question.

References

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