Gevrey class

An intermediate space between the spaces of smooth (i.e. $C ^ { \infty }$-) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first motivating example (see [a13], in which regularity estimates of the heat kernel are deduced).

Given $\Omega \subset \mathbf{R} ^ { n }$ and $s \geq 1$, the Gevrey class $G ^ { S } ( \Omega )$ (of index $s$) is defined as the set of all functions $f \in C ^ { \infty } ( \Omega )$ such that for every compact subset $K \subset \Omega$ there exists a $C = C _ { f , K} > 0$ satisfying

\begin{equation*} \operatorname { max } _ { x \in K } | \partial ^ { \alpha } f ( x ) | \leq C ^ { | \alpha | + 1 } ( | \alpha ! | ) ^ { s }, \end{equation*}

\begin{equation*} \alpha \in \mathbf{Z} _ { + } ^ { n } , | \alpha | = \alpha _ { 1 } + \ldots + \alpha _ { n }. \end{equation*}

For $s = 1$ one recovers the space of all real-analytic functions on $\Omega$, while for $s > 1$, $G _ { 0 } ^ { s } ( \Omega ) = G ^ { s } ( \Omega ) \cap C _ { 0 } ^ { \infty } ( \Omega )$ contains non-zero functions, $C ^ { \infty _ { 0 } } ( \Omega )$ being the set of all $C ^ { \infty } ( \Omega )$-functions with compact support. There are various equivalent ways to define $G ^ { S } ( \Omega )$ (cf. [a27]). Introducing the natural inductive topology on $G _ { 0 } ^ { S } ( \Omega )$, for $s > 1$, one can define the space $\mathcal{D} _ { s } ^ { \prime } ( \Omega )$ of Gevrey $s$-ultra-distributions as the dual to $G _ { 0 } ^ { S } ( \Omega )$. The space of $s$-ultra-distributions contains the Schwartz distributions (cf. also Generalized functions, space of). The Gevrey classes are the most simple case of classes of ultra-differentiable functions (or Denjoy–Carleman classes; see, e.g., [a17]). Since the scale of spaces $G ^ { S }$ starts from the analytic functions (for $s = 1$) and ends in the $C ^ { \infty }$-category (setting $s = \infty$), the Gevrey classes play an important role in various branches of partial and ordinary differential equations; namely, whenever the properties of certain differential operators (mappings) differ in the $C ^ { \infty }$ and in the analytic category, it is natural to investigate the behaviour of such operators (mappings) in the scale of Gevrey classes $G ^ { S }$ and, if possible, to find the critical value(s) of $s$, i.e. those for which a change of behaviour occurs. In particular, all weak solutions to the heat equation $( \partial _ { t } - \sum _ { j = 1 } ^ { n } \partial _ { x _ { j } } ^ { 2 } ) u = 0$ are $C ^ { \infty }$, while they are not real-analytic. In the scale of Gevrey spaces, the result is sharp; namely, $u \in G ^ { s } ( \Omega )$ for $s = 2$ (and hence for all $s \geq 2$), but, in general, $u \notin G ^ { S } ( \Omega )$ if $1 \leq s < 2$.

Applications.

The Gevrey classes $G ^ { S }$, $s > 1$, have numerous applications, a few of the main applications being listed below.

Gevrey micro-local analysis.

For $s > 1$ one says that a given $s$-ultra-distribution $u \in \mathcal{D} _ { s } ^ { \prime } ( \Omega )$ is (micro-locally) $G ^ { S }$-regular at a point $( x ^ { 0 } , \xi ^ { 0 } ) \in \Omega \times ( {\bf R} ^ { n } \backslash \{ 0 \} )$ if there exist a $\varphi \in G ^ { s_0 } ( \Omega )$, $\varphi ( x ^ { 0 } ) \neq 0$, an open cone $\mathcal{C} \ni \xi ^ { 0 }$ in $\mathbf{R} ^ { n } \backslash \{ 0 \}$, and a positive constant $c$ such that

\begin{equation*} | \widehat { \varphi u } ( \xi ) | \leq c ^ { - 1 } e ^ { - c | \xi | ^ { 1 / s } } \end{equation*}

for $\xi \in \mathcal{C}$, where $\hat { f } ( \xi ) = \int _ { {\bf R} ^ { n } } e ^ { - i x \xi } f ( x ) d x$ denotes the Fourier transform of $f$ and $x \xi : = x _ { 1 } \xi _ { 1 } + \ldots + x _ { n } \xi _ { n }$. This definition is independent of the choice of $\varphi$. The $G ^ { S }$ wave front set $\operatorname {WF} _ { s } u$ of $u \in \mathcal{D} ^ { \prime } ( \Omega )$ is the smallest closed conic subset $\Gamma$ of $\Omega \times ( \mathbf{R} ^ { n } \backslash \{ 0 \} )$ such that $u$ is $G ^ { S }$-regular at each $( x ^ { 0 } , \xi ^ { 0 } ) \notin \Gamma$. Here, being a conic subset $\Gamma \subset \Omega \times ( \mathbf{R} ^ { n } \backslash \{ 0 \} )$ means that $( x , \xi ) \in \Gamma$ implies $( x , t \xi ) \in \Gamma$ for all $t > 0$. For equivalent definitions, see [a14], [a27].

Let

\begin{equation*} P ( x , D ) = \sum _ { | \alpha | \leq m } p _ { \alpha } ( x ) D _ { x } ^ { \alpha } \end{equation*}

be a linear partial differential operator (cf. also Linear partial differential equation), with $p _ { \alpha } \in G ^ { s } ( \Omega )$, $D _ { x } ^ { \alpha } = D _ { x _ { 1 } } ^ { \alpha _ { 1 } } \ldots D _ { x _ { n } } ^ { \alpha _ { n } }$, $D _ { x _ { k } } = - i \partial _ { x _ { k } }$. The presence of the imaginary unit $i$ allows one to define $P ( x , D )$ via the Fourier transform, namely

\begin{equation*} P ( x , D ) u = ( 2 \pi ) ^ { - n } \int _ { {\bf R} ^ { n } } e ^ { i x \xi } p ( x , \xi ) \hat { u } ( \xi ) d \xi, \end{equation*}

with $p ( x , \xi ) = \sum _ { | \alpha | \leq m } p _ { \alpha } ( x ) \xi ^ { \alpha }$. This definition is valid for pseudo-differential operators (cf. Pseudo-differential operator) as well, where $p ( x , \xi )$ is a suitable symbol from the Hörmander classes $S _ { 1,0 } ^ { m }$ or from other classes (see [a27] for more details and references). The characteristic set $\Sigma _ { P }$ of $P$ is defined by

\begin{equation*} \Sigma _ { P } = \{ ( x , \xi ) \in \Omega \times ( \mathbf{R} ^ { n } \backslash \{ 0 \} ) : p _ { m } ( x , \xi ) = 0 \}, \end{equation*}

with $p _ { m } ( x , \xi ) = \sum _ { | \alpha | = m } p _ { \alpha } ( x ) \xi ^ { \alpha }$ standing for the principal symbol (cf. also Symbol of an operator; Principal part of a differential operator). The operator is called $G ^ { S }$-hypo-elliptic (respectively, $G ^ { S }$-micro-locally hypo-elliptic) in an open set $U \subset \Omega$ (respectively, in an open conic set $\Gamma \subset \Omega \times ( \mathbf{R} ^ { n } \backslash \{ 0 \} )$) if for every $u \in \mathcal{D} _ { s } ^ { \prime } ( U )$ satisfying $P ( x , D ) u \in G ^ { S } ( U )$ (respectively, $\operatorname{WF} _ { s } ( P ( x , D ) u ) \cap \Gamma = \emptyset$) necessarily $u \in G ^ { s } ( U )$ (respectively, $\operatorname{WF} _ { s } u \cap \Gamma = \emptyset$).

Recall that $P ( x , D )$ is called of principal type if $( x , \xi ) \in \Sigma _ { P }$ implies that $d_ {x , \xi} p _ { m } ( x , \xi )$ is not linearly dependent on $\sum _ { j = 1 } ^ { n } \xi _ { j } d x _ { j }$. The operator is called of multiple characteristics type if there exists a $( x , \xi ) \in \Sigma _ { p }$ such that $d_{x , \xi} p _ { m } ( x , \xi ) = 0$. The properties of operators of principal type are basically the same in the analytic-Gevrey category and the $C ^ { \infty }$-category. An essential difference occurs in the case of multiple characteristics. For operators with constant multiple real or complex characteristics, modelled by $P ( x , D ) = L ^ { m } + Q ( x , D )$, $Q$ being an operator of order $\leq m - 1$ while $L$ is a first-order operator modelled by $L = L _ { 1 } = D _ { x _ { 1 } }$ or $L = L _ { 2 } = D _ { x _ { 1 } } + i x _ { 1 } ^ { h } D _ { x _ { 2 } }$, $h \in \mathbf{N}$, the behaviour of $P$ in $G ^ { S }$, $1 < s < m / ( m - 1 )$, is governed by the operator $L$, independently of the lower-order terms in $Q$. However, if $s > m / ( m - 1 )$, then the lower-order terms affect both the $G ^ { S }$-hypo-ellipticity and the propagation of $G ^ { S }$ singularities, cf. [a5], [a21], [a27]. In fact, often one is interested in finding a critical index $s _ { 0 } > 1$ such that for $1 \leq s < s _ { 0 }$ and $s > s 0$ certain properties are complementary. In particular, if $L = L _ { 2 }$ and $h$ is even, there are examples of operators analytic and Gevrey $G ^ { S }$-hypo-elliptic for $1 \leq s \leq m / ( m - 1 )$ but not $C ^ { \infty }$- and Gevrey $G ^ { S }$-hypo-elliptic for large values of the Gevrey index $s$ (see [a21], [a27] for more details and references).

As to the $G ^ { S }$-hypo-ellipticity for operators of the form $P ( x , D ) = \sum _ { j = 1 } ^ { n } X _ { j } ^ { 2 }$, $X_j$ being analytic vector fields satisfying the Hörmander bracket hypothesis, a typical pattern of behaviour is the following one: There is a critical index $s_0$ such that for $s > s 0$, the $G ^ { S }$-hypo-ellipticity of $P ( x , D )$ holds, while for $1 \leq s < s _ { 0 }$ it does not (cf. [a2], [a7], [a23]).

Gevrey singularities appear in the study of initial-boundary value problems for hyperbolic equations in domains with analytic diffractive boundaries (cf. [a18] and the references therein). In particular, for the wave equation, Gevrey $G ^ { 3 }$-singularities along the diffractive analytic boundary appear, and this fact is used in scattering theory (cf. [a1]).

Gevrey solvability.

The operator $P ( x , D )$ is called (locally) $G ^ { S }$-solvable in $\Omega$ if for every $f \in G _ { 0 } ^ { s } ( \Omega )$ there exists a $u \in \mathcal{D} _ { s } ^ { \prime } ( \Omega )$ such that $P ( x , D ) u = f$.

Since $G^{S}$-solvability implies $G ^ { t }$-solvability for $t < s$, when $P$ is not solvable in the $C ^ { \infty }$-category one looks for an index $s _ { 0 } > 1$ such that the operator $P$ is $G ^ { S }$-solvable for $1 \leq s < s _ { 0 }$ and not for $s > s_0$. The model operators $L ^ { m } + Q$, with $L = L _ { 1 }$ or $L = L _ { 2 }$ and $h$ being even, are $G ^ { S }$-solvable for $1 < s \leq m / ( m - 1 )$, while for $s > m / ( m - 1 )$ they need not be $G ^ { S }$-solvable (cf. [a6], [a21], [a27]).

$G ^ { S }$-solvability for semi-linear partial differential operators, provided $1 < s \leq m / ( m - 1 )$, is proved in [a12].

Hyperbolic equations.

The Gevrey classes serve as a framework for the well-posedness of the Cauchy problem for weakly hyperbolic linear partial differential operators (cf. also Linear hyperbolic partial differential equation and system)

\begin{equation*} P ( t , x ; D _ { t } , D _ { x } ) u = \end{equation*}

\begin{equation*} = D _ { t } ^ { m } u + \sum _ { j = 1 } ^ { m } \sum _ { | \alpha | \leq m - j } p _ { j , \alpha } ( t , x ) D _ { t } ^ { j } D _ { x } ^ { \alpha } u = f ( t , x ) ,\; D _ { t } ^ { j } u ( 0 , x ) = u _ { j } ^ { 0 } ( x ) , \quad j = 0 , \ldots , m - 1. \end{equation*}

Weak hyperbolicity means that the roots of $p _ { m } ( t , x ; \tau , \xi ) = 0$ with respect to $\tau$ are real. If $d$ is the maximal multiplicity of the real roots in $\tau$, then the Cauchy problem is always well-posed in the framework of the Gevrey classes $G ^ { S }$, provided $1 \leq s \leq d / ( d - 1 )$. If $s > d / ( d - 1 )$, one can point out specific lower-order terms such that the existence fails. More subtle estimates for the critical Gevrey index are obtained by using the distance between the roots or via additional restrictions on the lower-order terms (so-called Levi-type conditions). See [a3], [a14], [a4], [a15], [a19], [a24], [a21] for more details and references. Local Gevrey well-posedness for weakly hyperbolic non-linear systems is shown in [a16] (see also [a12]). Goursat problems for Kirchoff-type equations in Banach spaces of Gevrey functions (cf. also Kirchhoff formula; Goursat problem) have been studied in [a10].

Divergent series and singular differential equations.

One may also define formal Gevrey spaces $G ^ { S }$, e.g. the set of all formal power series

\begin{equation*} \sum _ { \alpha \in \mathbf{Z} _+^ { n } } \frac { a _ { \alpha } } { ( | \alpha | ! ) ^ { s - 1 } } x ^ { \alpha }, \end{equation*}

where for some $C > 0$ the following estimates hold:

\begin{equation*} | a _ { \alpha } | \leq C ^ { | \alpha | + 1 } , \alpha \in \mathbf{Z} _ { + } ^ { n }. \end{equation*}

Such formal Gevrey spaces are used in the study of divergent series and singular ordinary linear differential equations with Gevrey coefficients (see [a26] and the references therein). The Fredholm property in such type of Gevrey spaces of certain singular analytic partial differential operators in $\mathbf{C} ^ { 2 }$ has been studied by means of Toeplitz operators (cf. [a22]).

Dynamical systems.

The framework of Gevery classes is used in the study of normal forms of analytic perturbations of (non-) integrable (non-) Hamiltonian systems. Roughly speaking, one obtains normal forms modulo exponentially small error terms of the type $e ^ { - 1 / \varepsilon ^ { \sigma } }$, where $\varepsilon > 0$ is small parameter, while $\sigma = 1 / ( s - 1 ) > 0$ is related to Gevrey-$G ^ { S }$-type estimates, or so-called Nekhoroshev-type estimates (see e.g. [a11] for Gevrey normal forms of billiard ball mappings and [a25] on normal forms of perturbations of Hamiltonian systems).

Evolution partial differential equations.

In the study of the analytic regularity of solutions of semi-linear evolution equations (Navier–Stokes, Kuramoto–Sivashinksi, Euler, the Ginzburg–Landau equation) with periodic boundary data for positive time, the term "Gevrey class" is used usually to denote the Banach space $G ^ { s } ( \mathcal{T} ^ { n } ; T )$, $T > 0$, (with $\mathcal{T} ^ { n } = \mathbf{R} ^ { n } / ( 2 \pi \mathbf{Z} ) ^ { n }$ being the $n$-dimensional torus) of smooth functions on $\mathcal{T} ^ { n }$ with the norm defined by means of the discrete Fourier transform

\begin{equation*} \| u \| _ { T } ^ { 2 } = \sum _ { \xi \in \mathbf{Z} ^ { n } } ( 1 + | \xi | ) ^ { 2 r } e ^ { 2 T | \xi | ^ { 1 / s } } | \hat { u } ( \xi ) | ^ { 2 }, \end{equation*}

for some $r > n / 2$. In these applications, the Gevrey index $s = 1$ (the analytic category). See [a8] for the Navier–Stokes equations; [a9] for recent results on semi-linear parabolic partial differential equations; and [a20] for a generalized Euler equation.

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