# Global wellposedness of
general nonlinear evolution equations
for distributions on the Fourier half space

Kenji Nakanishi  and  Baoxiang Wang  Research Institute for Mathematical
Sciences, Kyoto University, Kyoto 606-8502, Japan<EMAIL_ADDRESS>School of Sciences, Jimei University, Xiamen, 361021 and School of
Mathematical Sciences, Peking University, Beijing 100871, China
<EMAIL_ADDRESS>
###### Abstract.

The Cauchy problem is studied for very general systems of evolution equations,
where the time derivative of solution is written by Fourier multipliers in
space and analytic nonlinearity, with no other structural requirement. We
construct a function space for the Fourier transform embedded in the space of
distributions, and establish the global wellposedness with no size
restriction. The major restriction on the initial data is that the Fourier
transform is supported on the half space, decaying at the boundary in the
sense of measure. We also require uniform integrability for the orthogonal
directions in the distribution sense, but no other condition. In particular,
the initial data may be much more rough than the tempered distributions, and
may grow polynomially at the spatial infinity. A simpler argument is also
presented for the solutions locally integrable in the frequency. When the
Fourier support is slightly more restricted to a conical region, the
generality of equations is extremely wide, including those that are even
locally illposed in the standard function spaces, such as the backward heat
equations, as well as those with infinite derivatives and beyond the natural
boundary of the analytic nonlinearity. As more classical examples, our results
may be applied to the incompressible and compressible Navier-Stokes and Euler
equations, the nonlinear diffusion and wave equations, and so on. The major
drawback of the Fourier support restriction is that the solutions cannot be
real valued.

###### Key words and phrases:

Nonlinear evolution equations, Global wellposedness, Distributions

###### 2020 Mathematics Subject Classification:

35A01, 35A02, 35B60, 35F55, 35G55, 35K55, 35L60, 35L70, 35Q30, 35Q31, 35Q55,
46F05, 46F10

The first author (K. N.) was supported by JSPS KAKENHI Grant Numbers
JP20H01814 and JP22H01132, the second author (B. W.) was supported in part by
NSFC Grant Number 12171007.

###### Contents

  1. 1 Introduction
    1. 1.1 Global wellposedness results
    2. 1.2 Solutions locally integrable in frequency
    3. 1.3 Outline of the paper
    4. 1.4 Notations
  2. 2 Spaces of distributions for the Fourier transform of solutions
    1. 2.1 Characterization of $\mathcal{X}$
    2. 2.2 Local basis in $\mathcal{X}$
    3. 2.3 Bounded sets in $\mathcal{X}$
    4. 2.4 Compact sets in $\mathcal{X}$
  3. 3 Convolution for the distributions
  4. 4 Global wellposedness for distributions
  5. 5 Comparison with other solutions
    1. 5.1 Solutions beyond blow-up
    2. 5.2 Solutions beyond the natural boundary
  6. 6 Optimality of the conditions for the solution space
    1. 6.1 Frequency on the half space
    2. 6.2 Decay to the boundary
    3. 6.3 Integrability in the orthogonal directions
    4. 6.4 Cone support for unbounded coefficients
  7. 7 Global solutions with locally integrable Fourier transform
  8. 8 Examples of the equations
    1. 8.1 Incompressible Navier-Stokes and Euler equations
    2. 8.2 Compressible Navier-Stokes and Euler equations
    3. 8.3 Nonlinear wave and Klein-Gordon equations
    4. 8.4 Nonlinear Schrödinger equations
    5. 8.5 Completely integrable equations
    6. 8.6 Nonlinear hyperbolic conservation laws
    7. 8.7 Nonlinear parabolic equations

##  1\. Introduction

We consider general systems of nonlinear evolution equations in the form

$\displaystyle\partial_{t}u=L(D)u+N(D)\mathscr{H}(M(D)u),\ \ u(0,x)=u_{0}(x),$
(1.1)

for the unknown function $u(t,x):\mathbb{R}^{1+d}\to\mathbb{C}^{n_{1}}$, where
$L(D)=\mathcal{F}^{-1}\hat{L}(\xi)\mathcal{F}$, $M(D)$ and $N(D)$ are given
matrix-valued Fourier multipliers, defined with the Fourier transform
$\mathcal{F}$ on $\mathbb{R}^{d}$ and the symbols $\hat{L}(\xi)$,
$\hat{M}(\xi)$, $\hat{N}(\xi)$ of matrix size $n_{1}\times n_{1}$,
$n_{2}\times n_{1}$, $n_{1}\times n_{3}$ respectively, for some
$d,n_{1},n_{2},n_{3}\in\mathbb{N}$. $\mathscr{H}$ is a given analytic function
defined on some neighborhood of $0\in\mathbb{C}^{n_{2}}$ with values in
$\mathbb{C}^{n_{3}}$, satisfying $\mathscr{H}(z)=o(z)$ as $z\to 0$.

Each component of $L,M,N$ with smooth symbol may be regarded as a linear and
continuous map on $\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$,
namely the Fourier (inverse) image of the space of Schwartz distributions
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$, defined as the dual space of
$\mathcal{F}\mathscr{D}(\mathbb{R}^{d})\subset\mathscr{S}(\mathbb{R}^{d})$
(for the precise definition, see [13] where it is denoted by
$\mathbf{D}^{\prime}$, [16, Chapter II, Section 1.4] where it is denoted by
$Z^{\prime}$, or the end of this section).

The goal of this paper is to establish global wellposedness of the Cauchy
problem (1.1) for such a wide class of equations in a certain function space
embedded in $\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$, with no
size restriction, but restricting the Fourier support to the half space
$[0,\infty)\times\mathbb{R}^{d-1}$. Such a kind of Fourier support conditions
have definite meanings for the solutions of certain physical models. For
instance, the solutions of the Constantin–Lax–Majda equation can be reduced to
those of the complex heat equation which are supported in the Fourier half
space, cf. [8]. The form of equations (1.1) covers major nonlinear evolution
equations, such as the Euler equations and the Navier-Stokes equations, both
in the incompressible case and of the compressible models, and many other
evolutional PDEs of nonlinear hyperbolic, parabolic, and dispersive types,
though the complex analyticity of the nonlinear part $\mathscr{H}$ may well
require some extension or reformulation of the equations.

The advantage of the Fourier support restriction is to make sense of
convolution between two distributions in the Fourier space, corresponding to
the multiplication in the physical space. Without that restriction, it would
be highly non-trivial even to make sense of the equations in such generality,
and of course, one should not expect the global wellposedness in any way
compatible with the standard theories. Nevertheless, there have been many
attempts for general existence of weak solutions to nonlinear PDEs. In
particular, there is an extensive literature of studies based on algebras of
generalized functions, those introduced by Colombeau [9] and those by Rosinger
[31]. Without trying to be exhaustive, we just mention two results for
arbitrary types of equations: the global Cauchy-Kovalevskaya theory for
analytic equations and data [31, Chapter 2], and local existence for general
equations and data [10]. Their settings are more general than ours, as the
equations may contain variable coefficients and the initial data may contain
arbitrary frequencies, but they also require considerable departure from the
linear distribution theory, including the meanings of the generalized
functions, the derivatives and the equations.

This paper is a generalization of the previous ones by the second author and
others, where unique global existence of solutions was proven for the Navier-
Stokes equations [14], the complex-valued heat equations [8], the nonlinear
Schrödinger equations [7] and the nonlinear Klein-Gordon equation [34], in
very rough function spaces defined by negative exponential weight in the
Fourier space, into which all the standard Sobolev spaces are continuously
embedded. Roughly speaking, besides the time decaying behavior and the lower
regularity in Sobolev spaces, the results in [14, 8, 7, 34] may be summarized
as follows: If the initial data belongs to the negative exponential Sobolev
(or modulation) space and the Fourier transform is supported in the first
octant $[0,\infty)^{d}$, then those equations have unique global solutions in
similar function spaces with less regularity.

The generalization in this paper is threefold. First, the equations are very
general and not specialized, with no structural condition than those mentioned
above (except some regularity and boundedness of the symbols
$\hat{L},\hat{M},\hat{N}$). Second, the function space for the solutions is
much wider. In the previous papers, the Fourier transform was locally $L^{2}$
and growing at most exponentially. In this paper, it may be a distribution of
arbitrary growth. In particular, the solutions in this paper may decay, grow,
or be periodic in space, while the previous ones decay in average sense.
Another benefit of the larger space is the wellposedness: solutions stay in
the same space as the initial data, and are continuous for all time. That was
not the case in the previous papers, due to the regularity loss in time.
Third, the Fourier support is on the bigger set, namely the half space instead
of the octant. We will also consider subsets of cone shape with possibly
growing aperture, for which the conditions on $\hat{L},\hat{M},\hat{N}$ become
much less restrictive, and the octant is a special case.

Since the space $\mathcal{X}$ is rather complicated, we will also consider
smaller function spaces where the Fourier transform is locally integrable,
with a much simpler proof with more explicit information on the solutions.

###  1.1. Global wellposedness results

Now let us describe the main results in this paper about the global
wellposedness for (1.1). For any $\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$
and $a,b\in\mathbb{R}$, we denote

$\begin{split}\mathbb{R}^{d}_{<a}(\mathrm{v}):=\\{\xi\in\mathbb{R}^{d}\mid\mathrm{v}\cdot\xi<a\\},\quad\mathbb{R}^{d}_{(a,b)}(\mathrm{v}):=\\{\xi\in\mathbb{R}^{d}\mid
a<\mathrm{v}\cdot\xi<b\\},\end{split}$ (1.2)

and similarly define $\mathbb{R}^{d}_{\leq a}(\mathrm{v})$,
$\mathbb{R}^{d}_{\geq a}(\mathrm{v})$, etc. One may fix
$\mathrm{v}=(1,0,\ldots,0)$ without losing generality. The function space
$\mathcal{X}(\mathrm{v})\subset\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is
defined, to be used for the Fourier transform of solutions, as the set of
distributions $F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$ with the following
two properties (X-i)–(X-ii):

(X-i) Decay at the boundary as measure:

There exist $a\in(0,\infty)$ and a complex Borel measure $\mathfrak{m}$ on
$\mathbb{R}^{d}$, such that $F$ and $\mathfrak{m}$ coincide when restricted to
$\mathbb{R}^{d}_{<a}(\mathrm{v})$, and the total variation $|\mathfrak{m}|$
satisfies $|\mathfrak{m}|(\mathbb{R}^{d}_{\leq 0}(\mathrm{v}))=0$. (It implies
$\operatorname{supp}F\subset\mathbb{R}^{d}_{\geq 0}$.)

More concretely, there exist $a\in(0,\infty)$ and finite Borel measures
$\mathfrak{m}_{k}$ on $\mathbb{R}^{d}$ satisfying
$\mathfrak{m}_{k}(\mathbb{R}^{d}_{\leq 0}(\mathrm{v}))=0$ for $k=0,1,2,3$,
such that for all $\varphi\in\mathscr{D}(\mathbb{R}^{d})$ with
$\operatorname{supp}\varphi\subset\mathbb{R}^{d}_{<a}(\mathrm{v})$,

$\begin{split}F(\varphi)=\sum_{k=0}^{3}\int_{\mathbb{R}^{d}}i^{k}\varphi(x)d\mathfrak{m}_{k}(x).\end{split}$
(1.3)

(X-ii) Integrability for the orthogonal directions as distribution:

For each $a\in(0,\infty)$, there exists $m\in\mathbb{N}$ such that
$F|_{\mathbb{R}^{d}_{<a}(\mathrm{v})}\in
W^{-m,1}(\mathbb{R}^{d}_{<a}(\mathrm{v}))$, namely the $L^{1}$-based Sobolev
space of the negative regularity.

More concretely, for each $a\in(0,\infty)$, there exist finite subsets
$A\subset\mathbb{N}_{0}^{d}$ and $\\{f_{\alpha}\\}_{\alpha\in A}\subset
L^{1}(\mathbb{R}^{d})$ such that for all
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$ with
$\operatorname{supp}\varphi\subset\mathbb{R}^{d}_{<a}(\mathrm{v})$,

$\begin{split}F(\varphi)=\sum_{\alpha\in
A}\int_{\mathbb{R}^{d}}\partial_{x}^{\alpha}\varphi(x)f_{\alpha}(x)dx.\end{split}$
(1.4)

The space $\mathcal{X}(\mathrm{v})$ is equipped with a locally convex
Hausdorff topology such that
$\mathcal{X}(\mathrm{v})\subset\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is
continuous. See Section 2 for the definition and the properties of
$\mathcal{X}(\mathrm{v})$ (and Lemma 2.4 for consistency with the above
description). Examples of $u_{0}\in\mathcal{F}^{-1}\mathcal{X}(\mathrm{v})$
may be found in (8.21).

The condition (X-i) implies that
$u_{0}\in\mathcal{F}^{-1}\mathcal{X}(\mathrm{v})\subset\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$
can not be real-valued unless it is $0$, because $u_{0}$ is real-valued if and
only if $\mathcal{F}u_{0}(-\xi)=\overline{\mathcal{F}u_{0}(\xi)}$. So we can
not deal with any non-trivial real-valued solutions under that condition. This
is the major and huge disadvantage for applications, and it is the same
feature as in the previous papers [14, 8, 7, 34]. All the surprising
generality rests on this price, which also precludes potential contradictions
with the standard results.

For the symbols of the Fourier multipliers in (1.1), we assume

(C-s) Coefficients of semi-linear type:

$\hat{L},\hat{M},\hat{N}$ are smooth on $\mathbb{R}^{d}_{>0}(\mathrm{v})$ such
that $\partial_{\xi}^{\alpha}e^{t\hat{L}(\xi)}(t^{\theta}\hat{N}(\xi))^{j}$
and $\partial_{\xi}^{\alpha}\hat{M}(\xi)$ are bounded for $j\in\\{0,1\\}$,
$0<t<1$ and $\xi\in\mathbb{R}^{d}_{(1/n,n)}(\mathrm{v})$, for some
$\theta\in[0,1)$, every $n\in\mathbb{N}$ and every
$\alpha\in\mathbb{N}_{0}^{d}$. Without the derivative ($\alpha=0$), they are
required to be bounded also on $\mathbb{R}^{d}_{(0,1)}(\mathrm{v})$.

Note that the symbols are allowed to be discontinuous at the boundary
$\mathrm{v}\cdot\xi=0$.

######  Theorem 1.1.

Let $d\in\mathbb{N}$, $\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$ and assume
(C-s). Then the Cauchy problem of (1.1) is globally wellposed in
$(\mathcal{F}^{-1}\mathcal{X}(\mathrm{v}))^{n_{1}}$ on $t\geq 0$.

See Corollary 4.2 and Theorem 4.1 for the more precise versions.

The above result is general but also restrictive about the equation when
$d\geq 2$, since the symbols are required to be uniformly bounded in the
orthogonal directions. Roughly speaking, the nonlinear part of the equation is
required to have less derivatives in the orthogonal directions than the linear
smoothing effect of $L$. We may remove this limitation by imposing a stronger
condition on the Fourier support than the half space. For any closed subset
$\mathcal{R}\subset\mathbb{R}^{d}$, let

$\begin{split}\mathcal{X}_{\mathcal{R}}(\mathrm{v}):=\\{F\in\mathcal{X}(\mathrm{v})\mid\operatorname{supp}F\subset\mathcal{R}\\}.\end{split}$
(1.5)

Since $\mathcal{X}(\mathrm{v})\subset\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is
continuous, it is a closed subspace of $\mathcal{X}(\mathrm{v})$. In this
case, we may deal with much more general symbols, namely

(C-0) Coefficients bounded at $0$:

$\hat{L},\hat{M},\hat{N}$ are smooth on a neighborhood of
$\mathcal{R}\cap\mathbb{R}^{d}_{>0}(\mathrm{v})$ and bounded as $\xi\to 0$.

######  Theorem 1.2.

Let $d\in\mathbb{N}$, $\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$, and
$\mathcal{R}\subset\mathbb{R}^{d}_{\geq 0}(\mathrm{v})$ be a closed subset
such that $\mathcal{R}+\mathcal{R}\subset\mathcal{R}$ and that
$\mathcal{R}\cap\mathbb{R}^{d}_{<a}(\mathrm{v})$ is bounded for each
$a\in(0,\infty)$. Assume (C-0). Then the Cauchy problem of (1.1) is globally
wellposed in $(\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}(\mathrm{v}))^{n_{1}}$
on $t\in\mathbb{R}$.

It is proven after Corollary 4.2 as an immediate consequence. The simplest
choice of $\mathcal{R}$ is the cone
$\\{|\xi|\leq\lambda(\mathrm{v}\cdot\xi)\\}$ with a constant $\lambda>0$,
which approaches $\mathbb{R}^{d}_{\geq 0}(\mathrm{v})$ as $\lambda\to\infty$.
The left side $|\xi|$ may be replaced with any norm on $\mathbb{R}^{d}$.
Another example is the octant $[0,\infty)^{d}$ with
$\mathrm{v}\in(0,\infty)^{d}$. More generally, $\mathcal{R}$ may grow
arbitrarily fast as $\mathrm{v}\cdot\xi\to\infty$: For any $C^{1}$ function
$R:[0,\infty)\to[0,\infty)$ with $R(0)=0$, there exists
$\mathcal{R}\supset\\{|\xi|\leq R(\mathrm{v}\cdot\xi)\\}$ with the require
properties (cf. Lemma 4.4).

In the above results, the symbols may grow arbitrarily fast as
$|\xi|\to\infty$ on $\mathcal{R}$ (which is possible only as
$\mathrm{v}\cdot\xi\to\infty$), meaning that the equation may contain
arbitrary order of derivatives, including infinity. Hence the global
wellposedness in $\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}(\mathrm{v})$ holds
even for extremely bad PDEs such as

$\begin{split}u_{t}=-\Delta u+u^{3},\quad
u_{t}=e^{e^{-\Delta}}[e^{(e^{-\Delta}u)^{2}}-1],\end{split}$ (1.6)

where the sign in front of $\Delta$ is the opposite one from the usual
settings. Such equations are not even locally wellposed in the standard
function spaces, and the ODE versions are not globally wellposed either. See
Section 5.2 for another example, where the solutions may live outside of the
natural boundary of the analytic nonlinearity $\mathscr{H}$.

The above generality indicates how strong is the assumption of the Fourier
support, which essentially eliminates all possible nonlinear feedback. In
fact, the core of the proof is to construct weight functions in the Fourier
space, depending on time, which control the nonlinear flow of frequencies to
$\mathrm{v}\cdot\xi\to\infty$, so that the standard contraction argument can
work on the subspaces forming $\mathcal{X}$ characterized by the weights. The
main novelty compared with the previous work [14, 8, 7, 34] is that the
procedure works without using any explicit form of the equations or the
weights. This flexibility is crucial to get the wellposedness in the entire
function space $\mathcal{F}^{-1}\mathcal{X}$.

###  1.2. Solutions locally integrable in frequency

Because of the vast generality of the equation and the initial data, it is not
easy to extract concrete information on the behavior of solutions and the
weights from the proof in $\mathcal{X}$. Restricting the initial data to the
Fourier transform of locally integrable functions, we may simplify the
construction of solutions considerably, with more explicit information on the
behavior. This version is still more general than the previous work on the
equations and the initial data.

In this case, we first fix a closed $\mathcal{R}\subset\mathbb{R}^{d}_{\geq
0}(\mathrm{v})$ satisfying $\mathcal{R}+\mathcal{R}\subset\mathcal{R}$,
together with a continuous and super-additive function
$\mathfrak{p}:\mathcal{R}\to[0,\infty)$ (namely
$\mathfrak{p}(\xi)+\mathfrak{p}(\eta)\leq\mathfrak{p}(\xi+\eta)$). Then for
the Fourier multipliers in (1.1), we assume

(C-m) Measurable coefficients:

$\hat{L},\hat{M},\hat{N}$ are measurable on $\mathcal{R}$ satisfying

$\begin{split}\sigma_{\max}(\hat{L}+\hat{L}^{*})\leq
2\theta\mathfrak{p},\quad\max_{j,k}|\hat{N}_{j,k}|+|\hat{M}_{l,m}|\lesssim
1+\mathfrak{p},\end{split}$ (1.7)

for all $\xi\in\mathcal{R}$ and some constant $\theta\in(0,1)$, where
$\hat{L}^{*}$ is the adjoint of $\hat{L}$ and $\sigma_{\max}$ is the maximal
eigenvalue of the self-adjoint matrix.

If $\mathfrak{p}:\mathcal{R}\to[0,\infty)$ satisfies (1.7) with locally
bounded $|\xi|^{-1}\mathfrak{p}(\xi)$, then there exists
$\tilde{\mathfrak{p}}:\mathcal{R}\to[0,\infty)$ that is continuous and super-
additive satisfying $\tilde{\mathfrak{p}}\geq\mathfrak{p}$, by Lemma 4.4. Then
we may replace $\mathfrak{p}$ with $\tilde{\mathfrak{p}}$.

Next, for any $s<0$, we consider the weighted $L^{1}$ space

$\begin{split}&Z^{s}_{0}:=\\{\varphi\in
L^{1}_{loc}(\mathbb{R}^{d})\mid\operatorname{supp}\varphi\subset\mathcal{R},\
\|\varphi\|_{Z^{s}_{0}}:=\|(\mathfrak{p}^{-1}+\mathfrak{p})e^{s\mathfrak{p}}\varphi\|_{L^{1}(\mathcal{R})}<\infty\\}.\end{split}$
(1.8)

More generally, we may choose any Banach space $Z^{s}\subset
L^{1}_{loc}(\mathbb{R}^{d})$ satisfying

$\begin{split}\operatorname{supp}\varphi\subset\mathcal{R},\quad\sup_{c\in\mathcal{R}}\|\varphi(\xi-c)\|_{Z^{s}}+\|\varphi\|_{Z^{s}_{0}}\lesssim\|\varphi\|_{Z^{s}}=\||\varphi|\|_{Z^{s}}\end{split}$
(1.9)

for all $\varphi\in Z^{s}$. $Z^{s}_{0}$ is the maximal choice for $Z^{s}$.
Another example is the anisotropic space for
$\xi:=(\bar{\xi},\tilde{\xi})\in\mathbb{R}^{d_{0}}\times\mathbb{R}^{d-d_{0}}$
with

$\begin{split}&\mathfrak{p}(\xi):=\lambda(\mathrm{v}\cdot\xi+(\mathrm{v}\cdot\xi)^{m}),\quad\mathcal{R}:=\\{\xi\in\mathbb{R}^{d}\mid|\bar{\xi}|\leq\lambda\mathfrak{p}(\xi)\\},\\\
&Z^{s}:=\\{\varphi\in
Z^{s}_{0}\mid\|\varphi\|_{Z^{s}}:=\|(\mathfrak{p}^{-\sigma}+1)\varphi\|_{L^{r}_{\bar{\xi}}L^{1}_{\tilde{\xi}}}<\infty\\},\end{split}$
(1.10)

for some $d_{0}\in\\{1,\ldots,d\\}$, $\lambda>0$, $m\geq 1$, $r\in[1,\infty]$
and $\sigma>1-d_{0}/r^{\prime}$.

######  Theorem 1.3.

Let $d\in\mathbb{N}$, $\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$,
$\mathcal{R}\subset\mathbb{R}^{d}_{\geq 0}(\mathrm{v})$ be closed satisfying
$\mathcal{R}+\mathcal{R}\subset\mathcal{R}$, and let
$\mathfrak{p}:\mathcal{R}\to[0,\infty)$ be continuous and super-additive.
Assume (C-m). Let $s<0$ and let $Z^{s}\subset L^{1}_{loc}(\mathbb{R}^{d})$ be
a Banach space satisfying (1.9). Then for any $B\in(0,\infty)$, there exists
$b\in[1,\infty)$ such that for any
$u_{0}\in(\mathcal{F}^{-1}L^{1}_{loc}(\mathbb{R}^{d}))^{n_{1}}$ satisfying
$\operatorname{supp}\mathcal{F}u_{0}\subset\mathcal{R}$ and
$\|e^{s\mathfrak{p}^{2}}\mathcal{F}u_{0}\|_{Z^{s}}\leq B$, the Cauchy problem
of (1.1) has a unique global solution $u$ on $t\geq 0$ satisfying
$e^{b(s\mathfrak{p}^{2}-t\mathfrak{p})}\mathcal{F}u(t)\in
Z^{s}(L^{\infty}_{t}(0,\infty))^{n_{1}}$.

See Theorem 7.1 for the more precise version, including the case where
nonlinearity $\mathscr{H}$ is decaying at $0$ in higher order. Note that the
above is not a wellposedness result, because the solution is in a much rougher
space than the initial data, due to the loss of regularity. In the previous
case of $\mathcal{X}$, the wellposedness holds thanks to its topological
structure, namely the inductive limit, but huge loss of regularity is going on
inside $\mathcal{X}$.

###  1.3. Outline of the paper

In Section 2, we give the precise definition and some basic properties of the
space $\mathcal{X}$. It is a locally convex topological vector space, defined
as the inductive limit of an uncountable directed system of Banach spaces in
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$. We give a local basis at
$0\in\mathcal{X}$, as well as characterization of bounded subsets and compact
subsets of $\mathcal{X}$. The latter is crucial for the proof of wellposedness
in $\mathcal{F}^{-1}\mathcal{X}$.

In Section 3, we investigate the convolution in those Banach spaces forming
$\mathcal{X}$. Lemma 3.4 is crucial for the contraction argument to solve the
Cauchy problem, where the point is that arbitrarily small factors may be
gained from the convolution on the half space with the boundary decay, if the
weights are appropriately chosen.

In Section 4, we prove the global wellposedness in
$\mathcal{F}^{-1}\mathcal{X}$, including the cases of stringer support
conditions. Solutions periodic in space are also considered as the special
case where the support is a lattice.

In Section 5, we compare those solutions in $\mathcal{F}^{-1}\mathcal{X}$ with
others, observing for the simplest ODE and for the Burgers equation that they
go beyond the classical blow-up. The example shows also that the global
wellposedness can not hold within the tempered distributions. Another example
shows that the global solutions exist even if the initial data are completely
outside of the natural boundary of the nonlinearity $\mathscr{H}$.

In Section 6, we investigate optimality of the space $\mathcal{X}$ or the
conditions (X-i)–(X-ii), as well as the stronger support condition, for the
very general wellposedness to hold, using again the simple ODE.

In Section 7, we consider the solutions locally integrable in the Fourier
space, and give a much simpler argument for the unique global existence of
solutions.

Finally in Section 8, our results are applied to several examples of
equations.

###  1.4. Notations

The sets of non-negative integers are denoted by
$\mathbb{N}_{0}=\\{0,1,2,\ldots,\\}$ and $\mathbb{N}=\\{1,2,\ldots,\\}$.
$1_{A}$ denotes the characteristic function of any set $A$.

In any seminormed space $V$, open and closed balls are denoted by

$\begin{split}B_{V}(c,r):=\\{y\in
V\mid\|c-y\|_{V}<r\\},\quad\bar{B}_{V}(c,r):=\\{y\in V\mid\|c-y\|_{V}\leq
r\\},\end{split}$ (1.11)

with center $c\in V$ and radius $r>0$. For vectors $x\in V^{n}$ of finite
dimension $n\in\mathbb{N}$, the seminorm is defined by
$\|x\|_{V}:=[\sum_{k=1,\ldots,n}\|x_{k}\|_{V}^{2}]^{1/2}$. For two normed
spaces $X,Y$, the Banach space of linear bounded operators from $X$ to $Y$ is
denoted by $\mathcal{B}(X\to Y)$. The normed space of $m\times n$ complex
matrices is denoted by $M(m,n)=\mathcal{B}(\mathbb{C}^{n}\to\mathbb{C}^{m})$.
That of square matrices is abbreviated $M(n)=M(n,n)$. The adjoint and the
transpose of $A\in M(m,n)$ are respectively denoted by $A^{*},A^{\intercal}\in
M(n,m)$. For any vector $\xi\in\mathbb{R}^{d}$, the components are denoted by

$\begin{split}\xi=(\xi_{1},\ldots,\xi_{d})=(\xi_{1},\xi^{\perp})\in\mathbb{R}\times\mathbb{R}^{d-1}.\end{split}$
(1.12)

The $L^{2}$ inner product on $\mathbb{R}^{d}$ is denoted by

$\begin{split}{\langle
f|g\rangle}:=\int_{\mathbb{R}^{d}}f(x)\overline{g(x)}dx.\end{split}$ (1.13)

$W^{m,p}(\Omega)$ denotes the $L^{p}$-Sobolev space of regularity
$m\in\mathbb{R}$ on an open set $\Omega\subset\mathbb{R}^{d}$, and
$H^{s}(\mathbb{R}^{d})=W^{s,2}(\mathbb{R}^{d})$. $B^{s}_{p,q}(\mathbb{R}^{d})$
denotes the inhomogeneous $L^{p}$-Besov space of regularity $s\in\mathbb{R}$
and interpolation index $q\in[1,\infty]$. $\mathscr{D}(\Omega)$ is the space
of smooth test functions compactly supported on $\Omega$, and
$\mathscr{S}(\mathbb{R}^{d})$ denotes the Schwartz class.
$\mathscr{D}^{\prime}(\Omega)$ is the dual space or the space of
distributions, and $\mathscr{S}^{\prime}(\mathbb{R}^{d})$ is the space of
tempered distributions. All the function spaces are considered with complex
values.

For any $\mathrm{v}\in\mathbb{R}^{d}$ and $a,b\in\mathbb{R}$, we define
$\mathbb{R}^{d}_{<a}(\mathrm{v})$ and $\mathbb{R}^{d}_{(a,b)}(\mathrm{v})$ by
(1.2), and similarly $\mathbb{R}^{d}_{\leq a}(\mathrm{v})$,
$\mathbb{R}^{d}_{>a}(\mathrm{v})$, and $\mathbb{R}^{d}_{\geq a}(\mathrm{v})$.
When $\mathrm{v}=e_{1}:=(1,0,\ldots,0)$, it is omitted as
$\mathbb{R}^{d}_{<a}:=\mathbb{R}^{d}_{<a}(e_{1})$, etc. Similarly, for any
$Z\subset\mathscr{D}^{\prime}(\mathbb{R}^{d})$, we denote

$\begin{split}&Z_{<a}:=\\{f\in
Z\mid\operatorname{supp}f\subset\mathbb{R}^{d}_{<a}\\},\quad
Z_{(a,b)}:=\\{f\in
Z\mid\operatorname{supp}f\subset\mathbb{R}^{d}_{(a,b)}\\}\end{split}$ (1.14)

as well as for the other subscripts. More generally, for any
$O\subset\mathbb{R}^{d}$, we denote

$\begin{split}Z_{O}:=\\{f\in Z\mid\operatorname{supp}f\subset
O\\}.\end{split}$ (1.15)

When $O$ is an open set in $\mathbb{R}^{d}$, $\mathscr{D}_{O}$ is often
identified with $\mathscr{D}(O)$. Strictly speaking, they are homeomorphic by
the restriction to $O$ and the zero extension to the outside.

For cut-off arguments, we fix a $C^{\infty}$ function
$\chi:\mathbb{R}\to[0,1]$ satisfying

$\begin{split}t\leq 1\implies\chi(t)=1,\quad t\geq
2\implies\chi(t)=0.\end{split}$ (1.16)

For any $l>0$, define $\chi^{l}(t):=\chi(t/l)$ and $\chi_{l}(t):=1-\chi(t/l)$,
so that we have

$\begin{split}&\operatorname{supp}\chi^{l}\subset(-\infty,2l),\quad\operatorname{supp}\chi_{l}\subset(l,\infty).\end{split}$
(1.17)

The smooth cut-off operators in $\xi_{1}$ are defined by

$\begin{split}(X^{l}\varphi)(\xi):=\chi^{l}(\xi_{1})\varphi(\xi),\quad(X_{l}\varphi)(\xi):=\chi_{l}(\xi_{1})\varphi(\xi).\end{split}$
(1.18)

For mollifiers, we fix $\zeta\in C^{\infty}(\mathbb{R}^{d})$ satisfying

$\begin{split}0\leq\zeta(\xi)=\zeta(|\xi|),\quad\operatorname{supp}\zeta\subset
B_{\mathbb{R}^{d}}(0,1),\quad\|\zeta\|_{L^{1}(\mathbb{R}^{d})}=1,\end{split}$
(1.19)

and let $\zeta_{n}(\xi):=n^{d}\zeta(n\xi)\to\delta(\xi)$ in
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$ as $n\to\infty$.

For any $\delta\geq 0$ and any interval $I\subset[0,\infty)$, let

$\begin{split}&C^{\scaleto{\nearrow}{4pt}}:=\\{f:[0,\infty)\to[0,\infty)\mid\text{continuous
and non-decreasing}\\},\\\
&C^{\scaleto{\searrow}{4pt}}:=\\{f:[0,\infty)\to[0,\infty)\mid\text{continuous
and non-increasing}\\},\\\ &C^{\scaleto{\nearrow}{4pt}}_{I}:=\\{f\in
C^{\scaleto{\nearrow}{4pt}}\mid f([0,\infty))\subset I\\},\quad
C^{\scaleto{\searrow}{4pt}}_{I}:=\\{f\in C^{\scaleto{\searrow}{4pt}}\mid
f([0,\infty))\subset I\\},\\\ &C^{\scaleto{\nearrow}{4pt}}_{\delta}:=\\{f\in
C^{\scaleto{\nearrow}{4pt}}\mid f(0)=0,\
f([0,2\delta])\subset[0,\tfrac{1}{2}]\\},\\\
&C^{\scaleto{\nearrow}{4pt}}_{+}:=\\{f\in C^{\scaleto{\nearrow}{4pt}}\mid
f(0)=0,\ t>0\implies f(t)>0\\}.\end{split}$ (1.20)

$C^{\scaleto{\nearrow}{4pt}}_{\delta}$ is closed for $\max$, so it is a
directed set for the pointwise order. Note that
$C^{\scaleto{\nearrow}{4pt}}_{0}=\bigcup_{0<\delta\leq
1}C^{\scaleto{\nearrow}{4pt}}_{\delta}\supset
C^{\scaleto{\nearrow}{4pt}}_{+}$. We will use
$C^{\scaleto{\nearrow}{4pt}}_{0}$ as an index set for the inductive limit to
construct function spaces of distributions.

For topological vector spaces $X,Y$, the continuous embedding is denoted by
$X\hookrightarrow Y$. For any $m\in[0,\infty)$, the Banach space of bounded
$C^{m}$ functions on $\mathbb{R}^{d}$ is denoted by

$\begin{split}\mathcal{B}^{m}:=\\{\varphi\in
C^{m}(\mathbb{R}^{d};\mathbb{C})\mid\|\varphi\|_{m}:=\max_{|\alpha|\leq
m}\sup_{\xi\in\mathbb{R}^{d}}|\partial^{\alpha}\varphi(\xi)|<\infty\\}.\end{split}$
(1.21)

Note that $\mathcal{B}^{m}=\mathcal{B}^{m_{0}}$ for the largest
$m_{0}\in\mathbb{N}$ satisfying $m_{0}\leq m$. The trivial extension to non-
integer $m$ is for notational convenience. Let $\mathcal{B}^{0}_{0}$ be the
closure of $\mathscr{D}(\mathbb{R}^{d})$ in $\mathcal{B}^{0}$ and let
$\mathcal{M}:=(\mathcal{B}^{0}_{0})^{\prime}$ be its dual space. Each
$\mathfrak{m}\in\mathcal{M}$ may be regarded as a (bounded) complex-valued
Borel measure on $\mathbb{R}^{d}$, whose total variation is a finite Borel
measure on $\mathbb{R}^{d}$, denoted by $|\mathfrak{m}|$, satisfying

$\begin{split}|\mathfrak{m}|(\mathbb{R}^{d})/2\leq\|\mathfrak{m}\|_{(\mathcal{B}^{0}_{0})^{\prime}}=:\|\mathfrak{m}\|_{\mathcal{M}}\leq
2|\mathfrak{m}|(\mathbb{R}^{d}).\end{split}$ (1.22)

We also introduce the subspace vanishing on the left:

$\begin{split}\mathcal{M}_{+}:=\\{\mathfrak{m}\in\mathcal{M}\mid|\mathfrak{m}|(\mathbb{R}^{d}_{\leq
0})=0\\},\end{split}$ (1.23)

then the Dirichlet condition (X-i) with $\mathrm{v}=e_{1}$ is equivalent to
existence of $l>0$ such that $X^{l}F\in\mathcal{M}_{+}$. See also Definitions
2.1–2.2 for some other function spaces.

We define the Fourier transform for
$\varphi,\psi\in\mathscr{S}(\mathbb{R}^{d})$ by

$\begin{split}\hat{\varphi}(\xi)=\mathcal{F}\varphi(\xi):=\tfrac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\varphi(x)e^{-ix\xi}dx,\quad\mathcal{F}^{-1}\psi(x)=\int_{\mathbb{R}^{d}}\psi(\xi)e^{i\xi
x}d\xi\end{split}$ (1.24)

which is extended as usual to the tempered distributions
$F\in\mathscr{S}^{\prime}(\mathbb{R}^{d})$ by duality

$\begin{split}(\mathcal{F}F)(\varphi)=\hat{F}(\varphi):=F(\hat{\varphi}),\end{split}$
(1.25)

for $\varphi\in\mathscr{S}(\mathbb{R}^{d})$. The above definition of
$\mathcal{F}$ may be less conventional, but it simplifies the constant in the
convolution, namely, for any $f\in\mathscr{S}$ and any
$g\in\mathscr{S}^{\prime}$,

$\begin{split}\mathcal{F}(fg)=(\mathcal{F}f)*(\mathcal{F}g).\end{split}$
(1.26)

The Fourier inverse image of the entire space of distributions
$\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is defined as the dual
space of

$\begin{split}\mathcal{F}\mathscr{D}(\mathbb{R}^{d}):=\\{\mathcal{F}\varphi\mid\varphi\in\mathscr{D}(\mathbb{R}^{d})\\}\subset\mathscr{S}(\mathbb{R}^{d}),\end{split}$
(1.27)

which has the natural topology induced by $\mathcal{F}$ from
$\mathscr{D}(\mathbb{R}^{d})$, so that
$\mathcal{F}:\mathscr{D}(\mathbb{R}^{d})\to\mathcal{F}\mathscr{D}(\mathbb{R}^{d})$
is homeomorphic. These spaces have been studied from the early time of
distributions; $\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is the
same as $\mathbf{D}^{\prime}$ in [13] and $Z^{\prime}$ in [16, Chapter II].

The Fourier transform
$\mathcal{F}:\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})\to\mathscr{D}^{\prime}(\mathbb{R}^{d})$
is defined by (1.25) for $\varphi\in\mathscr{D}(\mathbb{R}^{d})$, satisfying
the same algebraic formulas as in $\mathscr{S}^{\prime}$. In particular,
(1.26) holds for $f\in\mathcal{F}\mathscr{D}$ and
$g\in\mathcal{F}^{-1}\mathscr{D}^{\prime}$. It is immediate from the
definition that
$\mathcal{F}:\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})\to\mathscr{D}^{\prime}(\mathbb{R}^{d})$
is homeomorphic, which justifies the notation
$\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$. More generally, for
any topological linear space
$X\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d})$, its Fourier inverse
image is denoted by

$\begin{split}\mathcal{F}^{-1}X:=\\{\mathcal{F}^{-1}f\mid f\in
X\\}\hookrightarrow\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d}),\end{split}$
(1.28)

with the natural topology induced by $\mathcal{F}^{-1}$, or the initial
topology for $\mathcal{F}$, so that $\mathcal{F}:\mathcal{F}^{-1}X\to X$ is
homeomorphic.

Since $\mathscr{D}\hookrightarrow\mathscr{S}$ is dense, so is
$\mathcal{F}\mathscr{D}\hookrightarrow\mathscr{S}$, hence we have
$\mathscr{S}^{\prime}(\mathbb{R}^{d})\hookrightarrow\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$,
which is also dense. Indeed, for the standard Littlewood-Paley decomposition
using a dyadic partition in the Fourier space
$\\{\varphi_{j}\\}_{j\in\mathbb{N}}\subset\mathcal{F}\mathscr{D}(\mathbb{R}^{d})$,
each piece $\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})\ni
F\mapsto\varphi_{j}*F\in\mathscr{S}^{\prime}(\mathbb{R}^{d})$ is continuous,
and $F=\sum_{j\in\mathbb{N}}\varphi_{j}*F$ on
$\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$. In such a way,
$\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is a natural space for
the analysis of PDE on $\mathbb{R}^{d}$, in contrast with
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$. In the latter, the Littlewood-Paley
decomposition is not well-defined, and both the heat and Schrödinger equations
are illposed in both time directions, whereas they are wellposed in
$\mathcal{F}^{-1}\mathscr{D}^{\prime}$ in both time directions.

Note that if $\mathcal{F}$ is replaced with $\mathcal{F}^{-1}$ in the above
definition, then we have
$\mathcal{F}\mathscr{D}=\mathcal{F}^{-1}\mathscr{D}\hookrightarrow\mathscr{S}$
including the topology, so we could naturally write
$\mathcal{F}^{-1}\mathscr{D}^{\prime}=\mathcal{F}\mathscr{D}^{\prime}$, but we
refrain from using this identification, to avoid possible ambiguity or
confusion. It is also worth noting that
$\mathcal{F}^{-1}\mathcal{X}\cap\mathcal{F}\mathcal{X}=\\{0\\}$ because of the
anisotropy.

##  2\. Spaces of distributions for the Fourier transform of solutions

Here we introduce some function spaces consisting of distributions in
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$, for the Fourier transform of the
solutions to the Cauchy problem (1.1). In the following, we fix
$\mathrm{v}:=e_{1}=(1,0,\ldots,0)$, without losing generality, since the other
case follows just by a linear transform of the spatial coordinates in
$\mathbb{R}^{d}$. The main purpose of this section is to define and
characterize the function space
$\mathcal{X}\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d})$.

######  Definition 2.1.

Let $d\in\mathbb{N}$. For $\delta,m\in[0,\infty)$, we define

$\begin{split}&\|\varphi\|_{m}^{+}:=\max_{|\alpha|\leq
m}\sup_{\xi\in\mathbb{R}^{d}_{\geq
0}}|\partial^{\alpha}\varphi(\xi)|,\quad\|\varphi\|_{m,\delta}:=\|X^{\delta}\varphi\|_{0}^{+}+\|X_{\delta}\varphi\|_{m}^{+},\\\
&\mathcal{B}^{m}_{+}:=\\{\varphi\in
C^{m}(\mathbb{R}^{d};\mathbb{C})\mid\|\varphi\|_{m}^{+}<\infty\\},\quad\mathcal{B}^{\infty}_{+}:=\bigcap_{m\in\mathbb{N}}\mathcal{B}^{m}_{+}.\end{split}$
(2.1)

For any $l,m\in[0,\infty)$, and $F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$, we
define

$\begin{split}\rho_{l}^{m}(F):=\sup\\{\mathop{\mathrm{Re}}F(\varphi)\mid\varphi\in\mathscr{D}_{<l},\
\|\varphi\|_{m}^{+}\leq 1\\}.\end{split}$ (2.2)

The functional $F\in\mathscr{D}^{\prime}$ may be extended to
$\mathcal{B}^{m}_{+<l}$ if $\rho^{m}_{l}(F)<\infty$, using that
$\mathscr{D}_{<l}\subset C^{m}_{<l}$ is dense for the $C^{m}$ topology (namely
for the local uniform convergence including the partial derivatives up to
order $m$). This extension is needed to define the convolution.

######  Lemma 2.1.

Let $F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$, $l>0$, $m\geq 0$ and
$\rho_{l}^{m}(F)<\infty$. Then $F\in\mathscr{D}^{\prime}_{\geq 0}$ and

$\begin{split}\lim_{R\to\infty}\rho_{l}^{m}(\chi_{R}(|\xi|)F)=0.\end{split}$
(2.3)

Hence $F$ is extended to a unique linear functional
$\mathcal{B}^{m}_{+<l}\to\mathbb{C}$ that is continuous with respect to the
$C^{m}$ topology. Moreover, for all $\varphi\in\mathcal{B}^{m}_{+<l}$,

$\begin{split}|F(\varphi)|\leq\rho_{l}^{m}(F)\|\varphi\|_{m}^{+}.\end{split}$
(2.4)

Note that the space $\mathcal{B}^{m}_{+}$ does not assume any decay or uniform
continuity for $|\xi|\to\infty$.

###### Proof.

For $\varphi\in\mathscr{D}_{<l}$, (2.4) is immediate from the definition. If
$\varphi\in\mathscr{D}_{<0}$, then $\|\varphi\|_{m}^{+}=0$ implies
$F(\varphi)=0$, hence $\operatorname{supp}F\subset\mathbb{R}^{d}_{\geq 0}$.
Next we prove (2.3). If it is false, then there exist sequences
$\varphi_{n}\in\mathscr{D}_{<l}$ and $R_{n}\to\infty$ such that
$\|\varphi_{n}\|_{m}^{+}\leq 1$, $F(\chi_{R_{n}}(|\xi|)\varphi_{n})=1$, and
$\operatorname{supp}\varphi_{n}\subset B_{\mathbb{R}^{d}}(0,R_{n+1})$. Let

$\begin{split}\psi_{N}:=\sum_{1\leq n\leq
N}\chi_{R_{n}}(|\xi|)\varphi_{n}\in\mathscr{D}_{<l}.\end{split}$ (2.5)

Since the summand is supported on $R_{n}<|\xi|<R_{n+1}$, which is mutually
disjoint, we have $\|\psi_{N}\|_{m}^{+}\lesssim 1$, whereas $F(\psi_{N})=N$,
contradicting (2.4) on $\mathscr{D}_{<l}$. Hence (2.3) is true. It implies
that $F|_{\mathscr{D}_{<l}}$ is continuous with respect to the $C^{m}$
topology. Then its unique extension to $\mathcal{B}^{m}_{+<l}$ is given by the
density of $\mathscr{D}_{<l}$. In fact, it may be defined by using mollifiers,
$F(\varphi):=\lim_{n\to\infty}F(\zeta_{n}*\chi^{n}(|\xi|)\varphi)$. ∎

######  Definition 2.2.

For $\mu,\nu\in C^{\scaleto{\nearrow}{4pt}}$, $l\in(0,\infty)$, and
$F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$, define

$\begin{split}&\rho^{\mu,\nu}_{l}(F):=\inf\\{B\geq 0\mid 0<t\leq
l\implies\rho^{\mu(t)}_{t}(F)\leq B\nu(t)\\},\quad(\inf\varnothing=\infty),\\\
&\rho^{\mu,\nu}(F):=\sup_{l>0}\rho^{\mu,\nu}_{l}(F),\quad\mathcal{X}^{\mu,\nu}:=\\{F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})\mid\rho^{\mu,\nu}(F)<\infty\\}.\end{split}$
(2.6)

$\mathcal{X}^{\mu,\nu}\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is
a Banach space. For any $\mu_{0},\mu_{1},\nu_{0},\nu_{1}\in
C^{\scaleto{\nearrow}{4pt}}$ satisfying
$(\mu_{0},\nu_{0})\leq(\mu_{1},\nu_{1})$ (pointwisely), we have
$\rho^{\mu_{1},\nu_{1}}(F)\leq\rho^{\mu_{0},\nu_{0}}(F)$ and
$\mathcal{X}^{\mu_{0},\nu_{0}}\hookrightarrow\mathcal{X}^{\mu_{1},\nu_{1}}$,
so that we may consider their union as the inductive limit in the category of
locally convex topological vector spaces, denoted by

$\begin{split}\mathcal{X}:=\bigcup_{\mu,\nu\in
C^{\scaleto{\nearrow}{4pt}}_{0}}\mathcal{X}^{\mu,\nu}\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d}).\end{split}$
(2.7)

The subscript $0$ in the index set $C^{\scaleto{\nearrow}{4pt}}_{0}$ requires
$\mu(0)=\nu(0)=0$, which is corresponding to the boundary condition (X-i) at
$\xi_{1}=0$, and crucial for our analysis of the PDE (1.1). When the direction
$e_{1}$ is replaced with a general
$\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$, it is defined by

$\begin{split}\mathcal{X}(\mathrm{v}):=\\{F(U_{\mathrm{v}}\xi)\mid
F\in\mathcal{X}\\},\end{split}$ (2.8)

where $U_{\mathrm{v}}$ is any invertible linear transform of $\mathbb{R}^{d}$
such that $U_{\mathrm{v}}\mathrm{v}=e_{1}$.

In this paper, we consider the inductive limits only in the category of
locally convex topological vector spaces. See e.g., [6] about basic properties
of them, though we will use only a few of them.
$\mathcal{X}\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d})$ ensures that
$\mathcal{X}$ is Hausdorff. Roughly speaking, $-\mu(l),\nu(l)$ are
respectively the regularity order and the size of $F$ on
$\mathbb{R}^{d}_{<l}$. By definition, we have for any
$F\in\mathcal{X}^{\mu,\nu}$, $l>0$ and $\varphi\in\mathscr{D}_{<l}$,

$\begin{split}|F(\varphi)|\leq\nu(l)\rho^{\mu,\nu}_{l}(F)\|\varphi\|_{\mu(l)}^{+},\end{split}$
(2.9)

where $\rho^{\mu,\nu}_{l}(F)$ is the optimal non-negative number satisfying
this inequality. Lemma 2.1 extends it to
$\varphi\in\mathcal{B}^{\mu(l)}_{+<l}$. If $\mu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ then $\mu\in
C^{\scaleto{\nearrow}{4pt}}_{\delta}$ for some $\delta>0$, $\mu(l)\leq 1/2<1$
for $l\leq 2\delta$, so that we may weaken the norm for $\varphi$ as

$\begin{split}|F(\varphi)|&\leq|F(X^{\delta}\varphi)|+|F(X_{\delta}\varphi)|\leq\nu(l)\rho^{\mu,\nu}_{l}(F)\|\varphi\|_{\mu(l),\delta}.\end{split}$
(2.10)

To change the size weight $\nu$, we have

######  Lemma 2.2.

For any $\kappa\in C^{\scaleto{\nearrow}{4pt}}$, $\mu,\nu\in
C^{\scaleto{\nearrow}{4pt}}$, $F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$ and
$l\in(0,\infty)$, we have

$\begin{split}\rho^{\mu,\nu}_{l}(F)\leq\kappa(l)\rho^{\mu,\kappa\nu}_{l}(F).\end{split}$
(2.11)

Let $\mu_{j},\nu_{j}\in C^{\scaleto{\nearrow}{4pt}}$ ($j=0,1$) and
$T\in\mathcal{B}(\mathcal{X}^{\mu_{1},\nu_{1}}\to\mathcal{X}^{\mu_{0},\nu_{0}})$
satisfy

$\begin{split}\rho^{\mu_{0},\nu_{0}}_{l}(TF)\leq
B(l)\rho^{\mu_{1},\nu_{1}}_{l}(F),\end{split}$ (2.12)

for some non-deacreasing $B:[0,\infty)\to[0,\infty)$, all
$F\in\mathcal{X}^{\mu_{1},\nu_{1}}$ and all $l\in(0,\infty)$. Then for any
bounded $\kappa\in C^{\scaleto{\nearrow}{4pt}}$, any
$F\in\mathcal{X}^{\mu_{1},\kappa\nu_{1}}$ and $l\in(0,\infty)$, we have

$\begin{split}\rho^{\mu_{0},\kappa\nu_{0}}_{l}(TF)\leq
B(l)\rho^{\mu_{1},\kappa\nu_{1}}_{l}(F).\end{split}$ (2.13)

###### Proof.

For any $0<t\leq l<\infty$ and $F\in\mathscr{D}^{\prime}$, we have

$\begin{split}\rho^{\mu(t)}_{t}(F)\leq\rho^{\mu,\kappa\nu}_{l}(F)\kappa(t)\nu(t)\leq\kappa(l)\rho^{\mu,\kappa\nu}_{l}(F)\nu(t),\end{split}$
(2.14)

which implies (2.11). Using (2.11), we have

$\begin{split}\rho^{\mu_{0}(t)}_{t}(TF)&\leq\nu_{0}(t)\rho^{\mu_{0},\nu_{0}}_{t}(TF)\leq\nu_{0}(t)B(t)\rho^{\mu_{1},\nu_{1}}_{t}(F)\leq\nu_{0}(t)B(l)\kappa(t)\rho^{\mu_{1},\kappa\nu_{1}}_{t}(F),\end{split}$
(2.15)

which implies (2.13). The boundedness of $\kappa$ was used merely to ensure
$F\in\mathcal{X}^{\mu_{1},\nu_{1}}$. ∎

In the analysis of the space $\mathcal{X}$, a basic tool is the smooth cut-off
$X_{\delta},X^{\delta}$ in $\xi_{1}$. Here we prepare a few estimates on them.
For the factor coming from the Leibniz rule, we define $M_{\delta}^{\chi}\in
C^{\scaleto{\nearrow}{4pt}}_{[1,\infty)}$ for all $\delta\in(0,1]$, first on
$[0,\tfrac{1}{2}]\cup\mathbb{N}$ by

$\begin{split}M_{\delta}^{\chi}(l)=\begin{cases}1+l,&(0\leq
l\leq\tfrac{1}{2}),\\\
\sum_{k=0}^{l}\tfrac{l!}{k!(l-k)!}\|\partial^{k}\chi_{\delta}\|_{L^{\infty}(\mathbb{R})},&(l\in\mathbb{N}),\end{cases}\end{split}$
(2.16)

and then extend it to $[0,\infty)$ as a continuous and increasing function,
e.g., the piecewise linear function.

######  Lemma 2.3.

Let $\delta,l\in(0,\infty)$, $m\in[0,\infty)$ and
$F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$. Then

$\begin{split}\rho^{m}_{l}(X_{\delta}F)\leq
M^{\chi}_{\delta}(m)\rho^{m}_{l}(F),\quad\rho^{m}_{l}(X^{\delta}F)\leq
M^{\chi}_{\delta}(m)\rho^{m}_{\min(l,2\delta)}(F).\end{split}$ (2.17)

Let $\mu,\nu,\tilde{\mu},\tilde{\nu}\in C^{\scaleto{\nearrow}{4pt}}$
satisfying $\mu\leq\tilde{\mu}$. Let $0<L,H,l<\infty$ and
$F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$.

  1. (1)

If $M^{\chi}_{L}(\mu)\nu\leq\tilde{\nu}$ on $[L,\infty)$, then
$\rho_{l}^{\tilde{\mu},\tilde{\nu}}(X_{L}F)\leq\rho_{l}^{\mu,\nu}(F)$.

  2. (2)

If $M^{\chi}_{H}(\mu)\nu\leq\tilde{\nu}$ on $(0,2H]$, then
$\rho_{l}^{\tilde{\mu},\tilde{\nu}}(X^{H}F)\leq\rho_{l}^{\mu,\nu}(F)$.

  3. (3)

If $M^{\chi}_{L}(\mu)M^{\chi}_{H}(\mu)\nu\leq\tilde{\nu}$ on $[L,2H]$, then
$\rho_{l}^{\tilde{\mu},\tilde{\nu}}(X_{L}X^{H}F)\leq\rho_{l}^{\mu,\nu}(F)$.

###### Proof.

By the Leibniz rule, we have for any $m\in[0,\infty)$ and
$\varphi\in\mathscr{D}$,

$\begin{split}\max(\|X^{\delta}\varphi\|_{m}^{+},\|X_{\delta}\varphi\|_{m}^{+})&\leq\sup_{|\alpha|\leq
m}\sum_{0\leq
k\leq\alpha_{1}}\tfrac{\alpha_{1}!}{(\alpha_{1}-k)!k!}\|\chi_{\delta}^{(k)}(\xi_{1})\|_{0}^{+}\|\partial^{\alpha-
ke_{1}}\varphi\|_{0}^{+}\\\ &\leq
M^{\chi}_{\delta}(m)\|\varphi\|_{m}^{+}.\end{split}$ (2.18)

Hence for any $\varphi\in\mathscr{D}_{<l}$ we have, using
$\operatorname{supp}X^{\delta}\varphi\subset\mathbb{R}^{d}_{<2\delta}$ as
well,

$\begin{split}&|X_{\delta}F(\varphi)|\leq\rho^{m}_{l}(F)\|X_{\delta}\varphi\|_{m}^{+}\leq\rho^{m}_{l}(F)M^{\chi}_{\delta}(m)\|\varphi\|_{m}^{+},\\\
&|X^{\delta}F(\varphi)|\leq\rho^{m}_{\min(l,2\delta)}(F)\|X^{\delta}\varphi\|_{m}^{+}\leq\rho^{m}_{\min(l,2\delta)}(F)M^{\chi}_{\delta}(m)\|\varphi\|_{m}^{+}.\end{split}$
(2.19)

from which the estimates in $\rho^{m}_{l}$ follow.

In the case of (1), we have $X_{L}F(\varphi)=0$ for $l\leq L$ and
$\varphi\in\mathscr{D}_{<l}$, while for $l\geq L$,

$\begin{split}\rho^{\tilde{\mu}(l)}_{l}(X_{L}F)\leq\rho^{\mu(l)}_{l}(X_{L}F)\leq
M^{\chi}_{L}(\mu(l))\rho_{l}^{\mu,\nu}(F)\nu(l)\leq\tilde{\nu}(l)\rho_{l}^{\mu,\nu}(F).\end{split}$
(2.20)

In the case of (2), we have similarly for all $l>0$ and
$l^{\prime}:=\min(l,2H)$

$\begin{split}\rho^{\tilde{\mu}(l)}_{l}(X^{H}F)\leq\rho^{\mu(l^{\prime})}_{l^{\prime}}(X^{H}F)&\leq
M^{\chi}_{H}(\mu(l^{\prime}))\rho^{\mu(l^{\prime})}_{l^{\prime}}(F)\leq\tilde{\nu}(l)\rho_{l}^{\mu,\nu}(F).\end{split}$
(2.21)

In the case of (3), we have similarly for $l\geq L$,

$\begin{split}\rho^{\tilde{\mu}(l)}_{l}(X_{L}X^{H}F)\leq
M^{\chi}_{L}(\mu(l^{\prime}))M^{\chi}_{H}(\mu(l^{\prime}))\rho^{\mu(l^{\prime})}_{l^{\prime}}(F)\leq\tilde{\nu}(l)\rho_{l}^{\mu,\nu}(F).\end{split}$
(2.22)

Thus we obtain the estimates in $\mathcal{X}^{\tilde{\mu},\tilde{\nu}}$ for
all the cases (1)–(3). ∎

###  2.1. Characterization of $\mathcal{X}$

Now we see that the definition of $\mathcal{X}$ in Section 1.1 is the same as
in this section.

######  Lemma 2.4.

For any $F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$, $F\in\mathcal{X}$ is
equivalent to (X-i)–(X-ii) with $\mathrm{v}=e_{1}$. Moreover, for any
$F\in\mathcal{X}$ and $\varphi\in\mathscr{D}(\mathbb{R}^{d})$, we have

$\begin{split}F(\varphi)=\lim_{l\to+0}F(X_{l}\varphi).\end{split}$ (2.23)

The above identity (2.23) allows us to ignore the test functions $\varphi$ on
$\mathbb{R}^{d}_{\leq 0}$. More precisely, let
$\mathscr{R}:\mathscr{D}^{\prime}(\mathbb{R}^{d})\to\mathscr{D}^{\prime}(\mathbb{R}^{d}_{>0})$
denote the restriction from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}_{>0}$, and for
any $G\in\mathscr{D}^{\prime}(\mathbb{R}^{d}_{>0})$ and
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$, let

$\begin{split}\mathscr{E}G(\varphi):=\lim_{l\to+0}G(X_{l}\varphi).\end{split}$
(2.24)

The limit does not converge for general
$G\in\mathscr{D}^{\prime}(\mathbb{R}^{d}_{>0})$, but when it does for all
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$, we have
$\mathscr{E}G\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$. (2.23) may be rewritten
as: $F=\mathscr{E}\mathscr{R}F$ for all $F\in\mathcal{X}$.

###### Proof.

For any $\delta>0$, $\mu\in C^{\scaleto{\nearrow}{4pt}}_{\delta}$, $\nu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ and any $F\in\mathcal{X}^{\mu,\nu}$, Lemma
2.1 implies $\operatorname{supp}F\subset\mathbb{R}^{d}_{\geq 0}$ and (2.9) for
any $l>0$ and $\varphi\in\mathcal{B}^{\mu(l)}_{+<l}$. In particular,

$\begin{split}|(X^{\delta}F)(\varphi)|\leq\nu(l)\rho^{\mu,\nu}(F)\|\varphi\|_{0},\end{split}$
(2.25)

for all $\varphi\in\mathcal{B}^{0}$, so we have $X^{\delta}F\in\mathcal{M}$
with the total variation satisfying

$\begin{split}|X^{\delta}F|(\mathbb{R}^{d}_{<l})\leq\nu(l)\rho^{\mu,\nu}(F)\to
0\quad(l\to+0).\end{split}$ (2.26)

Hence by the dominated convergence, $X^{\delta}F\in\mathcal{M}_{+}$, It also
implies $F(X^{l}\varphi)\to 0$ as $l\to+0$ for any $\varphi\in\mathscr{D}$,
hence $F(\varphi)=\mathscr{E}\mathscr{R}F(\varphi)$.

For larger $\xi_{1}$, it suffices to show for each $l>0$ that $X^{l}F\in
W^{-m,1}(\mathbb{R}^{d})$ for any $m\in\mathbb{N}$ satisfying
$m-1\geq\mu(2l)$. Henceforth we ignore the dependence of constants on $F,l,m$.
Define $\\{c_{\alpha}\\}_{|\alpha|\leq m}\subset\mathbb{Z}$ by

$\begin{split}(1-\Delta)^{m}=\sum_{|\alpha|\leq
m}c_{\alpha}\partial^{2\alpha},\end{split}$ (2.27)

and let
$\psi_{n}^{(\alpha)}:=(1-\Delta)^{-m}c_{\alpha}\partial^{\alpha}\zeta_{n}\in\mathscr{S}$
and $\psi_{n}^{(\alpha)\dagger}(\xi):=\psi_{n}^{(\alpha)}(-\xi)$. Then we have
$\sum_{|\alpha|\leq m}\partial^{\alpha}\psi_{n}^{(\alpha)}=\zeta_{n}$. Let
$f_{n}:=\zeta_{n}*X^{l}F$. Then $f_{n}\in C^{\infty}$, and we have from (2.9),

$\begin{split}|{\langle
f_{n}|\bar{\varphi}\rangle}|=|X^{l}F(\zeta_{n}*\varphi)|\lesssim\|\zeta_{n}*\varphi\|_{m-1}\lesssim
n^{m-1}\|\varphi\|_{0}\end{split}$ (2.28)

for all $\varphi\in\mathscr{D}$, which implies
$\|f_{n}\|_{L^{1}(\mathbb{R}^{d})}\lesssim n^{m-1}$. Similarly we have, for
$|\beta|\leq 1$,

$\begin{split}|{\langle\partial^{\beta}\psi_{n}^{(\alpha)}*f_{n}|\bar{\varphi}\rangle}|&=|X^{l}F(\partial^{\beta}\psi_{n}^{(\alpha)\dagger}*\zeta_{n}*\varphi)|\\\
&\lesssim\|\partial^{\beta}\psi_{n}^{(\alpha)\dagger}*\zeta_{n}*\varphi\|_{m-1}\lesssim\|\zeta_{n}*\varphi\|_{0}\lesssim\|\varphi\|_{0}.\end{split}$
(2.29)

Hence $\psi_{n}^{(\alpha)}*f_{n}$ is bounded in $W^{1,1}(\mathbb{R}^{d})$, and
so convergent along a subsequence in $L^{1}_{loc}(\mathbb{R}^{d})$ for
$|\alpha|\leq m$. Let $g_{\alpha}\in L^{1}(\mathbb{R}^{d})$ be the limit. Then

$\begin{split}X^{l}F(\varphi)=\lim_{n\to\infty}{\langle\zeta_{n}*f_{n}|\bar{\varphi}\rangle}&=\lim_{n\to\infty}\sum_{|\alpha|\leq
m}{\langle\psi_{n}^{(\alpha)}*f_{n}|(-\partial)^{\alpha}\bar{\varphi}\rangle}\\\
&=\sum_{|\alpha|\leq m}{\langle
g_{\alpha}|(-\partial)^{\alpha}\bar{\varphi}\rangle}.\end{split}$ (2.30)

Hence $X^{l}F=\sum_{|\alpha|\leq m}\partial^{\alpha}g_{\alpha}\in
W^{-m,1}(\mathbb{R}^{d})$. Thus (X-i)–(X-ii) are satisfied.

Conversely, suppose that $F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$ satisfies
(X-i)–(X-ii). By (X-i), we have $|F(\varphi)|\leq
2|\mathfrak{m}|(\mathbb{R}^{d}_{<l})\|\varphi\|_{0}^{+}$ for
$\varphi\in\mathscr{D}_{<l}$ and small $l>0$, and by (X-ii),
$|F(\varphi)|\leq\|F|_{\mathbb{R}^{d}_{<l}}\|_{W^{-m,1}}\|\varphi\|_{m}^{+}$
for any $l>0$, $\varphi\in\mathscr{D}_{(0,l)}$ and some $m=m(l)\in\mathbb{N}$.
Since
$|\mathfrak{m}|(\mathbb{R}^{d}_{<l})\to|\mathfrak{m}|(\mathbb{R}^{d}_{\leq
0})=0$ as $l\to+0$, we may find $\mu,\nu\in C^{\scaleto{\nearrow}{4pt}}_{0}$
such that (2.9) holds for all $l>0$ and $\varphi\in\mathscr{D}_{<l}$. Then
$F\in\mathcal{X}^{\mu,\nu}\subset\mathcal{X}$. ∎

###  2.2. Local basis in $\mathcal{X}$

Here we characterize the topology of $\mathcal{X}$ by constructing a local
basis at $0$. For any $m\in\mathbb{N}_{0}$ and $\nu\in
C_{+}^{\scaleto{\nearrow}{4pt}}$, let

$\begin{split}\mathscr{B}^{m}(\nu):=\\{F\in\mathcal{X}\mid\sup_{l>0}\rho^{m}_{l}(F)/\nu(l)\leq
1\\}.\end{split}$ (2.31)

For any $R_{0},R_{1}\in(0,\infty)$ and any
$N=\\{N_{m}\\}_{m\in\mathbb{N}}\subset C_{+}^{\scaleto{\nearrow}{4pt}}$, let

$\begin{split}&\mathscr{B}(N):=\bigcup_{M\in\mathbb{N}}\sum_{1\leq m\leq
M}\mathscr{B}^{m}(N_{m}),\\\
&\mathscr{B}(R_{0},R_{1},N):=\bar{B}_{\mathcal{M}_{+}}(0,R_{0})+\mathcal{X}_{\geq
R_{1}}+\mathscr{B}(N).\end{split}$ (2.32)

Since $\mathscr{B}(R_{0},R_{1},N)$ is the increasing limit of the sum of balls
and a subspace, it is an absolutely convex subset of $\mathcal{X}$. Let
$\mathscr{B}_{\mathcal{X}}$ be the collection of all such
$\mathscr{B}(R_{0},R_{1},N)$.

######  Proposition 2.5.

$\mathscr{B}_{\mathcal{X}}$ is a local basis at $0$ of $\mathcal{X}$. In other
words, a subset of $\mathcal{X}$ is a neighborhood of $0\in\mathcal{X}$ iff it
contains a member of $\mathscr{B}_{\mathcal{X}}$.

###### Proof.

First we show that every
$\mathscr{B}(R_{0},R_{1},N)\in\mathscr{B}_{\mathcal{X}}$ is a
$0$-neighborhood. Since it is absolutely convex, by definition of the
inductive limit $\mathcal{X}=\bigcup_{\mu,\nu}\mathcal{X}^{\mu,\nu}$, it is
equivalent that its intersection with $\mathcal{X}^{\mu,\nu}$ is a
$0$-neighborhood of $\mathcal{X}^{\mu,\nu}$ for every $\mu,\nu\in
C_{0}^{\scaleto{\nearrow}{4pt}}$.

Let $\delta\in(0,R_{1}/4)$, $\mu\in C_{\delta}^{\scaleto{\nearrow}{4pt}}$,
$\nu\in C_{0}^{\scaleto{\nearrow}{4pt}}$ and $F\in\mathcal{X}^{\mu,\nu}$. Let
$F_{0}:=X^{\delta}F$, $F_{2}:=X_{R_{1}}F$ and $F_{1}:=X_{\delta}X^{R_{1}}F$.
Then $F_{0}\in\mathcal{M}_{+}$ with
$\|F_{0}\|_{\mathcal{M}}\leq\nu(2\delta)\rho^{\mu,\nu}(F)$,
$F_{1}\in\mathcal{X}_{(\delta,2R_{1})}$ and $F_{2}\in\mathcal{X}_{\geq
R_{1}}$. By Lemma 2.3, $X_{\delta}X^{R_{1}}$ is bounded
$\mathcal{X}^{\mu,\nu}\to\mathcal{X}^{m,N_{m}}$ for $m:=\mu(2R_{1})$. Hence if
$\rho^{\mu,\nu}(F)$ is small enough then $\|F_{0}\|_{\mathcal{M}}\leq R_{0}$,
$\rho^{m,N_{m}}(F_{1})\leq 1$, and $F\in\mathscr{B}(R_{0},R_{1},N)$. So
$\mathscr{B}_{\mathcal{X}}$ consists of $0$-neighborhoods in $\mathcal{X}$.

Next let $U\subset\mathcal{X}$ be an absolutely convex $0$-neighborhood. If
$\bar{B}_{\mathcal{M}_{+}}(0,R_{0})\not\subset U$ for all $R_{0}>0$, then
there is a sequence $F_{n}\in\mathcal{M}_{+}\setminus U$ such that
$\|F_{n}\|_{\mathcal{M}}<1/n^{2}$. Let
$\nu_{n}(l):=\|X^{2l}F_{n}\|_{\mathcal{M}}\in C^{\scaleto{\nearrow}{4pt}}_{0}$
and $\nu:=\sup_{n\in\mathbb{N}}n\nu_{n}$. Then $n\nu_{n}\leq
n\|F_{n}\|_{\mathcal{M}}<1/n$ implies that $\nu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$, while
$\rho^{0}_{l}(F_{n})\leq\nu_{n}(l)\leq\tfrac{1}{n}\nu(l)$ for all $l>0$
implies $\rho^{0,\nu}(F_{n})\leq\tfrac{1}{n}\to 0$, so $F_{n}\to 0$ in
$\mathcal{X}^{0,\nu}\hookrightarrow\mathcal{X}$, contradicting $F_{n}\not\in
U$. Hence $\bar{B}_{\mathcal{M}_{+}}(0,R_{0})\subset U$ for some $R_{0}>0$.

Next, if there is no $R_{1}<\infty$ such that $\mathcal{X}_{\geq R_{1}}\subset
U$, then there are sequences $\mu_{n},\nu_{n}\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ and
$F_{n}\in\mathcal{X}^{\mu_{n},\nu_{n}}_{>n}\setminus U$. Define $\mu,\nu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ first at each $l\in\mathbb{N}$ by

$\begin{split}\mu(l)=\max_{1\leq n\leq l}\mu_{n}(l+1),\quad\nu(l)=\max_{1\leq
n\leq l}n\rho^{\mu_{n},\nu_{n}}(F_{n})\nu_{n}(l+1),\end{split}$ (2.33)

and then extend them to $[0,\infty)$. Then we have $\mu_{n}(l)\leq\mu(l)$ and
$n\rho^{\mu_{n},\nu_{n}}(F_{n})\nu_{n}(l)\leq\nu(l)$ for all $n\in\mathbb{N}$
and $l\geq n$, so
$\rho^{\mu(l)}_{l}(F_{n})\leq\nu_{n}(l)\rho^{\mu_{n},\nu_{n}}(F_{n})\leq\tfrac{1}{n}\nu(l)$,
while $\operatorname{supp}F_{n}$ implies $\rho^{\mu(l)}_{l}(F_{n})=0$ for
$l\leq n$. Hence $\rho^{\mu,\nu}(F_{n})\leq\tfrac{1}{n}\to 0$, so $F_{n}\to 0$
in $\mathcal{X}^{\mu,\nu}\hookrightarrow\mathcal{X}$, contradicting
$F_{n}\not\in U$. Hence $\mathcal{X}_{\geq R_{1}}\subset U$ for some
$R_{1}\in[1,\infty)$.

It remains to consider the $\mathscr{B}^{m}(N_{m})$ part. Let
$R_{j}:=2^{-j+1}R_{1}$ and $\nu_{j}(l):=\tfrac{l}{2R_{j}}\in
C_{+}^{\scaleto{\nearrow}{4pt}}$ for $j\in\mathbb{N}$. Since
$U\subset\mathcal{X}$ is a $0$-neighborhood, there is
$\\{\varepsilon_{m,j}\\}_{m,j\in\mathbb{N}}\subset(0,\infty)$ such that
$\bar{B}_{\mathcal{X}^{m\nu_{j},\nu_{j}}}(0,\varepsilon_{m,j})\subset U$. Take
any $\\{N_{m}\\}_{m\in\mathbb{N}}\subset C_{+}^{\scaleto{\nearrow}{4pt}}$
satisfying for all $m,j\in\mathbb{N}$

$\begin{split}N_{m}(2R_{j})\leq[M^{\chi}_{R_{j+1}}(m)M^{\chi}_{R_{j}}(m)]^{-1}2^{-j-3}\varepsilon_{m,j}.\end{split}$
(2.34)

For any $m,j\in\mathbb{N}$ and $F\in\mathscr{B}^{m}(N_{m})$, let
$F_{j}:=X_{R_{j+1}}X^{R_{j}}F$. Then Lemma 2.3 implies

$\begin{split}\rho^{m\nu_{j},\nu_{j}}(F_{j})\leq\nu_{j}(R_{j+1})^{-1}2^{-j-3}\varepsilon_{m,j}\rho^{m,N_{m}}(F)\leq
2^{-j-1}\varepsilon_{m,j},\end{split}$ (2.35)

so $F_{j}\in 2^{-j-1}U$, while $F\in\mathcal{X}$ implies
$\|X^{R_{J}}F\|_{\mathcal{M}}<R_{0}/4$ for some $J\in\mathbb{N}$. Then

$\begin{split}F=X^{R_{J}}F+X_{R_{1}}F+\sum_{1\leq j<J}F_{j}\in
U/4+U/4+\sum_{1\leq j<J}2^{-j-1}U\subset U,\end{split}$ (2.36)

by the absolute convexity of $U$, and similarly
$\mathscr{B}(R_{0}/4,R_{1},\\{2^{-m-1}N_{m}\\}_{m\in\mathbb{N}})\subset U$.
Hence $\mathscr{B}_{\mathcal{X}}$ is a local basis of $\mathcal{X}$. ∎

The local basis $\mathscr{B}_{\mathcal{X}}$ of $\mathcal{X}$ implies the
following embeddings. For $\delta>0$ and $m\in\mathbb{N}$, define the spaces
of distributions

$\begin{split}&\mathcal{X}_{L}:=\\{F\in\mathscr{D}^{\prime}_{\geq
0}\mid\\{X^{l}F\\}_{l>0}\subset L^{1}(\mathbb{R}^{d})\\},\\\
&\mathcal{X}_{\mathcal{M}}:=\\{F\in\mathscr{D}^{\prime}_{\geq
0}\mid\\{X^{l}F\\}_{l>0}\subset\mathcal{M}_{+}\\},\\\
&W^{-m,1}_{\delta}:=\\{F\in\mathscr{D}^{\prime}_{\geq\delta}\mid\\{X^{l}F\\}_{l>0}\subset
W^{-m,1}(\mathbb{R}^{d})\\},\end{split}$ (2.37)

equipped with the initial topology for the family of mappings
$\\{X^{l}\\}_{l>0}$. Then

$\begin{split}L^{1}_{\geq 0}\hookrightarrow\left\\{\begin{aligned}
&\mathcal{M}_{+}\\\
&\mathcal{X}_{L}\end{aligned}\right\\}\hookrightarrow\mathcal{X}_{\mathcal{M}}\hookrightarrow\mathcal{X},\quad
W^{-m,1}_{\delta}\hookrightarrow\mathcal{X}.\end{split}$ (2.38)

Indeed, $L^{1}(\mathbb{R}^{d})\subset\mathcal{M}$ is a closed subspace with
the same norm. If $F\in L^{1}(\mathbb{R}^{d})$ satisfies
$\operatorname{supp}F\subset\mathbb{R}^{d}_{\geq 0}$, then
$F\in\mathcal{M}_{+}$ follows from the dominated convergence, hence
$L^{1}_{\geq 0}\hookrightarrow\mathcal{M}_{+}$ and
$\mathcal{X}_{L}\hookrightarrow\mathcal{X}_{\mathcal{M}}$. $L^{1}_{\geq
0}\hookrightarrow\mathcal{X}_{L}$ and
$\mathcal{M}_{+}\hookrightarrow\mathcal{X}_{\mathcal{M}}$ are obvious.
$\mathcal{X}_{\mathcal{M}}\hookrightarrow\mathcal{X}$ because
$\mathscr{B}(R_{0},R_{1},N)\supset(X^{R_{1}})^{-1}B_{\mathcal{M}_{+}}(0,R_{0})$,
and $W^{-m,1}_{\delta}\hookrightarrow\mathcal{X}$ because

$\begin{split}\mathscr{B}(R_{0},R_{1},N)\supset\\{F\in\mathscr{D}^{\prime}_{\geq\delta}\mid\|X^{R_{1}}F\|_{W^{-m,1}}<N_{m}(\delta)\\},\end{split}$
(2.39)

which follows from the duality
$|G(\varphi)|\leq\|G\|_{W^{-m,1}}\|\varphi\|_{m}^{+}$ for
$G\in\mathscr{D}^{\prime}_{\geq\delta}$ and $\varphi\in\mathscr{D}$.

To justify very weak solutions in $\mathcal{F}^{-1}\mathcal{X}$, the density
of nice functions is crucial, for which the flexibility of weights $\mu,\nu$
is necessary. In fact, $\mathscr{D}(\mathbb{R}^{d})$ is not dense in any
$\mathcal{X}^{\mu,\nu}$, but we may construct approximate identities
$I_{n}:\mathcal{X}\to\mathscr{D}_{>0}$ as follows.

######  Proposition 2.6.

Let $\\{c_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ and
$\\{r_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ satisfy $|c_{n}|+r_{n}\to 0$
as $n\to\infty$ and $c_{n}\in\mathbb{R}^{d}_{>r_{n}}$ for all
$n\in\mathbb{N}$. Let $\varphi_{n}(\xi):=r_{n}^{-d}\zeta(r_{n}^{-1}(\xi-
c_{n}))$, $\psi_{n}(\xi):=\chi(r_{n}|\xi|)$ and
$I_{n}F:=\varphi_{n}*(\psi_{n}F)$. Then $I_{n}:\mathcal{X}\to\mathscr{D}_{>0}$
is continuous for each $n\in\mathbb{N}$, and $I_{n}F\to F$ in $\mathcal{X}$ as
$n\to\infty$ for all $F\in\mathcal{X}$. For any
$\mathcal{R}\subset\mathbb{R}^{d}_{\geq 0}$ such that
$\mathcal{R}+\mathcal{R}\subset\mathcal{R}$ and the closure of the interior
contains $0\in\mathbb{R}^{d}$, we may choose $c_{n},r_{n}$ such that $I_{n}$
preserves $\operatorname{supp}F\subset\mathcal{R}$.

Those $I_{n}$ are just the standard mollification moving the support to the
right.

###### Proof.

By definition, $\varphi_{n},\psi_{n}\in\mathscr{D}(\mathbb{R}^{d})$ and
$\operatorname{supp}\varphi_{n}\subset
B_{\mathbb{R}^{d}}(c_{n},r_{n})\subset\mathbb{R}^{d}_{>0}$. Hence
$F\in\mathscr{D}^{\prime}_{\geq 0}$ implies $I_{n}F\in\mathscr{D}_{>0}$. If
the closure of the interior of $\mathcal{R}$ contains $0$, then we may choose
$c_{n},r_{n}$ such that $B_{\mathbb{R}^{d}}(c_{n},r_{n})\subset\mathcal{R}$.
Then for $\operatorname{supp}F\subset\mathcal{R}$, we have
$\operatorname{supp}(I_{n}F)\subset\operatorname{supp}\varphi_{n}+\operatorname{supp}F\subset\mathcal{R}+\mathcal{R}\subset\mathcal{R}$.
The continuity of $I_{n}$ follows from that of
$I_{n}:\mathscr{D}^{\prime}(\mathbb{R}^{d})\to\mathscr{D}(\mathbb{R}^{d})$,
which is well-known.

It remains to prove $I_{n}F\to F$ in $\mathcal{X}$. Take any $F\in\mathcal{X}$
and $\mathscr{B}(R_{0},R_{1},N)\in\mathscr{B}_{\mathcal{X}}$. By Lemma 2.4,
there exist $\delta\in(0,R_{1}/2)$ and $m\in\mathbb{N}$ such that

$\begin{split}X^{\delta}F\in\mathcal{M}_{+},\quad\|X^{\delta}F\|_{\mathcal{M}}<\tfrac{1}{8}R_{0},\quad
X^{R_{1}}F\in W^{-m,1}(\mathbb{R}^{d}).\end{split}$ (2.40)

Let $F=F^{0}+F^{1}+F^{2}$ with $F^{0}:=X^{\delta}F$ and $F^{2}:=X_{R_{1}}F$.
Then for all $n\in\mathbb{N}$ we have

$\begin{split}I_{n}F^{0}\in\mathcal{M}_{+},\quad\|I_{n}F^{0}\|_{\mathcal{M}}\leq\|F^{0}\|_{\mathcal{M}}<\tfrac{1}{8}R_{0},\quad\operatorname{supp}I_{n}F^{2}\subset\mathbb{R}^{d}_{>R_{1}},\end{split}$
(2.41)

which implies
$I_{n}(F^{0}+F^{2})-(F^{0}+F^{2})\in\tfrac{1}{8}(U+U+U+U)\subset\tfrac{1}{2}U$.
On the other hand, $F^{1}\in W^{-m,1}(\mathbb{R}^{d})$ and
$\operatorname{supp}F^{1}\subset\mathbb{R}^{d}_{\geq\delta}$ imply that
$\operatorname{supp}I_{n}F^{1}\subset\mathbb{R}^{d}_{\geq\delta}$ and
$I_{n}F^{1}\to F^{1}$ in $W^{-m,1}(\mathbb{R}^{d})$, hence $I_{n}F^{1}\to
F^{1}$ in $W^{-m,1}_{\delta}\hookrightarrow\mathcal{X}$ by (2.38). In
particular $I_{n}F^{1}-F^{1}\in\tfrac{1}{2}U$ for large $n$, then $I_{n}F-F\in
U$. Hence $I_{n}F\to F$ in $\mathcal{X}$. ∎

###  2.3. Bounded sets in $\mathcal{X}$

Here we characterize the bounded subsets of $\mathcal{X}$. First recall that
the inductive limit $X=\bigcup_{n\in\mathbb{N}}X_{n}$ of a sequence of normed
spaces $X_{1}\hookrightarrow X_{2}\hookrightarrow\cdots$ is said to be regular
if every bounded set $B\subset X$ is a bounded set of some $X_{n}$. It is not
always true, but [15] gives some sufficient conditions.

For any $m\in\mathbb{N}_{0}$, we define the negative Sobolev spaces of
measure-type by

$\begin{split}&\mathcal{W}^{m}:=\\{F\in\mathscr{D}^{\prime}(\mathbb{R}^{d}):\|F\|_{\mathcal{W}^{m}}<\infty\\},\quad\mathcal{W}^{\infty}:=\bigcup_{m\in\mathbb{N}_{0}}\mathcal{W}^{m}\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d}),\\\
&\|F\|_{\mathcal{W}^{m}}:=\sup\\{\mathop{\mathrm{Re}}F(\varphi):\varphi\in\mathscr{D}(\mathbb{R}^{d}),\
\|\varphi\|_{m}\leq 1\\}.\end{split}$ (2.42)

Then $\mathcal{W}^{m}\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is a
Banach space, whose closed unit ball $\bar{B}_{\mathcal{W}^{m}}(0,1)$ is
closed in $\mathscr{D}^{\prime}(\mathbb{R}^{d})$ because of the definition by
test functions, and $\mathscr{D}^{\prime}(\mathbb{R}^{d})$ is reflexive. Hence
[15, Corollary 1] implies that the inductive limit $\mathcal{W}^{\infty}$ is
regular.

######  Proposition 2.7.

For any subset $B\subset\mathcal{X}$, its boundedness is equivalent to
existence of $\mu\in C_{0}^{\scaleto{\nearrow}{4pt}}$ and $\nu\in
C^{\scaleto{\nearrow}{4pt}}$ such that $\rho^{\mu(l)}_{l}(F)\leq\nu(l)$ for
all $F\in B$ and $l>0$.

######  Remark 2.1.

If $\nu\in C^{\scaleto{\nearrow}{4pt}}_{0}$ (namely $\nu(0)=0$) then $B$ is
bounded in $\mathcal{X}^{\mu,\nu}$, but we may not choose $\nu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ in general, which is easily observed already
in the subspace $L^{1}_{\geq
0}\hookrightarrow\mathcal{M}_{+}\hookrightarrow\mathcal{X}$: For any non-zero
$f\in\mathscr{D}_{>0}$, the concentrating sequence of mass
$\\{nf(n\xi_{1},\xi^{\perp})\\}_{n\in\mathbb{N}}$ is bounded in $L^{1}_{\geq
0}$, but in none of $\mathcal{X}^{\mu,\nu}$. It is convergent in
$\mathscr{D}^{\prime}_{\geq 0}$ to a measure on the boundary $\xi_{1}=0$,
which is not in $\mathcal{X}$. In short,
$\mathcal{X}=\bigcup_{\mu,\nu}\mathcal{X}^{\mu,\nu}$ fails to be regular
because of the decay condition (X-i) at $\xi_{1}=0$.

###### Proof.

Let $B\subset\mathcal{X}$ be bounded. For every $l\in\mathbb{N}$,
$X^{l}:\mathcal{X}\to\mathcal{W}^{\infty}$ is continuous, and
$\mathcal{W}^{\infty}$ is the regular inductive limit, so there exist some
$m_{l},n_{l}\in\mathbb{N}$ such that

$\begin{split}X^{l}(B)\subset B_{\mathcal{W}^{m_{l}}}(0,n_{l}).\end{split}$
(2.43)

Next we show that $X^{l}B$ is bounded in $\mathcal{M}_{+}$ for some $l>0$.
Otherwise, there exist sequences $F_{n}\in B$, $(0,1)\ni l_{n}\to 0$ and
$\varphi_{n}\in\mathscr{D}_{(0,l_{n})}$ such that $\|\varphi_{n}\|_{0}\leq
2^{-n}$ and $F_{n}(\varphi_{n})>2$. Then for each $m\in\mathbb{N}$, there
exists $N_{m}\in C_{+}^{\scaleto{\nearrow}{4pt}}$ satisfying
$N_{m}(l_{n})\|\varphi_{n}\|_{m}\leq 2^{-m-n}$ for all $n\in\mathbb{N}$, which
is possible because of $l_{n}\to 0$. For any $n\in\mathbb{N}$,
$G_{0}\in\bar{B}_{\mathcal{M}_{+}}(0,1)$, $G_{1}\in\mathcal{X}_{\geq 1}$ and
$G_{2}\in\mathscr{B}(N)$, we have

$\begin{split}&|G_{0}(\varphi_{n})|\leq 2^{-n},\quad
G_{1}(\varphi_{n})=0,\quad|G_{2}(\varphi_{n})|\leq\sum_{m\in\mathbb{N}}N_{m}(l_{n})\|\varphi_{n}\|_{m}\leq
2^{-n}.\end{split}$ (2.44)

Hence $|G(\varphi_{n})|\leq 2^{1-n}$ for any $G\in\mathscr{B}(1,1,N)$, so
$F_{n}\not\in 2^{n}\mathscr{B}(1,1,N)$, contradicting the boundedness of
$\\{F_{n}\\}_{n\in\mathbb{N}}\subset B$ in $\mathcal{X}$. Hence there exist
$\delta,C_{*}\in(0,\infty)$ such that $X^{\delta}B\subset
B_{\mathcal{M}_{+}}(0,C_{*})$. For $l>0$ and $\varphi\in\mathscr{D}_{<l}$, we
have, using (2.43) and Lemma 2.3,

$\begin{split}|F(\varphi)|\leq|F(X^{\delta}\varphi)|+|F(X_{\delta}\varphi)|&\leq
C_{*}\|\varphi\|_{0}^{+}+n_{2l}M^{\chi}_{\delta}(m_{k})\|\varphi\|_{m_{2l}}^{+}.\end{split}$
(2.45)

There exist $\mu\in C_{\delta}^{\scaleto{\nearrow}{4pt}}$ and $\nu\in
C^{\scaleto{\nearrow}{4pt}}$ satisfying $\nu(0)\geq C_{*}$ and

$\begin{split}\mu(l)\geq m_{2k},\quad\nu(l)\geq
C_{*}+M^{\chi}_{\delta}(m_{k})n_{2k}\end{split}$ (2.46)

for all $k\in\mathbb{N}_{\geq 2}$ and $\max(\delta,k-1)<l\leq k$. Then we have
$\rho^{\mu(l)}_{l}(F)\leq\nu(l)$ for all $F\in B$.

Conversely, suppose that such $\mu\in C_{\delta}^{\scaleto{\nearrow}{4pt}}$
and $\nu\in C^{\scaleto{\nearrow}{4pt}}$ exist. Let $\delta>0$ such that
$\mu(2\delta)<1$. There is $\tilde{\nu}\in C^{\scaleto{\nearrow}{4pt}}_{0}$
satisfying $M^{\chi}_{\delta}(\mu(l))\nu(l)\leq\tilde{\nu}(l)$ for
$l\geq\delta$. Then Lemma 2.3 implies that $X_{\delta}B$ is bounded in
$\mathcal{X}^{\mu,\tilde{\nu}}\hookrightarrow\mathcal{X}$, and that
$X^{\delta}B$ is bounded in $\mathcal{M}_{+}$, so it is absorbed by any
$B_{\mathcal{M}_{+}}(0,R_{0})\subset\mathscr{B}(R_{0},R_{1},N)\in\mathscr{B}_{\mathcal{X}}$,
meaning the boundedness in $\mathcal{X}$. Hence $B=X^{\delta}B+X_{\delta}B$ is
also bounded in $\mathcal{X}$. ∎

###  2.4. Compact sets in $\mathcal{X}$

Here we characterize compact subsets of $\mathcal{X}$. First we recall variant
notions of compactness. A topological space $X$ is sequentially compact iff
every sequence has a convergent subsequence. $X$ is countably compact iff
every sequence has an accumulation point. Countable compactness is weaker both
than compactness and than sequential compactness, but there is no order in
general between the latter two.

######  Lemma 2.8.

Let $\tau_{1},\tau_{2}$ be topologies on a set $X$. Let $\tau_{1}$ be
countably compact and stronger than $\tau_{2}$. Let $\tau_{2}$ be Hausdorff
and sequentially compact. Then so is $\tau_{1}$.

In particular, in any topological vector space embedded into
$\mathscr{D}^{\prime}$, countably compact subsets are sequentially compact.

###### Proof.

The Hausdorff property transfers to the stronger topology, so it suffices to
show the sequential compactness of $\tau_{1}$. For any sequence $S_{0}$, there
is a subsequence $S_{1}\subset S_{0}$ convergent to some $x_{\infty}$ in
$\tau_{2}$. If $S_{1}$ is not convergent to $x_{\infty}$ in $\tau_{1}$, then
there is a $\tau_{1}$-neighborhood $U$ of $x_{\infty}$ such that
$S_{2}:=S_{1}\setminus U$ is a subsequence, which has a
$\tau_{1}$-accumulation point $x_{*}$. Since $S_{2}\cap U=\varnothing$, we
have $x_{*}\not=x_{\infty}$. Then $S_{2}\to x_{\infty}$ and accumulating at
$x_{*}\not=x_{\infty}$ in $\tau_{2}$, contradicting the Hausdorff. Hence
$S_{1}\to x_{\infty}$ in $\tau_{1}$. ∎

Next we consider the compactness in $\mathcal{W}^{\infty}$.

######  Lemma 2.9.

$K\subset\mathcal{W}^{\infty}$ is countably compact iff it is compact in some
$\mathcal{W}^{m}$.

###### Proof.

Since $\mathcal{W}^{m}\hookrightarrow\mathcal{W}^{\infty}$ by definition, it
suffices to show the necessity. For the radial mollifiers
$\zeta_{n}=n^{d}\zeta(n\xi)$, we have

$\begin{split}&|\zeta_{n}*F(\varphi)|\leq\|F\|_{\mathcal{W}^{m}}\|\zeta_{n}*\varphi\|_{m}\leq\|F\|_{\mathcal{W}^{m}}n^{m}\|\zeta\|_{W^{1,m}}\|\varphi\|_{0},\\\
&|(F-\zeta_{n}*F)(\varphi)|\leq\|F\|_{\mathcal{W}^{m}}\|\varphi-\zeta_{n}*\varphi\|_{m}\leq\|F\|_{\mathcal{W}^{m}}n^{-1}\|\varphi\|_{m+1},\end{split}$
(2.47)

for all $m\in\mathbb{N}_{0}$, $\varphi\in\mathscr{D}$, $F\in\mathcal{W}^{m}$
and $n\in\mathbb{N}$. In particular,
$\zeta_{n}*:\mathcal{W}^{m}\to\mathcal{W}^{0}$ is bounded for all
$m\in\mathbb{N}_{0}$, so is
$\zeta_{n}*:\mathcal{W}^{\infty}\to\mathcal{W}^{0}$ by definition of the
inductive limit.

Let $K\subset\mathcal{W}^{\infty}$ be countably compact. Since
$\mathcal{W}^{\infty}\hookrightarrow\mathscr{D}^{\prime}$ and bounded subsets
of $\mathscr{D}^{\prime}$ are precompact, the above lemma implies that $K$ is
sequentially compact. Also, the regularity of $\mathcal{W}^{\infty}$ yields
some $m\in\mathbb{N}$ and $R<\infty$ such that $K\subset
B:=\bar{B}_{\mathcal{W}^{m}}(0,R)$.

To show that $K$ is compact in $\mathcal{W}^{m+1}$, take any sequence
$\\{F_{n}\\}_{n\in\mathbb{N}}\subset K$. Since $K$ is sequentially compact in
$\mathcal{W}^{\infty}$, we may replace it by its subsequence convergent to
some $F_{\infty}$ in $\mathcal{W}^{\infty}$. Let
$F_{n}^{\prime}:=F_{n}-F_{\infty}$. Since
$\varphi_{k}*:\mathcal{W}^{\infty}\to\mathcal{W}^{0}$ is continuous for any
$k\in\mathbb{N}$, we have $\|\zeta_{k}*F_{n}^{\prime}\|_{\mathcal{W}^{m+1}}\to
0$ as $n\to\infty$, while by the above estimate,

$\begin{split}\|F_{n}^{\prime}-\zeta_{k}*F_{n}^{\prime}\|_{\mathcal{W}^{m+1}}\leq\|F_{n}^{\prime}\|_{\mathcal{W}^{m}}k^{-1}\leq
2Rk^{-1}\to 0\end{split}$ (2.48)

as $k\to\infty$ uniformly for $n\in\mathbb{N}$. Thus we obtain
$\|F_{n}^{\prime}\|_{\mathcal{W}^{m+1}}\to 0$ as $n\to\infty$, and so $K$ is
compact in $\mathcal{W}^{m+1}$. ∎

It turns out that every compact subset of $\mathcal{X}$ is bounded in a single
$\mathcal{X}^{\mu,\nu}$, but the compactness does not hold for that norm,
losing regularity. By interpolation, the loss may be chosen arbitrarily small,
but to estimate convolutions, it is more convenient to use the derivatives as
follows. For any $\mu,\nu,\iota\in C_{0}^{\scaleto{\nearrow}{4pt}}$, let

$\begin{split}&\mathcal{X}^{\mu,\nu,\iota}_{\partial}:=\\{F_{0}+\sum_{1\leq
k\leq d}\partial_{k}F_{k}\in\mathscr{D}^{\prime}(\mathbb{R}^{d})\mid
F_{0}\in\mathcal{X}^{\mu,\nu},\
F_{1},\ldots,F_{d}\in\mathcal{X}^{\mu,\iota}\\},\\\
&\qquad\|G\|_{\mathcal{X}^{\mu,\nu,\iota}_{\partial}}:=\inf\\{\rho^{\mu,\nu}(F_{0})+\sum_{1\leq
k\leq d}\rho^{\mu,\iota}(F_{k})\mid G=F_{0}+\sum_{1\leq k\leq
d}\partial_{k}F_{k}\\}.\end{split}$ (2.49)

As shown below, the weight $\iota$ is arbitrary in characterizing the
compactness, but its freedom is important for boundedness of multipliers with
singularity at the boundary.

For any $G\in\mathcal{X}^{\mu,\nu,\iota}_{\partial}$, $l\in(0,\infty)$ and
$\varphi\in\mathscr{D}_{<l}$, we have

$\begin{split}|G(\varphi)|&\leq|F_{0}(\varphi)|+\sum_{1\leq k\leq
d}|F_{k}(\partial_{k}\varphi)|\\\
&\leq\max(\nu(l),\iota(l))\|\varphi\|_{\mu(l)+1}[\rho^{\mu,\nu}(F_{0})+\sum_{1\leq
k\leq d}\rho^{\mu,\iota}(F_{k})],\end{split}$ (2.50)

where $F_{0},\ldots,F_{d}$ are as in the above definition. Hence

$\begin{split}\rho^{\mu+1,\max(\nu,\iota)}(G)\leq\|G\|_{\mathcal{X}^{\mu,\nu,\iota}_{\partial}}\end{split}$
(2.51)

and in particular
$\mathcal{X}^{\mu,\nu,\iota}_{\partial}\hookrightarrow\mathscr{D}^{\prime}$.
Since $\mathcal{X}^{\mu,\nu}$ is Banach and the representation
$F=F_{0}+\sum_{k=1}^{d}\partial_{k}F_{k}$ is preserved when each $F_{k}$
converges in $\mathscr{D}^{\prime}$, we deduce that
$\mathcal{X}^{\mu,\nu,\iota}_{\partial}$ is also Banach. By definition, we
also have

$\begin{split}\|F\|_{\mathcal{X}^{\mu,\nu,\iota}_{\partial}}\leq\rho^{\mu,\nu}(F),\quad\max_{1\leq
k\leq
d}\|\partial_{k}F\|_{\mathcal{X}^{\mu,\nu,\iota}_{\partial}}\leq\rho^{\mu,\iota}(F).\end{split}$
(2.52)

######  Proposition 2.10.

For any subset $K\subset\mathcal{X}$, the following conditions are equivalent.

  1. (1)

$K$ is compact in $\mathcal{X}$.

  2. (2)

$K$ is sequentially compact in $\mathcal{X}$.

  3. (3)

$K$ is countably compact in $\mathcal{X}$.

  4. (4)

$K$ is bounded in $\mathcal{X}^{\mu,\nu}$ and compact in
$\mathcal{X}^{\mu,\nu,\iota}_{\partial}$ for some $\mu,\nu\in
C_{0}^{\scaleto{\nearrow}{4pt}}$ and all $\iota\in
C_{+}^{\scaleto{\nearrow}{4pt}}$.

  5. (5)

$K$ is bounded in $\mathcal{X}^{\mu,\nu}$ and compact in the
$\rho^{\mu+1,\nu}$ norm for some $\mu,\nu\in C_{0}^{\scaleto{\nearrow}{4pt}}$.

###### Proof.

(3) follows both from (1) and from (2) in general. Moreover, since
$\mathcal{X}\hookrightarrow\mathscr{D}^{\prime}$, by Lemma 2.8, (3) implies
(2). (4) implies (5) because of (2.51).

Next we show that (5) implies (1). Let $R<\infty$ such that $K\subset
B:=\bar{B}_{\mathcal{X}^{\mu,\nu}}(0,R)$. Then it suffices to show that
$B\subset\mathcal{X}$ is continuous for the $\rho^{\mu+1,\nu}$ metric on $B$.
Take any convergent sequence $B\ni F_{n}\to F_{\infty}\in B$ and any
$U:=\mathscr{B}(R_{0},R_{1},N)\in\mathscr{B}_{\mathcal{X}}$. Let
$F_{n}^{\prime}:=F_{n}-F_{\infty}$. There exists $\delta>0$ such that
$\mu(2\delta)<1$ and $\nu(2\delta)<\tfrac{R_{0}}{4R}$. Then for all
$n\in\mathbb{N}$, $\rho^{\mu,\nu}(F_{n}^{\prime})<2R$ implies
$X^{\delta}F_{n}^{\prime}\in B_{\mathcal{M}_{+}}(0,R_{0}/2)\subset U/2$. There
exist $\mu_{1},\nu_{1}\in C_{0}^{\scaleto{\nearrow}{4pt}}$ such that
$\mu(l)+1\leq\mu_{1}(l)$ and $M^{\chi}_{\delta}(\mu(l)+1)\nu(l)\leq\nu_{1}(l)$
for $l\geq\delta$. Then Lemma 2.3 implies
$\rho^{\mu_{1},\nu_{1}}(X_{\delta}F_{n}^{\prime})\leq\rho^{\mu+1,\nu}(F_{n}^{\prime})\to
0$, so $X_{\delta}F_{n}^{\prime}\to 0$ in $\mathcal{X}$. Thus
$X_{\delta}F_{n}^{\prime}\in U/2$ and $F_{n}^{\prime}\in U$ for large $n$.
Hence $B\subset\mathcal{X}$ is continuous for $\rho^{\mu+1,\nu}$, so $K$ is
compact also in $\mathcal{X}$.

It remains to show that (2) implies (4). Let $K\subset\mathcal{X}$ be
sequentially compact. Proposition 2.7 yields $\mu\in
C_{\delta}^{\scaleto{\nearrow}{4pt}}$ with some $\delta>0$ and $\nu_{0}\in
C^{\scaleto{\nearrow}{4pt}}$ such that $\rho^{\mu(l)}_{l}(F)\leq\nu_{0}(l)$
for all $F\in K$ and $l>0$.

Let $S(l):=\sup_{F\in K}\|X^{l}F\|_{\mathcal{M}}$ for $l>0$ and suppose that
$S(l)\not\to 0$ as $l\to+0$. Then there exist $\varepsilon>0$ and sequences
$F_{n}\in K$, $(0,\delta)\ni l_{n}\to 0$, and
$\varphi_{n}\in\mathscr{D}_{<l_{n}}$ such that $\|\varphi_{n}\|_{0}\leq 1$ and
$F_{n}(\varphi_{n})>\varepsilon$ for all $n\in\mathbb{N}$. By the sequential
compactness of $K$, we may assume that $F_{n}$ converges to some $F_{\infty}$
in $\mathcal{X}$. There exists $\\{N_{m}\\}_{m\in\mathbb{N}}\subset
C_{+}^{\scaleto{\nearrow}{4pt}}$ satisfying
$N_{m}(l_{n})\|\varphi_{n}\|_{m}\leq 2^{-m-3}\varepsilon$ for all
$m,n\in\mathbb{N}$. For any $n\in\mathbb{N}$,
$G_{0}\in\bar{B}_{\mathcal{M}_{+}}(0,\varepsilon/4)$,
$G_{1}\in\mathcal{X}_{\geq 1}$ and $G_{2}\in\mathscr{B}(N)$, we have

$\begin{split}&|G_{0}(\varphi_{n})|\leq\varepsilon/4,\quad|G_{1}(\varphi_{n})|=0,\quad|G_{2}(\varphi_{n})|\leq\sum_{m\in\mathbb{N}}N_{m}(l_{n})\|\varphi_{n}\|_{m}\leq\varepsilon/4,\end{split}$
(2.53)

so $|G(\varphi_{n})|\leq\varepsilon/2$ for any
$G\in\mathscr{B}(\varepsilon/4,1,N)$ and $n\in\mathbb{N}$. On the other hand,
$F_{\infty}\in\mathcal{X}$ and $l_{n}\to+0$ imply $F_{\infty}(\varphi_{n})\to
0$, hence $|(F_{n}-F_{\infty})(\varphi_{n})|>\varepsilon/2$ for large
$n\in\mathbb{N}$, then
$F_{n}-F_{\infty}\not\in\mathscr{B}(\varepsilon/4,1,N)$, contradicting
$F_{n}\to F_{\infty}$ in $\mathcal{X}$. Hence $S(l)\to 0$ as $l\to+0$, so that
we may find $\nu_{1}\in C_{0}^{\scaleto{\nearrow}{4pt}}$ such that $K\subset
B_{\mathcal{X}^{\mu,\nu_{1}}}(0,1)$.

Let $\nu\in C_{0}^{\scaleto{\nearrow}{4pt}}$ such that $\nu_{1}\leq\nu$,
$\nu_{1}(l)M^{\chi}_{l}(\mu(l))/\nu(l)\to 0$ as $l\to\infty$, and
$\nu_{1}(l)/\nu(l)\to 0$ as $l\to+0$. Take any sequence $F_{n}\in K$
convergent to some $F_{\infty}\in K$ in $\mathcal{X}$. It suffices to prove
$F_{n}\to F_{\infty}$ also in $\mathcal{X}^{\mu,\nu,\iota}_{\partial}$ for all
$\iota\in C_{+}^{\scaleto{\nearrow}{4pt}}$. Since we have, using Lemma 2.3,

$\begin{split}&\rho^{\mu,\nu}(X_{H}F)+\rho^{\mu,\nu}(X^{L}F)\leq\rho^{\mu,\nu_{1}}(F)\left[\sup_{l\geq
H}\tfrac{\nu_{1}(l)M^{\chi}_{l}(\mu(l))}{\nu(l)}+\sup_{0<l\leq
2L}\tfrac{\nu_{1}(l)}{\nu(l)}\right]\to 0\end{split}$ (2.54)

as $H\to\infty$ and $\delta\geq L\to+0$ uniformly for all $F\in K$, it
suffices to show that $F_{n}^{\prime}:=X_{L}X^{H}(F_{n}-F_{\infty})\to 0$ in
$\mathcal{X}^{\mu,\nu,\iota}_{\partial}$ for all $0<L<\delta<1<H<\infty$.

Since $X^{H}:\mathcal{X}\to\mathcal{W}^{\infty}$ is continuous, $X^{H}(K)$ is
also countably compact in $\mathcal{W}^{\infty}$. Then by Lemma 2.9, there
exists $m\in\mathbb{N}$ such that $X^{H}(K)$ is compact in $\mathcal{W}^{m}$.
Since $F_{n}\to F_{\infty}$ in $\mathscr{D}^{\prime}$, we deduce that
$\|F_{n}^{\prime}\|_{\mathcal{W}^{m}}\to 0$, while
$\\{F_{n}^{\prime}\\}_{n\in\mathbb{N}}$ is bounded in $\mathcal{X}^{\mu,1}$.
Let $\\{\varphi_{n}\\}_{n\in\mathbb{N}}$ be the mollifier in Proposition 2.6
and $\varphi_{n}^{\dagger}(\xi):=\varphi_{n}(-\xi)$. Then

$\begin{split}|\varphi_{k}*F_{n}^{\prime}(\varphi)|&\leq\|F_{n}^{\prime}\|_{\mathcal{W}^{m}}\|\varphi_{k}^{\dagger}*X_{L/2}\varphi\|_{m}\\\
&\leq\|F_{n}^{\prime}\|_{\mathcal{W}^{m}}r_{k}^{-m}\|\zeta\|_{W^{1,m}}M^{\chi}_{L/2}(\mu(L))\|\varphi\|_{\mu(L)}^{+},\end{split}$
(2.55)

for any $k\in\mathbb{N}$ and $\varphi\in\mathscr{D}$, while
$\varphi_{k}*F_{n}^{\prime}(\varphi)=0$ if $\varphi\in\mathscr{D}_{<l}$ for
some $l\leq L$. Hence if we choose $k=k(n)\to\infty$ sufficiently slowly
compared with $\|F_{n}^{\prime}\|_{\mathcal{W}^{m}}\to 0$ as $n\to\infty$,
then $\rho^{\mu,\nu}(\varphi_{k}*F_{n}^{\prime})\to 0$. On the other hand, we
have

$\begin{split}&F_{n}^{\prime}-\varphi_{k}*F_{n}^{\prime}=\nabla\cdot
F^{\prime}_{k,n},\quad
F^{\prime}_{k,n}:=\int_{0}^{1}\int_{|y-c_{k}|<r_{k}}y\varphi_{k}(y)F_{n}^{\prime}(x-\theta
y)dyd\theta,\end{split}$ (2.56)

and for any $l\in(0,\infty)$, using that
$\operatorname{supp}F_{n}^{\prime}\subset\mathbb{R}^{d}_{>L}$,

$\begin{split}\rho^{\mu(l)}_{l}(F^{\prime}_{k,n})\lesssim(|c_{k}|+r_{k})\rho^{\mu(l)}_{l}(F_{n}^{\prime})\lesssim\tfrac{|c_{k}|+r_{k}}{\iota(L)}\iota(l)\rho^{\mu,1}(F_{n}^{\prime}).\end{split}$
(2.57)

Hence $\rho^{\mu,\iota}(F^{\prime}_{k,n})\to 0$ as $k(n)\to\infty$ and
$n\to\infty$. Thus we obtain
$\|F_{n}^{\prime}\|_{X^{\mu,\nu,\iota}_{\partial}}\to 0$, hence $F_{n}\to
F_{\infty}$ in $\mathcal{X}^{\mu,\nu,\iota}_{\partial}$, which concludes the
proof. ∎

######  Remark 2.2.

Applying the above result to any sequence
$\\{F_{n}\\}_{n\in\mathbb{N}\cup\\{\infty\\}}\subset\mathcal{X}$, we obtain
characterization of convergence: $F_{n}\to F_{\infty}$ in $\mathcal{X}$ iff

$\begin{split}\sup_{n\in\mathbb{N}}\mathcal{X}^{\mu,\nu}(F_{n})<\infty,\quad\lim_{n\to\infty}\|F_{n}-F_{\infty}\|_{\mathcal{X}^{\mu,\nu,\iota}_{\partial}}=0,\end{split}$
(2.58)

for some $\mu,\nu\in C_{0}^{\scaleto{\nearrow}{4pt}}$ and all $\iota\in
C_{+}^{\scaleto{\nearrow}{4pt}}$. The latter suffices to hold for one
$\iota\in C_{+}^{\scaleto{\nearrow}{4pt}}$, and also the norm may be replaced
with $\rho^{\mu+1,\nu}$.

It is natural to look for simpler characterizations. On one hand, the
convergence in any $\mathcal{X}^{\mu,\nu}$ is too strong. A simple counter-
example is $F_{n}(\xi):=\xi_{1}e^{in\xi_{1}}$. It is bounded in
$\mathcal{X}^{0,\nu}$ with $\nu(l):=l$, and
$F_{n}=in^{-1}[(e^{in\xi_{1}}-1)-\partial_{1}\xi_{1}(e^{in\xi_{1}}-1)]$
implies $\rho^{1,\nu}(F_{n})\to 0$, hence $F_{n}\to 0$ in $\mathcal{X}$,
whereas
$\|F_{n}\|_{\mathcal{M}(\mathbb{R}^{d}_{<l})}=\|F_{1}\|_{\mathcal{M}(\mathbb{R}^{d}_{<l})}>0$
for all $l>0$ implies $\inf_{n}\|F_{n}\|_{\mathcal{X}^{\mu,\nu}}>0$ for any
choice of $\mu,\nu\in C_{0}^{\scaleto{\nearrow}{4pt}}$. On the other hand, the
bounded weak-* convergence is too weak:
$\sup_{n}\|F_{n}\|_{\mathcal{X}^{\mu,\nu}}<\infty$ and $F_{n}\to F_{\infty}$
in $\mathscr{D}^{\prime}(\mathbb{R}^{d})$ do not imply $F_{n}\to F_{\infty}$
in $\mathcal{X}$, if $d\geq 2$. A simple counter-example is
$F_{n}(\xi):=\varphi(\xi-(0,nv))$ for any non-zero
$\varphi\in\mathscr{D}_{>0}$ and $v\in\mathbb{R}^{d-1}$. Another candidate is
the weak convergence in a single $\mathcal{X}^{\mu,\nu}$, but it does not seem
convenient to handle.

The multiplication factor $\kappa$ is essential for the following multiplier
estimate. Let

$\begin{split}Y:=C^{\infty}(\mathbb{R}^{d}_{>0};\mathbb{C})\cap
L^{\infty}(\mathbb{R}^{d}_{(0,1)};\mathbb{C})\cap\bigcap_{m\geq
0}\bigcap_{n\geq
2}W^{m,\infty}(\mathbb{R}^{d}_{(1/n,n)};\mathbb{C}).\end{split}$ (2.59)

It consists of smooth functions on $\mathbb{R}^{d}_{>0}$ that are bounded as
$|\xi^{\perp}|\to\infty$, including all the derivatives, locally uniformly for
$0<\xi_{1}<\infty$, and also bounded (but not derivatives) uniformly as
$\xi_{1}\to+0$. $Y$ is a Fréchet space with the topology defined by those
seminorms of $L^{\infty}(\mathbb{R}^{d}_{(0,1)})$ and
$W^{m,\infty}(\mathbb{R}^{d}_{(1/n,n)})$. Thanks to (2.23), for any
$F\in\mathcal{X}$ and $f\in Y$ we may define the product
$fF\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$ by
$(fF)(\varphi):=\mathscr{E}\mathscr{R}F(f\varphi)$ for all
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$. It is indeed bounded on $\mathcal{X}$
as follows.

######  Lemma 2.11.

Let $n_{1},n_{2}\in\mathbb{N}$ and $A:\mathbb{R}^{d}_{>0}\to M(n_{1},n_{2})$
with all components in $Y$. Then for any $\delta>0$ and $\mu\in
C^{\scaleto{\nearrow}{4pt}}_{\delta}$, there exist $\beta,\gamma\in
C^{\scaleto{\nearrow}{4pt}}$ such that

$\displaystyle\|A^{\intercal}\varphi\|_{\mu(l),\delta}\leq\beta(l)\|\varphi\|_{\mu(l)}^{+},$
(2.60) $\displaystyle\sum_{k\in\mathbb{N}}\gamma(2^{2-k}\delta)\|(\nabla
A^{\intercal})\varphi\|_{L^{\infty}(1<2^{k}\xi_{1}/\delta<4)}+\gamma(l)\|X_{\delta}(\nabla
A^{\intercal})\varphi\|_{\mu(l)}\leq\beta(l)\|\varphi\|_{\mu(l)}^{+},$ (2.61)

for all $l\in(0,\infty)$ and $\varphi\in(\mathscr{D}_{<l})^{n_{1}}$.
$(\beta,\gamma)$ may be chosen uniformly for a set of $A$ whose components are
bounded in $Y$.

Moreover, for any $\beta\in C^{\scaleto{\nearrow}{4pt}}$ satisfying (2.60) and
any $\nu\in C^{\scaleto{\nearrow}{4pt}}$, the multiplication with $A$ is
bounded $\mathcal{X}^{\mu,\nu}\to\mathcal{X}^{\mu,\beta\nu}$. If in addition
$\gamma\in C^{\scaleto{\nearrow}{4pt}}$ satisfies (2.61) then it is also
bounded
$\mathcal{X}^{\mu,\nu,\gamma\nu}_{\partial}\to\mathcal{X}^{\mu,\beta\nu,\beta\gamma\nu}_{\partial}$.
The operator norm is bounded respectively as

$\begin{split}\|AF\|_{\mathcal{X}^{\mu,\beta\nu}}\leq\|F\|_{\mathcal{X}^{\mu,\nu}},\quad\|AF\|_{\mathcal{X}^{\mu,\beta\nu,\beta\gamma\nu}_{\partial}}\leq
2\|F\|_{\mathcal{X}^{\mu,\nu,\gamma\nu}_{\partial}}.\end{split}$ (2.62)

###### Proof.

Using the Leibniz rule and Lemma 2.3, we have

$\begin{split}\|A^{\intercal}\varphi\|_{\mu(l),\delta}&\leq\|A^{\intercal}\|_{L^{\infty}(\mathbb{R}^{d}_{(0,l)};M(n_{2},n_{1}))}\|\varphi\|_{0}^{+}\\\
&\quad+2^{\mu(l)}\sup_{|\alpha|\leq\mu(l)}\|\partial^{\alpha}A^{\intercal}\|_{L^{\infty}(\mathbb{R}^{d}_{(\delta,l)};M(n_{2},n_{1}))}M^{\chi}_{\delta}(\mu(l))\|\varphi\|_{\mu(l)}^{+},\end{split}$
(2.63)

where the factors multiplied with the seminorms of $\varphi$ are non-
increasing and locally bounded in $l\in(0,\infty)$, so that we may find
$\beta_{0}\in C^{\scaleto{\nearrow}{4pt}}$ dominating the sum of them. Then we
have

$\begin{split}\|A^{\intercal}\varphi\|_{\mu(l),\delta}\leq\beta_{0}(l)\|\varphi\|_{\mu(l)}^{+}.\end{split}$
(2.64)

Similarly, there exist $\\{a_{k}\\}_{k\in\mathbb{N}}\subset[1,\infty)$ and
$\beta_{1}\in C^{\scaleto{\nearrow}{4pt}}$ such that

$\begin{split}\|(\nabla
A^{\intercal})\varphi\|_{L^{\infty}(1<2^{k}\xi_{1}/\delta<4)}\leq
a_{k}\|\varphi\|_{0}^{+},\quad\|X_{\delta}(\nabla
A^{\intercal})\varphi\|_{\mu(l)}\leq\beta_{1}(l)\|\varphi\|_{\mu(l)}^{+}.\end{split}$
(2.65)

Then we may find $\gamma\in C^{\scaleto{\nearrow}{4pt}}$ satisfying
$\gamma(2^{2-k}\delta)a_{k}\leq 2^{-1-k}$ for all $k\in\mathbb{N}$ and
$\gamma(l)=\gamma(2\delta)\leq 1/4$ for all $l\geq 2\delta$. Choosing
$\beta:=\max(1,\beta_{0},\beta_{1})$ satisfies both (2.60) and (2.61).

Next we prove the boundedness assuming (2.60). Combining it with (2.10) yields

$\begin{split}|AF(\varphi)|=|F(A^{\intercal}\varphi)|\leq\nu(l)\rho^{\mu,\nu}_{l}(F)\|A^{\intercal}\varphi\|_{\mu(l),\delta}\leq\nu(l)\rho^{\mu,\nu}_{l}(F)\beta(l)\|\varphi\|_{\mu(l)}^{+},\end{split}$
(2.66)

for all $F\in\mathcal{X}^{\mu,\nu}$, $l>0$ and $\varphi\in\mathscr{D}_{<l}$.
Hence $\rho^{\mu,\beta\nu}_{l}(AF)\leq\rho^{\mu,\nu}_{l}(F)$.

Suppose further (2.61). For any
$F\in\mathcal{X}^{\mu,\nu,\gamma\nu}_{\partial}$, let
$F=F_{0}+\sum_{k}\partial_{k}F_{k}$. For the first two terms in
$AF=AF_{0}+\sum_{k}\partial_{k}(AF_{k})-\sum_{k}(\partial_{k}A)F_{k}$, we
have, using the above estimate,

$\begin{split}\rho^{\mu,\beta\nu}(AF_{0})\leq\rho^{\mu,\nu}(F_{0}),\quad\rho^{\mu,\beta\gamma\nu}(AF_{k})\leq\rho^{\mu,\gamma\nu}(F_{k}).\end{split}$
(2.67)

For the remaining term, let
$X_{\delta}^{(k)}:=X_{2^{-k}\delta}-X_{2^{1-k}\delta}$ for $k\in\mathbb{N}$
and $X_{\delta}^{(0)}:=X_{\delta}$. Using (2.61), we have for any $l>0$ and
$\varphi\in\mathscr{D}_{<l}$,

$\begin{split}&|(\partial_{k}A)F_{k}(\varphi)|=|\mathscr{E}\partial_{k}A\mathscr{R}F_{k}(\varphi)|\leq\sum_{k\in\mathbb{N}_{0}}|F_{k}(X_{\delta}^{(k)}\partial_{k}A^{\intercal}\varphi)|\\\
&\leq\rho^{\mu,\gamma\nu}(F_{k})\nu(l)\left[\sum_{k\in\mathbb{N}}\gamma(2^{2-k}\delta)\|X_{\delta}^{(k)}\partial_{k}A^{\intercal}\varphi\|_{0}+\gamma(l)\|X_{\delta}\partial_{k}A^{\intercal}\varphi\|_{\mu(l)}^{+}\right],\\\
&\leq\rho^{\mu,\gamma\nu}(F_{k})\nu(l)\beta(l)\|\varphi\|_{\mu(l)}^{+}.\end{split}$
(2.68)

Thus we obtain
$\rho^{\mu,\beta\nu}((\partial_{k}A)F_{k})\leq\rho^{\mu,\gamma\nu}(F_{k})$.
Summing the above three estimates yields

$\begin{split}\|AF\|_{\mathcal{X}^{\mu,\beta\nu,\beta\gamma\nu}_{\partial}}\leq\rho^{\mu,\nu}(F_{0})+2\sum_{1\leq
k\leq d}\rho^{\mu,\gamma\nu}(F_{k}),\end{split}$ (2.69)

and then the infimum over $F_{0},\ldots,F_{d}$ leads to the desired operator
bound. ∎

##  3\. Convolution for the distributions

The purpose of the following analysis is to quantify the propagation of
positive frequency by convolution as well as its decay near the zero
frequency, in the distribution sense. For $F,G\in\mathcal{X}$, the convolution
is formally defined by

$\begin{split}\mathscr{D}(\mathbb{R}^{d})\ni\forall\varphi,\quad(F*G)(\varphi):=F_{\xi}(G_{\eta}(\varphi(\xi+\eta))),\end{split}$
(3.1)

where the subscript $\eta$ of $G_{\eta}$ indicates the independent variable of
the test function. In general, for any
$G\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$ and
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$, we have
$G_{\eta}(\varphi(\xi+\eta))\in C^{\infty}(\mathbb{R}^{d})$ and

$\begin{split}\partial^{\alpha}_{\xi}G_{\eta}(\varphi(\xi+\eta))=G_{\eta}(\partial^{\alpha}\varphi(\xi+\eta)),\end{split}$
(3.2)

but we need some condition for $F\in\mathscr{D}^{\prime}$ to act on it.

######  Lemma 3.1.

Let $l>0$, $m\geq 0$, $G\in\mathscr{D}^{\prime}(\mathbb{R}^{d})$,
$\rho_{l}^{m}(G)<\infty$ and $\varphi\in\mathscr{D}_{<l}$. Then we have
$G_{\eta}(\varphi(\xi+\eta))\in\mathcal{B}^{\infty}_{+<l}$ and

$\begin{split}\forall
n\in\mathbb{N},\quad\|G_{\eta}(\varphi(\xi+\eta))\|_{n}^{+}\leq\rho_{l}^{m}(G)\|\varphi\|_{m+n}^{+}.\end{split}$
(3.3)

###### Proof.

Since $\operatorname{supp}\varphi$ is compact, we have
$l^{\prime}:=\max\\{\xi_{1}\mid\xi\in\operatorname{supp}\varphi\\}<l$. Then
for any $\xi\in\mathbb{R}^{d}_{>l^{\prime}}$, we have
$\operatorname{supp}_{\eta}\varphi(\xi+\eta)\subset\mathbb{R}^{d}_{<0}$, so
$\operatorname{supp}G\subset\mathbb{R}^{d}_{\geq 0}$ implies
$f(\xi):=G_{\eta}(\varphi(\xi+\eta))=0$. Hence
$\operatorname{supp}f\subset\mathbb{R}^{d}_{\leq
l^{\prime}}\subset\mathbb{R}^{d}_{<l}$. Moreover, $f\in C^{\infty}$ and for
any $\xi\in\mathbb{R}^{d}_{>0}$ and $\alpha\in\mathbb{N}_{0}^{d}$ with
$|\alpha|\leq n$ we have

$\begin{split}|\partial^{\alpha}_{\xi}f(\xi)|=|G_{\eta}(\partial^{\alpha}\varphi(\xi+\eta))|\leq\rho_{l}^{m}(G)\|\partial^{\alpha}\varphi\|_{m}^{+}\leq\rho_{l}^{m}(G)\|\varphi\|_{m+n}^{+}.\end{split}$
(3.4)

The supremum for $\xi$ and $\alpha$ yields the desired estimate. ∎

The above lemma makes sense of (3.1) for $F,G\in\mathcal{X}$, where $F$ is the
extension by Lemma 2.1 to $\mathcal{B}_{+<l}^{\infty}$ if
$\varphi\in\mathscr{D}_{<l}$. This definition is compatible with the
convolution for sign-definite locally integrable $F,G$. Moreover, we have the
following estimate on $F*G$.

######  Lemma 3.2.

Let $\delta\in(0,1]$, $\mu\in C^{\scaleto{\nearrow}{4pt}}_{\delta}$ and
$\nu\in C^{\scaleto{\nearrow}{4pt}}$. Then for any
$F,G\in\mathcal{X}^{\mu,\nu}$, $l\in(0,\infty)$, $\varepsilon\in(0,\delta/2]$
and $\varphi\in\mathscr{D}_{<l}$, we have

$\begin{split}\tfrac{|F*G(\varphi)|}{\rho^{\mu,\nu}_{l}(F)\rho^{\mu,\nu}_{l}(G)}\leq
3[\nu(2\varepsilon)\nu(l)&+1_{l>2\varepsilon}\nu(\min(2\delta,l/2))\nu(l-\varepsilon)]\|\varphi\|_{\mu(l)}^{+}\\\
&+1_{l>4\delta}[M_{\delta}^{\chi}(\mu(l-\delta))\nu(l-\delta)]^{2}\|\varphi\|_{2\mu(l-\delta)}^{+}.\end{split}$
(3.5)

###### Proof.

First we consider the case of $l\leq 4\delta$. For any
$\varphi\in\mathscr{D}_{<l}$, there is $l^{\prime}\in(0,l)$ such that
$\operatorname{supp}\varphi\subset\mathscr{D}_{<l^{\prime}}$. Let
$\tilde{\chi}\in C^{\infty}(\mathbb{R};[0,1])$ be a cut-off function
satisfying

$\begin{split}\operatorname{supp}\tilde{\chi}\subset(-\infty,l/2),\quad\tilde{\chi}((-\infty,l^{\prime}/2])=\\{1\\},\end{split}$
(3.6)

$\tilde{\chi}^{\varepsilon}:=\chi^{\varepsilon}\tilde{\chi}$, and
$\tilde{\chi}_{\varepsilon}:=\chi_{\varepsilon}\tilde{\chi}$. Then we
decompose

$\begin{split}(F*G)(\varphi)&=F_{\xi}G_{\eta}(\tilde{\chi}^{\varepsilon}(\xi_{1})\varphi(\xi+\eta))+G_{\eta}F_{\xi}(\tilde{\chi}^{\varepsilon}(\eta_{1})\varphi(\xi+\eta))\\\
&\quad-
F_{\xi}G_{\eta}(\tilde{\chi}^{\varepsilon}(\xi_{1})\tilde{\chi}^{\varepsilon}(\eta_{1})\varphi(\xi+\eta))\\\
&\quad+F_{\xi}G_{\eta}(\tilde{\chi}_{\varepsilon}(\xi_{1})\varphi(\xi+\eta))+G_{\eta}F_{\xi}(\tilde{\chi}_{\varepsilon}(\eta_{1})\varphi(\xi+\eta))\\\
&\quad-
F_{\xi}G_{\eta}(\tilde{\chi}_{\varepsilon}(\xi_{1})\tilde{\chi}_{\varepsilon}(\eta_{1})\varphi(\xi+\eta)),\end{split}$
(3.7)

using that
$(1-\tilde{\chi}(\xi_{1}))(1-\tilde{\chi}(\eta_{1}))\varphi(\xi+\eta)=0$ by
the support condition. Since $2\varepsilon\leq\delta$ and $\mu\in
C^{\scaleto{\nearrow}{4pt}}_{\delta}$ imply $\mu(2\varepsilon)<1$, the first
term is bounded by

$\begin{split}|F_{\xi}G_{\eta}(\tilde{\chi}^{\varepsilon}(\xi_{1})\varphi(\xi+\eta))|&\leq\rho_{l}^{\mu,\nu}(F)\nu(2\varepsilon)\|G_{\eta}(\varphi(\xi+\eta))\|_{0}^{+}\\\
&\leq\rho_{l}^{\mu,\nu}(F)\nu(2\varepsilon)\rho_{l}^{\mu,\nu}(G)\nu(l)\|\varphi\|_{\mu(l)}^{+}.\end{split}$
(3.8)

By symmetry, the second term is bounded exactly in the same way, while the
third term is easier. The fourth term is estimated similarly by

$\begin{split}|F_{\xi}G_{\eta}(\tilde{\chi}_{\varepsilon}(\xi_{1})\varphi(\xi+\eta))|&\leq\rho_{l}^{\mu,\nu}(F)\nu(l/2)\sup_{\xi_{1}>\varepsilon}|G_{\eta}(\varphi(\xi+\eta))|\\\
&\leq\rho_{l}^{\mu,\nu}(F)\nu(l/2)\rho_{l}^{\mu,\nu}(G)\nu(l-\varepsilon)\|\varphi\|_{\mu(l)}^{+}.\end{split}$
(3.9)

The fifth term is the same by symmetry, while the final term is bounded by

$\begin{split}|F_{\xi}G_{\eta}(\tilde{\chi}_{\varepsilon}(\xi_{1})\tilde{\chi}_{\varepsilon}(\eta_{1})\varphi(\xi+\eta))|&\leq\rho_{l}^{\mu,\nu}(F)\nu(l/2)\sup_{\xi_{1}>\varepsilon}|G_{\eta}(\tilde{\chi}_{\varepsilon}(\eta_{1})\varphi(\xi+y))|\\\
&\leq\rho_{l}^{\mu,\nu}(F)\nu(l/2)\rho_{l}^{\mu,\nu}(G)\nu(l-\varepsilon)\|\varphi\|_{0}^{+},\end{split}$
(3.10)

since $l/2\leq 2\delta$ implies $\mu(l/2)<1$. Thus we obtain

$\begin{split}\tfrac{|F*G(\varphi)|}{\rho_{l}^{\mu,\nu}(F)\rho_{l}^{\mu,\nu}(G)}&\leq
3[\nu(2\varepsilon)\nu(l)+1_{l>2\varepsilon}\nu(l/2)\nu(l-\varepsilon)]\|\varphi\|_{\mu(l)}^{+},\end{split}$
(3.11)

where the restriction by $1_{l>2\varepsilon}$ comes from
$\tilde{\chi}_{\varepsilon}=0$ for $l\leq 2\varepsilon$.

For the other case $4\delta<l$, we replace the above
$\tilde{\chi}_{\varepsilon}$ with
$\chi_{\varepsilon}^{\delta}:=\chi_{\varepsilon}\chi^{\delta}$ and decompose

$\begin{split}(F*G)(\varphi)&=F_{\xi}G_{\eta}(\chi^{\varepsilon}(\xi_{1})\varphi(\xi+\eta))+G_{\eta}F_{\xi}(\chi^{\varepsilon}(\eta_{1})\varphi(\xi+\eta))\\\
&\quad-
F_{\xi}G_{\eta}(\chi^{\varepsilon}(\xi_{1})\chi^{\varepsilon}(\eta_{1})\varphi(\xi+\eta))\\\
&\quad+F_{\xi}G_{\eta}(\chi_{\varepsilon}^{\delta}(\xi_{1})\varphi(\xi+\eta))+G_{\eta}F_{\xi}(\chi_{\varepsilon}^{\delta}(\eta_{1})\varphi(\xi+\eta))\\\
&\quad-
F_{\xi}G_{\eta}(\chi_{\varepsilon}^{\delta}(\xi_{1})\chi_{\varepsilon}^{\delta}(\eta_{1})\varphi(\xi+\eta))+F_{\xi}G_{\eta}(\chi_{\delta}(\xi_{1})\chi_{\delta}(\eta_{1})\varphi(\xi+\eta)).\end{split}$
(3.12)

The first six terms are estimated in the same way as in the previous case,
using
$\operatorname{supp}\chi_{\varepsilon}^{\delta}\subset\mathbb{R}^{d}_{(\varepsilon,2\delta)}$.
For the last remaining term, we have

$\begin{split}l^{\prime}:=l-\delta\implies\operatorname{supp}\chi_{\delta}(\xi_{1})\chi_{\delta}(\eta_{1})\varphi(\xi+\eta)\subset\\{\delta<\xi_{1},\eta_{1}<l^{\prime}\\},\end{split}$
(3.13)

and so

$\begin{split}&|F_{\xi}G_{\eta}(\chi_{\delta}(\xi_{1})\chi_{\delta}(\eta_{1})\varphi(\xi+\eta))|\leq\rho_{l}^{\mu,\nu}(F)\nu(l^{\prime})\|\chi_{\delta}(\xi_{1})G_{\eta}(\chi_{\delta}(\eta_{1})\varphi(\xi+\eta))\|_{\mu(l^{\prime})(\xi)}^{+}.\end{split}$
(3.14)

The last norm is bounded by

$\begin{split}&\max_{|\alpha|\leq\mu(l^{\prime})}\sup_{\delta<\xi_{1}<l^{\prime}}\sum_{0\leq
k\leq\alpha_{1}}\tfrac{\alpha_{1}!}{(\alpha_{1}-k)!k!}|\chi_{\delta}^{(k)}(\xi_{1})G_{\eta}(\chi_{\delta}(\eta_{1})\partial^{\alpha-
ke_{1}}\varphi(\xi+\eta))|\\\ &\leq
M_{\delta}^{\chi}(\mu(l^{\prime}))\rho_{l}^{\mu,\nu}(G)\nu(l^{\prime})\sup_{\xi_{1}>\delta}\max_{|\alpha|\leq\mu(l^{\prime})}\|\chi_{\delta}(\eta_{1})\partial^{\alpha}\varphi(\xi+\eta)\|_{\mu(l^{\prime})(\eta)}^{+},\end{split}$
(3.15)

where the last norm of (LABEL:exp_FG1) is bounded similarly by
$M_{\delta}^{\chi}(\mu(l^{\prime}))\|\varphi\|_{2\mu(l^{\prime})}^{+}$, and
thus we obtain the desired estimate. ∎

As an immediate consequence of the above lemma, we have

######  Lemma 3.3.

Let $\delta\in(0,1]$, $\mu\in C^{\scaleto{\nearrow}{4pt}}_{\delta}$,
$\nu,\tilde{\mu},\tilde{\nu}\in C^{\scaleto{\nearrow}{4pt}}$. Suppose

  1. (1)

$\mu\leq\tilde{\mu}$ and $2\mu(l-\delta)\leq\tilde{\mu}(l)$ for $l>4\delta$.

  2. (2)

For each $l>0$ there is $\varepsilon\in(0,\delta/2]$ such that
$9\nu(2\varepsilon)\nu(l)\leq\tilde{\nu}(l)$, and, if $2\varepsilon<l$ then
$9\nu(l-\varepsilon)^{2}\leq\tilde{\nu}(l)$.

  3. (3)

For $l>4\delta$,
$3[M_{\delta}^{\chi}(\mu(l-\delta))\nu(l-\delta)]^{2}\leq\tilde{\nu}(l)$.

Then we have for any $F,G\in\mathcal{X}^{\mu,\nu}$ and $l\in(0,\infty)$

$\begin{split}\rho_{l}^{\tilde{\mu},\tilde{\nu}}(F*G)\leq\rho_{l}^{\mu,\nu}(F)\rho_{l}^{\mu,\nu}(G).\end{split}$
(3.16)

Thus we have reduced the convolution estimate to the weight conditions. Note
that the condition on $\mu$ is independent of $\nu,\tilde{\nu}$. The point is
that the left side for $\tilde{\nu}$ has either a small factor or a smaller
argument $l-\varepsilon$ or $l-\delta$. Hence, if $\mu,\nu$ are accelerating
sufficiently fast, then $\mathcal{X}^{\mu,\nu}$ becomes a convolution algebra,
and the convolution may even gain arbitrary small factor. More precisely, we
have the following.

######  Lemma 3.4.

For each $\delta\in(0,1]$, we have two mappings

$\begin{split}&\mathfrak{C}_{\delta}^{1}:C^{\scaleto{\nearrow}{4pt}}_{\delta}\to
C_{\delta}^{\scaleto{\nearrow}{4pt}},\quad\mathfrak{C}_{\delta}^{2}:C^{\scaleto{\nearrow}{4pt}}_{\delta}\times
C^{\scaleto{\nearrow}{4pt}}_{0}\times C^{\scaleto{\searrow}{4pt}}_{(0,1]}\to
C_{0}^{\scaleto{\nearrow}{4pt}}\end{split}$ (3.17)

satisfying the following. Define
$\mathfrak{C}_{\delta}:C^{\scaleto{\nearrow}{4pt}}_{\delta}\times
C^{\scaleto{\nearrow}{4pt}}_{0}\times C^{\scaleto{\searrow}{4pt}}_{(0,1]}\to
C^{\scaleto{\nearrow}{4pt}}_{\delta}\times C_{0}^{\scaleto{\nearrow}{4pt}}$ by

$\begin{split}\mathfrak{C}_{\delta}(\mu_{0},\nu_{0},\kappa):=(\mathfrak{C}_{\delta}^{1}(\mu_{0}),\mathfrak{C}_{\delta}^{2}(\mathfrak{C}_{\delta}^{1}(\mu_{0}),\nu_{0},\kappa)).\end{split}$
(3.18)

Let $\mu_{1}:=\mathfrak{C}_{\delta}^{1}(\mu_{0})$ and
$\nu_{1}:=\mathfrak{C}_{\delta}^{2}(\mu_{0},\nu_{0},\kappa)$ for any
$\mu_{0},\nu_{0},\kappa$ in the domain. Then

  1. (1)

$\mu_{0}\leq\mu_{1}$, $\nu_{0}\leq\kappa\nu_{1}\in
C^{\scaleto{\nearrow}{4pt}}_{0}$.

  2. (2)

$\tilde{\mu}:=\mu:=\mu_{1}$ satisfies the condition (1) of Lemma 3.3.

  3. (3)

$\mu:=\mu_{0}$, $\nu:=\nu_{1}$ and $\tilde{\nu}:=\kappa\nu_{1}$ satisfy the
conditions (2)-(3) of Lemma 3.3. In particular, for
$(\mu_{1},\nu_{1}):=\mathfrak{C}_{\delta}(\mu_{0},\nu_{0},\kappa)$,
$F,G\in\mathcal{X}^{\mu_{1},\nu_{1}}$ and $l\in(0,\infty)$ we have

$\begin{split}\rho_{l}^{\mu_{1},\kappa\nu_{1}}(F*G)\leq\rho_{l}^{\mu_{1},\nu_{1}}(F)\rho_{l}^{\mu_{1},\nu_{1}}(G).\end{split}$
(3.19)

  4. (4)

$\mathfrak{C}_{\delta}^{1},\mathfrak{C}_{\delta}^{2}$ are non-decreasing for
the pointwise order.

  5. (5)

$\mathfrak{C}_{\delta}^{1},\mathfrak{C}_{\delta}^{2}$ are continuous in the
topology of locally uniform convergence.

  6. (6)

Let $b\in C^{\scaleto{\nearrow}{4pt}}_{[1,\infty)}$ satisfy $b(l)=b(0)$ as
long as $\nu_{0}(l)\leq\kappa$. Then
$\mathfrak{C}_{\delta}^{2}(\mu_{0},b\nu_{0},\kappa)\geq
b\mathfrak{C}_{\delta}^{2}(\mu_{0},\nu_{0},\kappa)$.

As a special case, for any $\mu_{0}\in C^{\scaleto{\nearrow}{4pt}}_{\delta}$
and $\nu_{0}\in C^{\scaleto{\nearrow}{4pt}}_{0}$, we obtain the Banach algebra
for convolution:
$\mathcal{X}^{\mu_{1},\nu_{1}}\supset\mathcal{X}^{\mu_{0},\nu_{0}}$ by
$(\mu_{1},\nu_{1}):=\mathfrak{C}_{\delta}(\mu_{0},\nu_{0},1)$. Hence their
union $\mathcal{X}$ is also a convolution algebra. In other words, the
pointwise multiplication is continuous on $\mathcal{F}^{-1}\mathcal{X}$ (after
continuous extension). A crucial point of the lemma for solving PDEs is that
we may include arbitrary small factors $\kappa$.

###### Proof.

For any $\mu_{0}\in C^{\scaleto{\nearrow}{4pt}}_{\delta}$, $\nu_{0}\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ and $\kappa\in
C^{\scaleto{\searrow}{4pt}}_{(0,1]}$, we will define
$\mu_{1}=\mathfrak{C}_{\delta}^{1}(\mu_{0})$ and
$\nu_{1}=\mathfrak{C}_{\delta}^{2}(\mu_{0},\nu_{0},\kappa)$. Henceforth,
monotonicity and continuity for (vector) functions are for the pointwise
ordering and for the local uniform convergence, respectively.

First we define $\mu_{1}$ by

$\begin{split}\mu_{1}(l):=2^{\max(l/\delta-3,0)}\mu_{0}(l).\end{split}$ (3.20)

Then $\mu_{0}\leq\mu_{1}\in C_{\delta}^{\scaleto{\nearrow}{4pt}}$ and for
$l\geq 4\delta$

$\begin{split}2\mu_{1}(l-\delta)=2^{l/\delta-3}\mu_{0}(l-\delta)\leq
2^{l/\delta-3}\mu_{0}(l)=\mu_{1}(l).\end{split}$ (3.21)

Moreover, $\mu_{0}\mapsto\mu_{1}$ is non-decreasing and continuous. Thus we
have obtained $\mathfrak{C}_{\delta}^{1}$.

Next we define $\nu_{1}$. Let $\hat{\nu}:=\kappa^{-1}\nu_{0}\in
C^{\scaleto{\nearrow}{4pt}}_{0}$. The mapping
$(\nu_{0},-\kappa)\mapsto\hat{\nu}$ is non-decreasing and continuous. For each
$l>0$, let

$\begin{split}\hat{\varepsilon}(l):=\sup\\{\varepsilon>0\mid
9\hat{\nu}(2\varepsilon)\leq\kappa(l)(1-2\varepsilon/\delta)\\}.\end{split}$
(3.22)

Note that the set is non-empty because $\hat{\nu}(0)=0<\kappa(l)$. The last
factor $2\varepsilon/\delta$ ensures $\hat{\varepsilon}(l)\leq\delta/2$, as
well as the uniform decay in $\varepsilon$ of the right hand side. Hence
$\hat{\varepsilon}:[0,\infty)\mapsto(0,\delta/2]$ is non-increasing and
continuous, so is the mapping $(\nu_{0},-\kappa)\mapsto\hat{\varepsilon}$. Let

$\begin{split}l_{0}:=\sup\\{l>0\mid l\leq
2\hat{\varepsilon}(l)\\}.\end{split}$ (3.23)

Then $0<l_{0}\leq\delta$, $2\hat{\varepsilon}(l_{0})=l_{0}$, $t\leq
2\hat{\varepsilon}(t)$ for $0<t\leq l_{0}$, and $0<2\hat{\varepsilon}(t)\leq
l_{0}$ for $t>l_{0}$, and the mapping $\hat{\varepsilon}\mapsto l_{0}$ is non-
decreasing and continuous. Let

$\begin{split}\gamma(l):=l-\hat{\varepsilon}(l).\end{split}$ (3.24)

Then $\gamma:[l_{0},\infty)\to[l_{0}/2,\infty)$ is an increasing
homeomorphism, so is the inverse
$\lambda:=\gamma^{-1}:[l_{0}/2,\infty)\to[l_{0},\infty)$. The mapping
$\hat{\varepsilon}\mapsto\lambda$ is non-decreasing and continuous.

To satisfy the condition (2) of Lemma 3.3 for $\nu=\nu_{1}$ and
$\tilde{\nu}=\kappa\nu_{1}$, we choose

$\begin{split}\varepsilon:=\min(\tfrac{l_{0}}{2},\hat{\varepsilon}(l)).\end{split}$
(3.25)

Define $\nu_{1}$ first on $[0,l_{0}]$ by $\nu_{1}(t):=\hat{\nu}(t)$. Since
$2\varepsilon\leq l_{0}$ and $\varepsilon\leq\hat{\varepsilon}(l)$, we have

$\begin{split}9\nu(2\varepsilon)\nu(l)=9\hat{\nu}(2\varepsilon)\nu_{1}(l)\leq
9\hat{\nu}(2\hat{\varepsilon}(l))\nu_{1}(l)\leq\kappa(l)\nu_{1}(l),\end{split}$
(3.26)

by the monotonicity of $\hat{\nu}$ and the definition of $\hat{\varepsilon}$.
Then it remains to satisfy

$\begin{split}&l>l_{0}\implies
9\nu_{1}(l-\hat{\varepsilon}(l))^{2}\leq\kappa(l)\nu_{1}(l),\\\
&l>4\delta\implies
3[M_{\delta}^{\chi}(\mu_{0}(l-\delta))\nu_{1}(l-\delta)]^{2}\leq\kappa(l)\nu_{1}(l),\end{split}$
(3.27)

as well as monotonicity of $\kappa\nu_{1}$, to apply Lemma 3.3.

We define $\nu_{1}$ to satisfy them iteratively starting from
$\nu_{1}=\hat{\nu}$ on $[0,l_{0}]$ which is already fixed above. Define the
sequence $l_{0}<l_{1}<\cdots$ by

$\begin{split}l_{n}:=\lambda(l_{n-1})\iff
l_{n-1}=\gamma(l_{n})=l_{n}-\hat{\varepsilon}(l_{n}).\end{split}$ (3.28)

Since $\hat{\varepsilon}$ is positive and continuous, this sequence diverges
$l_{n}\nearrow\infty$ as $n\to\infty$ (otherwise $\hat{\varepsilon}(l_{n})\to
0$ would contradict). Then we define $\nu_{1}$ on $(l_{n-1},l_{n}]$
inductively by

$\begin{split}&\nu_{1}(l):=\mathscr{M}(\hat{\nu}(l),\max(\nu_{L}(l),\nu_{H}(l))),\quad\mathscr{M}(\alpha,\beta):=\max(\alpha,\beta,\alpha\beta),\\\
&\quad\nu_{L}(l):=\tfrac{9\nu_{1}(\gamma(l))^{2}}{\kappa(l)},\quad\nu_{H}(l):=1_{l>3\delta}\tfrac{3[M_{\delta}^{\chi}(\mu_{0}(l-\delta))\nu_{1}(l-\delta)]^{2}}{\kappa(l)}.\end{split}$
(3.29)

Since $l_{n}-l_{n-1}=\hat{\varepsilon}(l_{n})<\delta$, the right hand side of
(LABEL:def_nu1) is well-defined if $\nu_{1}$ is already defined on
$[0,l_{n-1}]$. Thus $\nu_{1}:[0,\infty)\to[0,\infty)$ is well-defined. Since
$\mathscr{M}(\alpha,\beta)\geq\max(\alpha,\beta)$, we have
$\nu_{1}\geq\max(\hat{\nu},\nu_{L},\nu_{H})$, which implies (LABEL:cond_conv)
and $\kappa\nu_{1}\geq\nu_{0}$. In fact, we could define
$\nu_{1}=\max(\hat{\nu},\nu_{L},\nu_{H})$ to satisfy the properties (1)–(5),
but the factor $\alpha\beta$ of $\mathscr{M}$ is needed for the superlinear
property (6). The monotonicity of $\nu_{1},\kappa\nu_{1}$ is also immediate by
induction.

Similarly for the continuity, it suffices to check it at $l=l_{0}$ and
$3\delta$. Around $l=l_{0}<3\delta$, we have $\nu_{H}=0$ and
$\nu_{L}<\hat{\nu}$, because $\nu_{1}=\hat{\nu}$ for $l\leq l_{0}$ and

$\begin{split}9\hat{\nu}(\gamma(l_{0}))^{2}\leq
9\hat{\nu}(l_{0})^{2}\leq\kappa(l_{0})(1-l_{0}/\delta)\hat{\nu}(l_{0}).\end{split}$
(3.30)

Hence $\nu_{1}=\hat{\nu}$ around $l=l_{0}$, which is continuous. Around
$l=3\delta$, we have $\nu_{H}<\nu_{L}$, because

$\begin{split}3[M_{\delta}^{\chi}(\mu_{0}(2\delta))\nu_{1}(2\delta)]^{2}<9\nu_{1}(2\delta)^{2}<9\nu_{1}(\gamma(3\delta))^{2},\end{split}$
(3.31)

where we used $\mu_{0}(2\delta)\leq 1/2$, $M_{\delta}^{\chi}(1/2)^{2}<3$ and
$\hat{\varepsilon}(3\delta)<\delta$. Hence the definition of $\nu_{1}$ is the
same as the case of $l<3\delta$ around $l=3\delta$, which is continuous. Thus
we conclude $\nu_{1},\kappa\nu_{1}\in C^{\scaleto{\nearrow}{4pt}}_{0}$. The
monotonicity and continuity of the mapping
$(\mu_{0},\hat{\nu},-\kappa)\mapsto\nu_{1}$ follows also by induction.

Thus we have defined the mapping $\mathfrak{C}_{\delta}$ satisfying the
desired properties (1)–(5).

It remains to check the final (6), namely the superlinearity in $\nu$ for
multiplication. We follow the above procedure for the new input
$(\mu_{0},b\nu_{0},\kappa)$, distinguishing the modified output with subscript
$\bullet$. We have $\hat{\nu}_{\bullet}=b\hat{\nu}\geq\hat{\nu}$. Then
$\hat{\varepsilon}_{\bullet}\leq\hat{\varepsilon}$, $l_{0\bullet}\leq l_{0}$,
$\gamma_{\bullet}\geq\gamma$ and $\lambda_{\bullet}\leq\lambda$, by their
mapping monotonicity. Hence by induction $l_{n\bullet}\leq l_{n}$ for all
$n\geq 0$.

Let $l_{*}:=\sup\hat{\nu}^{-1}([0,1])$. Then by the assumption, $b\geq 1$ is a
constant on $l<l_{*}$, where $\nu_{1\bullet}\geq b\nu_{1}$ follows by
induction. For $l\geq l_{*}$, $\hat{\nu}_{0}\geq 1$ implies

$\begin{split}\nu_{1}=\mathscr{M}(\hat{\nu},\max(\nu_{L},\nu_{H}))=\hat{\nu}\max(1,\nu_{L},\nu_{H}),\end{split}$
(3.32)

hence

$\begin{split}\nu_{1\bullet}=\hat{\nu}_{\bullet}\max(1,\nu_{L\bullet},\nu_{H\bullet})=b\hat{\nu}\max(1,\nu_{L\bullet},\nu_{H\bullet}),\end{split}$
(3.33)

so $\nu_{1\bullet}\geq b\nu_{1}$ for $l\geq l_{*}$ follows from the
monotonicity of $\nu_{L},\nu_{H}$. ∎

The convolution estimate in $\mathcal{X}$ is transferred to
$\mathcal{X}^{\mu,\nu,\iota}_{\partial}$ by

######  Lemma 3.5.

Let $\mu_{k},\nu_{k}\in C^{\scaleto{\nearrow}{4pt}}$ for $k=0,1,2$ such that
for all $F,G\in\mathcal{X}$ and $l\in(0,\infty)$ we have
$\rho_{l}^{\mu_{0},\nu_{0}}(F*G)\leq\rho_{l}^{\mu_{1},\nu_{1}}(F)\rho_{l}^{\mu_{2},\nu_{2}}(G)$.
Then for any $\kappa\in C^{\scaleto{\nearrow}{4pt}}$, we have

$\begin{split}\|F*G\|_{\mathcal{X}^{\mu_{0},\nu_{0},\kappa\nu_{0}}_{\partial}}\leq\|F\|_{\mathcal{X}^{\mu_{1},\nu_{1},\kappa\nu_{1}}_{\partial}}\|G\|_{\mathcal{X}^{\mu_{2},\nu_{2}}}.\end{split}$
(3.34)

###### Proof.

Let $F\in\mathcal{X}^{\mu_{1},\nu_{1},\kappa\nu_{1}}_{\partial}$ and
$F=F_{0}+\partial_{1}F_{1}+\cdots+\partial_{d}F_{d}$ for some
$F_{0}\in\mathcal{X}^{\mu_{1},\nu_{1}}$ and
$F_{k}\in\mathcal{X}^{\mu_{1},\kappa\nu_{1}}$ ($k\geq 1$). Then, we have
$F*G=F_{0}*G+\sum_{1\leq k\leq d}\partial_{k}(F_{k}*G)$ with

$\begin{split}&\|F_{0}*G\|_{\mathcal{X}^{\mu_{0},\nu_{0}}}\leq\|F_{0}\|_{\mathcal{X}^{\mu_{1},\nu_{1}}}\|G\|_{\mathcal{X}^{\mu_{2},\nu_{2}}},\\\
&\|F_{k}*G\|_{\mathcal{X}^{\mu_{0},\kappa\nu_{0}}}\leq\|F_{k}\|_{\mathcal{X}^{\mu_{1},\kappa\nu_{1}}}\|G\|_{\mathcal{X}^{\mu_{2},\nu_{2}}},\end{split}$
(3.35)

by using Lemma 2.2. The infimum for $\\{F_{k}\\}_{0\leq k\leq d}$ yields the
conclusion. ∎

##  4\. Global wellposedness for distributions

Consider the Taylor expansion of
$\mathscr{H}=(\mathscr{H}_{1},\ldots,\mathscr{H}_{n_{3}})$ at
$z=0\in\mathbb{C}^{n_{2}}$

$\begin{split}\mathscr{H}_{k}(z)=\sum_{\alpha\in\mathbb{N}_{0}^{n_{2}},\
|\alpha|\geq
2}\tfrac{1}{\alpha!}\mathscr{H}_{k}^{(\alpha)}(0)z^{\alpha}\quad(k=1,\ldots,n_{3}),\end{split}$
(4.1)

which is assumed to be absolutely convergent in a $0$-neighborhood in
$\mathbb{C}^{n_{2}}$: There exists $R_{\mathscr{H}}>0$ such that for all
$r<R_{\mathscr{H}}$, we have

$\begin{split}\tilde{\mathscr{H}}(r):=\sum_{\alpha\in\mathbb{N}_{0}^{n_{2}},|\alpha|\geq
2}\tfrac{1}{\alpha!}\|\mathscr{H}^{(\alpha)}(0)\|_{\mathbb{C}^{n_{3}}}r^{|\alpha|-2}<\infty.\end{split}$
(4.2)

Note that $\mathscr{H}(0)=0$ is crucial for the global wellposedness, since
otherwise $\mathscr{H}$ could produce $0$ frequency in the solution, leading
to ODE blow-up in general (cf. Section 6.1). On the other hand,
$\mathscr{H}^{\prime}(0)=0$ is not an essential restriction, because the
linear part may be included in $L$.

The difference of $\mathscr{H}$ between two points may be estimated using

$\begin{split}\tilde{\mathscr{H}}^{(1)}(r):=\sum_{1\leq j\leq
n_{2}}\sum_{\alpha\in\mathbb{N}_{0}^{n_{2}},|\alpha|\geq
1}\tfrac{1}{\alpha!}\|\mathscr{H}^{(\alpha+e_{j})}(0)\|_{\mathbb{C}^{n_{3}}}r^{|\alpha|-1},\end{split}$
(4.3)

which is also finite for $r<R_{\mathscr{H}}$. The Fourier transform (in $x$)
of equation (1.1) is written for $\hat{u}:=\mathcal{F}u$ as

$\begin{split}&\hat{u}_{t}=\hat{L}\hat{u}+\hat{N}\hat{\mathscr{H}}(\hat{M}\hat{u}),\quad\hat{\mathscr{H}}_{k}(v):=\sum_{\alpha\in\mathbb{N}_{0}^{n_{2}}}\tfrac{1}{\alpha!}\mathscr{H}_{k}^{(\alpha)}(0)v^{*\alpha},\quad\hat{u}(0)=\hat{u}_{0},\end{split}$
(4.4)

where $v^{*\alpha}$ denotes the multi-index power in convolution, defined by

$\begin{split}v^{*\alpha}=v_{1}^{*\alpha_{1}}*\cdots*v_{n_{2}}^{*\alpha_{n_{2}}},\quad
v_{1}^{*(a+1)}=v_{1}^{*a}*v_{1},\quad v_{1}^{*0}=\delta,\end{split}$ (4.5)

and the Duhamel form is

$\begin{split}\hat{u}(t)=\mathcal{D}(\hat{u}_{0},\hat{u}):=e^{t\hat{L}(\xi)}\hat{u}_{0}+\int_{0}^{t}e^{(t-s)\hat{L}(\xi)}\hat{N}(\xi)\hat{\mathscr{H}}(\hat{M}(\xi)\hat{u}(s))ds.\end{split}$
(4.6)

Note that the right side is closed on $C(\mathbb{R};\mathcal{X})$. The idea is
to solve it by the fixed point theorem using some norm on
$C([0,\infty);\mathcal{X}^{\mu,\nu})$ for appropriate choice of $\mu,\nu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$, where $\nu$ depends on $t\geq 0$.

Fix $\delta\in(0,1]$ and $\mu\in C^{\scaleto{\nearrow}{4pt}}_{\delta}$
satisfying $2\mu(l-\delta)\leq\mu(l)$ for $l>4\delta$. If $\mu_{0}\in
C^{\scaleto{\nearrow}{4pt}}_{\delta}$ is given, then we may choose
$\mu:=\mathfrak{C}_{\delta}^{1}(\mu_{0})\geq\mu_{0}$ by Lemma 3.4. For the
coefficients $\hat{L},\hat{M},\hat{N}$, assume that they are smooth on
$\mathbb{R}^{d}_{>0}$ and that there exist $\theta\in[0,1)$ and
$c,\tilde{L},\tilde{M}\in C^{\scaleto{\nearrow}{4pt}}_{[1,\infty)}$ satisfying
for all $t,l>0$, $\varphi\in(\mathscr{D}_{<l})^{n_{1}}$, and
$\psi\in(\mathscr{D}_{<l})^{n_{2}}$,

$\begin{split}&\|e^{t\hat{L}^{\intercal}(\xi)}\varphi\|_{\mu(l),\delta}+\|t^{\theta}\hat{N}^{\intercal}(\xi)e^{t\hat{L}^{\intercal}(\xi)}\varphi\|_{\mu(l),\delta}\leq
c(l)e^{t\tilde{L}(l)}\|\varphi\|_{\mu(l)}^{+},\\\
&\|\hat{M}^{\intercal}(\xi)\psi\|_{\mu(l),\delta}\leq\tilde{M}(l)\|\psi\|_{\mu(l)}^{+}.\end{split}$
(4.7)

The factor $t^{\theta}$ is to exploit smoothing effect of $e^{t\hat{L}}$ when
it is available. Lemma 2.11 implies that (C-s) with $\mathrm{v}=e_{1}$ is a
sufficient condition for the above to hold for all $\mu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$. In particular, it is satisfied with
$\theta=0$ if all components of $\hat{L},\hat{M},\hat{N}$ are in $Y$ defined
in (2.59). For $d=1$, it is enough that $\hat{L},\hat{M},\hat{N}$ are smooth
on $\xi_{1}>0$ and bounded as $\xi_{1}\to+0$.

Now fix $\nu_{0}\in C^{\scaleto{\nearrow}{4pt}}_{0}$, and let
$l_{*}:=\sup\nu_{0}^{-1}([0,1])$. Without losing generality, we may assume
$l_{*}<\infty$ (otherwise we may multiply $\nu_{0}$ with a big constant
without changing the space $\mathcal{X}^{\mu,\nu_{0}}$). Then we may replace
$\tilde{L}$ in (LABEL:coeff_bd) by a larger function if necessary such that
$\tilde{L}(l)$ is a constant for $0<l<l_{*}$.

Fix $b_{0}>0$ and the initial data $\hat{u}_{0}\in\mathcal{X}^{\mu,\nu_{0}}$
such that $\rho^{\mu,\nu_{0}}(\hat{u}_{0})\leq b_{0}$. For any $\nu_{1}\in
C([0,\infty);C^{\scaleto{\nearrow}{4pt}}_{0})$, $\hat{u}\in
C([0,\infty);\mathcal{X}^{n_{1}})$ and $l>0$, we obtain from (2.10) and
(LABEL:coeff_bd)

$\begin{split}\rho^{\mu(l)}_{l}(\mathcal{D}(\hat{u}_{0},\hat{u}))&\leq
c(l)e^{t\tilde{L}(l)}\nu_{0}(l)\rho^{\mu,\nu_{0}}_{l}(\hat{u}_{0})\\\
&\quad+\int_{0}^{t}c(l)(t-s)^{-\theta}e^{(t-s)\tilde{L}(l)}\nu_{1}(s,l)\rho^{\mu,\nu_{1}(s)}_{l}(\hat{\mathscr{H}}(\hat{M}(\xi)\hat{u}(s)))ds.\end{split}$
(4.8)

Let $b_{1}:=2b_{0}$ and fix $\varepsilon>0$ small enough so that

$\begin{split}b_{1}\varepsilon<R_{\mathscr{H}},\quad
b_{1}\varepsilon\tilde{\mathscr{H}}(b_{1}\varepsilon)\Gamma(1-\theta)\leq
1/2,\quad
b_{1}\varepsilon\tilde{\mathscr{H}}^{(1)}(b_{1}\varepsilon)\Gamma(1-\theta)\leq
1/2.\end{split}$ (4.9)

Define $\nu_{*},\nu_{1},\nu_{2},\nu:[0,\infty)\to
C^{\scaleto{\nearrow}{4pt}}_{0}$ and $\kappa\in
C^{\scaleto{\searrow}{4pt}}_{(0,1]}$ by using Lemma 3.4

$\begin{split}&\nu_{*}(t,l):=\varepsilon^{-1}\nu_{0}(l)e^{t(1+\tilde{L}(l))},\quad\kappa:=1/(c\tilde{M}),\\\
&\nu_{2}(t):=\mathfrak{C}^{2}_{\delta}(\mu,\nu_{*}(t),\kappa),\quad\nu_{1}:=\kappa\nu_{2},\quad\nu(t):=\varepsilon\nu_{2}(t)/\tilde{M}=\varepsilon
c\nu_{1}(t).\end{split}$ (4.10)

At any fixed $t\geq 0$, let $v\in\mathcal{X}^{n_{1}}$ and $V:=\max_{1\leq
j\leq n_{2}}\rho^{\mu,\nu_{2}}(v_{j})\leq\rho^{\mu,\nu_{2}}(v)$. If
$V<R_{\mathscr{H}}$ then repeated use of (3.19) to the Taylor expansion of
$\hat{\mathscr{H}}$ yields

$\begin{split}\rho^{\mu,\nu_{1}}(\hat{\mathscr{H}}(v))\leq\sum_{|\alpha|\geq
2}\tfrac{1}{\alpha!}\|\mathscr{H}^{(\alpha)}(0)\|_{\mathbb{C}^{n_{3}}}V^{|\alpha|}\leq\tilde{\mathscr{H}}(V)V^{2}.\end{split}$
(4.11)

Note that every summand contains convolution as $|\alpha|\geq 2$. The
difference for any $v^{0},v^{1}\in\mathcal{X}^{n_{1}}$ is estimated in the
same way as above,

$\begin{split}&\tilde{V}:=\max_{k=0,1}\max_{j=1,\ldots,n_{2}}\rho^{\mu,\nu_{2}}(v^{k}_{j})<R_{\mathscr{H}}\\\
&\implies\rho^{\mu,\nu_{1}}(\hat{\mathscr{H}}(v^{0})-\hat{\mathscr{H}}(v^{1}))\leq\tilde{\mathscr{H}}^{(1)}(\tilde{V})\tilde{V}\rho^{\mu,\nu_{2}}(v^{0}-v^{1}).\end{split}$
(4.12)

Since $\tilde{M}\nu\leq\varepsilon\nu_{2}$, we have, again using (2.10) and
(LABEL:coeff_bd)

$\begin{split}\rho^{\mu,\nu_{2}}(\hat{M}\hat{u})\leq\varepsilon\rho^{\mu,\nu}(\hat{u}),\end{split}$
(4.13)

and, injecting it into (4.11),

$\begin{split}\rho^{\mu,\nu_{1}}(\hat{\mathscr{H}}(\hat{M}\hat{u}))\leq\tilde{\mathscr{H}}(\varepsilon\rho^{\mu,\nu}(\hat{u}))|\varepsilon\rho^{\mu,\nu}(\hat{u})|^{2},\end{split}$
(4.14)

for all $t\geq 0$, provided that
$\varepsilon\rho^{\mu,\nu}(\hat{u})<R_{\mathscr{H}}$. Injecting this into
(4.8), we obtain

$\begin{split}&\rho^{\mu(l)}_{l}(\mathcal{D}(\hat{u}_{0},\hat{u}))\leq
c(l)e^{t\tilde{L}(l)}\nu_{0}(l)b_{0}\\\
&\qquad+\int_{0}^{t}c(l)(t-s)^{-\theta}e^{(t-s)\tilde{L}(l)}\nu_{1}(s,l)\tilde{\mathscr{H}}(\varepsilon\rho^{\mu,\nu(s)}(\hat{u}(s)))|\varepsilon\rho^{\mu,\nu(s)}(\hat{u}(s))|^{2}ds.\end{split}$
(4.15)

Suppose that $\rho^{\mu,\nu(t)}(\hat{u}(t))\leq b_{1}$ for all $t\geq 0$.
Since $\nu=\varepsilon c\nu_{1}\geq\varepsilon c\nu_{*}\geq
c\nu_{0}e^{t\tilde{L}}$, we obtain

$\begin{split}\rho^{\mu,\nu(t)}(\mathcal{D}(\hat{u}_{0},\hat{u}(t)))\leq
b_{0}+\varepsilon\tilde{\mathscr{H}}(\varepsilon
b_{1})b_{1}^{2}\sup_{l>0}\int_{0}^{t}(t-s)^{-\theta}e^{s-t}\tfrac{\tilde{\nu}_{1}(s,l)}{\tilde{\nu}_{1}(t,l)}ds,\end{split}$
(4.16)

where

$\begin{split}\tilde{\nu}_{1}(t,l):=e^{-t(1+\tilde{L}(l))}\nu_{1}(t,l).\end{split}$
(4.17)

Now the superlinear property (6) of $\mathfrak{C}^{2}_{\delta}$ in Lemma 3.4
implies

$\begin{split}\tfrac{\tilde{\nu}_{1}(s)}{\tilde{\nu}_{1}(t)}=e^{(t-s)(1+\tilde{L})}\tfrac{\mathfrak{C}^{2}_{\delta}(\mu,\nu_{*}(s),\kappa)}{\mathfrak{C}^{2}_{\delta}(\mu,\nu_{*}(t),\kappa)}\leq\tfrac{\mathfrak{C}^{2}_{\delta}(\mu,e^{(t-s)(1+\tilde{L})}\nu_{*}(s),\kappa)}{\mathfrak{C}^{2}_{\delta}(\mu,\nu_{*}(t),\kappa)}=1,\end{split}$
(4.18)

where we used that $\tilde{L}$ is constant on $[0,l^{*}]$ and
$\nu_{*}(s,l^{*})\geq\nu_{0}(l^{*})=1$. Thus we obtain

$\begin{split}\sup_{t\geq 0}\rho^{\mu,\nu(t)}(\hat{u}(t))\leq
b_{1}\implies\rho^{\mu,\nu}(\mathcal{D}(\hat{u}_{0},\hat{u}))\leq
b_{0}+\varepsilon\tilde{\mathscr{H}}(\varepsilon
b_{1})b_{1}^{2}\Gamma(1-\theta),\end{split}$ (4.19)

and also

$\begin{split}&\max_{k=0,1}\sup_{t\geq 0}\rho^{\mu,\nu}(\hat{u}^{k})\leq
b_{1}\implies\\\ &\quad\sup_{t\geq
0}\rho^{\mu,\nu}(\mathcal{D}(\hat{u}_{0},\hat{u}^{0})-\mathcal{D}(\hat{u}_{0},\hat{u}^{1}))\leq\varepsilon
b_{1}\tilde{\mathscr{H}}^{(1)}(\varepsilon b_{1})\Gamma(1-\theta)\sup_{t\geq
0}\rho^{\mu,\nu}(\hat{u}^{0}-\hat{u}^{1}).\end{split}$ (4.20)

Hence by the choice of $\varepsilon$ in (4.9),
$\hat{u}\mapsto\mathcal{D}(\hat{u}_{0},\hat{u})$ is a contraction map on the
complete metric space

$\begin{split}\\{\hat{u}\in C([0,\infty);\mathcal{X}^{n_{1}})\mid\sup_{t\geq
0}\rho^{\mu,\nu(t)}(\hat{u}(t))\leq b_{1}=2b_{0}\\}\end{split}$ (4.21)

for the metric $\sup_{t\geq 0}\rho^{\mu,\nu}$, whose fixed point is a unique
global solution of (4.6). For any $T>0$, the integral in (4.6) may be defined
as the Bochner integral in $\mathcal{X}^{\mu,\nu_{T}}$ for $t<T$, for
$\nu_{T}:=\min_{0\leq t\leq T}\nu(t)\in C^{\scaleto{\nearrow}{4pt}}_{0}$, as
well as the Riemann integral in $\mathcal{X}$. In this sense, (4.6) is
equivalent to (LABEL:Feq). Thus we obtain a unique global solution $u$ of
(1.1) in the inverse Fourier transform of the above space (4.21).

######  Theorem 4.1.

Let $\delta\in(0,1]$ and $\mu\in C^{\scaleto{\nearrow}{4pt}}_{\delta}$ satisfy
$2\mu(l-\delta)\leq\mu(l)$ for $l>4\delta$. Suppose that there exist
$\theta\in[0,1)$ and $c,\tilde{L},\tilde{M}\in
C^{\scaleto{\nearrow}{4pt}}_{[1,\infty)}$ satisfying (LABEL:coeff_bd) for all
$t,l>0$, $\varphi\in(\mathscr{D}_{<l})^{n_{1}}$ and
$\psi\in(\mathscr{D}_{<l})^{n_{2}}$. Then for any $\nu_{0}\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ and any $b_{0}\in(0,\infty)$, there exists
$\nu\in C([0,\infty);C^{\scaleto{\nearrow}{4pt}}_{0})$ with
$\nu(t)\geq\nu_{0}$ for all $t\geq 0$ such that for any
$\hat{u}_{0}\in(\mathscr{D}^{\prime}(\mathbb{R}^{d}))^{n_{1}}$ satisfying
$\rho^{\mu,\nu_{0}}(\hat{u}_{0})\leq b_{0}$, there is a unique global solution
$u\in C([0,\infty);(\mathcal{F}^{-1}\mathcal{X})^{n_{1}})$ of (LABEL:Feq)
satisfying $\rho^{\mu,\nu(t)}(\hat{u}(t))\in L^{\infty}_{t}(0,\infty)$.

Note that the growth condition (LABEL:coeff_bd) on the coefficients
$\hat{L},\hat{M},\hat{N}$ depends on $\mu$ or the growth of initial condition
in $\xi_{1}$. If we do not care about the dependence, then it may be
simplified to the wellposedness in $\mathcal{F}^{-1}\mathcal{X}$. If the data
is supported on a frequency set closed under addition, then we may also
restrict the coefficients onto that set. In this paper, the global
wellposedness in a topological space $X$ for $t\geq 0$ is defined by

  1. (1)

For any $u(0)\in X$, there is a unique global solution $u\in C([0,\infty);X)$.

  2. (2)

For any sequence $\\{u_{n}(0)\\}_{n\in\mathbb{N}}\subset X$ convergent to some
$u_{\infty}(0)\in X$, let $u_{*}\in C([0,\infty);X)$ be the corresponding
global solutions. Then for any $T\in(0,\infty)$, $u_{n}(t)\to u_{\infty}(t)$
in $X$ uniformly for $0\leq t\leq T$ as $n\to\infty$.

The global wellposedness for $t\in\mathbb{R}$ is similarly defined, as well as
the local one. The above definition is indeed the standard one in the Hadamard
sense when $X$ is a metric space. However, if $X$ is not a sequential space,
then (2) implies only the sequential continuity for the solution map from the
initial data. Our space $\mathcal{F}^{-1}\mathcal{X}$ is not apparently
sequential, in which the topological continuity can be a challenging problem,
because the nonlinearity does not preserve the local convexity.

######  Corollary 4.2.

Let $\mathcal{R}\subset\mathbb{R}^{d}_{\geq 0}$ be a closed subset satisfying
$\mathcal{R}+\mathcal{R}\subset\mathcal{R}$. Suppose that
$\hat{L},\hat{M},\hat{N}$ are smooth on a neighborhood of
$\mathcal{R}\cap\mathbb{R}^{d}_{>0}$, satisfying (C-s) with
$\mathrm{v}=e_{1}$. Then the Cauchy problem (LABEL:Feq) is globally wellposed
in $(\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}})^{n_{1}}$ for $t\geq 0$.

The simplest choice is $\mathcal{R}=\mathbb{R}^{d}_{\geq 0}$, then
$\mathcal{X}_{\mathcal{R}}=\mathcal{X}$. As mentioned already, a sufficient
condition is that $\hat{L},\hat{M},\hat{N}\in Y$ (cf. (2.59)), and then the
global wellposedness holds for $t\in\mathbb{R}$, because the last condition is
invariant for the time inversion $t\mapsto-t$.

###### Proof.

Since the support property is closed in $\mathscr{D}^{\prime}(\mathbb{R}^{d})$
and $\mathcal{X}\hookrightarrow\mathscr{D}^{\prime}$,
$\mathcal{X}_{\mathcal{R}}\subset\mathcal{X}$ is also closed, and it is the
inductive limit of $\mathcal{X}_{\mathcal{R}}^{\mu,\nu}$. The closedness in
convolution is immediate from $\mathcal{R}+\mathcal{R}\subset\mathcal{R}$,
while the closedness is obvious for multiplication by the coefficients. Hence
all the operations in the construction of solutions, including the iteration
argument, are closed in $\mathcal{X}_{\mathcal{R}}$, so it suffices that the
coefficients $\hat{L},\hat{M},\hat{N}$ are defined around $\mathcal{R}$, and
we do not need their smoothness near $\xi_{1}=0$. Moreover, (C-s) with
$\mathrm{v}=e_{1}$ implies that (LABEL:coeff_bd) is satisfied for all $\mu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ and $t,l>0$.

For any $\delta\in(0,1]$ and $\mu_{0}\in
C^{\scaleto{\nearrow}{4pt}}_{\delta}$, let
$\mu:=\mathfrak{C}^{1}_{\delta}(\mu_{0})$. Then for any $\nu_{0}\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ and $b_{0}\in(0,\infty)$, by the above
result, there exists $\nu\in C([0,\infty);C^{\scaleto{\nearrow}{4pt}}_{0})$
such that for any $\hat{u}_{0}\in(\mathcal{X}_{\mathcal{R}})^{n_{1}}$
satisfying $\rho^{\mu,\nu_{0}}(\hat{u}_{0})\leq b_{0}$, there is a unique
global solution $u\in
C([0,\infty);(\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}})^{n_{1}})$ of (4.6)
satisfying $\rho^{\mu,\nu(t)}(\hat{u}(t))\in L^{\infty}(0,\infty)$. Thus we
obtain a global solution for any initial data
$u_{0}\in(\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}})^{n_{1}}$.

If we have two solutions $u_{1},u_{2}\in
C([0,\infty);(\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}})^{n_{1}})$ for the
same initial data $u_{0}$, then for any $T\in(0,\infty)$,
$\hat{u}_{1}([0,T])\cup\hat{u}_{2}([0,T])\subset(\mathcal{X}_{\mathcal{R}})^{n_{1}}$
is compact, so Proposition 2.10 yields some $\mu_{1},\nu_{1}\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ and $b_{1}\in(0,\infty)$ such that
$\rho^{\mu_{1},\nu_{1}}(\hat{u}_{j}(t))\leq b_{1}$ for $j=1,2$ and
$t\in[0,T]$. Then we may apply the uniqueness by contraction for some weight
$(\mu_{2},\nu_{2})\geq(\mu_{1},\nu_{1})$ and the size $b_{1}$ to deduce that
$u_{1}=u_{2}$ on $t\in[0,T]$. Since $T\in(0,\infty)$ is arbitrary, we obtain
the uniqueness in
$C([0,\infty);(\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}})^{n_{1}})$.

For the sequential continuity of the solution map, take any sequence of
initial data $\hat{u}_{0,n}\in(\mathcal{X}_{\mathcal{R}})^{n_{1}}$ convergent
to some $\hat{u}_{0,\infty}$ in $\mathcal{X}_{\mathcal{R}}$, and let $u_{n}$
be the corresponding sequence of solutions. Since
$\\{\hat{u}_{0,n}\\}_{n\in\mathbb{N}\cup\\{\infty\\}}\subset(\mathcal{X}_{\mathcal{R}})^{n_{1}}$
is compact, Proposition 2.10 yields some $\mu_{0},\nu_{0}\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ and $b_{0}\in(0,\infty)$ such that
$\\{\hat{u}_{0,n}\\}\subset B_{\mathcal{X}^{\mu_{0},\nu_{0}}}(0,b_{0}/4)$, and
that $\hat{u}_{0,n}\to\hat{u}_{0,\infty}$ in
$\mathcal{X}^{\mu_{0},\nu_{0},\iota}_{\partial}$ for any $\iota\in
C^{\scaleto{\nearrow}{4pt}}_{+}$. After replacing $\mu_{0}$ with
$\mu:=\mathfrak{C}^{1}_{\delta}(\mu_{0})$ as before, we use Lemma 2.11 to
obtain the weight $\gamma$ so that the coefficient multipliers are bounded
also on $\mathcal{X}^{\mu,\nu,\gamma\nu}_{\partial}$. For the convolutions in
the nonlinearity, we use Lemma 3.5. Thus we obtain, in the same way as
(LABEL:cont_diff_est), for $\hat{u}^{0},\hat{u}^{1}$ satisfying $\sup_{t\geq
0}\rho^{\mu,\nu}(\hat{u}^{k})\leq b_{1}$,

$\begin{split}&\|\mathcal{D}(\hat{u}_{0}^{0},\hat{u}^{0})-\mathcal{D}(\hat{u}_{0}^{1},\hat{u}^{1})\|_{L^{\infty}_{t}\mathcal{X}^{\mu,\nu,\gamma\nu}_{\partial}}\\\
&\quad\leq\rho^{\mu,\nu_{0}}(\hat{u}_{0}^{0}-\hat{u}_{0}^{1})+4\varepsilon
b_{1}\tilde{\mathscr{H}}^{(1)}(\varepsilon
b_{1})\Gamma(1-\theta)\|\hat{u}^{0}-\hat{u}^{1}\|_{L^{\infty}_{t}\mathcal{X}^{\mu,\nu,\gamma\nu}_{\partial}},\end{split}$
(4.22)

where the extra factor $4$ comes from the factor $2$ in Lemma 2.11 used twice.
Since $\rho^{\mu,\nu_{0}}(\hat{u}_{0,n})<b_{0}/4$, the contraction on (4.21)
implies that $\rho^{\mu,\nu}(\hat{u}_{n})\leq b_{1}/4=b_{0}/2$ for all
$n\in\mathbb{N}\cup\\{\infty\\}$ and $t\geq 0$. Then (LABEL:diff_est_Xp) with
the smallness of $b_{1}\varepsilon$ in (4.9) implies

$\begin{split}\|\hat{u}_{n}-\hat{u}_{\infty}\|_{L^{\infty}_{t}\mathcal{X}^{\mu,\nu,\gamma\nu}_{\partial}}\leq
2\rho^{\mu,\nu_{0}}(\hat{u}_{0,n}-\hat{u}_{0,\infty})\to 0,\end{split}$ (4.23)

as $n\to\infty$. Combining it with the uniform bound
$\rho^{\mu,\nu}(\hat{u}_{n})\leq b_{0}/2$, we obtain by Proposition 2.10 that
$\hat{u}_{n}(t)\to\hat{u}_{\infty}(t)$ in $\mathcal{X}$ locally uniformly for
all $t\geq 0$. ∎

A notable special case of $\mathcal{X}_{\mathcal{R}}$ is when $\mathcal{R}$ is
a lattice:

$\begin{split}\exists L\in(0,\infty)^{d},\quad\mathcal{R}\subset\prod_{1\leq
j\leq d}\tfrac{2\pi}{L_{j}}\mathbb{Z}.\end{split}$ (4.24)

In this case, $\mathcal{R}$ should avoid the boundary points on $\xi_{1}=0$,
because such distributions do not satisfy the decay condition at $\xi_{1}=0$,
(X-i). Under the above constraint, the solutions in
$\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$ are periodic in space: For any
$F\in\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$ and $k\in\mathbb{Z}^{d}$, we
have

$\begin{split}F(x_{1}+k_{1}L_{1},\ldots,x_{d}+k_{d}L_{d})=F(x)\end{split}$
(4.25)

in the distribution ($\mathcal{F}^{-1}\hat{\mathscr{D}}^{\prime}$) sense.

On the other hand, if $F\in\mathcal{F}^{-1}L^{1}_{loc}(\mathbb{R}^{d})$, then
$F$ is decaying at the spatial infinity in the average sense: By Riemann-
Lebesgue,

$\begin{split}\mathcal{F}\mathscr{D}(\mathbb{R}^{d})\ni\forall\varphi,\quad\lim_{|x|\to\infty}(\varphi*F)(x)=0.\end{split}$
(4.26)

For example, the Littlewood-Paley decomposition of $F$ satisfies the decay.
Thus the space $\mathcal{F}^{-1}\mathcal{X}$ contains both the periodic
solutions and the decaying solutions, which is in a sharp contrast with the
usual settings for the Cauchy problem, where the boundary condition at
infinity has to be distinguished by the solution space.

The condition (C-s) requires uniform boundedness of the coefficients
$\hat{L},\hat{M},\hat{N}$ in the orthogonal directions. For much more general
equations, it may be satisfied by restricting $\operatorname{supp}\hat{u}_{0}$
in the orthogonal directions, which is the case if
$\mathcal{R}\cap\mathbb{R}^{d}_{<l}$ is bounded for each $l\in(0,\infty)$,
then (C-0) implies (C-s) with $\theta=0$ for all $t\in\mathbb{R}$. Thus
Corollary 4.2 implies Theorem 1.2. Note that in this case (LABEL:Feq) makes
sense in $\mathcal{X}_{\mathcal{R}}$, with the classical time derivative in
the topology of $\mathcal{X}$. Moreover, $\mathcal{X}_{\mathcal{R}}$ is a bit
simplified:

######  Lemma 4.3.

If $\mathcal{R}\subset\mathbb{R}^{d}$ is closed such that
$\mathcal{R}\cap\mathbb{R}^{d}_{<l}$ is bounded for each $l>0$, then

$\begin{split}\mathcal{X}_{\mathcal{R}}=\\{F\in\mathscr{D}^{\prime}(\mathbb{R}^{d})\mid\operatorname{supp}F\subset\mathcal{R},\