1 Introduction
Wiener [Reference Wiener15] studied the class
$A(\mathbb{T})$
of all continuous functions on the torus
$\mathbb{T}$
with the absolutely convergent Fourier series, and proved that
$F(z)=1/z$
operates on
$A(\mathbb{T})$
. Lévy [Reference Lévy9] gave an extension of this result by showing that an analytic function operates on
$A(\mathbb{T})$
, which is called the Wiener–Lévy Theorem. After that, there are many papers about operating functions on the same function spaces with respect to Fourier series by Helson, Kahane, Katznelson, Rudin, and so forth (see [Reference Helson, Kahane, Katznelson and Rudin6, Reference Rudin10]).
In this paper, we give the characterization of operating functions on modulation spaces
$M^{p,1}(\mathbb{R})$
and Wiener amalgam spaces
$W^{p,1}(\mathbb{R})$
. The modulation spaces
$M^{p,q}$
and the Wiener amalgam spaces
$W^{p,q}$
are two function spaces introduced by Feichtinger [Reference Feichtinger, Krishna, Radha and Thangavelu4]. The precise definition of these spaces will be given in Section 2, but the main idea of these spaces is to consider the space variable and the variable of its Fourier transform simultaneously. Let
$F$
be a complex-valued function on
$\mathbb{R}^{2}$
and
$X=M^{p,1}(\mathbb{R})$
or
$W^{p,1}(\mathbb{R})$
. If
$F(\text{Re}f,\text{Im}f)\in X$
for every
$f\in X$
, then we say that
$F$
operates on
$X$
.
Concerning modulation spaces, Wiener amalgam spaces and operating functions, the following theorem is known.
Theorem A.
(Bhimani [Reference Bhimani2], Bhimani and Ratnakumar [Reference Bhimani and Ratnakumar3]) Let
$1\leqslant p<\infty$
and
$F$
be a complex-valued function on
$\mathbb{R}^{2}$
. Suppose
$X=M^{p,1}(\mathbb{R})$
or
$W^{p,1}(\mathbb{R})$
. If
$F$
operates on
$X$
, then
$F$
is a real analytic function on
$\mathbb{R}^{2}$
with
$F(0)=0$
. Conversely, if
$F$
is a real analytic on
$\mathbb{R}^{2}$
with
$F(0)=0$
, then
$F$
operates on
$M^{1,1}(\mathbb{R})(=W^{1,1}(\mathbb{R}))$
.
We remark that Theorem A answers negatively the open problem posed by Ruzhansky–Sugimoto–Wang [Reference Ruzhansky, Sugimoto and Wang11] about the general power type nonlinearity of the form
$|u|^{\unicode[STIX]{x1D6FC}}u$
. In [Reference Bhimani2] and [Reference Bhimani and Ratnakumar3], they also propose an open problem: is the condition in Theorem A sufficient or not for
$p>1$
? This paper gives an affirmative answer to this problem. Our result is as follows.
Theorem 1.1. Let
$1\leqslant p<\infty$
and
$F$
be a real analytic function on
$\mathbb{R}^{2}$
with
$F(0)=0$
. Suppose
$X=M^{p,1}(\mathbb{R})$
or
$W^{p,1}(\mathbb{R})$
. Then
$F$
operates on
$X$
.
Since
$F(s,t)=\frac{1}{(1+s^{2})(1+t^{2})}$
is a real analytic function on
$\mathbb{R}^{2}$
, we obtain
$F(\text{Re}f,\text{Im}f)\in X$
for all
$f\in X$
by Theorem 1.1.
Combining Theorems A and 1.1, we have the following characterization of operating functions on
$M^{p,1}(\mathbb{R})$
and
$W^{p,1}(\mathbb{R})$
.
Corollary 1.2. Let
$1\leqslant p<\infty$
and
$F$
be a complex-valued function on
$\mathbb{R}^{2}$
. Suppose
$X=M^{p,1}(\mathbb{R})$
or
$W^{p,1}(\mathbb{R})$
. Then
$F$
operates on
$X$
if and only if
$F$
is a real analytic function with
$F(0)=0$
.
2 Preliminaries
The following notation will be used throughout this article. We write
${\mathcal{S}}(\mathbb{R})$
to denote the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on
$\mathbb{R}$
and
${\mathcal{S}}^{\prime }(\mathbb{R})$
to denote the space of tempered distributions on
$\mathbb{R}$
, that is, the topological dual of
${\mathcal{S}}(\mathbb{R})$
. The Fourier transform is defined by
$\widehat{f}(\unicode[STIX]{x1D709})=\int _{\mathbf{R}}f(x)e^{-ix\cdot \unicode[STIX]{x1D709}}dx$
and the inverse Fourier transform by
$f^{\vee }(x)=(2\unicode[STIX]{x1D70B})^{-1}\widehat{f}(-x)$
. We also write
$C_{c}^{\infty }(\mathbb{R})$
to denote the set of all complex-valued infinitely differentiable functions on
$\mathbb{R}$
with compact support.
2.1 Real analytic function
A complex-valued function
$F$
on
$\mathbb{R}^{2}$
is said to be real analytic on
$\mathbb{R}^{2}$
if for each
$(s_{0},t_{0})\in \mathbb{R}^{2}$
,
$F$
has a power series expansion
$$\begin{eqnarray}F(s,t)=\mathop{\sum }_{m,n=0}^{\infty }a_{mn}(s-s_{0})^{m}(t-t_{0})^{n}\end{eqnarray}$$
which converges absolutely in a neighborhood of
$(s_{0},t_{0})$
.
2.2 Short-time Fourier transform
Let
$f\in {\mathcal{S}}^{\prime }(\mathbb{R})$
and
$g\in {\mathcal{S}}(\mathbb{R})$
. Then the short-time Fourier transform
$V_{g}f$
of
$f$
with respect to the window
$g$
is defined by
$$\begin{eqnarray}V_{g}f(x,\unicode[STIX]{x1D709})=\langle f(t),g(t-x)e^{it\cdot \unicode[STIX]{x1D709}}\rangle =\int _{\mathbb{ R}}f(t)\overline{g(t-x)}e^{-it\cdot \unicode[STIX]{x1D709}}\,dt.\end{eqnarray}$$
2.3 Modulation spaces
Let
$1\leqslant p,q\leqslant \infty$
and
$g\in {\mathcal{S}}(\mathbb{R})\setminus \{0\}$
. Then the modulation space
$M^{p,q}(\mathbb{R})=M^{p,q}$
consists of all
$f\in {\mathcal{S}}^{\prime }(\mathbb{R})$
such that the norm
$$\begin{eqnarray}\Vert f\Vert _{M^{p,q}(\mathbb{R})}=\left(\int _{\mathbb{R}}\left(\int _{\mathbb{R}}|V_{g}f(x,\unicode[STIX]{x1D709})|^{p}dx\right)^{q/p}d\unicode[STIX]{x1D709}\right)^{1/q}\end{eqnarray}$$
is finite (with usual modifications if
$p=\infty$
or
$q=\infty$
).
We note that since
$V_{g}\overline{f}(x,\unicode[STIX]{x1D709})=\overline{V_{\overline{g}}f(x,-\unicode[STIX]{x1D709})}$
, we have
$$\begin{eqnarray}\Vert \overline{f}\Vert _{M^{p,q}}=\Vert f\Vert _{M^{p,q}},~\Vert \text{Re}f\Vert _{M^{p,q}}\leqslant \Vert f\Vert _{M^{p,q}},~\Vert \text{Im}f\Vert _{M^{p,q}}\leqslant \Vert f\Vert _{M^{p,q}}.\end{eqnarray}$$
We collect basic properties of modulation spaces in the following lemma (see [Reference Feichtinger, Krishna, Radha and Thangavelu4, Reference Gröchenig5, Reference Ruzhansky, Sugimoto and Wang11–Reference Toft14] for more details).
Lemma 2.1.
(1) The space
$M^{p,q}(\mathbb{R})$
is a Banach space, whose definition is independent of the choice of
$g$
. More precisely, we have for$$\begin{eqnarray}\Vert f\Vert _{M_{[g_{0}](\mathbb{R})}^{p,q}}\leqslant C\Vert g\Vert _{M_{[g_{0}](\mathbb{R})}^{1,1}}\Vert f\Vert _{M_{[g](\mathbb{R})}^{p,q}}\end{eqnarray}$$
$f\in M^{p,q}(\mathbb{R})$
and
$g_{0},g\in {\mathcal{S}}(\mathbb{R}^{n})\setminus \{0\}$
, where $$\begin{eqnarray}\Vert f\Vert _{M_{[g]}^{p,q}(\mathbb{R})}=\left\Vert \Vert V_{g}f(x,\unicode[STIX]{x1D709})\Vert _{L^{p}(\mathbb{R}_{x})}\right\Vert _{L^{q}(\mathbb{R}_{\unicode[STIX]{x1D709}})}.\end{eqnarray}$$
(2)
$M^{p,min\{p,p^{\prime }\}}(\mathbb{R}^{n}){\hookrightarrow}L^{p}(\mathbb{R}^{n}){\hookrightarrow}M^{p,max\{p,p^{\prime }\}}(\mathbb{R}^{n}).$
In particular, we have
$M^{2,2}(\mathbb{R}^{n})=L^{2}(\mathbb{R}^{n})$
.(3)
$M^{p,1}(\mathbb{R})\subset C(\mathbb{R})$
, that is,
$f$
is continuous on
$\mathbb{R}$
if
$f\in M^{p,1}(\mathbb{R})$
.(4) If
$p_{1}\leqslant p_{2}$
and
$q_{1}\leqslant q_{2}$
, then
$M^{p_{1},q_{1}}(\mathbb{R}){\hookrightarrow}M^{p_{2},q_{2}}(\mathbb{R})$
.(5)
$(\text{Density and duality})$
If
$p,q<\infty$
, then
${\mathcal{S}}(\mathbb{R})$
is dense in
$M^{p,q}(\mathbb{R})$
and
$(M^{p,q}(\mathbb{R}))^{\prime }=M^{p^{\prime },q^{\prime }}(\mathbb{R})$
.(6)
$(\text{Multiplication})$
If
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}$
and
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}+1$
, then Moreover, we have$$\begin{eqnarray}\Vert fg\Vert _{M^{p,q}(\mathbb{R})}\leqslant C\Vert f\Vert _{M^{p_{1},q_{1}}(\mathbb{R})}\Vert g\Vert _{M^{p_{2},q_{2}}(\mathbb{R})},\quad f,g\in {\mathcal{S}}(\mathbb{R}).\end{eqnarray}$$
that is,$$\begin{eqnarray}\Vert fg\Vert _{M^{p,1}(\mathbb{R})}\leqslant C\Vert f\Vert _{M^{p,1}(\mathbb{R})}\Vert g\Vert _{M^{p,1}(\mathbb{R})},\quad f,g\in M^{p,1}(\mathbb{R}),\end{eqnarray}$$
$M^{p,1}(\mathbb{R})$
is a multiplication algebra.(7)
$(\text{Convolution})$
If
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}+1$
and
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$
, then $$\begin{eqnarray}\Vert f\ast g\Vert _{M^{p,q}(\mathbb{R})}\leqslant C\Vert f\Vert _{M^{p_{1},q_{1}}(\mathbb{R})}\Vert g\Vert _{M^{p_{2},q_{2}}(\mathbb{R})},\quad f,g\in {\mathcal{S}}(\mathbb{R}).\end{eqnarray}$$
(8)
$(\text{Dilation property})$
There exists constants
$C,C^{\prime }>0$
such that for$$\begin{eqnarray}\displaystyle \Vert f_{\unicode[STIX]{x1D706}}\Vert _{M^{\infty ,1}(\mathbb{R})} & {\leqslant} & \displaystyle C\Vert f\Vert _{M^{\infty ,1}(\mathbb{R})},\quad f\in M^{\infty ,1}(\mathbb{R}),\nonumber\\ \displaystyle \Vert (f_{\unicode[STIX]{x1D706}})^{\wedge }\Vert _{M^{1,\infty }(\mathbb{R})} & {\leqslant} & \displaystyle C^{\prime }\Vert \widehat{f}\Vert _{M^{1,\infty }(\mathbb{R})},\quad \widehat{f}\in M^{1,\infty }(\mathbb{R})\nonumber\end{eqnarray}$$
$0<\unicode[STIX]{x1D706}\leqslant 1$
. Here we denote
$f_{\unicode[STIX]{x1D706}}(x)=f(\unicode[STIX]{x1D706}x)$
.
2.4 Wiener amalgam spaces
Let
$1\leqslant p,q\leqslant \infty$
and
$g\in {\mathcal{S}}(\mathbb{R})\setminus \{0\}$
. Then the Wiener amalgam space
$W^{p,q}(\mathbb{R})=W^{p,q}$
consists of all
$f\in {\mathcal{S}}^{\prime }(\mathbb{R})$
such that the norm
$$\begin{eqnarray}\Vert f\Vert _{W^{p,q}(\mathbb{R})}=\left(\int _{\mathbb{R}}\left(\int _{\mathbb{R}}|V_{g}f(x,\unicode[STIX]{x1D709})|^{q}d\unicode[STIX]{x1D709}\right)^{p/q}dx\right)^{1/p}\end{eqnarray}$$
is finite (with usual modifications if
$p=\infty$
or
$q=\infty$
).
We remark that since
$V_{g}f(x,\unicode[STIX]{x1D709})=(2\unicode[STIX]{x1D70B})^{-1}e^{-ix\cdot \unicode[STIX]{x1D709}}V_{\widehat{g}}\widehat{f}(\unicode[STIX]{x1D709},-x)$
, we have
$$\begin{eqnarray}C_{1}\Vert \widehat{f}\Vert _{M^{q,p}(\mathbb{R})}\leqslant \Vert f\Vert _{W^{p,q}(\mathbb{R})}\leqslant C_{2}\Vert \widehat{f}\Vert _{M^{q,p}(\mathbb{R})}\end{eqnarray}$$
for some positive constants
$C_{1}$
and
$C_{2}$
. This implies that the definition of
$W^{p,q}$
is independent of the choice of
$g$
since the modulation space
$M^{q,p}$
is independent of the choice of
$g$
. For the same reason,
$W^{p,q}$
has the following properties.
Lemma 2.2. Let
$1\leqslant p,p_{1},p_{2},q,q_{1},q_{2}\leqslant \infty$
and
$\frac{1}{p}+\frac{1}{p^{\prime }}=1=\frac{1}{q}+\frac{1}{q^{\prime }}$
.
(1)
$M^{p,1}(\mathbb{R}){\hookrightarrow}W^{p,1}(\mathbb{R})$
.(2)
$W^{p,1}(\mathbb{R})\subset C(\mathbb{R})$
, that is,
$f$
is continuous on
$\mathbb{R}$
if
$f\in W^{p,1}(\mathbb{R})$
.(3) If
$p_{1}\leqslant p_{2}$
and
$q_{1}\leqslant q_{2}$
, then
$W^{p_{1},q_{1}}(\mathbb{R}){\hookrightarrow}W^{p_{2},q_{2}}(\mathbb{R})$
.(4)
$(\text{Multiplication})$
If
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}$
and
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}+1$
, then Moreover, we have$$\begin{eqnarray}\Vert fg\Vert _{W^{p,q}(\mathbb{R})}\leqslant C\Vert f\Vert _{W^{p_{1},q_{1}}(\mathbb{R})}\Vert g\Vert _{W^{p_{2},q_{2}}(\mathbb{R})},\quad f,g\in {\mathcal{S}}(\mathbb{R}).\end{eqnarray}$$
that is,$$\begin{eqnarray}\Vert fg\Vert _{W^{p,1}(\mathbb{R})}\leqslant C\Vert f\Vert _{W^{p,1}(\mathbb{R})}\Vert g\Vert _{W^{p,1}(\mathbb{R})},\quad f,g\in W^{p,1}(\mathbb{R}),\end{eqnarray}$$
$W^{p,1}(\mathbb{R})$
is a multiplication algebra.(5)
$(\text{Dilation property})$
There exists constant
$C>0$
such that Here we denote$$\begin{eqnarray}\displaystyle & \Vert f_{\unicode[STIX]{x1D706}}\Vert _{W^{\infty ,1}(\mathbb{R})}\leqslant C\Vert f\Vert _{W^{\infty ,1}(\mathbb{R})},\quad 0<\unicode[STIX]{x1D706}\leqslant 1,~f\in W^{\infty ,1}(\mathbb{R}). & \displaystyle \nonumber\end{eqnarray}$$
$f_{\unicode[STIX]{x1D706}}(x)=f(\unicode[STIX]{x1D706}x)$
.
We also recall the following characterization of modulation spaces
$M^{p,1}(\mathbb{R})$
and Wiener amalgam spaces
$W^{p,1}(\mathbb{R})$
.
Definition 2.3. Let
$1\leqslant p\leqslant \infty$
and
$f$
be a function defined on
$\mathbb{R}$
. Suppose
$X=M^{p,1}(\mathbb{R})$
or
$W^{p,1}(\mathbb{R})$
.
(1) Let
$x_{0}\in \mathbb{R}$
. If there exist a neighborhood
$V$
of
$x_{0}$
and a function
$g\in X$
satisfying
$f(x)=g(x)$
for every
$x\in V$
, then we say
$f$
belongs to
$X$
locally at a point
$x_{0}\in \mathbb{R}$
.(2) If there exist a compact set
$K\subset \mathbb{R}$
and
$h\in X$
satisfying
$f(x)=h(x)$
for all
$x\in \mathbb{R}\setminus K$
, then we say
$f$
belongs to
$X$
at
$\infty$
.
We denote by
$X_{loc}$
, the space of functions that are locally in
$X$
at each point
$x_{0}\in \mathbb{R}$
.
Lemma 2.4. (cf. [Reference Bhimani2, Lemma 4.3], [Reference Bhimani and Ratnakumar3, Propositions 3.12 and 3.13], [Reference Katznelson8])
Let
$1\leqslant p\leqslant \infty$
and
$f$
be a function defined on
$\mathbb{R}$
. Suppose
$X=M^{p,1}(\mathbb{R})$
or
$W^{p,1}(\mathbb{R})$
.
(1)
$f\in X_{loc}$
, if and only if
$\unicode[STIX]{x1D711}f\in X$
for every
$\unicode[STIX]{x1D711}\in C_{c}^{\infty }(\mathbb{R})$
.(2)
$f$
belongs to
$X$
at
$\infty$
, if and only if there exists a function
$\unicode[STIX]{x1D711}\in C_{c}^{\infty }(\mathbb{R})$
such that
$(1-\unicode[STIX]{x1D711})f\in X$
.(3) If
$f\in X_{loc}$
and
$f$
belongs to
$X$
at
$\infty$
, then
$f\in X$
.
2.5 The space
$A(\mathbb{T})$
Let
$\mathbb{T}=\mathbb{R}/2\unicode[STIX]{x1D70B}\mathbb{Z}$
be the torus. Then the space
$A(\mathbb{T})$
consists of all continuous function on
$\mathbb{T}$
having an absolutely convergent Fourier series, that is, the function
$f$
for which
$$\begin{eqnarray}\Vert f\Vert _{A(\mathbb{T})}:=\mathop{\sum }_{n=-\infty }^{\infty }|\widehat{f}(n)|<\infty \quad \text{with }\widehat{f}(n)=\frac{1}{2\unicode[STIX]{x1D70B}}\int _{0}^{2\unicode[STIX]{x1D70B}}f(t)e^{-int}dt.\end{eqnarray}$$
Lemma 2.5. [Reference Kahane7, pp. 56–57]
Let
$\unicode[STIX]{x1D706}\in (0,1)$
and define the
$2\unicode[STIX]{x1D70B}$
-periodic function
$\mathbf{V}_{\unicode[STIX]{x1D706}}\in C(\mathbb{T})$
by
$$\begin{eqnarray}\mathbf{V}_{\unicode[STIX]{x1D706}}(x)=2\unicode[STIX]{x1D6E5}_{2\unicode[STIX]{x1D706}}(x)-\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D706}}(x),\quad x\in [-\unicode[STIX]{x1D70B},\unicode[STIX]{x1D70B}],\end{eqnarray}$$
where
$\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D706}}(x)=\max \{0,1-\frac{|x|}{\unicode[STIX]{x1D706}}\}.$
Moreover, we define
$$\begin{eqnarray}\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}(x)=\mathbf{V}_{\unicode[STIX]{x1D706}}(x-x_{0})\end{eqnarray}$$
for
$x_{0}\in \mathbb{R}$
. Then for every
$g\in A(\mathbb{T})$
with
$g(x_{0})=0$
, we have
$\Vert \mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}g\Vert _{A(\mathbb{T})}\rightarrow 0$
as
$\unicode[STIX]{x1D706}\rightarrow 0$
.
We state the following lemma whose proof is almost a repetition of arguments in Bényi and Oh [Reference Bényi and Oh1, Proposition B.1] and Bhimani [Reference Bhimani2, Proposition 3.3].
Lemma 2.6. Let
$1\leqslant p\leqslant \infty$
and
$\unicode[STIX]{x1D719}\in C_{c}^{\infty }(\mathbb{R})$
with
$\text{supp }\unicode[STIX]{x1D719}\subset (k\unicode[STIX]{x1D70B},(k+2)\unicode[STIX]{x1D70B})$
for some
$k\in \mathbb{Z}$
. Suppose
$X=M^{p,1}(\mathbb{R})$
or
$W^{p,1}(\mathbb{R})$
. Then there exist positive constants
$C_{\unicode[STIX]{x1D719}}^{1}$
and
$C_{\unicode[STIX]{x1D719}}^{2}$
(which depend on
$\unicode[STIX]{x1D719}$
but do not depend on
$k$
) such that
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D719}f\Vert _{A(\mathbb{T})}\leqslant C_{\unicode[STIX]{x1D719}}^{1}\Vert f\Vert _{X},\quad f\in X,\end{eqnarray}$$
and
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D719}f\Vert _{X}\leqslant C_{\unicode[STIX]{x1D719}}^{2}\Vert f\Vert _{A(\mathbb{T})},\quad f\in A(\mathbb{T}).\end{eqnarray}$$
3 The Proof of Theorem 1.1
We first prove that if
$F$
is real analytic, then
$F(\text{Re}f,\text{Im}f)\in X_{loc}$
for every
$f\in X$
.
Proposition 3.1. Let
$1\leqslant p<\infty$
and
$F$
be a complex-valued real analytic function on
$\mathbb{R}^{2}$
with
$F(0)=0$
. If
$f\in X$
, then
$F(\text{Re}f,\text{Im}f)$
belongs to
$X$
locally at
$x_{0}$
for all
$x_{0}\in \mathbb{R}$
.
Proof. Let
$f\in X$
and
$x_{0}\in \mathbb{R}$
. Set
$f(x_{0})=s_{0}+it_{0}$
with
$s_{0},t_{0}\in \mathbb{R}$
. Since
$F$
is real analytic, for some
$\unicode[STIX]{x1D6FF}>0$
$$\begin{eqnarray}F(s,t)=\mathop{\sum }_{m,n=0}^{\infty }a_{mn}(s-s_{0})^{m}(t-t_{0})^{n}\end{eqnarray}$$
and
$$\begin{eqnarray}\mathop{\sum }_{m,n=0}^{\infty }|a_{mn}|~|s-s_{0}|^{m}|t-t_{0}|^{n}<\infty\end{eqnarray}$$
if
$|s-s_{0}|<\unicode[STIX]{x1D6FF}$
and
$|t-t_{0}|<\unicode[STIX]{x1D6FF}$
.
Let
$\unicode[STIX]{x1D719}\in C_{c}^{\infty }(\mathbb{R})$
such that
$\unicode[STIX]{x1D719}(x)=1$
near
$x_{0}$
and
$\text{supp }\unicode[STIX]{x1D719}\subset (x_{0}-\frac{1}{10},x_{0}+\frac{1}{10})$
, and define
$g_{j}$
$(j=1,2)$
by
$$\begin{eqnarray}g_{1}(x)=(\text{Re}f)(x)-s_{0},\quad g_{2}(x)=(\text{Im}f)(x)-t_{0}.\end{eqnarray}$$
We note that
$\unicode[STIX]{x1D719}g_{j}\in X$
and
$(\unicode[STIX]{x1D719}g_{j})(x_{0})=0$
for
$j=1,2$
. Thus by Lemma 2.6 we can easily see
$\unicode[STIX]{x1D719}g_{j}\in A(\mathbb{T})$
. So, by Lemma 2.5 we obtain
$\Vert \mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{j}\Vert _{A(\mathbb{T})}\rightarrow 0$
as
$\unicode[STIX]{x1D706}\rightarrow 0.$
Thus, for any
$\unicode[STIX]{x1D700}\in (0,\unicode[STIX]{x1D6FF})$
there exists
$\unicode[STIX]{x1D706}_{0}>0$
such that
$$\begin{eqnarray}\Vert \mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{j}\Vert _{A(\mathbb{T})}<\frac{\unicode[STIX]{x1D700}}{(1+C_{\unicode[STIX]{x1D719}}^{2})(1+C_{X})}\end{eqnarray}$$
for
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{0}$
, where
$C_{X}$
denotes the constant with
$$\begin{eqnarray}\Vert f\cdot g\Vert _{X}\leqslant C_{X}\Vert f\Vert _{X}\Vert g\Vert _{X},\quad f,g\in X.\end{eqnarray}$$
Hence by Lemma 2.6 again, we obtain
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D719}\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{j}\Vert _{X}\leqslant C_{\unicode[STIX]{x1D719}}^{2}\Vert \mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{j}\Vert _{A(\mathbb{T})}<\frac{\unicode[STIX]{x1D700}}{1+C_{X}}\end{eqnarray}$$
for
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{0}$
. Now we define
$$\begin{eqnarray}G(x)=\mathop{\sum }_{m,n=0}^{\infty }a_{mn}(\unicode[STIX]{x1D719}\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{1})^{m}(x)(\unicode[STIX]{x1D719}\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{2})^{n}(x)\end{eqnarray}$$
for
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{0}$
. Since
$\unicode[STIX]{x1D700}<\unicode[STIX]{x1D6FF}$
and
$a_{00}=0$
, by Lemmas 2.1 and 2.2 we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{m,n=0}^{\infty }\Vert a_{mn}(\unicode[STIX]{x1D719}\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{1})^{m}(\unicode[STIX]{x1D719}\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{2})^{n}\Vert _{X}\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant C_{X}\mathop{\sum }_{m,n=0}^{\infty }|a_{mn}|~\Vert (\unicode[STIX]{x1D719}\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{1})^{m}\Vert _{X}\Vert \unicode[STIX]{x1D719}(\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}\unicode[STIX]{x1D719}g_{2})^{n}\Vert _{X}\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant C_{X}\mathop{\sum }_{m,n=0}^{\infty }|a_{mn}|\unicode[STIX]{x1D700}^{m+n}<\infty ,\nonumber\end{eqnarray}$$
and thus
$G\in X.$
Moreover, since
$\unicode[STIX]{x1D719}(x)=1$
near
$x_{0}$
, we have
$$\begin{eqnarray}\unicode[STIX]{x1D719}(x)\mathbf{V}_{\unicode[STIX]{x1D706}}^{x_{0}}(x)\unicode[STIX]{x1D719}(x)g_{j}(x)=g_{j}(x)\end{eqnarray}$$
near
$x_{0}$
, and thus
$$\begin{eqnarray}\displaystyle G(x) & = & \displaystyle \mathop{\sum }_{m,n=0}^{\infty }a_{mn}(\text{Re}f(x)-s_{0})^{m}(\text{Im}f(x)-t_{0})^{n}\nonumber\\ \displaystyle & = & \displaystyle F(\text{Re}f,\text{Im}f)\nonumber\end{eqnarray}$$
near
$x_{0}$
. This completes the proof.◻
Next, we prove that if
$F$
is real analytic, then
$F(\text{Re}f,\text{Im}f)$
belongs
$X$
at
$\infty$
for every
$f\in X$
. For this, we prepare the following proposition.
Proposition 3.2. Let
$1\leqslant p<\infty$
and
$f\in X$
. For any
$\unicode[STIX]{x1D700}>0$
, there exists a real-valued function
$\unicode[STIX]{x1D6F9}\in C_{c}^{\infty }(\mathbb{R})$
such that
$$\begin{eqnarray}\Vert (1-\unicode[STIX]{x1D6F9})f\Vert _{X}<\unicode[STIX]{x1D700}.\end{eqnarray}$$
Proof. Let
$f\in X$
,
$\unicode[STIX]{x1D700}>0$
and
$\unicode[STIX]{x1D719}$
be a real-valued function in
$C_{c}^{\infty }(\mathbb{R})$
with
$\unicode[STIX]{x1D719}(0)=1$
. Since
${\mathcal{S}}(\mathbb{R})$
is dense in
$X$
, there exists
$g\in {\mathcal{S}}(\mathbb{R})$
such that
$$\begin{eqnarray}\Vert f-g\Vert _{X}<\frac{\unicode[STIX]{x1D700}}{2(1+C_{0}\Vert \unicode[STIX]{x1D719}\Vert _{X})},\end{eqnarray}$$
where
$C_{0}$
is decided later. We also recall the fact that for any
$g\in {\mathcal{S}}(\mathbb{R})(\subset M^{1,1}(\mathbb{R})=W^{1,1}(\mathbb{R}))$
, there exists
$\unicode[STIX]{x1D706}_{0}\in (0,1)$
such that
$$\begin{eqnarray}\Vert (1-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}})g\Vert _{M^{1,1}(\mathbb{R})}<\frac{\unicode[STIX]{x1D700}}{2}\end{eqnarray}$$
for
$\unicode[STIX]{x1D706}\in (0,\unicode[STIX]{x1D706}_{0})$
, where
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}}(x)=\unicode[STIX]{x1D719}(\unicode[STIX]{x1D706}x)$
(see for example [Reference Bhimani and Ratnakumar3, Proposition 3.14]).
Now we define
$\unicode[STIX]{x1D6F9}\in C_{c}^{\infty }(\mathbb{R})$
by
$\unicode[STIX]{x1D6F9}(x)=\unicode[STIX]{x1D719}(\unicode[STIX]{x1D706}x)$
. Then we have
$$\begin{eqnarray}\displaystyle \Vert (1-\unicode[STIX]{x1D6F9})f\Vert _{X} & {\leqslant} & \displaystyle \Vert (1-\unicode[STIX]{x1D6F9})(f-g)\Vert _{X}+\Vert (1-\unicode[STIX]{x1D6F9})g\Vert _{X}\nonumber\\ \displaystyle & = & \displaystyle \Vert f-g-\unicode[STIX]{x1D6F9}(f-g)\Vert _{X}+\Vert (1-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}})g\Vert _{X}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \Vert f-g\Vert _{X}+C\Vert \unicode[STIX]{x1D6F9}\Vert _{Y}\Vert f-g\Vert _{X}+\Vert (1-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}})g\Vert _{M^{1,1}(\mathbb{R})}\nonumber\\ \displaystyle & {<} & \displaystyle (1+C\Vert \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}}\Vert _{Y})\Vert f-g\Vert _{X}+\frac{\unicode[STIX]{x1D700}}{2},\nonumber\end{eqnarray}$$
for
$\unicode[STIX]{x1D706}\in (0,\unicode[STIX]{x1D706}_{0})$
, where
$Y=M^{\infty ,1}$
(if
$X=M^{p,1}$
) or
$=W^{\infty ,1}$
(if
$X=W^{p,1}$
). By Lemmas 2.1 and 2.2 we have
$$\begin{eqnarray}C\Vert \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}}\Vert _{Y}\leqslant C_{0}\Vert \unicode[STIX]{x1D719}\Vert _{Y}\end{eqnarray}$$
for
$\unicode[STIX]{x1D706}\in (0,\unicode[STIX]{x1D706}_{0})$
. Hence we have
$$\begin{eqnarray}\Vert (1-\unicode[STIX]{x1D6F9})f\Vert _{X}<\unicode[STIX]{x1D700}.\hspace{220.00034pt}\square\end{eqnarray}$$
Corollary 3.3. Let
$1\leqslant p<\infty$
and
$f_{1},\ldots ,f_{N}\in X$
. For any
$\unicode[STIX]{x1D700}>0$
, there exists a real-valued function
$\unicode[STIX]{x1D6F9}\in C_{c}^{\infty }(\mathbb{R})$
such that
$$\begin{eqnarray}\Vert (1-\unicode[STIX]{x1D6F9})f_{i}\Vert _{X}<\unicode[STIX]{x1D700},\quad i=1,\ldots ,N.\end{eqnarray}$$
Proposition 3.4. Let
$1\leqslant p<\infty$
and
$F$
be a real analytic function on
$\mathbb{R}^{2}$
with
$F(0)=0$
. If
$f\in X$
, then there exists
$H\in X$
such that
$$\begin{eqnarray}H(x)=F(\text{Re}f(x),\text{Im}f(x))\end{eqnarray}$$
except for some compact set in
$\mathbb{R}$
.
Proof. Let
$f\in X$
. Since
$F$
is real analytic, for some
$\unicode[STIX]{x1D6FF}>0$
$$\begin{eqnarray}F(s,t)=\mathop{\sum }_{m,n=0}^{\infty }a_{mn}s^{m}t^{n}\end{eqnarray}$$
and
$$\begin{eqnarray}\mathop{\sum }_{m,n=0}^{\infty }|a_{mn}|~|s|^{m}|t|^{n}<\infty\end{eqnarray}$$
if
$|s|<\unicode[STIX]{x1D6FF}$
and
$|t|<\unicode[STIX]{x1D6FF}$
. By Corollary 3.3 there exists a real-valued function
$\unicode[STIX]{x1D6F9}\in C_{c}^{\infty }(\mathbb{R})$
such that
$$\begin{eqnarray}\displaystyle \Vert (1-\unicode[STIX]{x1D6F9})\text{Re}f\Vert _{X}<\frac{\unicode[STIX]{x1D6FF}}{1+C_{X}},\quad \Vert (1-\unicode[STIX]{x1D6F9})\text{Im}f\Vert _{X}<\frac{\unicode[STIX]{x1D6FF}}{1+C_{X}}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$C_{X}$
denotes the constant with
$$\begin{eqnarray}\Vert f\cdot g\Vert _{X}\leqslant C_{X}\Vert f\Vert _{X}\Vert g\Vert _{X},\quad f,g\in X.\end{eqnarray}$$
From this and Lemmas 2.1 and 2.2,
$$\begin{eqnarray}H(x)=\mathop{\sum }_{m,n=0}^{\infty }a_{mn}((1-\unicode[STIX]{x1D6F9}(x))\text{Re}f(x))^{m}((1-\unicode[STIX]{x1D6F9}(x))\text{Im}f(x))^{n}\end{eqnarray}$$
converges in
$X.$
Since
$\unicode[STIX]{x1D6F9}\in C_{c}^{\infty }(\mathbb{R})$
,
$H(x)=F(\text{Re}f(x),\text{Im}f(x))$
except for some compact set in
$\mathbb{R}$
.◻
Acknowledgments
The authors wish to thank the anonymous referee for carefully reading their paper and for providing some helpful comments. This work was supported by JSPS KAKENHI, Grant Numbers JP16K17606, JP26400129.
