Besov spaces associated with the Harmonic oscillator

TL;DR

This paper introduces and thoroughly explores Besov spaces linked to the harmonic oscillator, detailing their fundamental properties, embeddings, and bilinear estimates, advancing the mathematical understanding of these function spaces.

Contribution

It provides a comprehensive analysis of Besov spaces associated with the harmonic oscillator, including new insights into their properties and estimates.

Findings

Detailed characterization of Besov spaces linked to the harmonic oscillator

Embedding properties and bilinear estimates established

Enhanced understanding of function space behavior in quantum harmonic analysis

Abstract

The Besov space associated with the harmonic oscillator is introduced and thoroughly explored in this paper. It provides a comprehensive summary of the fundamental concepts of the Besov spaces, their embedding properties, bilinear estimates, and related topics.

Equations

$$H=-\Delta +|x{|}^2.$$

$$\{\begin{array}{l}D(H)=\{f\in L^2|\Delta f,|x{|}^{2}f\in L^2\},\\ Hf=(-\Delta +|x{|}^2)f,f\in D(H).\end{array}$$

$$\begin{array}{rl}& \underset{\lambda \to \infty }{\lim }E(\lambda )f=f\text{ in }{L}^2,\text{for all }f\in L^2,\\ & Hf={\int }_{-\infty }^{\infty }\lambda dE(\lambda )f\text{ in }{L}^2,f\in D(H).\end{array}$$

$$\Vert Hf{\Vert }_{L^2}\simeq \Vert (1+H)f{\Vert }_{L^2}.$$

$$\Vert f{\Vert }_{L^2}\le \frac{1}{\inf \sigma (H)}\Vert Hf{\Vert }_{L^2}.$$

$$\mathrm{supp}{\varphi }_0\subset [2^{-1},2],\sum _{j\in \mathbb{Z}}{\varphi }_0(2^{-j}\lambda )=1\text{ for }\lambda >0,$$

$${\varphi }_j(\lambda ):={\varphi }_0(2^{-j}\lambda ),\text{ for }\lambda \in \mathbb{R}.$$

$$B_{p,q}^s(H):=\{f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)|\Vert f{\Vert }_{B_{p,q}^s(H)}<\infty \},$$

$$\|f\|_{B^{s}_{p,q}(H)}:=\Big{\|}\Big{\{}2^{sj}\|\phi_{j}(\sqrt{H})f\|_{L^{p}}\Big{\}}_{j=-1}^{\infty}\Big{\|}_{\ell^{q}}.$$

$$2^{j_0+1}\le \inf \sigma (H).$$

$$f=\sum _{j\in \mathbb{Z}}{\varphi }_j(\sqrt{H})f=\sum _{j=-1}^{\infty }{\varphi }_j(\sqrt{H})f,$$

$${\varphi }_j(\sqrt{H})f=0\text{ if }j<-1.$$

$$\Big{\|}\Big{\{}2^{sj}\|\phi_{j}(\sqrt{H})f\|_{L^{p}}\Big{\}}_{j=-1}^{\infty}\Big{\|}_{\ell^{q}}\simeq\|\psi(\sqrt{H})f\|_{L^{p}}+\Big{\|}\Big{\{}2^{sj}\|\phi_{j}(\sqrt{H})f\|_{L^{p}}\Big{\}}_{j\in\mathbb{N}}\Big{\|}_{\ell^{q}},$$

$$\|f\|_{B^{-s}_{p^{\prime},q^{\prime}}}\simeq\sup_{\|g\|_{B^{s}_{p,q}}=1}\Big{|}\sum_{j=-1}^{\infty}\int_{\mathbb{R}^{d}}\phi_{j}(\sqrt{H})f(x)\overline{\Phi_{j}(\sqrt{H})g(x)}~dx\Big{|},$$

$$\color[rgb]{0,0,0}\|g\|_{B^{s}_{p,q}}\leq C\sup_{f\in Q_{p^{\prime},q^{\prime}}^{-s}}|\langle f,g\rangle|.$$

$$C^{-1}\Vert f{\Vert }_{B_{p,\infty }^0(H)}\le \Vert f{\Vert }_{L^p}\le C\Vert f{\Vert }_{B_{p,1}^0(H)}$$

$$s-\frac{d}{p}=\theta\Big{(}-\frac{d}{r}\Big{)}+(1-\theta)\Big{(}s_{0}-\frac{d}{r_{0}}\Big{)},$$

$$-\frac{d}{r}\ne s_0-\frac{d}{r_0},\{\begin{array}{ll}s\le (1-\theta )s_0& \text{if }\max \{r,r_0\}\le p,\\ s<(1-\theta )s_0& \text{if }\min \{r,r_0\}\le p<\max \{r,r_0\}.\end{array}$$

$$\Vert f{\Vert }_{B_{p,1}^s(H)}\le \Vert f{\Vert }_{B_{r,\infty }^0(H)}^{\theta }\Vert f{\Vert }_{B_{r_0,\infty }^{s_0}(H)}^{1-\theta }.$$

$$\begin{array}{r}\underset{j\ge -1}{\sup }\Vert \varphi (2^{-j}\sqrt{H}){\Vert }_{L^p\to L^p}<\infty ,\end{array}$$

$$\begin{split}fg=&\sum_{k}\phi_{k}(\sqrt{H})f\sum_{l}\phi_{l}(\sqrt{H})g=\Big{(}\sum_{k\leq l-2}+\sum_{l\leq k-2}+\sum_{|k-l|\leq 1}\Big{)}(f_{k}g_{l}),\\ =&f\varolessthan g+f\varogreaterthan g+f\varodot g.\end{split}$$

$$\Vert f\text{varolessthan}g{\Vert }_{B_{p,q}^s(H)}\le C\Vert f{\Vert }_{L^{p_1}}\Vert g{\Vert }_{B_{p_2,q}^s(H)}.$$

$$\Vert f\text{varolessthan}g{\Vert }_{B_{p,q}^{s+r}(H)}\le C\Vert f{\Vert }_{B_{p_1,\infty }^s}\Vert g{\Vert }_{B_{p_2,q}^r(H)}.$$

$$\Vert f\text{varodot}g{\Vert }_{B_{p,q}^s(H)}\le C\Vert f{\Vert }_{B_{p_1,q_1}^{s_1}(H)}\Vert g{\Vert }_{B_{p_2,q_2}^{s_2}(H)}.$$

$$k\le l-2,l\le k-2,|k-l|\le 1$$

$$k<l-N_0,l>k-N_0,|k-l|\le N_0.$$

$$s>0,1\le p,p_1,p_2,p_3,p_4,q\le \infty ,\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p_3}+\frac{1}{p_4}.$$

$$\|fg\|_{B^{s}_{p,q}(H)}\leq C\Big{(}\|f\|_{B^{s}_{p_{1},q}(H)}\|g\|_{L^{P_{2}}}+\|f\|_{L^{p_{3}}}\|g\|_{B^{s}_{p_{4},q}(H)}\Big{)},$$

$$\Vert fg{\Vert }_{B_{p,q}^s}\le C\Vert f{\Vert }_{B_{p_1,q}^s}\Vert g{\Vert }_{B_{p_2,q}^r},$$

$$\Vert f\text{varolessthan}g{\Vert }_{B_{p,q}^s}\lesssim \Vert f\text{varolessthan}g{\Vert }_{B_{p,q}^{s+r}}\lesssim \Vert f{\Vert }_{B_{p_1,q}^s}\Vert g{\Vert }_{B_{p_2,q}^r},$$

Full text

1 Department of Mathematics, Faculty of Fundamental Science and Engineering, Waseda University,

169-8555 Tokyo, Japan;

2 Mathematical Institute, Faculty of Science, Tohoku University,

980-8578, Sendai, Japan;

1. Introduction

We study the Besov space based on the Littlewood-Paley decomposition associated with the harmonic oscillator on $\mathbb{R}^{d}$, for $d\geq 1$,

[TABLE]

The operator $H$ is one of the important operators in quantum mechanics. Moreover, when rigorously analyzing physically significant nonlinear equations, for example, the Gross-Pitaevskii equation [2, 3, 4, 10, Y-2004], the Sobolev spaces and Besov spaces based on this harmonic oscillator as the fundamental operator, as well as the bilinear estimates in these spaces, are extremely useful.

The eigenvalues of $H$ are well known, and the eigenfunctions are written explicitly using Hermite functions. In this paper, we decompose the spectrum of $H$ to introduce dyadic decomposition, and utilize the boundedness of the spectral multiplier to introduce Besov spaces associated with the operator $H$. The aim of this paper is to establish basic estimates in the Besov spaces associated with the operator $H$.

The Hermite Besov spaces have been introduced by Petrushev and Xu [PX-2008] (see also [BD-2015, BDY-2012]), in a different way from this paper, based on the Calderón reproducing formula for the identity operator. The spaces introduced by them are equivalent to ours (see Theorem 1.4 below). Since we prefer the setting better adapted to the analysis of partial differential equations, in this paper, we introduce Besov spaces associated with $H$ following the argument in [IMT-2019], whose key feature is that it deals with Besov spaces based on the Dirichlet Laplacian.

We see that $H$ has a self-adjoint realization on $L^{2}$, and can be written as follows.

[TABLE]

By applying the spectral theorem, the resolution of the identity $\{E(\lambda)\}_{\lambda\in\mathbb{R}}$ exists such that

[TABLE]

We denote $\sigma(H)$ the spectrum of $H$. It is known that $\inf\sigma(H)$ is strictly positive, which implies the equivalence between the two norms of the homogeneous and the non-homogeneous types,

[TABLE]

Remark that

[TABLE]

Because of this, the two norms define one function space, and we will use the left hand side to introduce Besov spaces. We take $\phi_{0}\in C_{0}^{\infty}(\mathbb{R})$ a non-negative function on $\mathbb{R}$ such that

[TABLE]

and $\{\phi_{j}\}_{j\in\mathbb{Z}}$ is defined by

[TABLE]

Definition 1.1**.**

For $s\in\mathbb{R}$, $1\leq p,q\leq\infty$, $B^{s}_{p,q}(H)$ is defined by

[TABLE]

where

[TABLE]

The first number $j=-1$ in the sequence is determined by $j_{0}\in\mathbb{Z}$ such that

[TABLE]

For simplicity, we will write the sum over $j\in\mathbb{Z}$, and explicitly indicate the sum over $j$ with $j\geq-1$ when a clarification is needed. On the partition of the unity it reads that

[TABLE]

since

[TABLE]

We notice that the positivity of the spectrum of $H$ implies the following equivalence.

[TABLE]

where $\psi\in C_{0}^{\infty}(\mathbb{R})$ satisfies $\psi+\sum_{j=1}^{\infty}\phi_{j}=1$.

Let us introduce the basic properties of the Besov space $B^{s}_{p,q}(H)$ in the following proposition.

Proposition 1.1**.**

Let $s,\alpha\in\mathbb{R}$ and $1\leq p,q,r\leq\infty$. The following (i)-(vii) hold:

1. (i)

$B^{s}_{p,q}(H)$* is a Banach space and enjoys $\mathcal{S}(\mathbb{R}^{d})\hookrightarrow B^{s}_{p,q}(H)\hookrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{d})$.* 2. (ii)

If $1\leq p,q<\infty$ and $1/p+1/p^{\prime}=1/q+1/q^{\prime}=1$, then the dual space of $B^{s}_{p,q}(H)$ is $B^{-s}_{p^{\prime},q^{\prime}}(H)$. Moreover, for any $p,q\in[1,\infty]$, we have the following norm equivalence.

[TABLE]

where $\Phi_{j}=\phi_{j-1}+\phi_{j}+\phi_{j+1}$. Denote $Q^{-s}_{p^{\prime},q^{\prime}}:=\{f\in\mathcal{S},\|f\|_{B^{-s}_{p^{\prime},q^{\prime}}}\leq 1\}$. If $g\in\mathcal{S}^{\prime}$, we then have

[TABLE] 3. (iii)

If $r\leq p$, then $B^{s+d(\frac{1}{r}-\frac{1}{p})}_{r,q}(H)\hookrightarrow B^{s}_{p,q}(H)$. 4. (iv)

For every $f\in B^{s+\alpha}_{p,q}(H)$, $H^{\frac{\alpha}{2}}f\in B^{s}_{p,q}(H)$. 5. (v)

If $s<\alpha$, the space $B^{\alpha}_{p,\infty}(H)$ is compactly embedded into $B^{s}_{p,1}(H).$ 6. (vi)

There exists a constant $C>0$ such that

[TABLE] 7. (vii)

Let $s,s_{0}>0$, $p,r,r_{0}\in[1,\infty]$, $\theta\in(0,1)$ satisfy

[TABLE]

[TABLE]

Then we have

[TABLE]

Remark that the above items (i)–(iv), where $H$ is replaced by the Dirichlet Laplacian, have been already established in [IMT-2019], and those arguments can be applied similarly for the case $H$. The equivalent norm in (ii) follows from the property of duality. We will thus give a brief proof only for (v), (vi) and (vii) in this paper, in Appendix.

We here mention that the uniform boundedness of the operators $\phi_{j}(\sqrt{H})$ in $j$ holds.

[TABLE]

for all $1\leq p\leq\infty$ and $\phi\in\mathcal{S}(\mathbb{R})$. This holds by the same reason as in Section 8 in [IMT-RMI]. Initially, $\phi_{j}(\sqrt{H})$ is defined on $L^{2}$ with an application of the spectral theorem and is a bounded operator on $L^{2}$. This uniform boundedness (1.2) on $L^{p}$, $1\leq p\leq\infty$ plays a very important role to establish the theory of Besov spaces, we thus give a brief proof in the appendix.

We write the Bony paraproduct formula.

[TABLE]

where $f_{k}=\phi_{k}(\sqrt{H})f$ and $g_{l}=\phi_{l}(\sqrt{H})g$. Then, we have the bilinear estimates as follows.

Proposition 1.2**.**

Let $s,r\in\mathbb{R},1\leq p,p_{1},p_{2},q\leq\infty$ and $1/p=1/p_{1}+1/p_{2}$.

• (i)

There exists a constant $C>0$ such that

[TABLE]

• (ii)

If $s<0$, then

[TABLE]

• (iii)

If $s=s_{1}+s_{2}>0$, $1\leq q_{1},q_{2}\leq\infty$ and $1/q=1/q_{1}+1/q_{2}$, then

[TABLE]

Remark. In the definition of the para product above we divided into the cases:

[TABLE]

but any number $N_{0}\in\mathbb{N}$ for this division works for the proof, for example we may consider

[TABLE]

As a simple application of this Proposition 1.2, we have the following bilinear estimates.

Theorem 1.1**.**

• (i)

Let

[TABLE]

Then there exists a positive constant $C$ such that

[TABLE]

for all $f\in B^{s}_{p_{1},q}(H)\cap L^{p_{3}},g\in L^{p_{2}}\cap B^{s}_{p_{4},q}(H)$.

• (ii)

Let $s<0<r$, $s+r>0$ and $1\leq p,p_{1},p_{2}\leq\infty$ with $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$ Then, we have

[TABLE]

for $f\in B^{s}_{p_{1},q}(H)$ and $g\in B^{r}_{p_{2},q}(H)$.

Remark. Indeed, by Proposition 1.2, we can estimate each paraproduct under the parameters’ condition of (ii) as follows.

[TABLE]

Remark. Only for the purpose to give a proof for Theorem 1.1, it is sufficient to use the decomposition $fg$ into two parts (see the proof):

[TABLE]

**Remark. ** The inequality (i) for the Besov spaces associated with the Laplacian is well-known (see e.g. [RS_1996]). In the Sobolev spaces associated to $H$, the existing estimate is as follows.

[TABLE]

where $s>0$, $1<p,p_{1},p_{2},p_{3},p_{4}<\infty$ and $1/p=1/p_{1}+1/p_{2}=1/p_{3}+1/p_{4}$ (see [10]). This is proved by the following equivalence between the norms ([7], also see Proposition 2.1),

[TABLE]

using the Hölder inequality and the bilinear estimate for the standard Laplacian $(-\Delta)^{s}$ (see e.g.,  [11]). We underline that in the Besov spaces $B^{s}_{p,q}(H)$, it is possible to include the indices $p=1$ and $\infty$, and the present paper gives a proof for this fact.

Following the similar arguments in the paper [Iw-2018], we have the following results about the smoothing effects of the semigroup $\{e^{-tH}\}_{t\geq 0}$.

Theorem 1.2**.**

*Let $t\geq 0$, $s,s_{1},s_{2}\in\mathbb{R}$, $1\leq p,p_{1},p_{2},q,q_{1},q_{2}\leq\infty$.

(i) $e^{-tH}$ is a bounded linear operator in $B^{s}_{p,q}(H)$, i.e., there exists a constant $C>0$ such that for any $f\in B^{s}_{p,q}(H)$*

[TABLE]

*for all $t\geq 0$.

(ii) If $s_{2}\geq s_{1}$, $p_{1}\leq p_{2}$ and*

[TABLE]

then there exists a constant $C>0$ such that

[TABLE]

for any $f\in B^{s_{1}}_{p_{1},q_{1}}(H)$.

We also have the continuity property of the semigroup in our Besov spaces associated with $H$ as well as the standard Besov spaces.

Theorem 1.3**.**

*Let $s\in\mathbb{R}$, $1\leq p,q\leq\infty$ and $1/p+1/p^{\prime}=1/q+1/q^{\prime}=1$.

(i) Assume that $q<\infty$ and $f\in B^{s}_{p,q}(H)$. Then*

[TABLE]

(ii)* Assume that $1<p\leq\infty$, $q=\infty$ and $f\in B^{s}_{p,\infty}(H)$. Then $e^{-tH}f$ converges to $f$ in the dual weak sense as $t\to 0$, namely,*

[TABLE]

for any $g\in B^{-s}_{p^{\prime},1}(H)$.

We have an equivalent norm of the Besov spaces by using the semigroup.

Theorem 1.4**.**

Let $s,s_{0}\in\mathbb{R}$, $s_{0}>s/2$ and $1\leq p,q\leq\infty$. Recall $j_{0}$, which was introduced in (1.1) (i.e. $j_{0}=-1$). Then there exists a constant $C>0$ such that

[TABLE]

for any $f\in B^{s}_{p,q}(H)$, where $X$ can be $L^{p}$ or $B^{0}_{p,r}(H)$ with $1\leq r\leq\infty$.

**Remark. ** Recalling $f_{j}=\phi_{j}(\sqrt{H})f$, we can estimate $\|(tH)^{s_{0}}e^{-tH}f_{j}\|_{L^{p}}$ by $(t2^{2j})^{s_{0}}e^{-t2^{2j}}\|f_{j}\|_{L^{p}}$, which leads us to

[TABLE]

On the other hand, the change of variable $t\mapsto 2^{-2j}t$ in the middle integral above implies

[TABLE]

Then, summing up in $j$ results in (1.7). Since $j_{0}=-1$ which is related to $\inf\sigma(H)>0$, the interval of the integral in the middle term of (1.7) is only a bounded interval near $t=0$. We may see from this fact that our case corresponds to the inhomogeneous case of the Besov space for the standard Laplacian.

The following theorem states the maximal regularity estimate for the semigroup.

Theorem 1.5**.**

Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Assume that $u_{0}\in B^{s+2-\frac{2}{q}}_{p,q}(H)$, $f\in L^{q}(0,\infty;B^{s}_{p,q}(H))$. Let $u$ be given by

[TABLE]

Then there exists a constant $C>0$ independent of $u_{0}$ and $f$ such that

[TABLE]

We finally mention a generalization of our results for the specific operator $H=-\Delta+|x|^{2}$ to more general Schrödinger operators with a potential $V$ that diverges at infinity, as studied in [Y-2004], where the potential $V$ is assumed to satisfy the following conditions for some $m>2$ :

1. (a)

There exist constants $R>0$ and $C_{1}>0$ such that

[TABLE] 2. (b)

For every multi-index $\alpha$, there exists a constant $C_{\alpha}>0$ such that

[TABLE]

It is possible to introduce the Besov spaces associated with $-\Delta+V(x)$, as was done in [IMT-2019]. We can then expect that the corresponding results stated in the introduction of this paper hold for these generalized operators. We also remark that the diverging property of the potential is crucial for showing the compact embedding in Proposition 1.1 (v).

This paper is organized as follows. Essentially, our tools for the bilinear estimates in Theorem 1.1 rely on the Leibnitz rules applied to the operator $H$, and commutative properties with the multiplication by $x$ and the derivatives $\nabla$. We prepare some lemmas to describe such practical results in Section 2. Section 3 is devoted to the proof of Theorem 1.1. Since Theorems 1.2-1.5 may be proved in a similar way in the existing literature, we will briefly add explanations on the proofs in the Appendix.

2. Preliminary

In this section, we prepare some useful lemmas for the proof of Theorem 1.1.

Proposition 2.1**.**

([7]) For any $p\in(1,\infty)$ and $\alpha>0$, there exists a constant $C>0$ such that

[TABLE]

Lemma 2.1**.**

For every multi-indices $\alpha,\beta$, there exists a constant $C>0$ such that

[TABLE]

for all $f\in L^{2}$ satisfying $H^{\frac{|\alpha|+|\beta|}{2}}f\in L^{2}$.

Proof.

If $\alpha=(0,\cdots,0)$ or $\beta=(0,\cdots,0)$, then Proposition 2.1 proves the inequality (2.1). It is sufficient to prove the case when $\alpha\not=0$ and $\beta\not=0$. Also it is sufficient to prove for $f\in\mathcal{S}(\mathbb{R}^{d})$ by the density argument.

When $|\alpha|=|\beta|=1$, we estimate $x_{j}\partial_{x_{k}}f$ ($j,k=1,2,\cdots,d$),

[TABLE]

Since $H\partial_{x_{k}}=\partial_{x_{k}}H-2x_{k}$, we have

[TABLE]

It follows by Proposition 2.1 and $\inf\sigma(H)>0$ that

[TABLE]

thus we obtain

[TABLE]

We apply the induction argument for the proof of the higher order cases. Let $M\geq 2$ be a natural number and we assume that

[TABLE]

Let us prove the estimate when $|\alpha|+|\beta|=M+1$.

If $|\alpha|$ is an even number, then by Proposition 2.1

[TABLE]

with $\frac{|\alpha|}{2}\in\mathbb{N}$. Since $H\nabla=\nabla H-2x$, there exist $\Lambda_{M}$ a subset consisting of indices $(\alpha^{\prime},\beta^{\prime})$ with the total order $|\alpha^{\prime}|+|\beta^{\prime}|$ less than $M$ and positive constants $C_{\alpha^{\prime},\beta^{\prime}}$ such that

[TABLE]

which proves that

[TABLE]

Proposition 2.1 and the assumption of the induction imply that

[TABLE]

We also know $\inf\sigma(H)>0$ and obtain the inequality (2.1).

If $|\alpha|$ is an odd number and $|\alpha|\geq 3$, then we write

[TABLE]

and by the integration by parts,

[TABLE]

Since $2+|\tilde{\alpha}|$ and $|\tilde{\alpha}|$ are even, we have by the Cauchy Schwarz inequality and the previous argument for even number polynomials that

[TABLE]

For the second term, we notice $|\tilde{\beta}|=|\beta|-1$ and that the order of the polynomial $\partial_{x_{k}}(x_{j}^{2}x^{\tilde{\alpha}}x^{\tilde{\alpha}})$ is $1+2|\tilde{\alpha}|=2|\alpha|-1$ at most, and we apply the multiplication of the polynomials of order $|\alpha|$ and $|\alpha|-1$ by $\nabla^{\tilde{\beta}}f$ and $\overline{\nabla^{\beta}f}$, respectively. We then write

[TABLE]

where $\Lambda^{\prime}_{M-1},\Lambda^{\prime\prime}_{M-1}$ are sets of multi-indices for polynomials and derivatives such that the sum of the two orders are $M-1$. We apply the assumption of the induction for $M-1$ to have that

[TABLE]

The above two inequalities proves the case when $|\alpha|+|\beta|=M+1$, and we conclude the estimate (2.1). ∎

The following lemma is fundamental for our argument and will be used several times in our proof. It is the uniform boundedness of the spectral multiplier with derivatives and multiplication by polynomials.

Lemma 2.2**.**

For multi-indices $\alpha$ and $\beta$, there exists a positive constant $C_{\alpha,\beta}$ such that for every $f\in L^{p}$ and $j\in\mathbb{Z}$,

[TABLE]

Let us give a comment on the proof of this lemma. In the case where $p=1$, we can apply the lemmas below and the argument in [IMT-RMI] to the operator with derivatives and polynomials to prove the inequality. The case where $p=\infty$ follows from the duality argument, and the case where $1<p<\infty$ is proved by interpolation.

To prove Lemma 2.2, we introduce a set $\mathscr{A}_{N}$ of some bounded operators on $L^{2}(\mathbb{R}^{d})$ and scaled amalgam spaces $\ell^{1}(L^{2})_{\theta}$ for $\theta>0$ to prepare a lemma. Hereafter, for $k\in\mathbb{Z}^{d}$, $C_{\theta}(k)$ denotes a cube with the center $\theta^{\frac{1}{2}}k$ and side length $\theta^{\frac{1}{2}}$, namely,

[TABLE]

and $\{\chi_{C_{\theta}(k)}\}_{k\in\mathbb{Z}^{d}}\subset C_{0}^{\infty}(\mathbb{R}^{d})$ is a partition of the unity such that

[TABLE]

[TABLE]

**Definition. ** For $N\in\mathbb{N}$, $\mathscr{A}_{N}$ denotes the set of all bounded operators $T$ on $L^{2}(\Omega)$ such that

[TABLE]

Remark. We remark that this partition of the unity consists of smooth functions, while in the reference [IMT-RMI] the authors use non-smooth functions to compose a partition of the unity. We need some smoothness of the partition to study the operators $T=x^{\alpha}\nabla^{\beta}\varphi(\sqrt{H})$ and $\varphi(\sqrt{H})x^{\alpha}\nabla^{\beta}$. The lemmas (Lemma 2.4 and Lemma 2.3) below hold also for our partition of the unity and is proved with suitable modification, but we omit the detail.

**Definition. ** The space $\ell^{1}(L^{2})_{\theta}$ is defined by letting

[TABLE]

where

[TABLE]

Lemma 2.3**.**

([IMT-2019])* The operator $\varphi(\sqrt{\theta H})$ with $\varphi\in\mathcal{S}(\mathbb{R})$ belongs to $\mathscr{A}_{N}$ for any $\theta>0$. Moreover, there exists a constant $C>0$ such that*

[TABLE]

Lemma 2.4**.**

([IMT-RMI])* (i) Let $N\in\mathbb{N}$ and $N>d/2$. Then there exists a constant $C>0$ such that*

[TABLE]

for any $T\in\mathscr{A}_{N}$ and $\theta>0$.

(ii)* Let $\beta$ be a real number satisfying $\beta>d/4$. Then there exists a constant $C>0$ such that*

[TABLE]

for any $\theta>0$.

Lemma 2.5**.**

Let $m,N\in\mathbb{N}$. For every multi-indices $\alpha,\beta$ with $|\alpha|+|\beta|\leq 2m$, there exists a constant $C>0$ such that

[TABLE]

[TABLE]

for any $\psi\in\mathcal{S}(\mathbb{R})$ with ${\rm supp\,}\psi\subset[0,\infty)$ and $\theta>0$.

Proof.

We write $R_{\theta}=(1+\theta H)^{-1}$ and notice that

[TABLE]

by which we can write for $j=1,2,\cdots,d$

[TABLE]

where $\Lambda_{|\beta|-1,N-1}$ is a subset of multi-indeces $(\beta^{\prime},k)$ such that $|\beta^{\prime}|\leq|\beta|-1$ and $k\leq N-1$, and we need to study the $L^{2}$-boudedness. Here we focus on the first term above, since the total orders of the derivatives and the polynomials are less and the second term can be handled similarly to the first term. We then consider the commutator for the first term,

[TABLE]

and the first term is handled by Lemma 2.1,

[TABLE]

where the supremum above is finite because of $\psi(0)=0$ and smooth at the origin. Next, we recall the formula

[TABLE]

from which it is sufficient to study

[TABLE]

We follow the argument with commutators (see the proof of Lemma 6.3 in [IMT-RMI]), but we only explain the different point. The problem for the commutator is reduced to $R_{\theta}^{m}$ instead of $R_{\theta}$ in [IMT-RMI], since

[TABLE]

We then see that there exist $\Lambda_{2k-1}$ a subset of $\alpha^{\prime},\beta^{\prime}$ with the total order $|\alpha^{\prime}|+|\beta^{\prime}|\leq 2k-1$ ($k=1,\cdots,m$) and constants $C_{\alpha^{\prime},\beta^{\prime}}$ such that

[TABLE]

We can then handle by the boundedness of $x^{\alpha}\nabla^{\beta}R_{\theta}^{m},x^{\alpha^{\prime}}\nabla^{\beta^{\prime}}R_{\theta}^{m}$ in $L^{2}$ proved by Lemma 2.2. Therefore we conclude that

[TABLE]

We explain how to prove the second inequality (2.6) by a similar argument to the proof above. We write

[TABLE]

and apply the boundedness of the operators $R_{\theta}^{m}x^{\alpha}\nabla^{\beta}$ in $L^{2}$, which is proved by Lemma 2.1 and the duality argument provided that $|\alpha|+|\beta|\leq 2m$. In fact, we may have

[TABLE]

for $f\in\mathcal{S}(\mathbb{R}^{d})$, where the above constant $C$ is independent of $n$ and $f$. A density argument implies that

[TABLE]

for all $n\in\mathbb{Z}^{d}$ and the constant $C$ is independent of $n$. We then obtain the second inequality (2.6). ∎

Proof of Lemma 2.2.

As explained below Lemma 2.2, we only prove the case when $p=1$. We also introduce $\theta$ such that $\theta=2^{-2j}$.

We write by the partition of the unity $\{\chi_{C_{\theta}(n)}\}_{n\in\mathbb{Z}^{d}}$,

[TABLE]

Given a positive real number $\gamma>\frac{d}{4}$, we choose $\widetilde{\varphi}\in\mathcal{S}(\mathbb{R})$ as

[TABLE]

By the definition of $\widetilde{\varphi}$, we have

[TABLE]

It follows from (2.3), (2.1) and (2.5) that for $T=x^{\alpha}\nabla^{\beta}\widetilde{\varphi}(\theta H)$

[TABLE]

We finally apply (2.4) and have that for $\theta=2^{-2j}$

[TABLE]

which proves Lemma 2.2. ∎

3. Proof of Theorem 1.1

In this section, we give a proof for Theorem  1.1. Note that the derivatives and multiplications of functions are taken in the $\mathcal{S}^{\prime}$ sense.

We start with the proof of Proposition 1.2, item (i). For each $j\in\mathbb{Z}$, we write

[TABLE]

We can handle the first case $|l-j|\leq 1$ in the same way as in standard Besov spaces associated with the Laplacian (see, for example, [1]). However, for the sake of completeness, we explain briefly here. For $j\in\mathbb{Z}$ given, using the boundedness of spectral multiplier (1.2) and the Hölder inequality, we get

[TABLE]

since

[TABLE]

In fact, by introducing $\psi_{l-2}=\sum_{k\leq l-2}\phi_{k}$, we can apply the uniform bound (1.2) to $\psi_{l-2}$ instead of $\phi_{j}$ to obtain (3.1). We then take the $l^{q}$ norm and apply the Young inequality.

[TABLE]

where the sum $\sum_{|j|\leq 1}2^{-sj}$ is finite. We point out that in the case of standard Laplacian $-\Delta$ we do not need to consider the case $|l-j|\geq 2$ since the supports of decomposition functions are disjoint, but in our case we need it. In this proof below, we will see that even if we consider $H=-\Delta+|x|^{2}$, since this case can be treated as a perturbation from the Laplacian $-\Delta$ case, the same bilinear estimates follow-namely the term of the case $|l-j|\geq 2$ should be small. Such an approach is inspired by the argument presented in [5], where the equivalence between the two Besov spaces with and without a potential is discussed.

Let us consider the second case $l-j\leq-2$. Take $m$ with $2m>|s|$ and fix. We see that

[TABLE]

and it follows by the uniform boundedness of the spectral multiplier (1.2) and the Leibniz rule with Lemma 2.2 (we do not use the equivalence of the Sobolev norm here because it excludes the cases $p=1,\infty$) that for each $l$

[TABLE]

Here we remark that

[TABLE]

In this inequality (3.3), we focused on the most important term $f_{k}H^{m}g_{l}$ in the right hand side of (3.2), and applied (3.1) for the term $\sum_{k\leq l-2}f_{k}$.111Writing $g_{l}=(\phi_{l-1}(\sqrt{H})+\phi_{l}(\sqrt{H})+\phi_{l+1}(\sqrt{H}))g_{l}$, we apply the spectral multiplier theorem to $(\phi_{l-1}(\sqrt{H})+\phi_{l}(\sqrt{H})+\phi_{l+1}(\sqrt{H}))$, and we can then keep $g_{l}$ in the inequality.

For the other terms, $f_{k}$ should have multiplication by the polynomials or derivatives of order one at least, and it follows by $2^{k}\leq 2^{l}$ that

[TABLE]

We apply this inequality, replace $l$ by $l+j$, and then the second case $l-j\leq-2$ can be estimated by

[TABLE]

Finally we treat the third case $l-j\geq 2$. We observe that the quantity \phi_{j}(\sqrt{H})\Big{(}\sum_{k\leq l-2}f_{k}g_{l}\Big{)} can be rewritten as follows.

[TABLE]

where $\Lambda_{2m}$ is a subset of indices such that $|\alpha^{\prime}|+|\alpha^{\prime\prime}|+|\beta^{\prime}|+|\beta^{\prime\prime}|\leq 2m$; indeed, when $m=1$, we can write as follows.

[TABLE]

Then we assume that (3.4) holds for $m$, and prove that the case $m+1$ holds. Indeed, we write the $m+1$ case as

[TABLE]

and use the result for $m=1$ case. Then we have

[TABLE]

Now we apply (3.4) with $m$ for $f_{k}\mapsto f_{k},-\Delta f_{k},\nabla f_{k}$ respectively and $g_{l}\mapsto H^{-1}g_{l}$. We may then see that the case $m+1$ also holds for (3.4).

Now, in the third case $l-j\geq 2$, dividing the sum in $k\leq l-2$ into two cases $k<j$, $k\geq j$, Lemma 2.2 yields that

[TABLE]

Indeed, the first term has been estimated as follows.

[TABLE]

where we have used Lemma 2.2 twice.

This implies that for the case of $k<j$,

[TABLE]

and for the case of $k\geq j$,

[TABLE]

The $l\leq k-2$ case, i.e. the product rule for $f\varogreaterthan g$ can be shown in the same way. Further, the item (ii) is also similarly proved.

Next we show (iii). First, as above we decompose for $j\geq-1$ fixed,

[TABLE]

We first consider (I). By (1.2),

[TABLE]

Thus we take the $l^{q}$ norm in $j$, and use the Young inequality and the Hölder inequality with $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$ to conclude

[TABLE]

where $\sum_{j\geq-2}2^{-(s_{1}+s_{2})j}<\infty$ if $s_{1}+s_{2}>0$. Next, for the term (II), as above, take $m$ such that $2m>s_{1}+s_{2}$. For a fixed $j\geq-1$, we write

[TABLE]

and apply a similar argument as in (3.2) with the condition that $|l-k|\leq 1$. By the Leibniz rule, we write

[TABLE]

where $\Lambda_{2m}$ is a set of multi-indices for polynomials and derivatives such that the total order $|\alpha|+|\beta|+|\alpha^{\prime}|+|\beta^{\prime}|$ is less than $2m$. We apply Lemma 2.2 and see that

[TABLE]

where the multi-indices must satisfy

[TABLE]

We can then write

[TABLE]

where $C^{\prime}_{m,n}$ ($n=0,1,2,\cdots,2m$) are appropriate real numbers depending on the subscript $m,n$.

Now, first, we take $l^{q}$ norm in $j$, then use the Young inequality,

[TABLE]

Again using the Young inequality in the last term we get

[TABLE]

Combining (i)-(iii), we obtain (iv). ∎

Appendix A proof of (1.2)

In this section, we give a brief proof for the uniform bound (1.2).

Proof of (1.2).

It is sufficient to show $L^{1}$-estimate for $\varphi(\sqrt{\theta H})$, then $L^{\infty}$-estimate follows by the duality argument. We then can make use of the Riesz-Thorin interpolation theorem to obtain $L^{p}$-estimates for $1\leq p\leq\infty$.

Recalling the definition $l^{1}(L^{2})_{\theta}$ in Section 2, we obtain

[TABLE]

where we have used the bound $|C_{\theta}(n)|^{1/2}\leq\theta^{d/4}.$

For $\beta>0$, we consider $\tilde{\varphi}\in\mathcal{S}(\mathbb{R})$ defined by

[TABLE]

Note that we may write

[TABLE]

Using Lemma 2.4, we get

[TABLE]

Remark that the bound in $L^{2}$ follows from

[TABLE]

for any $\theta>0.$ Moreover, thanks to Lemma 2.3, the right hand side is estimated by

[TABLE]

provided $\beta$ satisfies $\beta>d/4$. Summarizing those estimates, we find that

[TABLE]

Therefore, we conclude that

[TABLE]

for any $\theta>0$ and $f\in L^{1}$. ∎

Appendix B proof of Proposition 1.1 (v) (vi) and (vii)

Proof of Proposition 1.1 (v).

We begin by proving the continuous embedding $L^{p}\hookrightarrow B^{s}_{p,1}(H)$ for $s<0$. This embedding is a fundamental result in the theory of non-homogeneous Besov spaces, and the proof proceeds as follows. Since $s<0$, we have by (1.2)

[TABLE]

We prove the compact embedding of $B^{\alpha}_{p,\infty}(H)\subset L^{p}$ provided that $\alpha>0$.

Let $\{f_{n}\}_{n=1}^{\infty}$ be a bounded sequence in $B^{\alpha}_{p,\infty}$. For $R>0$, we have

[TABLE]

[TABLE]

which implies that

[TABLE]

Define

[TABLE]

Since $f_{n,J}$ is smooth, we have from the Arzelá-Ascoli theorem that for each $J$, $M$, a subsequence $\{f_{n_{k},J,M}\}_{k=1}^{\infty}$ exists such that it converges uniformly to an continuous function, which we denote by $f^{M}_{\leq J}$, on each compact set $\{|x|\leq M\}$ of $\mathbb{R}^{d}$. We may choose a subsequence satisfying the monotonicity $\{f_{n_{k},J,M+1}\}_{k=1}^{\infty}\subset\{f_{n_{k},J,M}\}_{k=1}^{\infty}$ with respect to $M$, and we then find $\{f_{n_{k},J}\}_{k=1}^{\infty}$ such that

[TABLE]

and $\{f_{n_{k},J}\}_{k=1}^{\infty}$ converges uniformly to the function $f^{M}_{\leq J}$ on the compact set $\{|x|\leq M\}$ for each $M$ and $J$. By the monotonicity with respect to $M$ and the uniform convergence,

[TABLE]

for all $M$. We introduce the function $f_{\leq J}$ such that

[TABLE]

and according to above arguments we see that $f_{n_{k},J}$ converges to $f_{\leq J}$ uniformly on $\{|x|\leq M\}$ for each $M\in\mathbb{N}$.

We easily see from Fatou’s lemma for $p<\infty$ and an elementary argument for $p=\infty$ that $f_{\leq J}\in L^{p}$. Moreover,

[TABLE]

and we may also have

[TABLE]

Let us define

[TABLE]

We then notice that

[TABLE]

We take an arbitrary positive number $\varepsilon$. For each $J$, we consider the subsequence $\{n_{k}\}_{k=1}^{\infty}$ associated with $f_{n_{k},J}$, and the norm of $f_{n_{k}}-f$. We write

[TABLE]

The first term is the norm for the high-spectrum part, and we have

[TABLE]

The third term is the norm for the high-spectrum part, and we have

[TABLE]

The fourth term is bounded by

[TABLE]

We here choose $J$ such that the first and the third terms are small, i.e.,

[TABLE]

and we may find $M$ such that the fourth term can be small as follows.

[TABLE]

We know that $f_{n_{k},J}$ converges to $f_{\leq J}$ uniformly on $\{|x|\leq M\}$ as $k\to\infty$, and then see that a natural number $k_{0}(J,M)$ exists such that for $k\geq k_{0}(J,M)$

[TABLE]

which implies that

[TABLE]

Therefore, we can find a subsequence of $\{f_{n}\}_{n=1}^{\infty}$ such that it converges to $f$ in $L^{p}$.

We turn to prove the compact embedding from $B^{\alpha}_{p,\infty}(H)$ to $B^{s}_{p,1}(H)$ provided that $s<\alpha$. By the lifting property, we can assume that $s<0<\alpha$. Let $\{f_{n}\}_{n=1}^{\infty}$ be a bounded sequence in $B^{\alpha}_{p,\infty}(H)$. By the previous proof, we can find a subsequence of $\{f_{n}\}_{n=1}^{\infty}$ which converges in $L^{p}$, and it is easy to see that the subsequence converges in $B^{s}_{p,1}(H)$ by the continuous embedding $L^{p}\hookrightarrow B^{s}_{p,1}(H)$ provided $s<0$, as we showed at the beginning of the proof. ∎

Proof of Proposition 1.1 (vi).

Since $\sum_{j\in\mathbb{Z}}\phi_{j}(\sqrt{H})$ is the identity operator, we have

[TABLE]

On the other hand,

[TABLE]

by (1.2). ∎

Proof of Proposition 1.1 (vii).

We follow the argument in Section 4 in [8]. To explain the idea, we only consider the case when $r,r_{0}\leq p$. We notice that

[TABLE]

We here recall the following inequality for $p\geq r$.

[TABLE]

which is a generalization of the boundedness (1.2) and we refer to Theorem 1.1 in [IMT-RMI] (see also the proof of Theorem 1.2 (ii) below).

For $N\in\mathbb{Z}$ we split the infinite series in the definition of the norm of Besov spaces into two series.

[TABLE]

since $s+d(1/r-1/p)>0,s+d(1/r_{0}-1/p)-s_{0}<0$. Choosing $N$ such that

[TABLE]

we obtain the inequality in (vii). ∎

**Remark. ** It is possible to prove a simpler inequality.

[TABLE]

where $\alpha,\alpha_{0},\alpha_{1}\in\mathbb{R}$, $\theta\in(0,1)$, $1\leq p,p_{0},p_{1},q,q_{0},q_{1}\leq\infty$,

[TABLE]

In fact, by the Hölder inequality, we get

[TABLE]

Therefore,

[TABLE]

Then we take $l^{q}$ norm, and apply the Hölder inequality.

Appendix C proofs of Theorems 1.2-1.5

Since the proof of Theorems 1.2–1.5 follows from the argument in [Iw-2018], we highlight only a few key points. To this end, we prepare a lemma, which is similar to Lemma 2.1 in [Iw-2018] for the Dirichlet Laplacian. Instead of the Dirichlet Laplacian, we consider the Hermite operator in this paper.

Lemma C.1**.**

Let $N>d/2$, $1\leq p\leq\infty$, $\delta>0$ and $a,b>0$. Then there exists a positive constant $C$, which depends on $N,\delta,a,b$, such that for any $\phi\in C_{0}^{\infty}(\mathbb{R})$ with ${\rm supp\,}\phi\subset[a,b]$, $G\in C^{\infty}((0,\infty))$

[TABLE]

for all $j\in\mathbb{Z}$, where $\|\phi\|_{H^{s}(\mathbb{R})}=\|(1+|\xi|^{2})^{\frac{s}{2}}\mathcal{F}[\phi]\|_{L^{2}(\mathbb{R})}$.

Proof of Lemma C.1.

The proof is similar to that of Lemma 2.1 in [Iw-2018], as the semigroup generated by the Hermite operator satisfies resolvent estimates and the following Gaussian upper bound (see, e.g., Proposition 3.1 in [IMT-RMI]). There exists a positive constant $C$ such that

[TABLE]

where $e^{-tH}(x,y)$ denotes the kernel of the operator $e^{-tH}$. We also refer to Section 6 in [IMT-RMI] for results on Schrödinger operators, including the Hermite operator. ∎

Proof of Theorem 1.2 .

(i) It is well known that the kernel of the semigroup $e^{-tH}$ satisfies the Gaussian upper bound (C.2). This implies $L^{p}$ boundedness for all $1\leq p\leq\infty$, which in turn proves boundedness of $e^{-tH}f$ in Besov spaces.

(ii) We consider only the case $q_{2}=1$ and $q_{1}=\infty$, as the embedding properties of Besov spaces allow us to deduce the other cases from this.

We define $\Phi_{j}=\phi_{j-1}+\phi_{j}+\phi_{j+1}$ and write

[TABLE]

The spectrum of the operator $e^{-tH}\Phi_{j}(\sqrt{H})$ is localized around a dyadic number, and Lemma C.1 with $G(\lambda)=e^{-t\lambda^{2}}$, $\lambda>0$ implies that there exists a positive constant $C$ such that

[TABLE]

Therefore, we obtain by Proposition 1.1 (iii) that

[TABLE]

and there exists a positive constant $C$ independent of $t$ such that

[TABLE]

Here, the convergence of the series follows from the positivity of $(s_{2}-s_{1})+d(1/p_{1}-1/p_{2})$. ∎

Proof of Theorem 1.3 .

The argument is similar to the proof of Theorem 1.2 in [Iw-2018]. We provide only a few comments on the proof.

(i) Since $1\leq q<\infty$, it is crucial that any function can be approximated by a finite sum of $\phi_{j}(\sqrt{H})f$ over a finite subset of $\mathbb{Z}$. We then establish the continuity by applying Lemma C.1 to the finite sum. A density argument completes the proof of (i).

(ii) When $q=\infty$, the continuity in the dual weak sense reduces to the case $q=1$, where the continuity has already been established in (i). ∎

Proof of Theorem 1.4 .

The argument is similar to the proof of Theorem 1.3 in [Iw-2018]. The starting point is to establish the following inequality.

Let $\alpha>0$, $s_{0}\in\mathbb{R}$, and $1\leq p\leq\infty$. Then, there exists a constant $C>0$ such that

[TABLE]

for any $t>0$, $j\in\mathbb{Z}$, and $f\in L^{p}(\mathbb{R}^{d})$.

The above inequality is established in the same manner as Lemma 5.1 in [Iw-2018], using Lemma C.1. We then proceed as in the proof presented in Section 5 of [Iw-2018]. ∎

Proof of Theorem 1.5 .

The proof follows the same argument as that of Theorem 1.4 in [Iw-2018] (see Section 6). ∎

Acknowledgement. The first author was supported by JSPS KAKENHI Grant Numbers 20K03669.