Hardy-Sobolev spaces of higher order associated to Hermite operator

1.   Introduction

The Hermite operator $L$ on $\mathbb R^{d}$ is defined by

The operator $L$ is positive and symmetric in $L^2(\mathbb R^{d})$, it can be decomposed as

where

When we study the problems associated with $L$^[1,2,3,4], the operators $A_i$ play the role of the partial differential operators $\frac{\partial}{\partial x_i}$ in the classical Euclidean case. For example, we can define the Riesz transform associated with Hermite operator by

Thangavelu ^[5] proved that $R_{i}^{L}$ and $R_{-i}^{L}$ were bounded on $L^p(\mathbb R^d)$ and used them to study the wave equations associated with $L$, where $1 < p < \infty$. Their boundedness on the local Hardy spaces ^[6] can be found in ^[4]. Moreover, whether we can characterize the local Hardy spaces by the Riesz transform associated with $L$? This problem was pointed out by Thangavelu in ^[4] and given a negative answer in ^[7]. In fact, the Riesz transform associated with $L$ can characterize a new space which is called Hardy space associated with $L$ ^[8]. Therefore, when we want to prove some results for $L$ similar to the classical case, we must introduce new function spaces for $L$. In ^[9], the authors defined the Sobolev spaces associated with $L$ and used them to study the Schrödinger equation for $L$. In ^[10,11], the authors defined the Besov spaces associated with $L$ and proved the boundedness of Riesz transforms on these spaces. In order to prove the endpoint version of the div-curl theorem for the Hermite operator, the Hardy-Sobolev space was defined in ^[12]. When we consider the equation $L^mF = f$ with $m$ is a positive integer and $f$ in the Hardy spaces associated to $L$, we need to define the higher-order Hardy-Sobolev spaces associated with $L$. In this paper, we will define and give several characterizations of these spaces.

In order to state our main results, we first introduce some notations. Let $H_k(x)$ denote the Hermite polynomials on $\mathbb R$, which can be defined as

The normalized Hermite functions are defined by

The higher-dimensional Hermite functions on $\mathbb{R}^d,$ can be defined in the following way: for $\alpha = (\alpha_1, \cdots, \alpha_d), \, \alpha_i\in \{0, 1, \cdots\}, \, x = (x_1, \cdots, x_d)\in \mathbb{R}^d$,

The Hermite functions $\{h_\alpha\}$ form a complete orthonormal basis of $L^2(\mathbb{R}^d)$. Let $|\alpha| = \alpha_1+\cdots+\alpha_d$. Then we have

Let $\{T_t^L\}_{t\geq 0}$ be the heat semigroup defined by

for $f\in L^2(\mathbb{R}^d)$ and

Then the Poisson semigroup is defined as

We define Hardy space $H_L^1(\mathbb{R}^d)$ for $d\geq 3$ as follows (cf. ^[8])

where $\mathcal{M}_Lf(x) = \sup_{t > 0}|(T_t^Lf)(x)|$.

The Riesz transforms of higher order can be defined as follows:

Definition 1.1. Let $m$ be a positive integer. The operator $L^{-\frac{m}{2}}$ is defined by

and the Riesz transform of order $m$ is defined by

where $1\leq |i_j|\leq d$ for $1\leq j\leq m$ and for any $\alpha\in\{0, 1, 2, \cdots\}^d$.

The first result of this paper is that we can characterize the Hardy space $H_L^1(\mathbb R^d)$ by $R^L_{i_1i_2\cdots i_m}$.

Theorem 1.2. $f\in H_L^1(\mathbb R^d)$ if and only if $R^L_{i_1i_2\cdots i_m} f\in L^1(\mathbb R^d)$ for all $1\leq|i_j|\leq d$ and $f\in L^1(\mathbb R^d)$, i.e., there exists $C > 0$ such that

Let $L_b = L+b$ with $b\in\mathbb{R}^+$ and $P_t^b$ be the semigroup with the infinitesimal generator $\sqrt{L_b}$. Then we can define the following version of higher-order Littlewood-Paley $g$-functions.

Definition 1.3. Let $m$ be a positive integer and $f\in L^p(\mathbb R^d)$. The Littlewood-Paley $g$-function of higher-order is defined by

The next result of this paper is that the Hardy space $H_{L}^{1}(\mathbb R^d)$ can be characterized by the higher order Littlewood-Paley g-function $g_{m, b}$.

Theorem 1.4. For $f\in L^1(\mathbb R^d)$, $f\in H_L^1(\mathbb R^d)$ if and only if $g_{m, b}(f)\in L^1(\mathbb R^d)$ and there exists $C > 0$ such that

Now, we introduce the Hardy-Sobolev space of higher order associated to $L$.

Definition 1.5. We define the Hardy-Sobolev space $H_L^{m, 1}(\mathbb{R}^d)$ of order $m$ as the set of functions $f\in L^1(\mathbb{R}^d)$ such that

with the norm

Definition 1.6. A locally integrable function $b$ is called a $(1, q)$-atom of $H_{L}^{m, 1}(\mathbb R^d)$ if it satisfies

The atomic quasi-norm in $H_{L}^{m, 1}(\mathbb R^{d})$ is defined by

where the infimum is taken over all decompositions $f = \sum c_{j}a_{j}$, where $a_{j}$ are $H_{L}^{m, 1}$-atoms. We can give the atomic decomposition of $H_{L}^{m, 1}(\mathbb R^d)$.

Theorem 1.7. The norms $\|\cdot\|_{H_{L}^{m, 1}}$ and $\|\cdot\|_{H_L^{m, 1}-atom}$ are equivalent, that is, there exists a constant $C > 0$ such that for $f\in H_L^{m, 1}(\mathbb R^d)$,

If we define the following version of the maximal function

then we have

Theorem 1.8. A function $f$ in $H_{L}^{m, 1}(\mathbb R^d)$ if and only if $M_{m, L}f\in H_{L}^{1}(\mathbb R^d)$ and $f\in L^{1}(\mathbb R^d)$. Moreover, there exists a constant $C > 0$ such that

The paper is organized as follows: in section 1, we will give several characterizations of the Hardy space $H_L^1(\mathbb R^d)$. The Hardy-Sobolev spaces will be studied in section 2.

Throughout the article, we will use $A$ and $C$ to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By $B_1 \sim B_2$, we mean that there exists a constant $C > 1$ such that ${C}^{-1} \leq {B_1}/{B_2}\leq C$ and $A\lesssim B$ means that there exists a positive constant $C$ such that $A\leq C B$.

2.   Square function characterizations of $H_L^1(\mathbb{R}^d)$

In this section, we will give several new characterizations of $H_L^1(\mathbb{R}^d)$.

Define (cf. [13,(1.5)])

The function $\rho_L(x)$ has the following propositions (cf. [13, Lemma 1.4]).

Proposition 2.1. There exists $k_0 > 0$ such that

In particular, $\rho_L(y) \sim \rho_L(x)$ if $\, |x-y| < C\rho_L(x)$.

We say $a(x)$ is an atom for the space $H_L^{1}(\mathbb R^d)$, if there exists a ball $B(x_0, r)$ such that

The atomic quasi-norm in $H_{L}^{1}(\mathbb R^{d})$ can be defined as

where the infimum is taken over all atomic decomposition of $f$.

In [8, Theorem 1.12], the author proved the following result.

Proposition 2.2. There exists $C > 0$ satisfying

For $b > 0$, since $\rho_L(x) = \rho_{L+b}(x)$, then by the atomic decomposition of $H_L^1(\mathbb R^d)$, we can obtain(cf. [7, Lemma 9])

Lemma 2.3. For $f\in L^1(\mathbb{R}^d)$, $f\in H_L^1(\mathbb R^d)$ is equivalent to $f\in H_{L+b}^1(\mathbb R^d)$ for $b > 0$.

The proof of the following lemma can be found in [9, Lemma 4].

Lemma 2.4. If $\beta\in\mathbb R$ and $f\in L^2(\mathbb R^d)$, then for $j = 1, 2, \cdots, d$

and for $j = -1, -2, \cdots, -d$

The boundedness of $R_{i_1, \cdots, i_m}^L$ on $L^p(\mathbb R^d)$ can be found in [1, Theorem B].

Proposition 2.5. The Riesz transforms associated to $L$ of higher order $R_{i_1, \cdots, i_m}^L$ are bounded on $L^p(\mathbb R^d)$, where $1 < p < \infty$.

The boundedness of Riesz transforms on Hardy spaces has been proved in [7, Theorem 2] and [14, Theorem 2].

Proposition 2.6. (1) $f\in H_L^1(\mathbb R^d)$ if and only if $R_i^L f\in L^1(\mathbb R^d)$ for $1\leq |i|\leq d$ and $f\in L^1(\mathbb R^d)$. Moreover, the operators $R_i^L$ are bounded on $H_{L}^{1}(\mathbb R^{d})$, that is, there exists $C > 0$ satisfying

(2) The Riesz transforms of higher order $R_{i_1, i_2, \cdots, i_m}^L$ are also bounded on $H_{L}^{1}(\mathbb R^{d})$, i.e., there exists $C > 0$ satisfied

where $1\leq|i_j|\leq d$ and $j = 1, 2, \cdots, m$.

Now, we can prove Theorem 1.2. In the following, we let $\Sigma(i) = 2$ for $i > 0$ and $\Sigma(i) = -2$ for $i < 0$. We use $\Sigma(i_1, i_2, \cdots, i_m)$ to denote $\Sigma(i_1)+\Sigma(i_2)+\cdots\Sigma(i_m)$.

Proof of Theorem 1.2. First, we let $f\in H_L^1(\mathbb R^d)$. Then, by Proposition 2.6, we obtain

For the reverse, by Lemma 2.3, it is sufficient to prove $f\in H_{L+b}^1(\mathbb R^d)$ for some $b > 0$. We will prove this by induction. If $m = 1$, this can be given by Proposition 2.6. We assume $m = n-1$ holds, then for $m = n$, by Lemma 2.4

Therefore, we can choose $b\in\mathbb R^+$ such that $b+\Sigma(i_2, \cdots, i_n) > 0$, then

Therefore $f\in H_{L}^1(\mathbb R^d)$ follows from the inductive assumption, and Theorem 1.2 is proved.

Let

Then, we can characterize $H_{L}^{1}(\mathbb R^d)$ by $g_{m, b}^0$ [15, Theorem 4] and $g_{1, b}$ (cf. [14, Theorem 1]).

Proposition 2.7. (a) For $f\in L^1(\mathbb R^d)$, $f\in H_L^1(\mathbb R^d)$ if and only if $g_{m, b}^0(f)\in L^1(\mathbb R^d)$ and there exists $C > 0$ such that

(b) For $f\in L^1(\mathbb R^d)$, $f\in H_L^1(\mathbb R^d)$ if and only if $g_{1, b}(f)\in L^1(\mathbb R^d)$ and there exists $C > 0$ such that

In the following, we will prove Theorem 1.4.

Proof of Theorem 1.4. If $f\in H_{L}^1(\mathbb R^d)$, then by Proposition 2.2, we have $f = \sum_{k = 1}^\infty \lambda_ka_k,$ where $a_k$ are atoms.

By Lemma 2.4, we know

then

where $a_k$ are atoms for $f$.

Therefore

Then, by Proposition 2.7 and Proposition 2.6

For the reverse, we assume $g_{m, b}(f)\in L^1(\mathbb R^d)$, then we will prove the theorem by induction of $m$. When $m = 1$, it follows from Proposition 2.7. We assume the case of $m$ holds, then we will prove $m+1$ holds. We first prove

This can be proved by changing variables as follows:

Let $\mathbf{K}$ be the Hilbert space defined as $h\in\mathbf{K}$ if and only if $h = \{h_{i_1, \cdots, i_m}(t)\}$, where $-d\leq i_1, \cdots, i_m\leq d$ and $0 < t < \infty$ with

Let $h = \{t^mA_{i_1}\cdots A_{i_m}(P_t^bf(x))\}$. Then, by the inductive assumption, we know $h\in\mathbf{K}$, and (2.2) shows

If we use $H_\mathbf{K}^1(\mathbb R^d)$ to denote the $\mathbf{K}-$valued Hardy spaces associated to $L$, then $h\in H_{\mathbf{K}}^1(\mathbb R^d)\subset L_{\mathbf{K}}^1(\mathbb R^d),$ i.e.,

Therefore, by the inductive assumption, we know $f\in H_L^1(\mathbb R^d)$ and Theorem 1.4 is proved.

3.   Hardy-Sobolev spaces

We first prove that $H_L^{m, 1}(\mathbb{R}^d)$ is a Banach space. In order to do that, we need the following lemma (cf. p.122 in ^[16]).

Lemma 3.1. Let $1\leq p < \infty$, $f\in W^{k, p}(\mathbb{R}^d)$ and $\{f_n\}$ be a sequence such that $\|f_n-f\|_p\rightarrow 0$. Then, for any $|\alpha|\leq k$, we have

where $W^{k, p}$ is the classical Sobolev spaces.

By Lemma 3.1, we can prove

Proposition 3.2. $H_L^{m, 1}(\mathbb{R}^d)$ is a Banach space.

Proof. Let $\{f_n\}$ be a Cauchy sequence in $H_L^{m, 1}(\mathbb{R}^d)$. Then $\{x_i^\mu\partial_j^\nu f_n\}_{\mu+\nu = m}$ is a Cauchy sequence in $H_L^1(\mathbb{R}^d)$. Since $H_L^1(\mathbb{R}^d)$ is a Banach space, there exists $g\in H_L^1(\mathbb{R}^d)$ such that

Let $f$ be the limit of $\{f_n\}$ in $L^1(\mathbb{R}^d)$. Then, by Lemma 2.3,

By $(3.1)$ and $(3.2)$, we obtain $g = x_i^\mu\partial_j^\nu f$. This proves $\|A_{i_1}A_{i_2}\cdots A_{i_m}f_n-A_{i_1}A_{i_2}\cdots A_{i_m}f\|_{H_L^1}\rightarrow0$ for $1\leq |i_j|\leq d$, i.e., $\|f_n-f\|_{H_L^{m, 1}}\rightarrow0$, then we get $H_L^{m, 1}(\mathbb{R}^d)$ is a Banach space. □

Now, we give an equivalent characterization of $H_L^{m, 1}(\mathbb{R}^d)$.

Definition 3.3. Let $\mathcal{H}_L^{m, 1}(\mathbb{R}^d) = L^{-\frac{m}{2}}(H_L^1(\mathbb{R}^d))$ or

with the norm $\|f\|_{\mathcal{H}_L^{m, 1}} = \|L^{\frac{m}{2}}f\|_{H_L^1}+\|f\|_{L^1}$.

Theorem 3.4. The norms $\|\cdot\|_{H_L^{m, 1}}$ and $\|\cdot\|_{\mathcal{H}_L^{m, 1}}$ are equivalent, that is, there exists a constant $C > 0$ such that for $f\in H_L^{m, 1}(\mathbb R^d)$,

Proof. Let $f\in H_L^{m, 1}(\mathbb{R}^d)$. Then, by Theorem 1.2,

i.e., $f\in \mathcal{H}_L^{m, 1}(\mathbb{R}^d)$.

If $f\in \mathcal{H}_L^{m, 1}(\mathbb{R}^d)$, by Proposition 2.6,

This gives the proof of Theorem 3.4. □

In the following, we consider the atomic decomposition of $H_L^{m, 1}(\mathbb{R}^d)$. Given $a > 0$, we define the operator

where $f\in\mathcal{S}(\mathbb R^d)$. Then, we have (cf. [9, Proposition 2])

Lemma 3.5. The operator $L^{-a}$ has the integral representation

for $f\in\mathcal{S}(\mathbb R^d)$. Moreover, there exists $\Phi_a\in L^1(\mathbb R^d)$ and a constant $C > 0$ such that

Let $G_t(x, y)$ denote the heat kernel of $L$, i.e.,

Fayman-Kac formula gives

where $h_t(x)$ is the Gauss kernel.

The heat kernel $G_t^b(x, y)$ of the semigroup $\{e^{-t(L+b)}\}$ is

It is easy to know

Therefore, we have the following estimations for $G_t^b(x, y)$ (cf. [17, Proposition 2-3]).

Lemma 3.6. (a) For $N \in \mathbb N$, there exists $C_{N} > 0$ such that

(b) For every $N > 0$, there are $C_{N} > 0$ and $C > 0$ such that for all $|h|\leq\frac{|x-y|}{2}$,

In order to prove the atomic decomposition of $H_L^{m, 1}(\mathbb R^d)$, we need the following lemma.

Lemma 3.7. Let $a(x)$ be an $(1, q)$-atom associated to ball $B(x_0, r)$ of $H_L^1(\mathbb R^d)$. Then

for $|x-x_0|\geq 2r$.

Proof. For $f\in\mathcal{S}(\mathbb R^d)$, we have

Therefore

Then, by Lemma 3.6 and note that $\rho_L(x)\leq 1$, when $|h|\leq \frac{|x-y|}{2}$, we have

If $r < \rho_L(x_0)$, then $a$ satisfies the vanishing condition, so

If $r\geq\rho_L(x_0)$, by Proposition 2.1, we can obtain $\rho(x)\leq Cr$ for $x\in B(x_0, r)$. Then, following from Lemma 3.6, we have

When $y\in B(x_0, r)$ and $|x-x_0| > 2r$, we obtain

Therefore

This gives the proof of Lemma 3.7.

Now we can give the proof of Theorem 1.7.

Proof of Theorem 1.7. To show $f = \sum\lambda_ib_i\in H_L^{m, 1}(\mathbb{R}^d)$, it suffices to prove that for any $(1, q)$-atom $b$, we have $\|b\|_{H_L^{m, 1}}\leq C$ with $C$ independent of $b$. By Theorem 3.4 and Proposition 2.5,

For the reverse, if $f\in H_L^{m, 1}(\mathbb{R}^d)$, there exists $g\in H_L^1(\mathbb{R}^d)$ such that $f = L^{-m/2}g$. Since $g = \sum\lambda_ia_i$, where $a_i$ are $(1, q)$-atoms in $H_L^1(\mathbb R^d)$, we get $f = \sum\lambda_iL^{-m/2}a_i$ with $\sum|\lambda_j| < \infty$. Since $L^{-m/2}a_i$ does not have compact support, it is not an atom for $H_L^{m, 1}(\mathbb{R}^d)$.

Let $a$ be a $(1, q)$-atom of $H_L^1(\mathbb R^d)$ such that $supp \, a\subset B(x_0, r)$ and $b(x) = L^{-m/2}a$. We choose a smooth partition of unity $1 = \phi_0+\sum_{j = 1}^\infty\phi_j$, where $\phi_0\equiv 1$ and $\phi_1\equiv 0$ on $|x-x_0| < 2r$.

and $\phi_j(x) = \phi_1(2^{1-j}x)$ for $j\geq 2$. Then $b(x) = \phi_0b+\sum_{j = 1}^\infty\phi_jb$. We will show $\phi_jb = \lambda_jb_j$ for appropriate scalars $\lambda_j$, where $b_j$ are $(1, q)$-atoms in $H_L^{m, 1}(\mathbb{R}^d)$ and $\sum|\lambda_j| < C$.

It is obvious, $supp\, b_j\subset B(x_0, 2^{4+j}r)$. Let

For $j = 0$, since $\|L^{m/2}b\|_{L^q} = 1$, we get $\|L^{m/2}\phi_0b\|_{L^q}\leq C$. For $j\geq 1$, since $L$ is self-adjoint and Lemma 3.7, we have

So $\lambda_j\leq C 2^{-j}$, which gives $\sum|\lambda_j|\leq C$.

In order to give the proof of Theorem 1.8, we need the following Poisson maximal function characterization of $H_L^1(\mathbb R^d)$(cf. [18, Theorem 8.2]).

Lemma 3.8. For $f\in L^1(\mathbb R^d)$, we have $f\in H_L^1(\mathbb R^d)$ if and only if $M_P(f)\in L^1(\mathbb R^d)$, where

Moreover, there exists $C > 0$ such that

Proof of Theorem 1.8. By Theorem 3.4 and Lemma 3.8, we obtain

This completes the proof of Theorem 1.8.

Author contributions

All authors have the same contribution to the paper.

Acknowledgements

Jizheng Huang is supported by Fundamental Research Funds for the Central Universities (# 500423101) and Beijing Natural Science Foundation of China(#1232023).

Conflict of interest

The authors declare there is no conflict of interest.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.