Orlicz estimates for parabolic Schrödinger operators with non-negative potentials on nilpotent Lie groups

1.   Introduction

Orlicz spaces have been widely studied since they were introduced by Orlicz ^[1], see for instance Orlicz space ^[2,3], weighted Orlicz space ^[4] and fractional order Orlicz space ^[5]. The regularity estimates of operators play an important role in various fields of analysis. Several boundedness properties of integral operators are applied to the study of the regularity of elliptic and parabolic equations with discontinuous coefficients^[6,7]. Moreover, Abdalmonem and Scapellato ^[8] have considered some Morrey-Herz spaces with variable exponents and have examined some boundedness properties of intrinsic square functions and their commutators in this framework. Additionally, in ^[9], Deringoz et al. studied the Calderon-Zygmund operators and their commutators on generalized weighted Orlicz-Morrey spaces. Recently, Orlicz estimates for some operators have also been obtained ^[10,11,12].

The analytical study of nonlinear partial differential equations is one of the most interesting fields of research for many researchers in recent years^{[13,14,15,16,17,18]}. In this paper, we are interested in the regularity estimates in Orlicz spaces for the parabolic Schrödinger operator with non-negative potentials on nilpotent Lie groups. In Euclidean space, many scholars have obtained many regularity estimates for elliptic and parabolic Schrödinger operators, such as $L^p$ estimates ^[19,20,21] and Orlicz estimates ^[22,23,24]. Li ^[25] obtained the $L^p$ estimates for the Schrödinger operators on nilpotent Lie groups, which generalizes the results in Euclidean space ^[20]. Liu, Huang and Dong ^[26] obtained the $L^p$ estimates for the parabolic Schrödinger operators on nilpotent Lie groups. Yang and Li ^[27] established the boundedness and compactness of commutators related with Schrödinger operators on Heisenberg groups.

Acerbi and Mingione ^[28] proposed a new domain decomposition approach, and applied it to study the local Sobolev estimates for the degenerate parabolic $p$-Laplacian systems. Wang, Yao, Zhou and Jia ^[29] encoded and simplified the iteration-covering procedure used in ^[28] and extended it to the whole space, and obtained the Orlicz estimates for the Poisson equation. Moreover, Yao ^[22,23] improved the method in ^[28,29], and obtained the Orlicz estimates for the Schrödinger operators in Euclidean space. We will extend the method of ^[22,23] to the nilpotent Lie groups, and apply it to obtain the Orlicz estimates for the parabolic Schrödinger operator with non-negative potentials on nilpotent Lie groups, which extend the $L^p$ estimates in ^[26].

Let ${\Bbb G}$ be a simply connected nilpotent Lie group, and the corresponding Lie algebra is $g$. Assume $X = \{ {X_1}, ..., {X_m}\}$ is a Hörmander system of left invariant vector fields on ${\Bbb G}$. In this paper, we consider the following parabolic Schrödinger operators on ${\Bbb G}$

where ${\Delta _X} = \sum\nolimits_{i = 1}^m {X_i^2}$ is the sub-Laplacian on ${\Bbb G}$, and the potential $V(x)$ belongs to the reverse Hölder class on ${\Bbb G}$: $V\left(x \right) \in R{H_q}\; (1 < q < +\infty)$ if $V\left(x \right) \in L_{loc}^q\left({\Bbb G} \right)$, $V(x) > 0$ almost everywhere, and there is a positive constant $C$ such that for all metric balls of ${\Bbb G}$,

The smallest constant $C$ that makes (1.2) true is called the $RH_q$ constant of $V$. If $q = +\infty$, then the left-hand side of (1.2) is essential supremum of $V$ on $B_r$, i.e.,

It's clear that $V \in R{H_\infty }$ implies that $V \in R{H_q}$ for $1 < q < +\infty$.

Inspired by ^[22,23] and based on the $L^p$ estimate in ^[26], we will study the Orlicz estimates for the operator (1.1) on nilpotent Lie groups. When $V \in R{H_q}$ ($1 < q < +\infty$) according to the method in ^[22,23], it is difficult to obtain a result similar to (3.7) in ^[22] or (2.24) in ^[23]. Therefore, we need to improve the domain decomposition method and the measure estimation of level sets in ^[22,23].

The main results of this paper are as follows.

Theorem 1. Assume $\phi \in {\Delta _2} \cap {\nabla _2}$ and $V \in R{H_q}, \; q > \max \{ D/2, {\alpha _1}\}$. Then for any $u \in C_0^\infty ({\Bbb G}\times (0, + \infty))$, we have $Vu \in {L^\phi }\left({{\Bbb G} \times (0, + \infty)} \right)$ and

where the positive constant $c$ is independent of $V$ and $u$. See Section 2 for the young function $\phi$, the global ${\Delta _2}$ condition, the global ${\nabla _2}$ condition, the constant ${\alpha _{\text{1}}}$ and the dimension $D$.

The proof of Theorem 1 is based on the following local estimate.

Theorem 2. Suppose that $V \in R{H_q}, \, q > \frac{{\mathit{\text{D}}}}{{\mathit{\text{2}}}}$. If $h(x, t)$ satisfies ${\partial _t}h - {\Delta _X}h + Vh = 0$ in ${Q_{4r}}({z_0})$, then there is a constant $c > 0$ independent of $V, \; h, \; r, \; {z_0}$ such that

where ${z_0} = ({x_0}, {t_0}) \in {\Bbb G} \times (0, +\infty)$, ${Q_r}({z_0})$ is the parabolic cylinders in ${\Bbb G} \times (0, +\infty)$, and ${B_r}({x_0})$ is the metric ball of center at $x_0$ and radius $r$ in ${\Bbb G}$ (see Section 2), $V\left({{B_{4r}}({x_0})} \right) = \int_{{B_{4r}}({x_0})} {V(x)} dx$.

Using Theorem 1 and the approximation method, it immediately gets the following corollary.

Corollary 3. Assume that $\phi \in {\Delta _2} \cap {\nabla _2}$ and $V \in R{H_q}, \; q > \max \{ D/2, {\alpha _1}\}$. Then for any $u \in W_{\phi, 0}^{1, 2}({\Bbb G} \times (0, + \infty))$, there exists is a positive constant $c$ independent of $V$ and $u$ such that

See Definition 11 for the definition of $W_{\phi, 0}^{1, 2}({\Bbb G} \times (0, + \infty))$.

Remark 4. Theorem 1 generalizes the $L^p$ estimate for the Schrödinger operator in ^[26] to the Orlicz estimate. In fact, due to ${t^p} \in {\Delta _2} \cap {\nabla _2}, \; t > 0, \; p > 1$, letting $\phi (t) = {t^p}, \; t > 0, \; p > 1$, we immediately obtain the $L^p$ estimates for (1.1) which have been obtained in ^[26] by the pointwise estimate for the heat kernel of Schrödinger operators. Here we generalizes the condition $V\in RH_{\infty}$ to the condition $V \in RH_q$, which is different from ^[22,23].

This paper is organized as follows. In Section 2, we introduce the definitions and related conclusions of nilpotent Lie groups and Orlicz spaces. In Section 3, inspired by ^[22,23], the level set $E_\lambda(1)$ (see (3.5)) is decomposed into a family of disjoint parabolic cylinders by using the covering lemma in homogeneous spaces (see Lemma 14). We prove Theorem 1, Theorem 2 and Corollary 3 in Section 4. In Section 5, we state the main conclusions of this paper.

2.   Preliminaries

This subsection introduces the relevant results of nilpotent Lie groups, and the proofs and more properties and examples can be found in ^[30].

2.1.   Background on nilpotent Lie groups

Let $\Bbb{G}$ be a simply connected nilpotent Lie group, and the corresponding Lie algebra is $g$. Assume $X = \{ {X_1}, \; {X_2}, ..., {X_m}\}$ is a Hörmander system of left invariant vector fields on $\Bbb{G}$. ${X_1}, \; {X_2}, ..., {X_m}$ satisfy Hörmander's condition ^[31]. It can be seen from ^[32] that the left invariant vector field can induce the Carnot-Carathéodory distance $d_c$: for any $\delta > 0$, let $A(\delta)$ be the set of absolutely continuous curves $\gamma :\; [0, 1] \to {\Bbb G}$ such that

Then for any $\xi, \; \eta \in {\Bbb G}$,

Let

be the metric ball of center at $x$ and radius $r$ in $\Bbb{G}$. Let $dx$ be the Haar measure on $\Bbb{G}$. For any measurable set $A\subseteq\Bbb{G}$, $\left| A \right|$ denotes the measure of $A$. Assume that $e$ is the unit element of $\Bbb{G}$, then for any $x\in \Bbb{G}$ and $r > 0$, $\left| {B(e, r)} \right| = \left| {B(x, r)} \right|$. $d$ and $D$ denote the local dimension and the dimension at infinity of $\Bbb{G}$, respectively. Let $D \geqslant d \geqslant 2$. According to ^[30], there is a positive constant $C_1$ such that

Moreover, there exist positive constants ${C_2} = {C_2}({C_1}, d, D) > 0$ and $C_3$ such that

The parabolic metric $d_p$ in ${\Bbb G} \times (0, +\infty)$ is defined by

where $z = (x, t), \; {z_0} = ({x_0}, \; {t_0}) \in {\Bbb G} \times (0, +\infty)$. For any ${z_0} = ({x_0}, {t_0}) \in {\Bbb G} \times (0, \infty)$ and $r > 0$, let

be the parabolic cylinders of center at $z_0$ and radius $r$ in ${\Bbb G} \times (0, +\infty)$.

2.2.   Orlicz spaces on nilpotent Lie groups

Here for the readers convenience, we give some definitions and related lemmas in Orlicz spaces, and more properties and proofs can be found in ^{[23,33,34,35]}.

We denote by $\Phi$ the function class that consists of all monotonically increasing convex functions $\phi:\; {\text{[0, + }}\infty {\text{)}} \to {\text{[0, + }}\infty {\text{)}}$.

Definition 5. (^[23]) A function $\phi \in \Phi$ is called a Young function if

Definition 6. (^[23]) A Young function $\phi$ is said to satisfy the global ${\Delta _2}$ condition, denoted by $\phi \in {\Delta _2}$, if there exists a constant $K > 0$ such that for any $t > 0$,

Definition 7. (^[23]) A Young function $\phi$ is said to satisfy the global ${\nabla _2}$ condition, denoted by $\phi \in {\nabla_2}$, if there exists a constant $a > 1$ such that for any $t > 0$,

The following lemma can be easily obtained from (2.2) and (2.3). For example, see ^[35].

Lemma 8. Let $\phi$ be a Young function. If $\phi \in {\Delta _2} \cap {\nabla _2}$, then for any $0 < {\theta _2} \leqslant 1 \leqslant {\theta _1} < +\infty$,

where ${\alpha _1} = \log _2^K, \; {\alpha _2} = \log _a^2 + 1, \; {\alpha_1} > {\alpha_2} > 1.$

Definition 9. (Orlicz spaces, ^[33,34]) Let $\phi$ be a Young function. Then the Orlicz class ${K^\phi }\left({{\Bbb G} \times (0, + \infty)} \right)$ is a set of all measurable functions $g:\; {\Bbb G} \times (0, +\infty) \to \mathbb{R}$ satisfying

Definition 10. (Orlicz-Sobolev spaces, ^[33,34]) The Orlicz-Sobolev space $W_\phi ^{1, 2}({\Bbb G} \times (0, + \infty))$ is the set of all functions $u$ satisfying ${u_t}, \; {X^\alpha }u \in {L^\phi }({\Bbb G} \times (0, \infty)), \, 0 \leqslant \left| \alpha \right| \leqslant 2$ with the norm defined by

where $Xu = \left({{X_1}u, ..., {X_m}u} \right)$, ${\left\| {Xu} \right\|_{{L^\phi }({\Bbb G} \times (0, + \infty))}} = \sum\limits_{i = 1}^m {{{\left\| {{X_i}u} \right\|}_{{L^\phi }({\Bbb G} \times (0, + \infty))}}}$, ${X^2}u = \{ {X_i}{X_j}u\} _{i, j = 1}^m$, ${\left\| {{X^2}u} \right\|_{{L^\phi }({\Bbb G} \times (0, + \infty))}} = \sum\limits_{i, j = 1}^m {{{\left\| {{X_i}{X_j}u} \right\|}_{{L^\phi }({\Bbb G} \times (0, + \infty))}}}$.

Bramanti et al. ^[21] defined the function space $W_V^{2, p}\left({{{\Bbb R}^n}} \right)$ is the closure of $C_0^\infty \left({{{\Bbb R}^n}} \right)$ in the norm

Inspired by this, we introduce the following function space.

Definition 11. The function space $W_{\phi, 0}^{1, 2}({\Bbb G} \times (0, + \infty))$ is the closure of $C_0^\infty ({\Bbb G} \times (0, + \infty))$ in the norm

Using the proof method of Lemma 2.1 in ^[23], it can be proved that if $g \in {L^\phi }\left({{\Bbb G} \times (0, + \infty)} \right)$, then

Similar to Byun and Ryu ^[36], it obtains the following lemma.

Lemma 12. Let $\Omega \subset {\Bbb G} \times (0, + \infty)$ be a bounded domain. If $\phi \in {\Delta _2} \cap {\nabla _2}$, then

where $\alpha_1$ and $\alpha_2$ are the constant in Lemma 8.

3.   Domain decomposition method

Denoting

then we have $1 < p < \alpha_2$, where $\alpha_2$ is the constant in Lemma 8. Assume that $u \in C_0^\infty ({\Bbb G} \times (0, + \infty))$. Some notations are given below for convenience. Denote

where $\varepsilon \in (0, 1)$ is a small enough constant which will be determined later, $f = Lu$, $dz = dxdt$. Let

Then it infers $L{u_\lambda } = {f_\lambda }$. Additionally, for any parabolic cylinder $Q$ in ${\Bbb G} \times (0, + \infty)$, we write

and the level set as

Lemma 13. (Covering lemma, ^[37]) Let $E$ be a bounded measurable set in the homogeneous space $(S, d, \mu)$. If $\left\{ {B\left({x, {\rho _x}} \right)} \right\}$ is any family of spheres with bounded radius covering $E$, then there exists at most countable disjoint subfamily $\left\{ {B\left({{x_i}, {\rho _i}} \right)} \right\}$ such that $\left\{ {B\left({{x_i}, {k_0}{\rho _i}} \right)} \right\}$ covering $E$ with

where the constants ${k_0} \geqslant 1$ and $c > 0$ only depend on $S$.

Inspired by ^[22,23], we decompose the level set $E_\lambda(1)$ into a family of disjoint parabolic cylinders.

Lemma 14. For any $\lambda > 0$, there is a family of disjoint parabolic cylinders ${\left\{ {Q\left({{z_i}, {\rho _i}} \right)} \right\}_{i \geqslant 1}}$ with ${z_i} = ({x_i}, {t_i}) \in {E_\lambda }(1)$, ${\rho _i} = \rho_{z_i} ({\lambda }) > 0$ such that

and

where ${k_0} \geqslant 1$ is a constant. Moreover, we have

Proof of Lemma 14: For any fixed $z = (x, t) \in {\Bbb G} \times (0, +\infty)$ and $\rho\left(\lambda \right) \geqslant {\rho _0}\left(\lambda \right) > 0$ with ${\lambda ^p}\left| {{Q_{{\rho _0}}}(z)} \right| = 1$, it follows from ^[33,34,35] that

Thus it infers

Then for a.e. $z = (x, t) \in {E_\lambda }\left(1 \right)$, by (3.4) and Lebesgue's differential theorem we get

From (3.10) we know that there is $\rho(\lambda) > 0$ such that

By using the above formula and (3.9), it can be seen that there is ${\rho _z} \in (0, {\rho _0}]$ such that

Thus,

To sum up, for a.e. $z \in {E_\lambda }\left(1 \right)$, there is a family of parabolic cylinders $Q_{\rho_{z}}(z)$ constructed as above. $\left({{\Bbb G} \times (0, + \infty), {d_p}, dxdt} \right)$ is a homogeneous space, therefore, according to Lemma 13, there is a family of countable disjoint parabolic cylinders ${\left\{ {{Q_{{\rho _i}}}\left({{z_i}} \right)} \right\}_{i \geqslant 1}}$ such that (3.6) and (3.7) hold.

Moreover, from (3.6) we obtain

It follows from the above formula that

Then (3.8) is immediately obtained. This completes our proof.

4.   Proof of the main results

4.1.   Proof of Theorem 2

To prove Theorem 2, three useful lemmas are first given.

Lemma 15. ([38, Chapter Ⅰ]) If $V\in RH_q, \; q > 1$, then there exist $1 \leqslant {p_0} < +\infty$ and a constant $c > 0$ such that for any nonnegative function $g$ and all parabolic cylinders $Q_r$,

where $V\left({{B_r}} \right) = \int_{{B_r}} {Vdx}.$

Lemma 16. ([39, Lemma 1.4]) Let $E$ be an open subset of the homogeneous space $(S, d, \mu)$. $\mathcal{F}(E)$ is the set of all metric spheres in $E$. If for $0 < q_1 < p$ and $0 \leqslant f \in L_{loc}^p(\mu)$, there exist the constants $A > 1, \; 1 < {\sigma _0} \leqslant {\sigma _0}^\prime$ such that

then for $0 < r < q_1$ and $1 < \sigma \leqslant \sigma ' \leqslant {\sigma _0}^\prime$, there exists a constant $A' > 1$ such that

For ${z_0} = ({x_0}, {t_0}) \in {\Bbb G} \times (0, +\infty)$ and $r > 0$, we write

Lemma 17. ([26, Lemma 2.9]) Assume that $V \in R{H_q}, q > \frac{D}{2}$. If $h(x, t)$ satisfies the homogeneous equation

then there exists a positive constant $c$ such that

Now we begin to prove Theorem 2.

Proof of Theorem 2: Denoting ${Q_1} = Q(({x_0}, {t_0} + {r^2}), \sqrt 2 r)$, ${\tilde Q_1} = \tilde Q(({x_0}, {t_0} + {r^2}), \sqrt 2 r)$, $2{Q_1} = Q(({x_0}, {t_0} + {r^2}), 2\sqrt 2 r)$, $2{\tilde Q_1} = \tilde Q(({x_0}, {t_0} + {r^2}), 2\sqrt 2 r)$, then

It follows from Lemma 17 that

Using (4.1) and Lemma 16, we obtain

When $l > 2$, by (4.1) and Hölder's inequality, it infers

Thus,

Letting $\frac{1}{l} = {p_0}$, from (4.2) and Lemma 15 we get

This completes our proof.

4.2.   Proof of Theorem 1

In this subsection, based on Theorem 2 and the domain decomposition method, we give the proof of Theorem 1.

Proof of Theorem 1: For $u \in C_0^\infty ({\Bbb G} \times (0, +\infty))$, there exists a parabolic cylinder $Q_{R_0}$ such that $\text{spt}\, (u) \subset {Q_{{R_0}}}$. Combining $V \in R{H_q}$, $q > \max \{ D/2, {\alpha _1}\}$, (2.2) and (2.3), we obtain

that is, $Vu \in {L^\phi }\left({{\Bbb G} \times (0, +\infty)} \right)$. It follows from (2.7) that

where $N_0$ is taken as

and the constants $p$ and $\alpha_1$ are the same as those in (3.1) and Lemma 8. In fact, from the proof of the following Theorem 1, we only need to take $N_0 > 1$ to satisfy $\mathop {\lim }\limits_{\varepsilon \to 0} {\varepsilon ^p}N_0^{{\alpha _1} - p} = 0$.

Now we begin to estimate $\left| {\left\{ {z \in {\Bbb G} \times (0, +\infty):\left| {Vu} \right| > 2{N_0}{\lambda _0}\lambda } \right\}} \right|$.

For any $i\geqslant 1$, from Lemma 14 and (3.4) we deduce that

and

where $k_0$ is the constant in Lemma 14. Let $w$ satisfy

By [26, Theorem 3.1], we have

Combined with (4.6) and (4.7),

Let $h = u_{\lambda}-w$. Then $h$ satisfies

By virtue of (4.5) and (4.8), we get

Denoting $\mu = {\lambda _0}\lambda$, we deduce that

Next we estimate $I_1$ and $I_2$. From (4.8) we find that

Using (1.2), Theorem 2, Hölder's inequality and (4.9), we conclude that

Therefore, we have

It follows from (3.7), (4.10) and (3.8) that

Furthermore, by the above inequality and (4.3) we have

where the constant $c$ is independent of $\varepsilon, \; N_0$. Now we estimate $I_3$ and $I_4$. According to Fubini's theorem and integration by parts, we deduce that

By using (2.5) and $1 < p < \alpha_2$, it infers

Moreover, using Lemma 8, we have

Note that $1 < p < \alpha_2$ must also be required in the above integral calculation process. Therefore, from the above analysis and (2.4), we observe that

where the constant $c$ is independent of $\varepsilon, \; N_0$.

By applying Fubini's theorem, integration by parts and (2.5), $I_4$ becomes

Setting ${\theta _2} = \frac{{\varepsilon \mu }}{{3\left| f \right|}}$, then $0 < {\theta _2} < 1.$ It follows from (2.5) that

By the above inequality, we obtain

Additionally, using (2.4), we get

where the constant $c$ is independent of $\varepsilon, \; N_0$.

Therefore, from (4.11) and (4.12) we obtain

where the constant $c > 0$ is independent of $\varepsilon$. Finally, using $1 < p < {\alpha _2} < {\alpha _1} < q$ and choosing a suitable $\varepsilon > 0$ such that

we obtain (1.4). This completes our proof.

4.3.   Proof of Corollary 3

Proof of Corollary 3: For $u \in W_{\phi, 0}^{1, 2}({\Bbb G} \times (0, \infty))$, according to Definition 11, there exists a sequence of $\{u_k\}$ of functions in $C_0^\infty ({\Bbb G} \times (0, \infty))$ such that

Therefore, from the above formula, Theorem 1, the convexity and monotonicity of $\varphi$, and (2.4) and (2.5), Corollary 3 is immediately proved. We complete the proof of Corollary 3.

5.   Conclusions

In this paper, we study the regularity estimates in Orlicz space for the parabolic Schrödinger operator $L = {\partial _t} - {\vartriangle _X} + V$ on nilpotent Lie groups. There are many essential differences between partial differential operators on nilpotent Lie groups and partial differential operators on Euclidean space. For example, the sub-Laplace operator ${\Delta _X} = \sum\limits_{i = 1}^m {X_i^2}$ on a nilpotent Lie group is a degenerate elliptic operator, while the Laplace operator $\Delta = \sum\limits_{i = 1}^m {\frac{{{\partial ^2}}}{{\partial x_i^2}}}$ in Euclidean space is a uniformly elliptic operator. Acerbi and Mingione ^[28] invented a new domain decomposition approach, which is completely free from harmonic analysis. Wang et al. ^[22,23,29] simplified and improved this approach, and obtained Orlicz estimates for some operators in Euclidean space. We extend the method of ^[22,23] to nilpotent Lie groups, and in order to generalize the condition $V\in RH_{\infty}$ to the condition $V \in RH_q$, we have appropriately improved the domain decomposition method and the measure estimation of level sets in ^[22,23]. By using this approach, we obtain the Orlicz estimates for the parabolic Schrödinger operator with non-negative potentials on a nilpotent Lie groups, which extend the $L^p$ estimates in ^[26]. Because this method needs to rely on the estimate of the metric sphere measure, this method is applicable to the Heisenberg group, Carnot group, etc, but it may not be suitable for the Hörmander's vector fields.

Use of AI tools declaration

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

Acknowledgments

The author wishes to thank the anonymous referees for offering valuable suggestions to improve the expressions. The author would like to thank Professor Pengcheng Niu for his insightful guidance and earnest help about regularity theory. This work was supported by the Guangxi Natural Science Foundation (Grant No. 2017GXNSFBA198130), and Guangxi Key Laboratory of Cryptography and Information Security (No. GCIS202134).

Conflict of interest

The author declares that there is no conflicts of interest regarding the publication of this paper.