Hierarchical volumetric transformer with comprehensive attention for medical image segmentation

1.   Introduction

In this paper, we consider the Schrödinger operators

where $\Delta = \sum_{i = 1}^n\frac{\partial^2}{\partial^2_{x_i}}$ and $V(x)$ is a nonnegative potential belonging to the reverse Hölder class $RH_{q}$ for some $q\geq \frac{n}{2}$. Assume that $f$ is a nonnegative locally $L^{q}(\mathbb{R}^n)$ integrable function on $\mathbb{R}^{n}$, then we say that $f$ belongs to $RH_{q}$ ($1 < q\leq \infty$) if there exists a positive constant $C$ such that the reverse Hölder's inequality

holds for $x$ in $\mathbb{R}^{n}$, where $B(x, r)$ denotes the ball centered at $x$ with radius $r < \infty$ ^[1]. For example, the nonnegative polynomial $V\in RH_\infty$, in particular, $|x|^2\in RH_\infty$.

Let the potential $V\in RH_q$ with $q \geq \frac{n}{2}$, and the critical radius function $\rho(x)$ is defined as

We also write $\rho(x) = \frac{1}{m_V(x)}, x\in \mathbb{R}^n$. Clearly, $0 < m_V(x) < \infty$ when $V\not = 0$, and $m_V(x) = 1$ when $V = 1$. For the harmonic oscillator operator (Hermite operator) $H = -\Delta +|x|^2$, we have $m_V(x)\sim (1+|x|)$.

Thanks to the heat diffusion semigroup $e^{-t\mathcal{L}}$ for enough good function $f$, the negative powers $\mathcal{L}^{-\frac{\alpha}{2}}(\alpha > 0)$ related to the Schrödinger operators $\mathcal{L}$ can be written as

Applying Lemma 3.3 in ^[2] for enough good function $f$ holds that

and the kernel $K_{\alpha}(x, y)$ satisfies the following inequality

Moreover, we have $K_{\alpha}(x, y)\leq\frac{C}{|x-y|^{n-\alpha}}, 0 < \alpha < n$.

Shen ^[1] obtained $L^{p}$ estimates of the Schrödinger type operators when the potential $V\in RH_q$ with $q\geq \frac{n}{2}$. For Schrödinger operators $\mathcal{L} = -\Delta+V$ with $V\in RH_q$ for some $q\geq \frac{n}{2}$, Harboure et al. ^[3] established the necessary and sufficient conditions to ensure that the operators $\mathcal{L}^{-\frac{\alpha}{2}}(\alpha > 0)$ are bounded from weighted strong and weak $L^p$ spaces into suitable weighted $BMO_{\mathcal{L}}(w)$ space and Lipschitz spaces when $p\geq \frac{n}{\alpha}$. Bongioanni Harboure and Salinas proved that the fractional integral operator $\mathcal{L}^{-\alpha/2}$ is bounded form $L^{p, \infty}(w)$ into $BMO^{\beta}_{\mathcal{L}}(w)$ under suitable conditions for weighted $w$ ^[4]. For more backgrounds and recent progress, we refer to ^[5,6,7] and references therein.

Ramseyer, Salinas and Viviani in ^[8] studied the fractional integral operator and obtained the boundedness from strong and weak $L^{p(\cdot)}$ spaces into the suitable Lipschitz spaces under some conditions on $p(\cdot)$. In this article, our main interest lies in considering the properties of fractional integrals operator $\mathcal{L}^{-\frac{\alpha}{2}}(\alpha > 0)$, related to $\mathcal{L} = -\Delta+V$ with $V\in RH_q$ for some $q\geq \frac{n}{2}$ in variable exponential spaces.

We now introduce some basic properties of variable exponent Lebsegue spaces, which are used frequently later on.

Let $p(\cdot):\Omega\rightarrow [1, \infty)$ be a measurable function. For a measurable function $f$ on $\mathbb{R}^n$, the variable exponent Lebesgue space $L^{p(\cdot)}(\Omega)$ is defined by

where $s$ is a positive constant. Then $L^{p(\cdot)}(\Omega)$ is a Banach space equipped with the follow norm

We denote

Let $\mathcal{P}(\mathbb{R}^{n})$ denote the set of all measurable functions $p$ on $\mathbb{R}^{n}$ that take value in $[1, \infty)$, such that $1 < p_{-}(\mathbb{R}^{n})\leq p(\cdot)\leq p_{+}(\mathbb{R}^{n}) < \infty.$

Assume that $p$ is a real value measurable function $p$ on $\mathbb{R}^n$. We say that $p$ is locally log-Hölder continuous if there exists a constant $C$ such that

and we say $p$ is log-Hölder continuous at infinity if there exists a positive constant $C$ such that

where $p(\infty): = \lim\limits_{|x|\rightarrow \infty}p(x)\in \mathbb{R}$.

The notation $\mathcal{P}^{\log}(\mathbb{R}^{n})$ denotes all measurable functions $p$ in $\mathcal{P}(\mathbb{R}^{n})$, which states $p$ is locally log-Hölder continuous and log-Hölder continuous at infinity. Moreover, we have that $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$, which implies that $p^{\prime}(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$.

Definition 1.1. ^[8] Assume that $p(\cdot)$ is an exponent function on $\mathbb{R}^n$. We say that a measurable function $f$ belongs to $L^{p(\cdot), \infty}(\mathbb{R}^n)$, if there exists a constant $C$ such that for $t > 0$,

It is easy to check that $L^{p(\cdot), \infty}(\mathbb{R}^n)$ is a quasi-norm space equipped with the following quasi-norm

Next, we define $Lip^{\mathcal{L}}_{\alpha, p(\cdot)}$ spaces related to the nonnegative potential $V$.

Definition 1.2. Let $p(\cdot)$ be an exponent function with $1 < p^{-}\leq p^{+} < \infty$ and $0 < \alpha < n$. We say that a locally integrable function $f\in Lip^{\mathcal{L}}_{\alpha, p(\cdot)}(\mathbb{R}^n)$ if there exist constants $C_1, C_2$ such that for every ball $B\subset\mathbb{R}^{n}$,

and for $R\geq\rho(x)$,

where $m_{B}f = \frac{1}{|B|}\int_{B}f$. The norm of space $Lip^{\mathcal{L}}_{\alpha, p(\cdot)}(\mathbb{R}^{n})$ is defined as the maximum value of two infimum of constants $C_1$ and $C_2$ in (1.4) and (1.5).

Remark 1.1. $Lip^{\mathcal{L}}_{\alpha, p(\cdot)}(\mathbb{R}^n)\subset \mathfrak{L}_{\alpha, p(\cdot)}(\mathbb{R}^n)$ is introduced in ^[8]. In particular, when $p(\cdot) = C$ for some constant, then $Lip^{\mathcal{L}}_{\alpha, p(\cdot)}(\mathbb{R}^n)$ is the usual weighted $BMO$ space $BMO^{\beta}_{\mathcal{L}}(w)$, with $w = 1$ and $\beta = \alpha-\frac{n}{p}$ ^[4].

Remark 1.2. It is easy to see that for some ball $B$, the inequality (1.5) leads to inequality (1.4) holding, and the average $m_Bf$ in (1.4) can be replaced by a constant $c$ in following sense

In 2013, Ramseyer et al. in ^[8] studied the Lipschitz-type smoothness of fractional integral operators $\mathcal{I}_\alpha$ on variable exponent spaces when $p^{+} > \frac{\alpha}{n}$. Hence, when $p^{+} > \frac{\alpha}{n}$, it will be an interesting problem to see whether or not we can establish the boundedness of fractional integral operators $\mathcal{L}^{-\frac{\alpha}{2}}(\alpha > 0)$ related to Schrödinger operators from Lebesgue spaces $L^{p(\cdot)}$ into Lipschitz-type spaces with variable exponents. The main aim of this article is to answer the problem above.

We now state our results as the following two theorems.

Theorem 1.3. Let potential $V \in RH_{q}$ for some $q\geq n/2$ and $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^{n})$. Assume that $1 < p^{-}\leq p^{+} < \frac{n}{(\alpha-\delta_{0})^{+}}$ where $\delta_{0} = \min\{1, 2-n/q\}$, then the fractional integral operator $\mathcal{I}_{\alpha}$ defined in (1.2) is bounded from $L^{p(\cdot)}(\mathbb{R}^{n})$ into $Lip^{\mathcal{L}}_{\alpha, p(\cdot)}(\mathbb{R}^{n})$.

Theorem 1.4. Let the potential $V \in RH_{q}$ with $q\geq n/2$ and $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^{n})$. Assume that $1 < p^{-}\leq p^{+} < \frac{n}{(\alpha-\delta_{0})^{+}}$ where $\delta_{0} = \min\{1, 2-n/q\}$. If there exists a positive number $r_{0}$ such that $p(x)\leq p^{\infty}$ when $|x| > r_{0}$, then the fractional integral operator $\mathcal{I}_{\alpha}$ defined in (1.2) is bounded from $L^{p(\cdot), \infty}(\mathbb{R}^{n})$ into $Lip^{\mathcal{L}}_{\alpha, p(\cdot)}(\mathbb{R}^{n})$.

To prove Theorem 1.3, we first need to decompose $\mathbb{R}^n$ into the union of some disjoint ball $B(x_k, \rho(x_k))(k\geq1)$ according to the critical radius function $\rho(x)$ defined in (1.1). According to Lemma 2.6, we establish the necessary and sufficient conditions to ensure $f\in Lip^{\mathcal{L}}_{\alpha, p(\cdot)}(\mathbb{R}^n)$. In order to prove Theorem 1.3, by applying Corollary 1 and Remark 1.2, we only need to prove that the following two conditions hold:

(ⅰ) For every ball $B = B(x_{0}, r)$ with $r < \rho(x_{0})$, then

(ⅱ) For any $x_{0}\in\mathbb{R}^{n}$, then

In order to check the conditions (ⅰ) and (ⅱ) above, we need to find the accurate estimate of kernel $K_\alpha(x, y)$ of fractional integral operator $\mathcal{I_\alpha}$ (see Lemmas 2.8 and 2.9, then use them to obtain the proof of this theorem; the proof of the Theorem 1.4 proceeds identically).

The paper is organized as follows. In Section 2, we give some important lemmas. In Section 3, we are devoted to proving Theorems 1.3 and 1.4.

Throughout this article, $C$ always means a positive constant independent of the main parameters, which may not be the same in each occurrence. $B(x, r) = \{y\in\mathbb{R}^{n} : |x-y| < r\}$, $B_{k} = B(x_{0}, 2^{k}R)$ and $\chi_{B_{k}}$ are the characteristic functions of the set $B_{k}$ for $k\in \mathbb{Z}$. $|S|$ denotes the Lebesgue measure of $S$. $f\sim g$ means $C^{-1}g\leq f\leq Cg$.

2.   Some useful lemmas

In this section, we give several useful lemmas that are used frequently later on.

Lemma 2.1. ^[9] Assume that the exponent function $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})$. If $f\in L^{p(\cdot)}(\mathbb{R}^{n})$ and $g\in L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})$, then

where $r_{p} = 1+1/{p^{-}}-1/{p^{+}}.$

Lemma 2.2. ^[8] Assume that $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^{n})$ and $1 < p^{-}\leq p^{+} < \infty$, and $p(x)\leq p(\infty)$ when $|x| > r_{0} > 1$. For every ball $B$ and $f\in L^{p(\cdot), \infty}$ we have

where the constant $C$ only depends on $r_{0}$.

Fo the following lemma see Corollary 4.5.9 in ^[10].

Lemma 2.3. Let $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$, then for every ball $B\subset\mathbb{R}^{n}$ we have

and

Lemma 2.4. Assume that $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^{n})$, then for all balls $B$ and all measurable subsets $S: = B(x_{0}, r_{0})\subset B: = B(x_{1}, r_{1})$ we have

Proof. We only prove the first inequality in (2.1), and the second inequality in (2.1) proceeds identically. We consider three cases below by applying Lemma 2.3, and it holds that

1) if $|S| < 1 < |B|$, then $\dfrac{\|\chi_{S}\|_{p^{\prime}(\cdot)}}{\|\chi_{B}\|_{p^{\prime}(\cdot)}}\sim\dfrac{|S|^{\frac{1}{p^{\prime}(x_{S})}}} {|B|^{\frac{1}{p^{\prime}(\infty)}}}\leq\Big(\dfrac{|S|}{|B|}\Big)^{\frac{1}{(p^{\prime})^{+}}} = \Big(\dfrac{|S|}{|B|}\Big)^{1-\frac{1}{p^{-}}};$

2) if $1\leq |S| < |B|$, then $\dfrac{\|\chi_{S}\|_{p^{\prime}(\cdot)}}{\|\chi_{B}\|_{p^{\prime}(\cdot)}}\sim\dfrac{|S|^{\frac{1}{p^{\prime}(\infty)}}} {|B|^{\frac{1}{p^{\prime}(\infty)}}}\leq\Big(\dfrac{|S|}{|B|}\Big)^{\frac{1}{(p^{\prime})^{+}}} = \Big(\dfrac{|S|}{|B|}\Big)^{1-\frac{1}{p^{-}}};$

3) if $|S| < |B| < 1$, then $\dfrac{\|\chi_{S}\|_{p^{\prime}(\cdot)}}{\|\chi_{B}\|_{p^{\prime}(\cdot)}}\sim\dfrac{|S|^{\frac{1}{p^{\prime}(x_{S})}}} {|B|^{\frac{1}{p^{\prime}(x_{S})}}}|B|^{\frac{1}{p^{\prime}(x_{S})}-\frac{1}{p^{\prime}(x_{B})}}\leq C\Big(\dfrac{|S|}{|B|}\Big)^{\frac{1}{(p^{\prime})^{+}}} = C\Big(\dfrac{|S|}{|B|}\Big)^{1-\frac{1}{p^{-}}}$, where $x_{S}\in S$ and $x_{B}\in B$.

Indeed, since $|x_{B}-x_{S}|\leq 2r_{1}$, by using the local-Hölder continuity of $p^{\prime}(x)$ we have

We end the proof of this lemma.

Remark 2.1. Thanks to the second inequality in (2.1), it is easy to prove that

Lemma 2.5. ^[1] Suppose that the potential $V\in B_{q}$ with $q\geq n/2$, then there exists positive constants C and $k_{0}$ such that

$\textrm{1)}$ $\rho(x)\sim \rho(y)$ when $|x-y|\leq C\rho(x)$;

$\textrm{2)}$ $C^{-1}\rho(x)\Big(1+\frac{|x-y|}{\rho(x)}\Big)^{-k_{0}}\leq\rho(y)\leq C\rho(x)\Big(1+\frac{|x-y|}{\rho(x)}\Big)^{k_{0}/(k_{0}+1)}$.

Lemma 2.6. ^[11] There exists a sequence of points $\{x_{k}\}^{\infty}_{k = 1}$ in $\mathbb{R}^n$ such that $B_{k}: = B\big(x_{k}, \rho(x_{k})\big)$ satisfies

$\textrm{1)}$ $\mathbb{R}^{n} = \bigcup_{k} B_{k}$,

$\textrm{2)}$ For every $k\geq 1$, then there exists $N\geq 1$ such that card $\{j: 4B_{j}\cap4B_{k}\neq\emptyset\}\leq N$.

Lemma 2.7. Assume that $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})$ and $0 < \alpha < n$. Let sequence $\{x_{k}\}^{\infty}_{k = 1}$ satisfy the propositions of Lemma $\textrm{2.6}$. Then a function $f\in Lip^{\mathcal{L}}_{\alpha, p(\cdot)}(\mathbb{R}^n)$ if and only if $f$ satisfies $\textrm{(1.4)}$ for every ball, and

Proof. Let $B: = B(x, R)$ denote a ball with center $x$ and radius $R > \rho(x)$. Noting that $f$ satisfies (1.4), and thanks to Lemma 2.6 we obtain that the set $G = \{k: B\cap B_{k}\neq\emptyset\}$ is finite.

Applying Lemma 2.5, if $z\in B_{k}\cap B$, we get

Thus, for every $k\in G$, we have $B_{k}\subset CB$.

Thanks to Lemmas 2.4 and 2.6, it holds that

The proof of this lemma is completed.

Corollary 1. Assume that $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})$ and $0 < \alpha < n$, then a measurable function $f \in Lip^{\mathcal{L}}_{\alpha, p(\cdot)}$ if and only if $f$ satisfies $\textrm{(1.4)}$ for every ball $B(x, R)$ with radius $R < \rho(x)$ and

Let $k_{t}(x, y)$ denote the kernel of heat semigroup $e^{-t\mathcal{L}}$ associated to $\mathcal{L}$, and $K_\alpha(x, y)$ be the kernel of fractional integral operator $\mathcal{I}_{\alpha}$, then it holds that

Some estimates of $k_{t}$ are presented below.

Lemma 2.8. ^[12] There exists a constant $C$ such that for $N > 0$,

Lemma 2.9. ^[13] Let $0 < \delta < \min(1, 2-\frac{n}{q})$. If $|x-x_{0}| < \sqrt{t}$, then for $N > 0$ the kernel $k_t(x, y)$ defined in (2.4) satisfies

for all $x, y$ and $x_{0}$ in $\mathbb{R}^{n}$.

3.   Proof of theorems

In this section, we are devoted to the proof of Theorems 1.3 and 1.4. To prove Theorem 1.3, thanks to Corollary 1 and Remark 1.2, we only need to prove that the following two conditions hold:

(ⅰ) For every ball $B = B(x_{0}, r)$ with $r < \rho(x_{0})$, then

(ⅱ) For any $x_{0}\in\mathbb{R}^{n}$, then

We now begin to check that these conditions hold. First, we prove (ⅱ).

Assume that $B = B(x_{0}, R)$ and $R = \rho(x_{0})$. We write $f = f_{1}+f_{2}$, where $f_{1} = f\chi_{2B}$ and $f_{2} = f\chi_{\mathbb{R}^{n}\setminus2B}$. Hence, by the inequality (1.3), we have

Applying Tonelli theorem, Lemma 2.1 and Remark 1.2, we get the following estimate

To deal with $f_{2}$, let $x\in B$ and we split $\mathcal{I}_{\alpha}f_{2}$ as follows:

For $I_{1}$, if $x\in B$ and $y\in \mathbb{R}^{n}\setminus 2B$, we note that $|x_{0}-y| < |x_{0}-x|+|x-y| < C|x-y|$. By Lemma 2.8, it holds that

where the constant $C$ only depends the constant $M$.

Applying Lemma 2.1 to the last integral, we get

By using Lemma 2.4, we arrive at the inequality

Here, the series above converges when $M > n-\frac{n}{p^{+}}$. Hence, for such $M$,

For $I_{2}$, thanks to Lemma 2.8, we may choose $M$ as above and $N\geq M$, then it holds that

As $x\in B$, thanks to Lemma 2.5, $\rho(x)\sim\rho(x_{0}) = R$. Hence we have

Since $M+\alpha-n-N < 0$, the integral above for variable $t$ converges, and by applying inequality (3.2) we have

thus we have proved (ⅱ).

We now begin to prove that the condition (ⅰ) holds. Let $B = B(x_{0}, r)$ and $r < \rho(x_{0})$. We set $f = f_{1}+f_{2}$ with $f_{1} = f\chi_{2B}$ and $f_{2} = f\chi_{\mathbb{R}^{n}\setminus2B}$. We write

Thanks to (3.1), it holds that

Let $x\in B$ and we split $\mathcal{I}_{\alpha}f_{2}(x)$ as follows:

For $I_{3}$, by the same argument it holds that

For $I_{4}$, by Lemma 2.9 and (3.3), it follows that

Let $s = \frac{|x-y|^{2}}{t}$, then we obtain the following estimate

Notice that the integral above for variable $s$ is finite, thus we only need to compute the integral above for variable $y$. Thanks to inequality (3.2), it follows that

so (ⅰ) is proved.

Remark 3.1. By the same argument as the proof of Theorem 1.3, thanks to Lemma 2.2 we immediately obtained that the conclusions of Theorem 1.4 hold.

Use of AI tools declaration

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

Acknowledgments

Ping Li is partially supported by NSFC (No. 12371136). The authors would like to thank the anonymous referees for carefully reading the manuscript and providing valuable suggestions, which substantially helped in improving the quality of this paper. We also thank Professor Meng Qu for his useful discussions.

Conflict of interest

The authors declare there are no conflicts of interest.