Anisotropic Moser-Trudinger type inequality in Lorentz space

1.   Introduction

Let $\Omega$ be a domain with finite measure in Euclidean $n$-space $\mathbb{R}^n$ with $n\geq 2$. When $1\leq p < n$, by the Sobolev embedding theorem, $W_{0}^{1, p}(\Omega)\subset L^q(\Omega)$, $1\leq q \leq \frac{np}{n-p}$. Moreover, for the critical situation $p = n$, $W_{0}^{1, n}(\Omega)\subset L^q(\Omega)$, $\forall q\geq 1$. However, we can show by many examples that $W_{0}^{1, n}(\Omega) \nsubseteq L^\infty(\Omega)$ ^[1,2]. For the anisotropic Sobolev inequalities, we refer to ^[3,4,5,6].

In 1971, Moser ^[2] established Trudinger's inequality

for any $\alpha\leq\alpha_n = n\omega_{n-1}^{\frac{1}{n-1}}$, where $\omega_{n-1}$ is the area of the surface of the unit $n$-ball. This constant $\alpha_n$ is sharp in the sense that, if $\alpha > \alpha_n$, then the above inequality (1.1) can no longer hold with some $C$ independent of $u$.

Furthermore, Alvino, Ferone, and Trombetti ^[7] proved the following Moser-Trudinger type inequality in Lorentz space. They obtained that if

then there exists a constant $C$, depending only on $n$ and $q$, such that

where $q'$ is the conjugate index of $q$, i.e., $q' = \frac{q}{q-1}$ and $C_n$ is the measure of unit ball in $\mathbb{R}^n$, and the constant $\beta_q$ is sharp.

There have been many generalizations related to the Moser-Trudinger inequality, see ^{[1,8,9,10,11,12,13,14,15,16,17,18]}, etc. These inequalities play a key role in Geometry analysis, calculus of variations and PDEs, see ^{[19,20,21,22,23,24,25,26,27]}, etc.

Recently, many authors have intended to establish the Moser-Trudinger type inequality under the anisotropic norm. Let $F\in C^2(\mathbb{R}^n\backslash \{0\})$ be a positive, convex, and homogeneous function, and the polar $F^o(x)$ of which represents a Finsler metric on $\mathbb{R}^n$. By calculating the Euler-Lagrange equation of the minimization problem

we obtain an operator which is called Finsler $p$-Laplacian operator:

The Finsler $p$-Laplacian becomes the standard $p$-Laplacian when $F$ is the Euclidean modulus, as well as the pseudo-$p$-Laplacian when $F(\xi) = (\sum_{i = 1}^{n}|\xi_i|^p)^{\frac{1}{p}}$. The Finsler $p$-Laplacian operator has been studied in several papers, see ^{[28,29,30,31,32,33,34,35]}, etc. More properties of $F(x)$ will be given in Section 2.

The first work involving the anisotropic Moser-Trudinger type inequality was that of Wang and Xia ^[35]. They replaced the Dirichlet norm $(\int_{\Omega}|\nabla u|^ndx)^{\frac{1}{n}}$ with the anisotropic norm $(\int_{\Omega}F^n(\nabla u)dx)^{\frac{1}{n}}$ and proved the following inequality:

where $\lambda\leq\lambda_n = n^{\frac{n}{n-1}}\kappa_n^{\frac{1}{n-1}}$, $\kappa_n = |x\in \mathbb{R}^n|F^o(x)\leq 1 |$ is the volume of the unit Wulff ball in $\mathbb{R}^n$, and the constant $\lambda_n$ is sharp. Clearly, this is a generation result of (1.1).

Along this line, in this paper we consider the anisotropic Moser-Trudinger type inequality in Lorentz space $L(n, q)$, $1\leq q\leq \infty$. The definition and properties of Lorentz space can be seen in Section 2. Now, we state main results in the paper.

Theorem 1.1. Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $n \geq 2$, and let $u \in W_0^{1, n}(\Omega)$ be a function such that

We conclude that:

(i) If $q = 1$, then

(ii) If $1 < q < \infty$, then there exists a constant $C$, depending only on $n$ and $q$, such that

What is more, the constant $\bar{\lambda}_q$ is sharp in the sense that, for any $\lambda > \bar{\lambda}_q$, inequality (1.6) can no longer hold with any $C$ independent of $u$.

(iii) If $q = \infty$, then

What is more, the constant $\bar{\lambda}_\infty$ is sharp in the sense that, for any $\lambda \geq \bar{\lambda}_\infty$, inequality (1.7) can no longer hold with any $C$ independent of $u$.

2.   Preliminaries

In this section, we provide some preliminaries on the Finsler-Laplacian and Lorentz space.

Let $F(x)$ be a function of class $C^{2}(\mathbb{R}^n\backslash\{0\})$, which is convex and even. $F(x)$ has positively homogenous of degree $1$, i.e., for any $t\in \mathbb{R}$, $\xi\in \mathbb{R}^n$,

A classical example is $F(\xi) = (\sum_{i}|\xi_i|^{q})^{\frac{1}{q}}$, $q\geq 1$. We further assume that

By the property of the homogeneity of $F$, we can find two positive constants $0 < a_1\leq a_2 < \infty$ to have

The image of the map $\phi(\xi) = F_{\xi}(\xi)$, $\xi\in S^{n-1}$, is a smooth and convex hypersurface in $\mathbb{R}^n$, which is called the Wulff shape of $F$. The support function $F^{o}(x)$ of $F(x)$ is defined by $F^{o}(x): = \sup\limits_{\xi\in U}\langle x, \xi\rangle$, where $U = \{x\in \mathbb{R}^n:F(x)\leq 1\}$. We can check that $F^{o}:\mathbb{R}^n\mapsto [0, +\infty)$ is also a function of class $C^{2}(\mathbb{R}^n\backslash\{0\})$. Besides, $F^{o}(x)$ is also a convex and homogeneous function. Furthermore, $F^{o}(x)$ is dual to $F(x)$ in the sense that

Define

which is called the Wulff ball of center at $x_0$ with radius $r$. Also for convenience, we denote the unit Wulff ball of center at origin as

and

which is the the volume of $\mathcal{W}_{1}$.

By the assumptions of $F(x)$, we have some conclusions of the function $F(x)$, see ^{[34,36,37,38,39,40]}.

Lemma 2.1. We have

(i) $|F(x)-F(y)|\leq F(x+y)\leq F(x)+F(y)$;

(ii) $\frac{1}{C}\leq|\nabla F(x)|\leq C$, and $\frac{1}{C}\leq|\nabla F^{o}(x)|\leq C$ for some $C > 0$ and any $x\neq 0$;

(iii) $\langle x, \nabla F(x)\rangle = F(x), \langle x, \nabla F^{o}(x)\rangle = F^{o}(x)$ for any $x\neq 0$;

(iv) $F(\nabla F^{o}(x)) = 1$, $F^{o}(\nabla F(x)) = 1$ for any $x\neq 0$;

(v) $F^{o}(x) F_{\xi}(\nabla F^{o}(x)) = x$ for any $x\neq 0$;

(vi) $F_\xi(t\xi) = \mathit{\text{sgn}}(t)F_{\xi}(\xi)$ for any $\xi\neq 0$ and $t\neq 0$.

Now, we give the co-area formula and isoperimetric inequality with respect to $F$. For a domain $\Omega\subset \mathbb{R}^n$, $K\subset \Omega$ and a bounded variation function $u\in BV(\Omega)$, the anisotropic bounded variation of $u$ with respect to $F$ is defined by

and the anisotropic perimeter of $K$ with respect to $F$ is defined by

where $\mathcal{X}_{K}$ is the characteristic function of the set $K$. Then, we have the co-area formula

and the isoperimetric inequality

see ^[33]. Moreover, the equality in (2.3) holds if and only if $K$ is a Wulff ball.

In the following, let $\Omega^{\sharp}$ be the homothetic Wulff ball in $\mathbb{R}^n$ centered at the origin, which satisfies

where $|\cdot|$ denotes the volume. For a real-valued function $u: \Omega\rightarrow \mathbb{R}$, the distribution function $\mu_u(t): [0, +\infty)\rightarrow [0, +\infty]$ of $u$ is defined as

The decreasing rearrangement $u^*$ of $u$ is defined as

Clearly the support of $u^*$ satisfies $supp u^*\subseteq [0, |\Omega|]$.

Furthermore, the convex symmetrization $u^{\sharp}$ of $u$ with respect to $F$ is defined as

Next, we recall some properties of Lorentz space $L(p, q)$.

A function $u$ belongs to Lorentz space $L(p, q)$, $1 < p < \infty$, $1 \leq q \leq \infty$, if the quantity

is finite. In particular, we note that $L(p, p) = L^p(\Omega)$ and $L(p, \infty) = M^p$, which is called the Marcinkiewicz space. Another important property of Lorentz space is the intermediate property between $L^p$ space. Precisely, for $1 < q < p < r < \infty$, the following conclusion holds:

And, we have

When $q > p$, it is easy to check that the quantity (2.4) is not a norm. Letting

the quantity

is a norm for any $p$ and $q$. Besides, it is proved in ^[41] that quantity (2.6) is equivalent to the quantity (2.4)

where $C\geq 1$ is a constant depending only on $p$ and $q$. What is more, under the norm (2.6), $L(p, q)$ is a Banach space. We refer to ^{[41,42,43,44]} for more information involving the Lorentz space $L(p, q)$.

Now, we give a relationship between two nonnegative functions in $L^1(\Omega)$. We say that $u$ is dominated by $v$, which is written by $u \prec v$, if

Many properties about the relationship are given, for example, in ^[45]. For later use, we recall the following property:

Lemma 2.2. ^[45] The following conclusions are equivalent:

(i) $u \prec v$;

(ii) for all nonnegative functions $\omega \in L^\infty(\Omega)$,

(iii) for all nonnegative functions $\omega \in L^\infty(\Omega)$,

Now, we state a key method to construct a function $\Psi$, which is dominated by a function $\psi$, see ^[45]. Let $D(s)$, $s \in [0, |\Omega|]$, be a family of subsets of $\Omega$ which have the following properties:

(i) $|D(s)| = s$;

(ii) $D(s_1) \subset D(s_2)$, if $s_1 < s_2$;

(iii) $D(s) = \{x \in \Omega: |u(x)| > t\}, \; \; \; \; \text{if} \; \; \; \; s = \mu_{u}(t)$.

We see that this means that $D(s)$ is the family of the level sets of $|u(x)|$. For a nonnegative function $\psi \in L^1(\Omega)$, we define $\Psi(t)$ as the function such that

For (2.8), we say that $\Psi$ is built from $\psi$ on the level sets of $|u|$. It is shown in ^[45] that

3.   Proof of Theorem 1.1

In this section, we complete the proof of Theorem 1.1. The proof of Theorem 1.1 is an adaptation of ones given in ^[7]. We first give some key lemmas. Let $u$ be a measurable function in $\Omega$ such that

We let $G(t)$ be the function built from $g$ on the level sets of $u$, as in (2.8). Then, we have the following result:

Lemma 3.1. The estimate

holds.

Proof. By (2.8) and (3.1), we have

where $\mu(t) = \mu_{u}(t)$. By the co-area formula (2.2) and isoperimetric inequality (2.3), we have

Then, we get

Thus, the lemma is obtained by direct integration. □

By Lemma 3.1, for the purpose of the estimate $u(x)$, we can estimate the $H$-symmetric and decreasing function

By the following lemma, we can estimate $u(x)$ by a function involving $g^\ast$.

Lemma 3.2. Let $g \in L^1(\Omega)$. For any nonnegative function $G$ defined in $[0, |\Omega|]$ such that $G \prec g$, we let $v$ be the function defined in (3.3). Then, we obtain

Proof. By (3.3), we have

where

Clearly, for any fixed $s$, $h(m, s)$ is decreasing with respect to $m$. Then, by Lemma 2.2 and the property $G \prec g$, we obtain (3.4). □ For the aim to prove Theorem 1.1, we need the following lemma proved by Adams ^[46].

Lemma 3.3. ^[46] Let $a(s, t)$ be a nonnegative measurable function in $\mathbb{R} \times [0, \infty)$, and for some $q \in (1, \infty)$, $q' = \frac{q}{q-1}$,

and

Assume that $\Phi(s) \geq 0$ and

Then, there exists a constant $C$, depending only on $q$ and $\nu$ such that

where

Now, it is sufficient to prove Theorem 1.1.

Proof of Theorem 1.1. We complete the proof by distinguishing three cases.

Case (i) $q = 1$. By Lemma 3.1, we have that

Then, by $G \prec g$ and Lemma 2.2, we have that

Then, (1.5) holds.

Case (ii) $1 < q < \infty$. By Lemma 3.2, we have

For the convenience, we denote $n'$ as the conjugate index of $n$, i.e., $n' = \frac{n}{n-1}$. Then,

where

and

It is obvious that (3.5) holds. Next, for any $1 < q < \infty$, we obtain

Then, we get (3.6) by choosing $\nu = (\frac{n'}{q'})^\frac{1}{q'}$.

Finally, by (1.4), we have that

This means that (3.7) holds. Then, by Lemma 3.3, we have

which means that

Furthermore, by the fact $u^*(s) \leq \bar{u}(s)$, we obtain

Case (iii) $q = \infty$. By (3.8) and (1.4), we obtain

It follows that

Clearly, the right hand side is finite if and only if $\lambda < n\kappa_n^\frac{1}{n} = \bar{\lambda}_\infty$. Then, we get (1.7).

At last, we prove the sharpness of (1.5)–(1.7).

We easily see that equality (1.5) holds if $u(x) = u^\sharp(x)$ and $F(\nabla u) = F(\nabla u)^\sharp \in L(n, 1)$.

The proof of sharpness for (1.3) is more complicated. If $1 < q < \infty$, for any $\lambda > \bar{\lambda}_q$, we will construct a sequence of functions $u_k$ such that $\|F(\nabla u_k)\|_{n, q} \leq 1$ and

Define

Then, by direct calculation, using Lemma 2.1, we have that the decreasing rearrangement of $F(\nabla {u_k})$ is

We consider $1 < q < n$, $q = n$, and $n < q < \infty$ separately.

When $1 < q < n$, making the change of variable $m = 1+se^k$, then

We let

Then,

Hence, we have $\lim\limits_{k\rightarrow \infty}\beta_k = 1$.

Now, we set

Clearly, $\|F(\nabla {v_k})\|_{n, q} = 1$. However, when $\lambda > \bar{\lambda}_q = (n\kappa_n^\frac{1}{n})^{q'}$, as $k\rightarrow +\infty$,

When $q = n$, the proof is similar to that in ^[2]. We have

Then, when $\lambda > \bar{\lambda}_{n, n} = n^\frac{n}{n-1}\kappa_n^\frac{1}{n-1}$, as $k\rightarrow +\infty$,

When $n\leq q < \infty$, from (2.5) and (3.11) we have

Then, as the case of $q = n$, it is easy to prove that

When $q = \infty$, we construct a function $u$ such that $\|F(\nabla u)\|_{n, \infty} \leq 1$, and for any $\lambda \geq \bar{\lambda}_\infty = n\kappa_n^\frac{1}{n}$,

Let

By direct calculation, using Lemma 2.1, we obtain

and then

Thus, when $\lambda \geq \bar{\lambda}_\infty = n\kappa_n^\frac{1}{n}$, by the co-area formula (2.2), we have

The proof is completed. □

4.   Conclusions

In this paper, we mainly study the anisotropic Moser-Trudinger type inequality in Lorentz space $L(n, q)$, $1 \leq q \leq \infty$. It is a generation result of Moser-Trudinger type inequality in Lorentz space. The extremal function of such inequality is closely related to existence of solutions of Finsler-Liouville type equation. We believe that the sharp inequality will be the key tool to study the existence of solutions for some quasi-linear elliptic equations, such as Finsler-Laplacian equation.

Use of AI tools declaration

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

Acknowledgments

The work of the first author is supported by NSFC of China (No. 12001472).

Conflict of interest

The authors declare there is no conflict of interest.