Sharp Adams type inequalities in Lorentz-Sobole space

1.   Introduction

Sharp Moser-Trudinger inequality and its high-order form (which is called Adams inequality) have received a lot of attention due to their wide applications to problems in geometric analysis, partial differential equations, spectral theory and stability of matter ^{[2,3,5,8,9,10,11,12,24,25,26,27]}. This paper is concerned with the problem of finding optimal Adams type inequalities in Lorentz-Sobolev space.

The Trudinger inequality, which can be seen as the critical case of the Sobolev imbedding, was first obtained by Trudinger ^[30]. More precisely, Trudinger employed the power series expansion to prove that there exists $\beta > 0$, such that

where $\Omega\subset \mathbb{R}^n$ is a bounded smooth domain and $W^{1, p}_0(\Omega)$ denotes the usual Sobolev space on $\Omega$, i.e., the completion of $C_{0}^{\infty}(\Omega)$(the space of all functions being infinity-times continuously differential in $\Omega$ with compact support) with the norm

Let $\Omega\subset\mathbb{R}^{n}$ be an open domain with finite measure. It is well known that for a positive integer $k < n$ and $1\leq p < \frac{n}{k}$, the Sobolev space $W^{k, p}_0\left(\Omega\right)$ embeds continuously into $L^{\frac{np}{n-kp}}\left(\Omega\right)$, but in the borderline case $p = \frac{n}{k}$, $W^{k, \frac{n}{k}}_0\left(\Omega\right) \varsubsetneq L^{\infty}\left(\Omega\right)$, unless $k = n$. For the case $k = 1$, Yudovich ^[31] and Trudinger ^[30] have shown that

and the function $E_\beta$ is continuous on $W^{1, n}_0\left(\Omega\right)$. In 1971, Moser sharped the Trudinger inequality and gave the sharp constant $\beta = nw_{n-1}^{\frac{1}{n-1}}$ of (1.1) by using the technique of the symmetry and rearrangement in ^[20].

Theorem A. ^[20] Let $\Omega\subset\mathbb{R}^{n}$ be an open domain with finite measure. Then, there exists a sharp constant $\beta_n = n\left(\frac{n\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})+1}\right)^{\frac{1}{n-1}}$, such that

for any $\beta\leq\beta_n$ and any $f\in C_0^\infty(\Omega)$ with $\int_\Omega|\bigtriangledown f|^ndx\leq1$. The constant $\beta_n$ is sharp in the sense that the above inequality can no longer hold with some $C_0$ independent of $f$ if $\beta > \beta_n$.

Theorem A has been extended in many directions, one of which states that

for any $\beta\leq\beta_{n} = n\omega_{n-1}^{\frac{1}{n-1}}$, plays an important role in analysis, where $\omega_{n-1}$ is the surface measure of the unit ball in $\mathbb{R}^{n}$. In fact, the constant $\beta_{n}$ is sharp in the sense that if $\beta > \beta_n$, the supremum is infinity.

Since the Polyá-Szegö inequality, on which the technique of the symmetry and rearrangement depends, is not valid on the high-order Sobolev space, many challenges arise in the research of high-order Trudinger-Moser inequalities. In 1988, Adams ^[1] utilized the method of representative formulas and potential theory to establish the sharp Adams inequalities on bounded domains.

Theorem B. ^[1] Let $\Omega$ be an open and bounded set in $\mathbb{R}^n$. If $m$ is a positive integer less than $n$, then there exists a constant $C_0 = C(n, m) > 0$ such that for any $u\in W_0^{m, \frac{n}{m}}(\Omega)$ with $\|\nabla^m u\|_{L^\frac{n}{m}(\Omega)}\leq 1$,

where

Furthermore, the constant $\beta(n, m)$ is best possible in the sense that for any $\beta > \beta(n, m)$, the integral can be made as large as possible. In the case of Sobolev space with homogeneous Navier boundary conditions $W_N^{m, \frac{n}{m}}(\Omega)$, the Adams inequality was extended by Cassani and Tarsi in ^[6]. It is easy to check that $W_N^{m, \frac{n}{m}}(\Omega)$ contains $W_0^{m, \frac{n}{m}}(\Omega)$ as a closed subspace.

Adimurthi and Sandeep proved a singular Moser-Trudinger inequality with the sharp constant in ^[2]. Since then, Moser's results for the first order derivatives and Adams' result for the high order derivatives were extended to the unbounded domain case. Earlier research of the Moser-Trudinger inequalities on the whole space goes back to Cao's work in ^[7]. Later, Li and Ruf ^[19,23] improved Cao's result and established the following result

where proof relies on the rearrangement argument and the Polyá-Szegö inequality. For more on the rearrangement argument, see ^[21,29]. In 2013, Lam and Lu ^[17] used a symmetrization-free approach to give a simple proof for the sharp Moser-Trudinger inequalities in $W^{1, n}(\mathbb{R}^n)$. It should be pointed out that this approach is surprisingly simple and can be easily applied to other settings where symmetrization argument does not work. Furthermore, they also developed a new tool to establish the Moser-Trudinger inequalities on the Heisenberg group and the Fractional Adams inequalities in $W^{s, \frac{n}{s}}(\mathbb{R}^n)$ ($0 < s < n$) (^[16]). For more applications of the symmetrization-free method, see also ^[18,32]. The Adams type inequality on $W_0^{m, \frac{n}{m}}(\Omega)$ when $\Omega$ has infinite volume and $m$ is an even integer was studied recently by Ruf and Sani in ^[22].

In ^[22], Ruf and Sani used the norm $\left\Vert u\right\Vert_{m, n} = \left\Vert (-\triangle+I)^{\frac{m}{2}}u\right\Vert_{\frac{n}{m}}$, which is equivalent to the standard Sobolev norm

In particular, if $u\in W_0^{m, \frac{n}{m}}(\Omega)$ or $u\in W^{m, \frac{n}{m}}(\mathbb{R}^n),$ then $\left\Vert u\right\Vert _{W^{m, \frac{n}{m}}}\leq \left\Vert u\right\Vert _{m, n}.$ Since Ruf and Sani only considered the case when $m$ is even, it leaves an open question if Ruf and Sani$^,$s result is still right when $m$ is odd. Recently, the authors of ^[17] solved the problem and proved the results of Adams type inequalities on unbounded domains when $m$ is odd.

We notice that when $\Omega$ has infinite volume, the usual Moser-Truding inequality become meaningless. In the case $|\Omega| = +\infty$, a modified Moser-Truding type inequality was established in ^[13].

Theorem C. ^[13] Assume $n\geq2$, $\beta > 0, -\infty < s\leq\alpha < n$ and $u\in L^n(\mathbb{R}^n; |x|^{-s}dx)\cap W^{1, n}(\mathbb{R}^{n})$, there esists a positive constant $C = C(n, s, \alpha, \beta)$ such that the inequality

Furthermore, for all $\beta\leq(1-\frac{\alpha}{n})\beta_n$, there holds

where $\phi(t) = e^t-\sum_{j = 0}^{n-2}\frac{t^j}{j!}$ and $L^n(\mathbb{R}^n; |x|^{-s}dx)$ denotes the weighted Lebesgue space endowed with the norm

Moreover the constant $(1-\frac{\alpha}{n})\beta_n$ is sharp in the sense that if $\beta > (1-\frac{\alpha}{n})\beta_n$, the supremum is infinity.

When $\alpha = 0$, Ruf in ^[23] and Li-Ruf in ^[19] proved the above modified Moser-Truding type inequality in $\mathbb{R}^{2}.$ Such type of inequality on unbounded domains in the subcritical case $(\beta < \beta_n, \ \alpha = 0)$ was first established by Cao in ^[7] for $n = 2$ and Adachi Tanaka in ^[4] for $n\geq3$ in high dimension.

In this paper, we will consider some sharp Adams type inequalities in Lorentz-Sobolev space $W^{\alpha}_{\frac{n}{m}, q}(\Omega\subseteq \mathbb{R}^n)$ with $q\neq n$ (If $q = n,$ the Lorentz norm becomes the $L^n(\mathbb{R}^n)$ domain norm). Let $1 < p < +\infty$ and $1\leq{q} < +\infty$. Then we recall the Lorentz space $L_{p, q}(\mathbb{R}^n)$ as: $\psi\in L_{p, q}(\mathbb{R}^n)$ if

It is well known that $\|\cdot\|_{p, q}^*$ is not a norm, and

is a norm for any $p$ and $q$. However, they are equivalent in the sense that

The Sobolev-Lorentz space (^[15])

equipped with the norm

for $0 < \alpha < n, m < n, 1 < q < \infty.$ For simplicity of notation, we write

for any $\Omega \subseteq \mathbb{R}^n$. Then we can formulate our main results as follows.

Theorem 1. Let $m\leq n$ be an integer, $0\leq\alpha < n$, $1 < q < +\infty$ and $A$ be a positive real number. Then for any bounded domain $\Omega\subset\mathbb{R}^n$ with $|\Omega|\geq A > 0,$ we have

$(1)$ $\underset{u\in \overline{W_{\frac{n}{m}, q}^{m}(\Omega)}}{\sup}\frac{1}{|\Omega|} \int_{\Omega }\exp({\beta_{n, m, q}\left\vert u\right\vert^{\frac{q}{q-1}}})dx\leq C_{m, n, q}.$

Additionally, the constant $\beta_{n, m, q} = \left(\frac{n}{\omega_{n-1}}\right)^{q^\prime\frac{n-m}{n}}K_{m, n}^{-q^\prime }$ is sharp in the sense that the supremum is infinity if $\beta > \beta_{n, m, q}$, where $K_{m, n} = \frac{\Gamma(\frac{n-m}{2})}{\pi^{\frac{n}{2}}2^m\Gamma(\frac{m}{2})}$.

$(2)$ $\underset{u\in \overline{W_{\frac{n}{m}, q}^{m}(\Omega)}}{\sup} \int_{\Omega}\frac{\exp[\beta_{n, m, q}(1-\frac{\alpha}{n})|u|^{\frac{q}{q-1}}]}{|x|^\alpha}\leq{C_{m, n, q, \alpha}}.$

Additionally, the constant $\beta_{n, m, q}$ is sharp in the sense that the supremum is infinity if $\beta > \beta_{n, m, q}$.

For the unbounded domain, we take $\mathbb{R}^n$ for example to have the following inequalities.

Theorem 2. Let $m, q, \alpha$ be the same as in Theorem 1. Then we have

and

where $\Phi(x) = e^x-\sum_{j = 0}^{k_{0}}\frac{x^j}{j!}, k_{0} = [\frac{q-1}{q}\frac{n}{m}]$ and $\beta_{n, m, q}$ is sharp in the sense that the supremum is infinity if $\beta > \beta_{n, m, q}$.

2.   Proofs of the main results

We begin this section with some preparations which are necessary for the proofs of our main results. Let $f:\mathbb{R}^n\rightarrow {\mathbb{R}}$ such that

for every $t > 0$. Its distribution function $d_{f}(t)$ and its decreasing rearrangement $f^{\ast}$ are defined by

and

respectively. Now, define $f^\sharp:\mathbb{R}^n\rightarrow {\mathbb{R}}$ by

where $v_{n}$ is the volume of the unit ball in $\mathbb{R}^n$. Then for every continuous increasing function $\Psi:[0, +\infty)\rightarrow {[0, +\infty)}$, it follows from ^[14] that

Since $f^{\ast}$ is nonincreasing, the maximal function of $f^{\ast}$, which is defined by

is also nonincreasing and $f^{\ast}\leq{f^{\ast\ast}}$. For more properties of the rearrangement, we refer the reader to ^[14,28].

Lemma 2.1. Let $0 < \alpha\leq{1}, 1 < p < \infty$ and $a(s, t)$ be a non-negative measurable function on $(-\infty, \infty)\times{[0, \infty]}$ such that

Then there is a constant $c_{0} = c_{0}(p, b, \alpha)$ such that if

then

Proof. The integral in (2.1) can be written as

where $F_{\alpha}(t)\leq\lambda$ and $E_{\alpha \lambda} = \int_{\Omega}e^{\alpha \lambda|u|\frac{n}{n-1}}dx$.

We first show that there is a constant $C = C(p, b, \alpha) > 0$ such that $F_{\alpha}(t)\geq-C$ for all $t\geq0.$ To do so, we claim that if $E_{\alpha\lambda}\neq\emptyset,$ then $\lambda\geq-C,$ and furthermore that if $t\in E_{\alpha\lambda},$ then there are $A_1 > 0$ and $B_1 > 0$ such that

In fact, if $E_{\alpha\lambda}\neq\emptyset,$ and $t\in E_{\alpha\lambda},$ then

Hence the desired result can be obtained by repeating the argument as in the proof of [1, Lemma 1].

The second is to prove that $|E_{\alpha\lambda}|\leq A|\lambda|+B$ for constants $A$ and $B$ depending only on $p, b$ and $\alpha$, which is straightforward via modifying the argument of [1, Lemma 1]. Thus, we complete the proof of Lemma 2.1.

Lemma 2.2. ^[15] There exists a constant $K_{n, m}$ depending only on $m$ and $n$ such that

for all $u\in{W^{\alpha}_{\frac{n}{m}, q}(\mathbb{R}^n)}$ and $1 < q\leq{+\infty}$.

Having disposed of the above lemmas, we can now turn to the proofs of Theorems 1 and 2.

2.1.   Proof of Theorem 1

Since $u\in{W^{m}_{\frac{n}{m}, q}(\mathbb{R}^n)}$, there exists a function $f\in{L_{\frac{n}{m}, q}(\mathbb{R}^n)}$ with $u = (I-\Delta)^{-\frac{m}{2}}f$ and $\|f\|_{\frac{n}{m}, q}\leq{1}$. Then $u = G_{m}\ast{f}$, where

It follows from O'Neil's lemma ^[21] that for all $t\geq{0}$,

Since $G_{m}$ is radial and decreasing, $G^{\ast}_{m}(r) = G_{m}(v_{n}^{\frac{1}{n}}r^{\frac{1}{n}})$. Therefore, by taking

and using the Hardy-Littlewood inequality, we find

where

Hence,

Set

Since

and $|\Omega| > A > 0$, we get

and

It's easy to check that when $0 < s < t$, $a(s, t)\leq{1}$. This, along with Lemma 2.1 gives $\int_{0}^{+\infty}\exp[-F(s)]ds\leq C_0$. Therefore, we have obtained

Next, we show the sharpness of $\beta_{n, m, q}$ according to Adams method in ^[1]. The equivalent form of Theorem 1(1) is

We need to prove that $\left(\frac{n}{\omega_{n-1}}\right)^{q^\prime\frac{(n-m)}{n}}$ is the best one for $\Omega = B$ (the unit ball centered at the origin). Choose $f\geq0$ such that $G_m\ast f\geq1$ for $x\in B_r: = \{x\in \mathbb{R}:|x|\leq r\}$ with $0 < r < 1.$ The equivalent form gives

and hence

thereby finding

with $Cap_{W^mL^{\frac{n}{m}, q}}(B_r, B) = \inf\|f\|_{L^{\frac{n}{m}, q}}^{q^\prime}(B).$ Here the infimum is taken over all $f > 0$ vanishing on the complement of $B$, and $G_m\ast f(x)\geq1$ on $E.$ It follows from the proof of [1, Theorem 2] that for any $\varepsilon > 0,$ one can find $0 < r < 1$ small enough such that

with

and

Then the domain of $h^*(t)$ is $(r^n\frac{\omega_{n-1}}{n}, \ \infty)$, where

Consequently,

This gives

Finally, a simple computation yields

which complete the proof of (1).

The statement (2) can be proved similarly as that of (1), we only pay attention to the difference arguments as follows. The Hardy-Littlewood inequality shows that

where

Let

Then

and

Since $a(s, t)\leq{1}$ for $0 < s < t$, we have $\int_{0}^{+\infty}\exp[-F_{1-\frac{\beta}{n}}(s)]ds$ by Lemma 2.1. Hence

What is left is to show the sharpness of $(1-\frac{\alpha}{n})\beta_{n, m, q}$, which also inspired by ^[1]. Since the equivalent form of (2) is

we only need to prove that $\left(\frac{n}{\omega_{n-1}}\right)^{q^\prime\frac{n-m}{n}}$ is the best one for $\Omega = B$. Similarly analysis as that of (1), we choose $f\geq0$ such that $G_m\ast f\geq1$ for $x\in B_r$ with $0 < r < 1,$ it follows from (1) that

and

Hence, $\beta\leq n\lim_{r\rightarrow0}(\log\frac{1}{r})[Cap_{\dot{w}L^{\frac{n}{m}, q}}(B_r; B)]^{q'},$ with $Cap_{\dot{w}L^{\frac{n}{m}, q}}(E; B) = \inf\|f\|_{L^{\frac{n}{s}, q}(B)},$ and $E$ is a compact subset of $B,$ where the infimum is taken over all $f\geq0$ vanishing on the complement of $B$, and $G_m*f(x)\geq1\ \text{on}\ \ E.$ Analysis similar as that of (1), for any $\varepsilon > 0,$ we can choose $0 < r < 1$ small enough such that

with

Consequently, we get

This shows

which gives

as desired.

2.2.   Proof of Theorem 2

For any $u\in{W^{m}_{\frac{n}{m}, q}(\mathbb{R}^n)}$ with $\|(I-\Delta)^{\frac{m}{2}}u\|_{\frac{n}{m}, q}\leq{1}$, set $A(u) = \|u\|_{w_{\frac{n}{m}, q}}$ and $\Omega = \{x\in{\mathbb{R}^n}:|u| > A(u)\}$. Then it is clear that $A(u)\leq1.$ By the property of the rearrangement, we know that for any $t\in{[0, |\Omega|)}$,

At the same time, Lemma 2.2 shows

Combining (2.3) with (2.4), we have $t\leq{K_{n, m}^{\frac{n}{m}}}$ for any $t\in{[0, |\Omega|)}.$ Therefore $|\Omega|\leq{K_{n, m}^{\frac{n}{m}}}.$ Write

where

Choose $\Omega^\prime$ such that $\Omega\subset\Omega^\prime$ and $|\Omega^\prime| = K_{n, m}^{\frac{n}{m}}.$ Then by Theorem B, we have

thereby finding

For the term $I_2,$ since $\mathbb{R}^n \setminus\Omega\subset\{|u(x)| < 1\}$ and $(k_0+1)\frac{q}{q-1} = ([\frac{q}{q-1}\frac{n}{m}]+1)\frac{q}{q-1} > \frac{n}{m},$ the Hardy-Littlewood inequality and Lemma 2.2 shows that

This is the first desired result.

The second inequality of Theorem 2 can be proved similarly via Theorem 1 and the above arguments, we omit its proof here.

3.   Conclusions

We deal mainly with several sharp weighted Adams type inequalities in Lorentz-Sobolev spaces. In particular, the sharpness of these inequalities were also obtained by constructing a proper test sequence. Moreover, we discuss the boundedness of partial fractional integral operators.

Use of AI tools declaration

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

Acknowledgments

The authors are supported by National Natural Science Foundation of China (Grant No.12271232) and Natural Science Foundation of Shandong Province (Grant No. ZR2022MA18).

Conflict of interest

The authors declare no conflicts of interest in this paper.