1. Introduction

Let $\Omega$ be a bounded domain of $\mathbb {R}^{n}$ containing the origin, $n\geq 2$ and $p>1$, the classical p-Hardy inequality asserts that

(1.1)\begin{equation} \int_{\Omega}|\nabla u|^{p}\,{\rm d}x\geq \left|\frac{n-p}{p}\right|^{p}\int_{\Omega}\frac{|u|^{p}}{|x|^{p}}\,{\rm d}x, \forall \ u\in C_0^{\infty}(\Omega), \end{equation}

with $\left |\frac {n-p}{p}\right |^{p}$ being the best constant and never achieved [Reference Brezis and Marcus6, Reference Brezis, Marcus and Shafrir7, Reference Davies10, Reference Hardy, Littlewood and Pólya15, Reference Opic and Kufner20, Reference Vazquez and Zuazua24]. Many improvements of Hardy inequality can be obtained by adding the error term in the right side of (1.1) [Reference Brezis and Vazquez8, Reference Gazzola, Grunau and Mitidieri12]. The first improvement was obtained by Brezis and Vazquez [Reference Brezis and Vazquez8]. When $p=2$, they have shown that (1.1) can be improved by adding subcritical Sobolev term $\int _{\Omega }|u|^{q}\,\textrm {d}x (1\leq q<2^{*}=\frac {2n}{n-2})$. After that, Chaudhuri and Ramaswamy [Reference Chaudhuri and Ramaswamy9] improved inequality (1.1) by introducing a subcritical Hardy–Sobolev term $\int _{\Omega }\frac {|u|^{q}}{|x|^{\beta }}\,\textrm {d}x \ \ (0\leq \beta <2$, $1\leq q<2^{*}_{\beta }:=\frac {2(n-\beta )}{n-2})$ [Reference Adimurthi and Tertikas1]. Later, Adimurthi, Chaudhuri and Ramaswamy [Reference Adimurthi and Ramaswamy2] extended their results to general $L_p$ Hardy inequality for $2\leq p< n$. In [Reference Filippas and Tertikas11], Filippas and Tertikas pointed out that the critical Sobolev type improvement for $p=2$ could be established by adding a logarithmic term. Their result is as follows.

Let $\Omega$ be a bounded domain in $\mathbb {R}^{n}(n\geq 3)$ containing the origin, $R_{\Omega }:=\sup _{x\in \Omega }|x|$, then for any $u\in H^{1}_0(\Omega )$ and $R\geq R_{\Omega }$, there exists a constant $C_n>0$ depending only on $n$, such that

(1.2)\begin{equation} \int_{\Omega}|\nabla u|^{2}\,{\rm d}x-\left(\frac{n-2}{2}\right)^{2}\int_{\Omega}\frac{|u|^{2}}{|x|^{2}}\,{\rm d}x \geq C_{n}\left(\int_{\Omega}\left(|u|^{2^{*}}X_1^{1+\frac{2^{*}}{2}}\left(\frac{|x|}{R}\right)\right)\,{\rm d}x\right)^{\frac{2}{2^{*}}}. \end{equation}

Here

(1.3)\begin{equation} X_1(t)=(1-logt)^{{-}1}, \quad t\in (0,1]. \end{equation}

Inequality (1.2) was sharp in the sense that $X_1^{1+\frac {2^{*}}{2}}$ cannot be replaced by a smaller power of $X_1$.

In [Reference Filippas and Tertikas11], the authors also established the series expansion of Hardy inequality. Their results were extended to the following general $L_p$ ($p\neq n$) Hardy inequality [Reference Barbatis, Filippas and Tertikas5].

Let $\Omega$ be a bounded domain in $\mathbb {R}^{n}(n\geq 3)$ containing the origin, $R_{\Omega }:=\sup _{x\in \Omega }|x|$, then for any $u\in W^{1,p}_0(\Omega \setminus \{0\})$ and $R\geq R_{\Omega }$, there holds

(1.4)\begin{align} \int_{\Omega}|\nabla u|^{p}\,{\rm d}x& \geq \left|\frac{n-p}{p}\right|^{p}\int_{\Omega}\frac{|u|^{p}}{|x|^{p}}\,{\rm d}x+\frac{p-1}{2p}\left|\frac{n-p}{p}\right|^{p-2}\nonumber\\ & \quad \times \sum\limits_{i=1}^{\infty}\int_{\Omega}\frac{|u|^{p}}{|x|^{p}}X_1^{2}\left(\frac{|x|}{R}\right)X_2^{2}\left(\frac{|x|}{R}\right)\cdots X_i^{2}\left(\frac{|x|}{R}\right)\,{\rm d}x. \end{align}

Here

(1.5)\begin{equation} X_k(t)=X_1(X_{k-1}(t)), \quad k\geq 2. \end{equation}

In [Reference Filippas and Tertikas11], the authors also proved the following series expansion of Hardy inequality for $p=2$ with critical sobolev term.

(1.6)\begin{align} \int_{\Omega}|\nabla u|^{2}\,{\rm d}x& \geq \left(\frac{n-2}{2}\right)^{2}\int_{\Omega}\frac{|u|^{2}}{|x|^{2}}\,{\rm d}x+\frac{1}{4}\sum\limits_{i=1}^{k}\int_{\Omega}\frac{|u|^{2}}{|x|^{2}}\prod\limits_{j=1}^{i}X_j^{2}\left(\frac{|x|}{R}\right)\,{\rm d}x\nonumber\\ & \quad+C_n\left(\int_{\Omega}|u|^{2^{*}}\prod\limits_{i=1}^{k+1}X_i^{1+\frac{2^{*}}{2}}\left(\frac{|x|}{R}\right)\,{\rm d}x\right)^{\frac{2}{2^{*}}}. \end{align}

The exponent $1+\frac {2^{*}}{2}$ on $X_{k+1}$ cannot be decreased.

Recently, Gkikas and Psaradakis [Reference Gkikas and Psaradakis13] generalized inequality (1.6) to the general case $1< p< n$ and $p>n$. When $1< p< n$, by adding an optimally weighted critical Sobolev norm, they obtained the following results.

Let $\Omega$ be a bounded domain in $\mathbb {R}^{n}$ containing the origin, $n\geq 2$ and $1< p< n$, $R_{\Omega }:=\sup _{x\in \Omega }|x|$, there exist constants $C_n>0$ depending only on $n$ and $B:=B(n,\,p)\geq 1$, such that for any $u\in W^{1,p}_0(\Omega )$, $R\geq B R_{\Omega }$ and $k\in \mathbb {N}$, there holds

(1.7)\begin{align} \int_{\Omega}|\nabla u|^{p}\,{\rm d}x& \geq \left(\frac{n-p}{p}\right)^{p}\int_{\Omega}\frac{|u|^{p}}{|x|^{p}}\,{\rm d}x+\frac{p-1}{2p}\left(\frac{n-p}{p}\right)^{p-2}\nonumber\\ & \quad\times \sum\limits_{i=1}^{k}\int_{\Omega}\frac{|u|^{p}}{|x|^{p}}\prod\limits_{j=1}^{i}X_j^{2}\left(\frac{|x|}{R}\right)\,{\rm d}x\nonumber\\ & \quad+C_n\left(\int_{\Omega}|u|^{p^{*}}\prod\limits_{i=1}^{k+1}X_i^{1+\frac{p^{*}}{p}}\left(\frac{|x|}{R}\right)\,{\rm d}x\right)^{\frac{p}{p^{*}}}. \end{align}

The exponent $1+\frac {p^{*}}{p}$ on $X_{k+1}$ cannot be decreased. When $p>n$, they established the series expansion of $L_p$ Hardy inequality by adding the optimally weighted Hölder seminorm.

All the previous results we mentioned are concerning about the case $p\neq n$. When $p=n$, Hardy inequality can be stated as follows [Reference Adimurthi and Ramaswamy2–Reference Barbatis, Filippas and Tertikas4, Reference Leray16].

Let $\Omega$ be a bounded domain in $\mathbb {R}^{n} (n\geq 2)$, containing the origin, then for any $R\geq R_{\Omega }$ and $u\in W_0^{1,n}(\Omega )$, one has

(1.8)\begin{equation} \int_{\Omega}|\nabla u|^{n}\,{\rm d}x\geq \left(\frac{n-1}{n}\right)^{n}\int_{\Omega}\frac{|u|^{n}}{|x|^{n}}X_1\left(\frac{|x|}{R}\right)\,{\rm d}x. \end{equation}

Barbatis, Filippas and Tertikas [Reference Barbatis, Filippas and Tertikas5] established the following series expansion of Hardy inequality for the case $p=n$.

Let $\Omega$ be a bounded domain in $\mathbb {R}^{n} (n\geq 2)$ containing the origin, then for any $R>R_{\Omega }$ and for all $u\in W^{1,n}_0(\Omega \setminus \{0\})$, one has

(1.9)\begin{align} \int_{\Omega}|\nabla u|^{n}\,{\rm d}x& \geq \left(\frac{n-1}{n}\right)^{n}\int_{\Omega}\frac{|u|^{n}}{|x|^{n}}X_1^{n}\left(\frac{|x|}{R}\right)\,{\rm d}x+\frac{1}{2}\left(\frac{n-1}{n}\right)^{n-1}\nonumber\\ & \quad\times \sum\limits_{i=2}^{\infty}\int_{\Omega}\frac{|u|^{n}}{|x|^{n}}X_1^{n}\left(\frac{|x|}{R}\right)X_2^{2}\left(\frac{|x|}{R}\right)\cdots X_i^{2}\left(\frac{|x|}{R}\right)\,{\rm d}x. \end{align}

In analogy with inequality (1.1), it is natural to ask whether similar critical Sobolev term can be added into inequality (1.8). Since the limit case of critical Sobolev inequality is Moser–Trudinger inequality [Reference Li and Ruf17, Reference Moser18, Reference Ruf22, Reference Tintarev23], the natural substitute of critical Sobolev term is some exponential function. Recently, Psaradakis and Spector [Reference Psaradakis and Spector21] established the following Leray–Trudinger inequality.

Let $\Omega$ be a bounded domain in $\mathbb {R}^{n} (n\geq 2)$ containing the origin, then for any $\epsilon >0$ and $R\geq R_{\Omega }$, there exist positive constants $A_{n,\epsilon }$ and $B_{n}$, such that for all $u\in W_0^{1,n}(\Omega )$ satisfying $I_1(u)\leq 1$, one has

(1.10)\begin{equation} \int_{\Omega}e^{A_{n,\epsilon}\left(|u|X_1^{\epsilon}\left(\frac{|x|}{R}\right)\right)^{\frac{n}{n-1}}}\,{\rm d}x\leq B_n{\rm vol}(\Omega), \end{equation}

where $I_1(u)$ is defined by

(1.11)\begin{equation} I_1(u):=\int_{\Omega}|\nabla u|^{n}\,{\rm d}x-\left(\frac{n-1}{n}\right)^{n}\int_{\Omega}\frac{|u|^{n}}{|x|^{n}}X_1^{n}\left(\frac{|x|}{R}\right)\,{\rm d}x. \end{equation}

Moreover, inequality (1.10) failed for $\epsilon =0$.

Inequality (1.10) is closely related with Hardy inequality and Moser–Trudinger inequality. Subsequently, Mallick and Tintarev [Reference Mallick and Tintarev19] extended inequality (1.10) to the following form:

Let $\Omega$ be a bounded domain in $\mathbb {R}^{n} (n\geq 2)$ containing the origin, then for any $\beta \geq \frac {2}{n}$ and $R\geq R_{\Omega }$, there exist positive constants $A_n$ and $B_n$, such that for any $0< c< A_n$ and for all $u\in W_0^{1,n}(\Omega )$ satisfying $I_1(u)\leq 1$, one has

(1.12)\begin{equation} \int_{\Omega}e^{c\left(|u|X_2^{\beta}\left(\frac{|x|}{R}\right)\right)^{\frac{n}{n-1}}}\,{\rm d}x\leq B_n {\rm vol}(\Omega),\end{equation}

where $X_2(t):=X_1(X_1(t))$. Moreover, inequality (1.12) failed if $\beta <\frac {1}{n}$ for any $c>0$.

The relationship of inequality (1.10) and inequality (1.12) motivates us to investigate whether inequality (1.12) can be improved to be series expansion. In this paper, we establish the following series expansion of Leray–Trudinger inequality. Our main result is as follows.

Theorem 1.1 Let $\Omega$ be a bounded domain in $\mathbb {R}^{n}$ containing the origin, $n\geq 2$ and $R_{\Omega }:=\sup _{x\in \Omega }|x|$. Then for any $k\in \mathbb {N},$ $k\geq 1$ and $R\geq R_{\Omega },$ there exist constants $A(k,\,n)$ and $B(k,\,n),$ such that for any $0< C< A(k,\,n)$ and $u\in W_0^{1,n}(\Omega )$ satisfying $I_k(u)\leq 1,$ one has

(1.13)\begin{equation} \int_{\Omega}e^{C\left(\left|u(x)\right|\prod\limits_{i=2}^{k+1}X_i^{\frac{2}{n}}\left(\frac{|x|}{R}\right)\right)^{\frac{n}{n-1}}}\,{\rm d}x\leq B(k,n)Vol(\Omega), \end{equation}

where $I_1(u)$ is defined by (1.11) and for $k\geq 2,$ $I_k(u)$ is defined by

(1.14)\begin{equation} I_k(u):=I_{k-1}(u)-\frac{1}{2}\left(\frac{n-1}{n}\right)^{n-1}\int_{\Omega}\frac{|u|^{n}}{|x|^{n}}X_1^{n}\left(\frac{|x|}{R}\right)X_2^{2}\left(\frac{|x|}{R}\right)\cdots X_k^{2}\left(\frac{|x|}{R}\right)\,{\rm d}x. \end{equation}

Moreover, if replacing $X_{k+1}^{2}$ by $X_{k+1}^{\beta },$ one has that inequality (1.13) holds for any $\beta \geq \frac {2}{n}$.

To prove the main result, we follow closely Trudinger's original proof (see [Reference Gilbarg and Trudinger14]), which has been used in [Reference Psaradakis and Spector21] and [Reference Mallick and Tintarev19]. Our main steps are as follows. Firstly, we find a suitable function (2.6), which is a supersolution of some Laplace equation (lemma 2.5). By this function, we define corresponding transform to obtain $L^{q}$ estimate (proposition 3.1). After that, we obtain the exponential integrability.

This paper is organized as follows. In § 2, we establish some important preliminaries. In § 3, we give the proof of theorem 1.1.