Some Applications of $L^1$-Estimates of Fractional Integral Operators in Lorentz Spaces

Abstract

In this paper, we establish some application of $L^1$-estimates for the Riesz potentials of order $\alpha$ in some Lorentz spaces. We use this estimate to improve certain Olsen-type inequalities in Lorentz spaces. In addition, some endpoint vector-valued inequalities for the Riesz potentials in Lebesgue spaces are obtained.

1 Introduction

Let f measurable function on ${\mathbb {R}}^n,n\ge 1,$ and $0<\alpha <n$. The fractional integral operator or Riesz potential of f of order $\alpha$ is defined as

$$\begin{aligned}I_\alpha f(x)=\frac{1}{\gamma (\alpha )}\int _{{\mathbb {R}}^n}\frac{f(y)}{|x-y|^{n-\alpha }}\text {d}y,x\in {\mathbb {R}}^n\end{aligned}$$

with $\displaystyle \gamma (\alpha ):=\frac{\pi ^{n/2}2^\alpha \Gamma (\frac{\alpha }{2})}{\Gamma (\frac{n-\alpha }{2})}$. Especially for $n>2$ and $\alpha =2$, solution of Poisson equation $\Delta u=f$ for $f\in C^\infty _c$ is $u=-I_2f$. One of the most important results related to $I_\alpha$ is the Hardy–Littlewood–Sobolev theorem; namely, the fractional integral operator $I_\alpha$ is bounded form $L^p({\mathbb {R}}^n)$ ($1<p<\frac{n}{\alpha }$) to $L^q({\mathbb {R}}^n)$ if and only if $\displaystyle \frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$. However, the operator $I_\alpha$ is unbounded form $L^{1}({\mathbb {R}}^n)$ to $L^\frac{n}{n-\alpha }({\mathbb {R}}^n)$. One of the direct consequences of the Hardy–Littlewood–Sobolev theorem is the Sobolev embedding theorem, namely the inclusion of the Sobolev spaces $W^{1,p}({\mathbb {R}}^n)$ to $L^q({\mathbb {R}}^n)$ for $1<p<n$ and $\frac{1}{q}=\frac{1}{p}-\frac{1}{n}$. This inclusion was later improved so that it applies for $p=1$ and $\displaystyle q=\frac{n}{n-1}$. See [1,2,3,4] for future studies about fractional integral operators in Lebesgue spaces.

In 1960, Stein and Weiss [5] constructed a subset of the Lebesgue space which was later named the Hardy space $H^p({\mathbb {R}}^n)$ and succeeded in showing the boundedness of $I_\alpha$ of $H^p({\mathbb {R}}^n)$ to $L^{\frac{np}{n-\alpha p}}({\mathbb {R}}^n)$ for $1\le p<n/\alpha$. There are some equivalent definitions for $H^p$. For this paper, we use the definition

$$\begin{aligned}H^p({\mathbb {R}}^n)=\{f\in L^p({\mathbb {R}}^n): Rf=\nabla I_1 f\in L^p({\mathbb {R}}^n,{\mathbb {R}}^n)\}.\end{aligned}$$

Note that $Rf=\nabla I_1 f$ can be written as $(1-n)(R_1f,\dots ,R_nf)$ with

$$\begin{aligned}R_jf(x)=\frac{1}{\gamma (\alpha )}\text {p.v.}\int _{{\mathbb {R}}^n}\frac{x_j-y_j}{|x-y|^{n+1}}f(y)\text {d}y, j=1,\dots ,n.\end{aligned}$$

For $1<p<\infty$, $H^p=L^p$, but for $p=1$, $H^1$ is strictly contained in $L^1$. Here is Stein and Weiss result about the boundedness of fractional integral operators in $H^p({\mathbb {R}}^n)$.

Theorem 1.1

Let $0<\alpha <n$ and $1\le p<n/\alpha$. There exists $C=C(\alpha ,n,p)>0$ such that

$$\begin{aligned}\Vert I_\alpha f\Vert _{L^{np/(n-\alpha p)}({\mathbb {R}}^n)}\le C(\Vert f\Vert _{L^p({\mathbb {R}}^n)}+\Vert Rf\Vert _{L^p({\mathbb {R}}^n,{\mathbb {R}}^n)})\end{aligned}$$

for all $f\in H^p({\mathbb {R}}^n)$.

From now on, we abbreviate $A\le C B$ by $A\lesssim B$.

In 2017, Schikora et al. [6] prove the special case for Theorem 1.1; namely,

$$\begin{aligned} \Vert I_\alpha f\Vert _{L^{n/(n-\alpha )}({\mathbb {R}}^n)}\lesssim \Vert Rf\Vert _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)} \end{aligned}$$

(1.1)

hold for all $f\in C^\infty _c({\mathbb {R}}^n)$ such that $Rf\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$. Two years later, Spector [7] improve inequalities (1.1) to Lorentz spaces. To state this result, we need to recall the definition of Lorentz spaces. The Lorentz space $L^{p,q}({\mathbb {R}}^n)$ is defined to be the set of all measurable functions f such that

$$\begin{aligned}\Vert f\Vert _{L^{p,q}({\mathbb {R}}^n)}:=p^{\frac{1}{q}}\left( \int _0^{\infty }\left[ t|\{x\in {\mathbb {R}}^n:|f(x)|>t\}|^\frac{1}{p}\right] ^q\frac{\text {d}t}{t}\right) ^\frac{1}{q}<\infty \end{aligned}$$

with $1\le p\le \infty ,1\le q\le \infty$. Now, we ready to state an improvement of (1.1). In [7], Spector proves the following theorem.

Theorem 1.2

[7] Let $n\ge 2$ and $0<\alpha <n$. Then

$$\begin{aligned}\Vert I_\alpha f\Vert _{L^{n/(n-\alpha ),1}({\mathbb {R}}^n)}\lesssim \Vert Rf\Vert _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)}\end{aligned}$$

for all $f\in C^\infty _c({\mathbb {R}}^n)$ such that $Rf\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$.

For Lorentz spaces, we have inclusion $L^p\subset L^{p,1}$. In the other words, Theorem 1.2 is the improvement of the inequality (1.1).

As an application of Theorem 1.2, we investigate some endpoint case of Olsen-type inequalities. To formulate these inequalities, let us recall the definition of Morrey spaces. For $1\le p<\infty$ and $0\le \lambda \le n$, the (classical) Morrey space $L^{p}_\lambda =L^{p}_\lambda ({\mathbb {R}}^n)$ is defined to be the space of all functions $f\in L^p_{loc}({\mathbb {R}}^n)$ for which

$$\begin{aligned}\Vert f\Vert _{L^p_\lambda ({\mathbb {R}}^n)}=\sup _{B(a,r)}\left( \frac{1}{r^\lambda }\int _{B(a,r)}|f(y)|^p\text {d}y\right) ^{1/p}<\infty ,\end{aligned}$$

where B(a, r) denotes the (open) ball centered at $a\in R$ with radius $r>0$. Suppose $1<p<\frac{n}{\alpha },0\le \lambda <n-\alpha p$, Olsen proved the following results (see [8, Theorem 2])

$$\begin{aligned}\Vert g\cdot I_\alpha f\Vert _{L^p_\lambda ({\mathbb {R}}^n)}\lesssim \Vert g\Vert _{L^{(n-\lambda )/\alpha }_\lambda ({\mathbb {R}}^n)}\Vert f\Vert _{L^p_\lambda ({\mathbb {R}}^n)}.\end{aligned}$$

In particular, for $\lambda =0$, we have

$$\begin{aligned} \Vert g\cdot I_\alpha f\Vert _{L^p({\mathbb {R}}^n)}\lesssim \Vert g\Vert _{L^\frac{n}{\alpha }({\mathbb {R}}^n)}\Vert f\Vert _{L^p({\mathbb {R}}^n)}.\end{aligned}$$

(1.2)

This inequality and Theorem 1.2 motivate us to find out what happen if $p=1$. Before we state the main result of this paper, we will recall the definition of fractional maximal operators. Suppose $0<\alpha <n$ and f is locally integrable, define

$$\begin{aligned}M_\alpha (f)(x)=\sup _{r>0}\frac{1}{r^{n-\alpha }}\int _{|y|<r}|f(x-y)|\text {d}y.\end{aligned}$$

Note that if $\alpha =0$, the definition above becomes Hardy–Littlewood maximal operators. Here is the main result of this paper.

Theorem 1.3

• 1. Suppose that $g\in L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)$ and $f\in C_c^\infty ({\mathbb {R}}^n)$ such that $Rf\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$. Then

  $$\begin{aligned} \Vert g\cdot I_\alpha f\Vert _{L^1({\mathbb {R}}^n)}\lesssim \Vert g\Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)}.\Vert Rf\Vert _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)}. \end{aligned}$$

  (1.3)

• 2. Suppose that $g\in L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)$ and $f\in C_c^\infty ({\mathbb {R}}^n)$ such that $R(|f|)\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$. Then

  $$\begin{aligned} \Vert g\cdot M_\alpha f\Vert _{L^1({\mathbb {R}}^n)}\lesssim \Vert g\Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)}.\Vert R(|f|)\Vert _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)}. \end{aligned}$$

  (1.4)

Remark 1.4

• 1. We can replace $L^1$ in the left-hand side of inequality (1.3) and (1.4) by $L^{1,\frac{n}{n-\alpha }}$. This statement can be proved using Theorem 1.2 and inclusion in Lorentz spaces.

• 2. The term in left-hand side can be seen as integral $I_\alpha f$ with weight g. In particular, if $g(x)=|x|^{-\alpha }$, then $g\in L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)$ and $g\notin L^{\frac{n}{\alpha }}({\mathbb {R}}^n)$. Therefore, (1.3) can be viewed as an extension of (1.2). We will give the proof of Theorem 1.3 in Sect. 2. (see [9] for the classical results on the boundedness of fractional integral operators on weighted Lebesgue spaces).

• 3. The characterization of $f\in C_c^\infty ({\mathbb {R}}^n)$ such that $R(|f|)\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$ is given as follows. From our observation, f should not be a nonnegative function and $|f|\notin C^\infty _c({\mathbb {R}}^n)$. To prove this, let $u=|f|\in C^\infty _c({\mathbb {R}}^n)$ and assume that $Ru\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$. Then, u satisfies the assumption of Theorem 1.2. However, there is a function $u_0\in C^\infty _c({\mathbb {R}}^n)$ and that $u_0$ is nonnegative, such that $R u_0\notin L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$.

Now, we will give vector-valued version for Theorems 1.2 and 1.3. In this case, we assume that $\vec {f}\in \ell ^r$ such that $\Vert \vec {f}\Vert _{\ell ^r}\in C^\infty _c({\mathbb {R}}^n)$ with $R(\Vert \vec {f}\Vert _{\ell ^r})\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$. Naturally we can just change f in Theorem 1.2 and 1.3 by $\Vert \vec {f}\Vert _{\ell ^r}$. But, we have some better results related to inequalities in Theorem 1.2 and 1.3 for vector-valued functions. Here is the vector version for Theorem 1.2 and 1.3.

Theorem 1.5

Let $r\ge 1$ and $0<\alpha <n$ then

$$\begin{aligned}\left\| \Vert I_\alpha \vec {f}\Vert _{\ell ^r}\right\| _{L^{n/(n-\alpha ),1}({\mathbb {R}}^n)}\lesssim \left\| R(\Vert \vec {f}\Vert _{\ell ^r})\right\| _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)}\end{aligned}$$

for every $\vec {f}$ with $\Vert \vec {f}\Vert _{\ell ^r}\in C^\infty _c({\mathbb {R}}^n)$ such that $R(\Vert \vec {f}\Vert _{\ell ^r})\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$.

Theorem 1.6

Suppose that $0<\alpha <n$ and that $r\ge 1$ satisfy $\frac{1}{r}+\frac{1}{r'}=1$. Then

$$\begin{aligned}\left\| \vec {g}\cdot I_\alpha \vec {f}\right\| _{L^1({\mathbb {R}}^n)}\lesssim \Big \Vert \Vert \vec {g}\Vert _{\ell ^{r'}}\Big \Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)} \left\| R(\Vert \vec {f}\Vert _{\ell ^{r}})\right\| _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)}\end{aligned}$$

for every $\vec {f}$ and $\vec {g}$ with $\Vert \vec {g}\Vert _{\ell ^{r'}}\in L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n),\Vert \vec {f}\Vert _{\ell ^{r}}\in C^\infty _c({\mathbb {R}}^n)$ such that $R(\Vert \vec {f}\Vert _{\ell ^r})\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$.

Theorems 1.5 and 1.6 will be proved in Sect. 3. This theorem can be proved using duality and Theorem 1.3.

2 Olsen-Type Inequalities in Lorentz Spaces

In this section, we provide some theorems that will be very helpful in proving the main result. First, let us recall the Hölder inequality in Lebesgue spaces.

Theorem 2.1

Suppose $p,p_1,p_2\in [1,\infty ]$ satisfy $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$. Then

$$\begin{aligned}\Vert f\cdot g\Vert _{L^p({\mathbb {R}}^n)}\le \Vert f\Vert _{L^{p_1}({\mathbb {R}}^n)}.\Vert g\Vert _{L^{p_2}({\mathbb {R}}^n)}\end{aligned}$$

for all $f\in L^{p_1}({\mathbb {R}}^n)$ and $g\in L^{p_2}({\mathbb {R}}^n).$

For the Lorentz space, we have the following generalization of Hölder inequality in Lorentz spaces due to O’Neil [10].

Theorem 2.2

Suppose $0<p_1,p_2,p<\infty$ and $0<q_1,q_2,q\le \infty$ satisfy $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}$. Then

$$\begin{aligned}\Vert fg\Vert _{L^{p,q}({\mathbb {R}}^n)}\lesssim \Vert f\Vert _{L^{p_1,q_1}({\mathbb {R}}^n)}\Vert g\Vert _{L^{p_2,q_2}({\mathbb {R}}^n)}\end{aligned}$$

for all $f\in L^{p_1,q_1}({\mathbb {R}}^n)$ and $g\in L^{p_2,q_2}({\mathbb {R}}^n)$.

We are now ready to prove Theorem 1.3 as follows.

Proof of Theorem 1.3

Suppose that $f\in C_c^\infty ({\mathbb {R}}^n)$, with $Rf\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$ and $g\in L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)$. By Hölder’s inequality in Lorentz spaces, we have

$$\begin{aligned} \Vert g\cdot I_\alpha f\Vert _{L^1({\mathbb {R}}^n)}\lesssim \Vert g\Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)}\Vert I_\alpha f\Vert _{L^{\frac{n}{n-\alpha },1}({\mathbb {R}}^n)}. \end{aligned}$$

Apply Theorem 1.2 to the right-hand side, we get

$$\begin{aligned}\Vert g\Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)}\Vert I_\alpha f\Vert _{L^{\frac{n}{n-\alpha },1}({\mathbb {R}}^n)} \lesssim \Vert g\Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)}\Vert Rf\Vert _{L^1({\mathbb {R}}^n, {\mathbb {R}}^n)}.\end{aligned}$$

For the second part, suppose that $f\in C_c^\infty ({\mathbb {R}}^n)$, with $R(|f|)\in L^1({\mathbb {R}}^n,{\mathbb {R}}^n)$ and $g\in L^{\frac{n}{\alpha },\infty }$. We already know that

$$\begin{aligned}M_\alpha f(x)\lesssim I_\alpha (|f|)(x).\end{aligned}$$

Combining this inequality with Theorems  1.2 and 2.2, we have

$$\begin{aligned}F \Vert g\cdot M_\alpha (f)\Vert _{L^1({\mathbb {R}}^n)}&\lesssim \Vert g\Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)}\Vert I_\alpha (|f|)\Vert _{L^{\frac{n}{n-\alpha },1}({\mathbb {R}}^n)} \\ {}&\lesssim \Vert g\Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)}\Vert R(|f|)\Vert _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)}. \end{aligned}$$

This completes the proof. $\square$

3 Vector-Valued Olsen-Type Inequalities

In this section, we will give the proof of Theorems 1.5 and 1.6.

3.1 Proof of Theorem 1.5

Let $r\ge 1$ and $0<\alpha <n$. First, we will proof the inequality

$$\begin{aligned}\Vert I_\alpha \vec {f}(x)\Vert _{\ell ^r}\lesssim I_\alpha (\Vert \vec {f}\Vert _{\ell ^r})(x).\end{aligned}$$

Actually, this proof is inspired by [11]. By using duality on the left-hand side, we get

$$\begin{aligned} \Vert I_\alpha \vec {f}(x)\Vert _{\ell ^r}&=\sup _{a=\{a_j(x)\}_{j=1}^\infty \in \ell ^{r'}:\Vert a\Vert _{\ell ^{r'}}=1}\left| \sum _{j=1}^{\infty }a_j(x)I_\alpha f_j(x)\right| \\ {}&\le \sup _{a=\{a_j(x)\}_{j=1}^\infty \in \ell ^{r'}:\Vert a\Vert _{\ell ^{r'}}=1}\sum _{j=1}^{\infty }|a_j(x)I_\alpha f_j(x)| \\ {}&\lesssim \sup _{a=\{a_j(x)\}_{j=1}^\infty \in \ell ^{r'}:\Vert a\Vert _{\ell ^{r'}}=1}\sum _{j=1}^{\infty }\int _{{\mathbb {R}}^n}\frac{|a_j(x)f_j(y)|}{|x-y|^{n-\alpha }}\text {d}y. \end{aligned}$$

Observe that

$$\begin{aligned}\sum _{j=1}^k\int _{{\mathbb {R}}^n}\frac{|a_j(x)f_j(y)|}{|x-y|^{n-\alpha }}\text {d}y=\int _{{\mathbb {R}}^n}\sum _{j=1}^k\frac{|a_j(x)f_j(y)|}{|x-y|^{n-\alpha }}\text {d}y\le \int _{{\mathbb {R}}^n}\sum _{j=1}^\infty \frac{|a_j(x)f_j(y)|}{|x-y|^{n-\alpha }}\text {d}y.\end{aligned}$$

To simplify the notation, we write

$$\begin{aligned}\sup _{a(x)=\{a_j(x)\}_{j=1}^\infty \in \ell ^{r'}:\Vert a(x)\Vert _{\ell ^{r'}}=1}\end{aligned}$$

as $\sup \limits _{\Vert a\Vert _{\ell ^{r'}}=1}$. From the above observation, we have

$$\begin{aligned} \sup _{\Vert a\Vert _{\ell ^{r'}}=1}\sum _{j=1}^{\infty }\int _{{\mathbb {R}}^n}\frac{|a_j(x)f_j(y)|}{|x-y|^{n-\alpha }}\text {d}y&\le \sup _{\Vert a\Vert _{\ell ^{r'}}=1}\int _{{\mathbb {R}}^n}\sum _{j=1}^{\infty }\frac{|a_j(x)f_j(y)|}{|x-y|^{n-\alpha }}\text {d}y \\ {}&\le \int _{{\mathbb {R}}^n}\sup _{\Vert a\Vert _{\ell ^{r'}}=1}\sum _{j=1}^{\infty }\frac{|a_j(x)f_j(y)|}{|x-y|^{n-\alpha }}\text {d}y. \end{aligned}$$

Again, by duality

$$\begin{aligned}\int _{{\mathbb {R}}^n}\sup _{\Vert a\Vert _{\ell ^{r'}}=1}\sum _{j=1}^{\infty }\frac{|a_j(x)f_j(y)|}{|x-y|^{n-\alpha }}\text {d}y\lesssim \frac{1}{\gamma (\alpha )} \int _{{\mathbb {R}}^n}\frac{\Vert \vec {f}(y)\Vert _{\ell ^r}}{|x-y|^{n-\alpha }}\text {d}y=I_\alpha (\Vert \vec {f}\Vert _{\ell ^r})(x).\end{aligned}$$

Finally, we have

$$\begin{aligned} \Vert I_\alpha \vec {f}(x)\Vert _{\ell ^r}\le I_\alpha (\Vert \vec {f}\Vert _{\ell ^r})(x). \end{aligned}$$

(3.1)

By Theorem 1.2 and the inequality (3.1), we conclude that

$$\begin{aligned}\Big \Vert \Vert I_\alpha \vec {f}(x)\Vert _{\ell ^r}\Big \Vert _{L^{n/(n-\alpha ),1}({\mathbb {R}}^n)}\lesssim \Big \Vert R(\Vert \vec {f}\Vert _{\ell ^r})\Big \Vert _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)}\end{aligned}$$

as desired.

3.2 Proof of Theorem 1.6

We know from Hölder inequality

$$\begin{aligned}|\vec {g}(x)\cdot I_\alpha \vec {f}(x)|\le \Vert \vec {g}(x)\Vert _{\ell ^{r'}}\Vert I_\alpha \vec {f}(x)\Vert _{\ell ^{r}}.\end{aligned}$$

From inequality (3.1), we get

$$\begin{aligned}|\vec {g}(x)\cdot I_\alpha \vec {f}(x)|\le \Vert \vec {g}(x)\Vert _{\ell ^{r'}}I_\alpha (\Vert \vec {f}\Vert _{\ell ^{r}})(x).\end{aligned}$$

Applying Theorem 1.3, we obtain

$$\begin{aligned} \left\| |\vec {g}(\cdot )\cdot I_\alpha \vec {f}(\cdot )|\right\| _{L^1({\mathbb {R}}^n)}&\le \left\| \Vert \vec {g}(\cdot )\Vert _{\ell ^{r'}}.I_\alpha (\Vert \vec {f}\Vert _{\ell ^{r}})(\cdot )\right\| _{L^1({\mathbb {R}}^n)} \\ {}&\lesssim \Vert \Vert \vec {g}(\cdot )\Vert _{\ell ^{r'}}\Vert _{L^{\frac{n}{\alpha },\infty }({\mathbb {R}}^n)}.\Vert R(\Vert \vec {f}\Vert _{\ell ^{r}})\Vert _{L^1({\mathbb {R}}^n,{\mathbb {R}}^n)}. \end{aligned}$$