Norm estimates for Bessel-Riesz operators on generalized Morrey spaces

TL;DR

This paper refines the understanding of Bessel-Riesz operators' boundedness on generalized Morrey spaces, providing new norm estimates and revisiting fractional integral operators within this context.

Contribution

It introduces improved proofs and norm estimates for Bessel-Riesz and fractional integral operators on generalized Morrey spaces, enhancing theoretical understanding.

Findings

Established boundedness of Bessel-Riesz operators on generalized Morrey spaces.

Provided explicit norm estimates for these operators.

Reproved boundedness of fractional integral operators with new norm bounds.

Abstract

We revisit the properties of Bessel-Riesz operators and refine the proof of the boundedness of these operators on generalized Morrey spaces using Young's inequality. We also obtain an estimate for the norm of these operators on generalized Morrey spaces. In addition, we reprove the boundedness of fractional integral operators on generalized Morrey spaces and estimate their norm.

Equations

$$I_{\alpha ,\gamma }f\left(x\right):={\int }_{{\mathbb{R}}^n}{K}_{\alpha ,\gamma }(x-y)f(y)dy=K_{\alpha ,\gamma }\ast f(x),x\in {\mathbb{R}}^n,$$

$$\Vert f{\Vert }_{L^{p,\varphi }}:=\underset{B=B(a,r)}{\sup }\frac{1}{\varphi (r)}{\left(\frac{1}{|B|}{\int }_B|f(x){|}^{p}dx\right)}^{\frac{1}{p}}<\infty ,$$

$$\Vert f{\Vert }_{L^{p,q}}:=\underset{B=B(a,r)}{\sup }|B{|}^{\frac{1}{q}-\frac{1}{p}}{\left({\int }_B|f(x){|}^{p}dx\right)}^{\frac{1}{p}}.$$

$$\Vert I_{\alpha ,\gamma }f{\Vert }_{L^q}\le \Vert K_{\alpha ,\gamma }{\Vert }_{L^t}\Vert f{\Vert }_{L^p},f\in L^p({\mathbb{R}}^n),$$

$$|B{|}^{\frac{s}{t}-1}{\int }_B{K}_{\alpha }^s(x)dx\le |B(0,R){|}^{\frac{s}{t}-1}{\int }_{B(0,R)}|x{|}^{(\alpha -n)s}dx\le C{R}^{n(\frac{s}{t}-1)}{R}^{n(1-\frac{s}{t})}=C.$$

$$\Vert I_{\alpha ,\gamma }f{\Vert }_{L^{p_2,q_2}}\le C\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}\Vert f{\Vert }_{L^{p_1,q_1}},f\in L^{p_1,q_1}({\mathbb{R}}^n),$$

$$\Vert I_{\alpha ,\gamma }f{\Vert }_{L^{p_2,\psi }}\le C\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}\Vert f{\Vert }_{L^{p_1,\varphi }},f\in L^{p_1,\varphi }({\mathbb{R}}^n),$$

$$f:=f_1+f_2:=f_{{\chi }_{\tilde{B}}}+f_{{\chi }_{{\tilde{B}}^c}},$$

$$|I_{\alpha ,\gamma }{f}_1(x)|\le$$

$$=$$

$$\le$$

$${\int }_{\tilde{B}}{K}_{\alpha ,\gamma }^{\frac{p_2-s}{p_2-1}}(x-y)|f(y){|}^{\frac{p_2-p_1}{p_2-1}}dy\le {\left({\int }_{\tilde{B}}{K}_{\alpha ,\gamma }^s(x-y)dy\right)}^{p_2^{\prime }(\frac{1}{s}-\frac{1}{p_2})}{\left({\int }_{\tilde{B}}|f(y){|}^{p_1}dy\right)}^{\frac{p_2^{\prime }}{s^{\prime }}}.$$

$$|I_{\alpha ,\gamma }{f}_1(x)|\le$$

$$\le$$

$${\int }_B|I_{\alpha ,\gamma }{f}_1(x){|}^{p_2}dx\le$$

$$\times {\left(C{R}^{n(1-\frac{s}{t})(\frac{1}{s}-\frac{1}{p_2})+\frac{n}{s^{\prime }}}{\varphi }^{\frac{p_1}{s^{\prime }}}(2R)\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}^{1-\frac{s}{p_2}}\Vert f{\Vert }_{L^{p_1,\varphi }}^{\frac{p_1}{s^{\prime }}}\right)}^{p_2}.$$

$${\int }_B|I_{\alpha ,\gamma }{f}_1(x){|}^{p_2}dx\le$$

$$\times {\left(C{R}^{n(1-\frac{s}{t})(\frac{1}{s}-\frac{1}{p_2})+\frac{n}{s^{\prime }}}{\varphi }^{\frac{p_1}{s^{\prime }}}(2R)\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}^{1-\frac{s}{p_2}}\Vert f{\Vert }_{L^{p_1,\varphi }}^{\frac{p_1}{s^{\prime }}}\right)}^{p_2}$$

$$\le$$

$$\times {\left(R^{n(1-\frac{s}{t})(\frac{1}{s}-\frac{1}{p_2})+\frac{n}{s^{\prime }}}{\varphi }^{\frac{p_1}{s^{\prime }}}(2R)\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}^{1-\frac{s}{p_2}}\Vert f{\Vert }_{L^{p_1,\varphi }}^{\frac{p_1}{s^{\prime }}}\right)}^{p_2}$$

$$\le$$

$$\times {\left(R^{n(1-\frac{s}{t})(\frac{1}{s}-\frac{1}{p_2})+\frac{n}{s^{\prime }}}{\varphi }^{\frac{p_1}{s^{\prime }}}(2R)\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}^{1-\frac{s}{p_2}}\Vert f{\Vert }_{L^{p_1,\varphi }}^{\frac{p_1}{s^{\prime }}}\right)}^{p_2}$$

$$\le$$

$$\frac{1}{\psi (R)}{\left(\frac{1}{|B|}{\int }_B|I_{\alpha ,\gamma }{f}_1(x){|}^{p_2}dx\right)}^{\frac{1}{p_2}}\le C\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}\Vert f{\Vert }_{L^{p_1,\varphi }}.$$

$$|I_{\alpha ,\gamma }{f}_2(x)|\le$$

$$\le$$

$$=$$

$$\le$$

$$K_{\alpha ,\gamma }(2^{k}R)\le C(2^{k}R{)}^{-\frac{n}{s}}{\left({\int }_{2^{k}R\le |x-y|<2^{k+1}R}{K}_{\alpha ,\gamma }^s(x-y)dy\right)}^{\frac{1}{s}}\le C(2^{k}R{)}^{-\frac{n}{t}}\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}.$$

$$|I_{\alpha ,\gamma }{f}_2(x)|\le$$

$$\le$$

$$=$$

$$\int_{B}|I_{\alpha,\gamma}f_{2}(x)|^{p_{2}}dx\leq C\,\bigl{(}\|K_{\alpha,\gamma}\|_{L^{s,t}}\|f\|_{L^{p_{1},\phi}}\bigr{)}^{p_{2}}\psi^{p_{2}}(R)|B|,$$

$$\frac{1}{\psi (R)}{\left(\frac{1}{|B|}{\int }_B|I_{\alpha ,\gamma }{f}_2(x){|}^{p_2}dx\right)}^{\frac{1}{p_2}}\le C\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}\Vert f{\Vert }_{L^{p_1,\varphi }}.$$

$$\frac{1}{\psi (R)}{\left(\frac{1}{|B|}{\int }_B|I_{\alpha ,\gamma }f(x){|}^{p_2}dx\right)}^{\frac{1}{p_2}}\le C\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}\Vert f{\Vert }_{L^{p_1,\varphi }}.$$

$$\Vert I_{\alpha ,\gamma }f{\Vert }_{L^{p_2,\psi }}\le C\Vert K_{\alpha ,\gamma }{\Vert }_{L^{s,t}}\Vert f{\Vert }_{L^{p_1,\varphi }},$$

Full text

NORM ESTIMATES FOR BESSEL-RIESZ OPERATORS ON GENERALIZED MORREY SPACES

Mochammad Idris1, Hendra Gunawan2, and Eridani3

1Department of Mathematics, Institut Teknologi Bandung,

Bandung 40132, Indonesia

[Permanent Address: Department of Mathematics, Lambung Mangkurat University,

Banjarbaru Campus, Banjarbaru 70714, Indonesia]

2Department of Mathematics, Institut Teknologi Bandung,

Bandung 40132, Indonesia

3Department of Mathematics, Airlangga University,

Campus C Mulyorejo, Surabaya 60115, Indonesia

E-mail addresses: 1idemath@gmail.com, 2hgunawan@math.itb.ac.id, and 3eridani.dinadewi@gmail.com

Abstract. We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent 1, and obtain a new estimate for their norm.

Key words: Bessel-Riesz operators, fractional integral operators, generalized Morrey spaces.

MSC 2000: Primary 42B20; Secondary 26A33, 42B25, 26D10.

1 Introduction

Integral operators such as maximal operators and fractional integral operators have been studied extensively in the last four decades. Here we are interested in Bessel-Riesz operators, which are related to fractional integral operators. Let $0<\alpha<n$ and $\gamma\geq 0$. The operator $I_{\alpha,\gamma}$ which maps every $f\in L^{p}_{\text{l}oc}(\mathbb{R}^{n}),\ 1\leq p<\infty$, to

[TABLE]

where $K_{\alpha,\gamma}(x):=\frac{|x|^{\alpha-n}}{(1+|x|)^{\gamma}}$, is called Bessel-Riesz operator, and the kernel $K_{\alpha,\gamma}$ is called Bessel-Riesz kernel. The boundedness of these operators on Morrey spaces and on generalized Morrey spaces was studied in [11, 10].

Let $1\leq p<\infty$ and $\phi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be of class $\mathcal{G}_{p}$, that is $\phi$ is almost decreasing [$\exists\,C>0$ such that $\phi(r)\geq C\phi(s)$ for $r\leq s$] and $\phi^{p}(r)r^{n}$ is almost increasing [$\exists\,C>0$ such that $\phi^{p}(r)r^{n}\leq C\phi^{p}(s)s^{n}$ for $r\leq s$]. Clearly if $\phi$ is of class $\mathcal{G}_{p}$, then $\phi$ satisfies the doubling condition, that is, there exists $C>0$ such that $\frac{1}{C}\leq\frac{\phi(r)}{\phi(s)}\leq C$ whenever $1\leq\frac{r}{s}\leq 2$. We define the generalized Morrey space $L^{p,\phi}(\mathbb{R}^{n})$ to be the set of all functions $f\in L^{p}_{\rm loc}(\mathbb{R}^{n})$ for which

[TABLE]

where $|B|$ denotes the Lebesgue measure of $B$. (Recall that the Lebesgue measure of $B=B(a,r)$ is $|B(a,r)|=C_{n}r^{n}$ for every $a\in\mathbb{R}^{n}$ and $r>0$, where $C_{n}>0$ depends only on $n$.)

If $1\leq p\leq q<\infty$ and $\phi(r):=C_{n}r^{-\frac{n}{q}}\ (r>0)$, then $L^{p,\phi}(\mathbb{R}^{n})$ is the classical Morrey space $L^{p,q}(\mathbb{R}^{n})$, which is equipped by

[TABLE]

Particularly, for $p=q$, $L^{p,p}(\mathbb{R}^{n})$ is the Lebesgue space $L^{p}(\mathbb{R}^{n})$.

In [11], we know that for $\gamma>0$, $K_{\alpha,\gamma}$ is a member of $L^{t}(\mathbb{R}^{n})$ spaces for some values of $t$ depending on $\alpha$ and $\gamma$. It follows from Young’s inequality [4] that

[TABLE]

whenever $1\leq p<t^{\prime},\ \frac{1}{q}=\frac{1}{p}-\frac{1}{t^{\prime}}$ (where $t^{\prime}$ denotes the dual exponent of $t$) and $\frac{n}{n+\gamma-\alpha}<t<\frac{n}{n-\alpha}$. This tells us that $I_{\alpha,\gamma}$ is bounded from $L^{p}(\mathbb{R}^{n})$ to $L^{q}(\mathbb{R}^{n})$ with $\|I_{\alpha,\gamma}\|_{L^{p}\to L^{q}}\leq\|K_{\alpha,\gamma}\|_{L^{t}}$. In [10], it is also shown that $I_{\alpha,\gamma}$ is bounded on generalized Morrey spaces but without a good estimate for its norm as on Morrey spaces. We shall now refine the results, by estimating the norms of the operators more carefully through the membership of $K_{\alpha}$ in Morrey spaces.

Note that for $\gamma=0,$ $I_{\alpha,0}=I_{\alpha}$ is the fractional integral operator with kernel $K_{\alpha}(x):=|x|^{\alpha-n}$. Hardy-Littlewood [7, 8] and Sobolev [17] proved the boundedness of $I_{\alpha}$ on Lebesgue spaces. The boundedness of $I_{\alpha}$ on Morrey spaces is proved by Spanne [16], and improved by Adams [1] and Chiarenza-Frasca [2]. Later, Nakai [13] obtained the boundedness of $I_{\alpha}$ on generalized Morrey spaces, which can be viewed as an extension of Spanne’s result. In 2009, Gunawan-Eridani [5] proved the boundedness of $I_{\alpha}$ on generalized Morrey spaces which extends Adams’ and Chiarenza-Frasca’s results.

In this paper, we give a new proof of the boundedness of $I_{\alpha,\gamma}$ on generalized Morrey spaces. At the same time, an upper bound for the norm of the operators is obtained. As a consequence of our result, we have an estimate for the norm of $I_{\alpha}$ (from a generalized Morrey space to another) in terms of the norm of $K_{\alpha}$ on the associated Morrey space. A lower bound for the norm of the operators is discussed in §3.

2 The Boundedness of $I_{\alpha,\gamma}$ on Generalized Morrey Spaces

We begin with a lemma about the membership of $K_{\alpha}$ in some Morrey spaces. Note that throughout this paper, the letters $C$ and $C_{k}$ denote constants which may change from line to line.

Lemma 2.1

If $0<\alpha<n$, then $K_{\alpha}\in L^{s,t}(\mathbb{R}^{n})$ where $1\leq s<t=\frac{n}{n-\alpha}$.

Proof. Let $0<\alpha<n$. Take an arbitrary $B=B(a,R)$ where $a\in\mathbb{R}^{n}$ and $R>0$. For $1\leq s<t=\frac{n}{n-\alpha}$, we observe that

[TABLE]

By taking the supremum over $B=B(a,R)$, we obtain $\|K_{\alpha}\|_{L^{s,t}}^{s}\leq C$. Hence $K_{\alpha}\in L^{s,t}(\mathbb{R}^{n})$.

Remark. For $0<\alpha<n$ and $\gamma>0$, we know that $K_{\alpha,\gamma}\in L^{t}(\mathbb{R}^{n})$ for $\frac{n}{n+\gamma-\alpha}<t<\frac{n}{n-\alpha}$ [11]. By the inclusion property of Morrey spaces (see [6]), we have $K_{\alpha,\gamma}\in L^{t}(\mathbb{R}^{n})=L^{t,t}(\mathbb{R}^{n})\subseteq L^{s,t}(\mathbb{R}^{n}),$ for $1\leq s\leq t$ and $\frac{n}{n+\gamma-\alpha}<t<\frac{n}{n-\alpha}$. Moreover, because $K_{\alpha,\gamma}(x)\leq K_{\alpha}(x)$ for every $x\in\mathbb{R}^{n}$, $K_{\alpha,\gamma}$ is also contained in $L^{s,t}(\mathbb{R}^{n})$ for $1\leq s<t=\frac{n}{n-\alpha}$.

As a counterpart of the results in [10, 11], we have the following theorem on the boundedness of $I_{\alpha,\gamma}$ on Morrey spaces. Note particularly that the estimate holds for $p_{1}=1$.

Theorem 2.2

If $0<\alpha<n$ and $\gamma\geq 0$, then $I_{\alpha,\gamma}$ is bounded from $L^{p_{1},q_{1}}$ to $L^{p_{2},q_{2}}$ with

[TABLE]

whenever $1\leq p_{1}\leq q_{1}<\frac{n}{\alpha},\ \frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{1}{s^{\prime}}$, and $\frac{1}{q_{2}}=\frac{1}{q_{1}}-\frac{1}{t^{\prime}}$, with $1\leq s<t=\frac{n}{n-\alpha}$ (for $\gamma\geq 0$) or $1\leq s\leq t$ and $\frac{n}{n+\gamma-\alpha}<t<\frac{n}{n-\alpha}$ (for $\gamma>0$).

Theorem 2.2 is in fact a special case of the boundedness of $I_{\alpha,\gamma}$ on generalized Morrey spaces, which is stated in the following theorem.

Theorem 2.3

Let $0<\alpha<n$ and $\gamma\geq 0$. If $\phi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is of class $\mathcal{G}_{p_{1}}$ such that $\int_{R}^{\infty}\phi(r)r^{\frac{n}{t^{\prime}}-1}dr\leq C\,\phi(R)R^{\frac{n}{t^{\prime}}}$ for every $R>0$, then $I_{\alpha,\gamma}$ is bounded from $L^{p_{1},\phi}$ to $L^{p_{2},\psi}$ where $\psi(r):=\phi(r)r^{\frac{n}{t^{\prime}}}$, with

[TABLE]

whenever $1\leq p_{1}<\frac{n}{\alpha}$ and $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{1}{s^{\prime}}$, with $1\leq s<t=\frac{n}{n-\alpha}$ (for $\gamma\geq 0$) or $1\leq s\leq t$ and $\frac{n}{n+\gamma-\alpha}<t<\frac{n}{n-\alpha}$ (for $\gamma>0$).

Proof. Suppose that $\gamma>0$ and all the hypotheses hold. For $f\in L^{p_{1},\phi}(\mathbb{R}^{n})$ and $B=B(a,R)$ where $a\in\mathbb{R}^{n}$ and $R>0$, write

[TABLE]

where $\widetilde{B}=B(a,2R)$ and $\widetilde{B}^{c}$ denotes its complement. To estimate $I_{\alpha,\gamma}f_{1}$, we observe that for every $x\in B$, Hölder’s inequality gives

[TABLE]

Meanwhile, we have

[TABLE]

Therefore we obtain

[TABLE]

We take the $p_{2}$-th power and integrate both sides over $B$ to get

[TABLE]

By Fubini’s theorem, we have

[TABLE]

whence

[TABLE]

Next, we estimate $I_{\alpha,\gamma}f_{2}$. For every $x\in B=B(a,R)$, we observe that

[TABLE]

For every $k\in\mathbb{Z}$, we have

[TABLE]

Since $\int_{R}^{\infty}\phi(r)r^{\frac{n}{t^{\prime}}-1}dr\leq C\,\phi(R)R^{\frac{n}{t^{\prime}}}$, we get

[TABLE]

Raising to the $p_{2}$-th power and integrating over $B$, we obtain

[TABLE]

whence

[TABLE]

Combining the two estimates for $I_{\alpha,\gamma}f_{1}$ and $I_{\alpha,\gamma}f_{2}$, we obtain

[TABLE]

Since this inequality holds for every $a\in\mathbb{R}^{n}$ and $R>0$, it follows that

[TABLE]

as desired.

We may repeat the same argument and use Lemma 2.1 to obtain the same inequality for the case where $\gamma=0$ and $1\leq s<t=\frac{n}{n-\alpha}$.

Remark. Theorems 2.2 and 2.3 give us upper estimates for the norm of the Bessel-Riesz operators (from one Morrey space to another). In particular, for $\gamma=0$, we have an estimate for the norm of the fractional integral operator $I_{\alpha}$ in terms of the norm of its kernel (on the associated Morrey space), which follows from the inequality

[TABLE]

for $1\leq p_{1}<\frac{n}{\alpha}$ and $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{1}{s^{\prime}}$, with $1\leq s<t=\frac{n}{n-\alpha}$.

In the following section, we discuss lower estimates for the norm of the operators in terms of the norm of the Bessel-Riesz kernel (on some Morrey spaces).

3 An Estimate for the Norm of the Operators

Recall that if $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ are normed spaces and that $T:(X,\|\cdot\|_{X})\rightarrow(Y,\|\cdot\|_{Y})$ is a linear operator, then the norm of $T$ (from $X$ to $Y$) is defined by

[TABLE]

Knowing that the Bessel-Riesz operator $I_{\alpha,\gamma}$ is a linear operator on Morrey spaces, we would like to estimate the norm of $I_{\alpha,\gamma}$ from a (generalized) Morrey space to another. We obtain the following result.

Theorem 3.1

Let $0<\alpha<n$, $\gamma\geq 0$, and $\phi$ is of class $\mathcal{G}_{p_{1}}$ where $1\leq p_{1}<\frac{n}{\alpha}$. If $\phi(r)r^{n}$ is almost increasing and for every $R>0$ we have (i) $\int_{R}^{\infty}\phi(r)r^{\frac{n}{t^{\prime}}-1}\,dr\leq C_{1}\phi(R)R^{\frac{n}{t^{\prime}}}$, (ii) $\int_{0}^{R}\phi^{p_{1}}(r)r^{n-1}\,dr\leq C_{2}\,\phi^{p_{1}}(R)R^{n}$, and (iii) $\int_{0}^{R}\frac{r^{n-1}}{\phi^{s^{\prime}}(r)r^{ns^{\prime}}}\,dr\leq\frac{C_{3}R^{n}}{\phi^{s^{\prime}}(R)R^{ns^{\prime}}}$, where $1\leq p_{1}<t$ and $1<s<t=\frac{n}{n-\alpha}$ (for $\gamma\geq 0$) or $1\leq p_{1}\leq t$, $1<s\leq t$, and $\frac{n}{n+\gamma-\alpha}<t<\frac{n}{n-\alpha}$ (for $\gamma>0$), then we have

[TABLE]

whenever $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{1}{s^{\prime}}$ and $\psi(r):=\phi(r)r^{\frac{n}{t^{\prime}}}$. In particular, for $\gamma=0$, $1\leq p_{1}<t$, and $1<s<t=\frac{n}{n-\alpha}$, we have

[TABLE]

whenever $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{1}{s^{\prime}}$ and $\psi(r):=\phi(r)r^{\frac{n}{t^{\prime}}}$.

Proof. Suppose that $\gamma>0$ and all the hypotheses hold. By Theorem 2.3, we already have

[TABLE]

To prove the lower estimate, put $\rho(r):=\phi(r)r^{n}$. Let $B=B(a,R)$ where $a\in\mathbb{R}^{n}$ and $R>0$. By our assumptions on $\phi$, we have

[TABLE]

Now take $f_{0}(x):=\phi(|x|)$. Here $\|f_{0}\|_{L^{p_{1},\phi}}\approx 1$. Moreover, one may compute that

[TABLE]

for every $x\in\mathbb{R}^{n}$. It follows that

[TABLE]

Next, by Hölder’s inequality, we have

[TABLE]

whence

[TABLE]

By taking the supremum over $B=B(a,R)$, we conclude that

[TABLE]

as desired.

The same argument applies for the case where $\gamma=0$, with $1\leq p_{1}<t$ and $1<s<t=\frac{n}{n-\alpha}$.

Remark. One may observe that the constants $C_{4}$ and $C_{5}$ in Theorem 3.1 depend on $\phi,n,p_{1},s$, and $t$, but not on $\alpha$ and $\gamma$. Although the lower and the upper bound are not comparable, we may still get useful information from these estimates, especially for the norm of the operator $I_{\alpha}$ from $L^{p_{1},\phi}$ to $L^{p_{2},\psi}$. Observe that for $1\leq p_{1}<t=\frac{n}{n-\alpha}$, we have $\|K_{\alpha}\|_{L^{p_{1},t}}^{p_{1}}=\frac{C}{(\alpha-n)p_{1}+n}\geq\frac{C}{\alpha}.$ Hence, if all the hypotheses in Theorem 3.1 hold for the case where $\gamma=0$, then we obtain $\|I_{\alpha}\|_{L^{p_{1},\phi}\to L^{p_{2},\psi}}\geq\frac{C}{\alpha}$, which blows up when $\alpha\to 0^{+}$. For $\phi(r):=r^{-\frac{n}{q_{1}}}$ with $1\leq p_{1}<q_{1}<\min\{s,\frac{n}{\alpha}\}$ and $1<s<\frac{n}{n-\alpha}$, our result reduces to the estimate $\|I_{\alpha}\|_{L^{p_{1},q_{1}}\to L^{p_{2},q_{2}}}\geq\frac{C}{\alpha}$ where $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{1}{s^{\prime}}$ and $\frac{1}{q_{2}}=\frac{1}{q_{1}}-\frac{\alpha}{n}$. A similar behavior of the norm of $I_{\alpha}$ from $L^{p_{1}}$ to $L^{p_{2}}$ for $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{\alpha}{n}$ when $\alpha\to 0^{+}$ is observed in [12, Chapter 4].

Acknowledgements. The first and second authors are supported by ITB Research & Innovation Program 2016. All authors would like to thank the anonymous referee for his/her careful reading and useful comments on the earlier version of this paper.