Skip to main content

Note on norm of an m-linear integral-type operator between weighted-type spaces

Abstract

We find a necessary and sufficient condition for the boundedness of an m-linear integral-type operator between weighted-type spaces of functions, and calculate norm of the operator, complementing some results by L. Grafakos and his collaborators. We also present an inequality which explains a detail in the proof of the boundedness of the linear integral-type operator on \(L^{p}({\mathbb {R}}^{n})\) space.

Introduction

Throughout this note the set of natural numbers is denoted by \({\mathbb {N}}\), the set of reals by \({\mathbb {R}}\), the set of positive reals by \({\mathbb {R}}_{+}\), the Euclidean n-dimensional space with the norm \(|x|= (\sum_{j=1}^{n}x_{j}^{2} )^{1/2}\), \(x=(x_{1},\ldots ,x_{n})\), by \({\mathbb {R}}^{n}\), the unit sphere in \({\mathbb {R}}^{n}\) by \(\mathbb{S}\), the \(n-1\)-dimensional surface measure by \(d\sigma (\zeta )\), \(\sigma _{n}=\sigma (\mathbb{S})\), the normalized surface measure \(d\sigma (\zeta )/\sigma _{n}\) is denoted by \(d\sigma _{N}(\zeta )\), the open unit ball in \({\mathbb {R}}^{n}\) by \({\mathbb {B}}\), the open unit ball in \({\mathbb {R}}^{n}\) centered at a and with radius r by \(B(a,r)\), the Lebesgue volume measure on \({\mathbb {R}}^{n}\) by \(dV(x)\), \(v_{n}=V({\mathbb {B}})\), the normalized Lebesgue volume measure \(dV(x)/v_{n}\) is denoted by \(dV_{N}(x)\), whereas \(L^{p}_{\alpha }(\Omega )\) denotes the weighted Lebesgue space on a domain \(\Omega \subseteq {\mathbb {R}}^{n}\) with the weight \(w(x)=|x|^{\alpha }\), that is,

$$ L_{\alpha }^{p}(\Omega )= \biggl\{ f : \Vert f \Vert _{L_{\alpha }^{p}(\Omega )}:= \biggl( \int _{\Omega } \bigl\vert f(x) \bigr\vert ^{p} \,dV_{\alpha }(x) \biggr)^{1/p}< +\infty \biggr\} , $$

where \(1\le p<+\infty \), \(\alpha >-n\), and \(dV_{\alpha }(x)=|x|^{\alpha }\,dV(x)\) (for \(\alpha =0\), the space is reduced to the standard \(L^{p}\) space on the domain; see, e.g., [1]). If \(k,l\in {\mathbb {N}}\) are such that \(k\le l\), then the notation \(j=\overline {k,l}\) stands for the set of all \(j\in {\mathbb {N}}\) such that \(k\le j\le l\).

The weighted-type space \(H_{\alpha }^{\infty }({\mathbb {R}}^{n})\), \(\alpha >0\), consists of all measurable functions f such that

$$ \Vert f \Vert _{H_{\alpha }^{\infty }}:=\operatorname{ess}\sup_{x\in {\mathbb {R}}^{n}} \vert x \vert ^{\alpha }\bigl\vert f(x) \bigr\vert < \infty . $$

The functional \(\|\cdot \|_{H_{\alpha }^{\infty }}\) is a norm on the space, where, as usual, we identify functions which are dV almost everywhere equal. Weighted-type spaces on various domains frequently appear in the literature and are quite suitable for investigations (see, e.g., [27] and the references therein).

The following integral-type operator

$$\begin{aligned} L(f) (x)=\frac{1}{x} \int _{0}^{x} f(t)\,dt, \quad x\ne 0, \end{aligned}$$
(1)

is a basic linear operator which has been studied on many spaces of functions. From the main result in [8] (see also [9]) we have that the operator is bounded on \(L^{p}({\mathbb {R}}_{+})\) space when \(p>1\). This result was later improved in [10] by proving the following formula:

$$ \Vert L \Vert _{L^{p}_{\alpha }({\mathbb {R}}_{+})\to L^{p}_{\alpha }({\mathbb {R}}_{+})}= \frac{p}{p-\alpha -1} $$
(2)

for \(p>\alpha +1\) in nowadays terminology.

Although there are many linear operators whose norms can be calculated (see, e.g., [1, 428] and the related references therein), they are, in fact, quite rare since for many more other operators the norms can be only estimated by some quantities. Some of these operators are integral-type ones, a topic of a considerable recent interest (see, e.g., [11, 1321, 2325, 2834] and the related references therein).

Operator (1) was generalized in [35] by introducing the following n-dimensional integral-type operator:

$$ {\mathcal{H}}(f) (x)=\frac{1}{V(B(0, \vert x \vert ))} \int _{B(0, \vert x \vert )}f(y)\,dV(y),\quad x\in {\mathbb {R}}^{n}\setminus \{0\}, $$
(3)

for nonnegative locally integrable functions on \({\mathbb {R}}^{n}\).

In [11] it was shown that the norm of the operator \({\mathcal{H}}:L^{p}({\mathbb {R}}^{n})\to L^{p}({\mathbb {R}}^{n})\) can be calculated. Namely, the following formula holds:

$$ \Vert {\mathcal{H}} \Vert _{L^{p}({\mathbb {R}}^{n})\to L^{p}({\mathbb {R}}^{n})}= \frac{p}{p-1}, $$
(4)

which matches the formula in (2) with \(\alpha =0\).

Note that \({\mathcal{H}}(f)(x\zeta )={\mathcal{H}}(f)(x)\) for every \(\zeta \in {\mathbb{S}}\), that is, the function \({\mathcal{H}}(f)(x)\) is radial for every f, and

$$ {\mathcal{H}}(f) (x)=\frac{1}{V(B(0, \vert x \vert ))} \int _{{\mathbb {B}}}f\bigl( \vert x \vert z\bigr) \vert x \vert ^{n}\,dV(z)= \int _{{\mathbb {B}}}f\bigl( \vert x \vert z\bigr)\,dV_{N}(z), $$

where we have used the change of variables \(y=|x|z\) and the fact that \(V(B(0,|x|))=|x|^{n}V({\mathbb {B}})\).

In [13] the following m-linear extension of operator (3) was introduced:

$$ {\mathcal{H}}^{m}(f_{1},\ldots ,f_{m}) (x)= \frac{1}{v_{mn} \vert x \vert ^{mn}} \int _{ \vert (y_{1}, \ldots ,y_{m}) \vert < \vert x \vert }\prod_{j=1}^{m}f_{j}(y_{j})\,dV(y_{1}) \cdots \,dV(y_{m}), $$
(5)

where \(m\in {\mathbb {N}}\), \(x\in {\mathbb {R}}^{n}\setminus \{0\}\), \(y_{1},\ldots ,y_{m}\in {\mathbb {R}}^{n}\), \(y_{j}=(y_{j}^{1},\ldots ,y_{j}^{n})\), \(j=\overline {1,m}\),

$$ \bigl\vert (y_{1},\ldots ,y_{m}) \bigr\vert = \Biggl(\sum_{j=1}^{m} \vert y_{j} \vert ^{2} \Biggr)^{1/2}= \Biggl(\sum _{j=1}^{m}\sum _{l=1}^{n}\bigl(y_{j}^{l} \bigr)^{2} \Biggr)^{1/2}, $$

\(f_{j}\), \(j=\overline {1,m}\), are nonnegative locally integrable functions on \({\mathbb {R}}^{n}\), and the norm of the m-linear operator was calculated from the product of weighted Lebesgue spaces \(\prod_{j=1}^{m} L^{p_{j}}_{\alpha _{j}p_{j}/p}({\mathbb {R}}^{n})\) to \(L_{\alpha }^{p}({\mathbb {R}}^{n})\), under some conditions posed on the parameters p, α, \(p_{j}\), and \(\alpha _{j}\), \(j=\overline {1,m}\).

For the definition of norm of an m-linear operator and some basic examples, see, for example, [36, pp. 51–55].

Note also that \({\mathcal{H}}^{m}(f)(x\zeta )={\mathcal{H}}^{m}(f)(x)\) for every \(\zeta \in {\mathbb{S}}\), that is, the function \({\mathcal{H}}^{m}(f)(x)\) is radial for every f, and

$$ {\mathcal{H}}^{m}(f_{1},\ldots ,f_{m}) (x)= \frac{1}{v_{mn}} \int _{ \vert (z_{1}, \ldots ,z_{m}) \vert < 1}\prod_{j=1}^{m} f_{j}\bigl( \vert x \vert z_{j} \bigr)\,dV(z_{1})\cdots \,dV(z_{m}), $$
(6)

where we have used the change of variables \(y_{j}=|x|z_{j}\), \(j=\overline {1,m}\).

Our main aim here is to complement the results in [13] by calculating the norm of the operator \({\mathcal{H}}^{m}:\prod_{j=1}^{m} H_{\alpha _{j}}^{\infty }({\mathbb {R}}^{n}) \to H_{\alpha }^{\infty }({\mathbb {R}}^{n})\) in the case \(\alpha =\sum_{j=1}^{m}\alpha _{j}\). We also explain a detail appearing in the proof of the main result in [11].

The following known formula, which transforms integrals in the Descartes coordinates to the polar ones in \({\mathbb {R}}^{n}\), will be frequently used in the section that follows:

$$ \int _{{\mathbb {R}}^{n}}f(x) \,dV(x)= \int _{0}^{\infty } \int _{\mathbb{S}}f( \rho \zeta )\,d\sigma (\zeta )\rho ^{n-1}\,d\rho , $$
(7)

where f is a nonnegative measurable function (see, e.g., [1, pp. 149–150]).

Main results

This section presents and proves our main results in this note.

Boundedness of operator (5) between weighted-type spaces

First we consider operator (3). We consider it separately since its proof explains the first main step in the proof of the general case and does not request complex calculation.

Theorem 1

Let \(\alpha \in (0,n)\). Then the operator \({\mathcal{H}}\) is bounded on \(H_{\alpha }^{\infty }({\mathbb {R}}^{n})\). Moreover, the following formula holds:

$$ \Vert {\mathcal{H}} \Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})\to H_{\alpha }^{\infty }({\mathbb {R}}^{n})}= \frac{n}{n-\alpha }. $$
(8)

Proof

Let

$$ f_{\alpha }(x)= \textstyle\begin{cases} 1/ \vert x \vert ^{\alpha }, & x\ne 0, \\ 0, & x=0. \end{cases} $$
(9)

Then it is clear that

$$ \Vert f_{\alpha }\Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})}=1. $$
(10)

Further, we have

$$\begin{aligned} \vert x \vert ^{\alpha }\biggl\vert \int _{{\mathbb {B}}}f_{\alpha }\bigl( \vert x \vert y \bigr)\,dV_{N}(y) \biggr\vert &= \vert x \vert ^{\alpha }\biggl\vert \int _{{\mathbb {B}}}\frac{dV_{N}(y)}{( \vert x \vert \vert y \vert )^{\alpha }} \biggr\vert \\ &=n \int _{\mathbb{S}}d\sigma _{N}(\zeta ) \int _{0}^{1}\rho ^{n-1- \alpha }\,d\rho = \frac{n}{n-\alpha } \end{aligned}$$

for each \(x\ne 0\), from which it follows that

$$ \bigl\Vert {\mathcal{H}}(f_{\alpha }) \bigr\Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})}= \operatorname{ess} \sup_{x\in {\mathbb {R}}^{n}} \vert x \vert ^{\alpha }\biggl\vert \int _{{\mathbb {B}}}f_{\alpha }\bigl( \vert x \vert y \bigr)\,dV_{N}(y) \biggr\vert =\frac{n}{n-\alpha }. $$
(11)

Equalities (10) and (11) imply

$$ \Vert {\mathcal{H}} \Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})\to H_{\alpha }^{\infty }({\mathbb {R}}^{n})}\ge \frac{n}{n-\alpha }. $$
(12)

On the other hand, we have

$$\begin{aligned} \bigl\Vert {\mathcal{H}}(f) \bigr\Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})}&=\operatorname{ess}\sup _{x \in {\mathbb {R}}^{n}} \vert x \vert ^{\alpha }\biggl\vert \int _{{\mathbb {B}}}f\bigl( \vert x \vert y\bigr)\,dV_{N}(y) \biggr\vert \\ &\le \Vert f \Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})}\sup_{x\in {\mathbb {R}}^{n} \setminus \{0\}} \vert x \vert ^{\alpha }\biggl\vert \int _{{\mathbb {B}}} \frac{dV_{N}(y)}{( \vert x \vert \vert y \vert )^{\alpha }} \biggr\vert \\ &=\frac{n}{n-\alpha } \Vert f \Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})} \end{aligned}$$

for every \(f\in H_{\alpha }^{\infty }({\mathbb {R}}^{n})\), from which it follows that

$$ \Vert {\mathcal{H}} \Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})\to H_{\alpha }^{\infty }({\mathbb {R}}^{n})}\le \frac{n}{n-\alpha }, $$
(13)

and consequently the boundedness of the operator when \(\alpha \in (0,n)\). From (12) and (13) the equality in (8) follows. □

Remark 1

Note that (10) holds for each \(\alpha >0\). However, if \(\alpha \ge n\), then by using formula (7), we have

$$ \vert x \vert ^{\alpha }\biggl\vert \int _{{\mathbb {B}}}f_{\alpha }\bigl( \vert x \vert y \bigr)\,dV_{N}(y) \biggr\vert =n \int _{0}^{1}\rho ^{n-1-\alpha }\,d\rho =+ \infty $$

for each \(x\ne 0\), from which it follows that \({\mathcal{H}}(f_{\alpha })\notin H_{\alpha }^{\infty }({\mathbb {R}}^{n})\). Hence, in this case the operator is not bounded on \(H_{\alpha }^{\infty }({\mathbb {R}}^{n})\). From this and Theorem 1 we obtain the following corollary.

Corollary 1

Let \(\alpha >0\). Then the operator \({\mathcal{H}}\) is bounded on \(H_{\alpha }^{\infty }({\mathbb {R}}^{n})\) if and only if \(\alpha < n\). Moreover, if \(\alpha \in (0,n)\), then formula (8) holds.

The following theorem deals with the boundedness of the m-linear operator defined in (5).

Theorem 2

Let \(m\in {\mathbb {N}}\), \(\alpha _{j}>0\), \(j=\overline {1,m}\), and

$$ \alpha =\sum_{j=1}^{m}\alpha _{j}. $$
(14)

Then the operator \({\mathcal{H}}^{m}:\prod_{j=1}^{m} H_{\alpha _{j}}^{\infty }({\mathbb {R}}^{n}) \to H_{\alpha }^{\infty }({\mathbb {R}}^{n})\) is bounded if and only if

$$ \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} \vert y_{j} \vert ^{-\alpha _{j}}\prod _{j=1}^{m}dV(y_{j})< \infty , $$
(15)

where \(\vec{y}=(y_{1},y_{2},\ldots ,y_{m})\).

Moreover, if condition (15) is satisfied, then the following formula for the norm of the operator holds:

$$ \bigl\Vert {\mathcal{H}}^{m} \bigr\Vert _{\prod _{j=1}^{m} H_{\alpha _{j}}^{\infty }( {\mathbb {R}}^{n})\to H_{\alpha }^{\infty }({\mathbb {R}}^{n})}=\frac{1}{v_{nm}} \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} \vert y_{j} \vert ^{-\alpha _{j}}\prod _{j=1}^{m}dV(y_{j}), $$
(16)

where \(\prod_{j=1}^{m}\,dV(y_{j}):=dV(y_{1})\cdots \,dV(y_{m})\).

Proof

Let the family of functions \(f_{\alpha }\) be defined in (9). Then clearly relation (10) holds. By using the condition in (14), after some simple calculation it follows that

$$\begin{aligned} \vert x \vert ^{\alpha }\Biggl\vert \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} f_{\alpha _{j}}\bigl( \vert x \vert y_{j}\bigr) \prod _{j=1}^{m}dV(y_{j}) \Biggr\vert &= \vert x \vert ^{\alpha }\Biggl\vert \int _{ \vert \vec{y} \vert < 1} \prod_{j=1}^{m}{ \bigl( \vert x \vert \vert y_{j} \vert \bigr)^{-\alpha _{j}}} \prod_{j=1}^{m}dV(y_{j}) \Biggr\vert \\ &= \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} \vert y_{j} \vert ^{-\alpha _{j}}\prod _{j=1}^{m}dV(y_{j}) \end{aligned}$$
(17)

for each \(x\ne 0\).

From (6), (17) and by using the definition of norm on the space \(H_{\alpha }^{\infty }({\mathbb {R}}^{n})\), it follows that

$$\begin{aligned} \bigl\Vert {\mathcal{H}}(f_{\alpha _{1}},\ldots ,f_{\alpha _{m}}) \bigr\Vert _{H_{\alpha }^{\infty }({\mathbb {R}}^{n})}&=\operatorname{ess}\sup_{x\in {\mathbb {R}}^{n}} \frac{ \vert x \vert ^{\alpha }}{v_{mn}} \Biggl\vert \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} f_{ \alpha _{j}}\bigl( \vert x \vert y_{j}\bigr)\prod _{j=1}^{m}dV(y_{j}) \Biggr\vert \\ &=\frac{1}{v_{mn}} \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} \vert y_{j} \vert ^{-\alpha _{j}} \prod _{j=1}^{m}dV(y_{j}). \end{aligned}$$
(18)

Equality (10) implies

$$ \prod_{j=1}^{m} \Vert f_{\alpha _{j}} \Vert _{H_{\alpha _{j}}^{\infty }}=1, $$

from which along with (18) it follows that

$$ \bigl\Vert {\mathcal{H}}^{m} \bigr\Vert _{\prod _{j=1}^{m} H_{\alpha _{j}}^{\infty }( {\mathbb {R}}^{n})\to H_{\alpha }^{\infty }({\mathbb {R}}^{n})}\ge \frac{1}{v_{mn}} \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} \vert y_{j} \vert ^{-\alpha _{j}}\prod _{j=1}^{m}dV(y_{j}). $$
(19)

On the other hand, we have

$$\begin{aligned} \begin{aligned} \bigl\Vert {\mathcal{H}}^{m}(f_{1},\ldots ,f_{m}) \bigr\Vert _{H_{\alpha }^{\infty }( {\mathbb {R}}^{n})}&=\operatorname{ess}\sup _{x\in {\mathbb {R}}^{n}} \frac{ \vert x \vert ^{\alpha }}{v_{mn}} \Biggl\vert \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} f_{j}\bigl( \vert x \vert y_{j}\bigr) \prod _{j=1}^{m}dV(y_{j}) \Biggr\vert \\ &\le \prod_{j=1}^{m} \Vert f_{j} \Vert _{H_{\alpha _{j}}^{\infty }}\sup_{x\in {\mathbb {R}}^{n}\setminus \{0\}} \frac{ \vert x \vert ^{\alpha }}{v_{mn}} \Biggl\vert \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m}{ \bigl( \vert x \vert \vert y_{j} \vert \bigr)^{-\alpha _{j}}} \prod_{j=1}^{m}dV(y_{j}) \Biggr\vert \\ &=\prod_{j=1}^{m} \Vert f_{j} \Vert _{H_{\alpha _{j}}^{\infty }}\frac{1}{v_{mn}} \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} \vert y_{j} \vert ^{-\alpha _{j}}\prod _{j=1}^{m}dV(y_{j}) \end{aligned} \end{aligned}$$

for every \((f_{1},\ldots ,f_{m})\in \prod_{j=1}^{m} H_{\alpha _{j}}^{\infty }( {\mathbb {R}}^{n})\), from which by taking the supremum over the unit balls in \(H_{\alpha _{j}}^{\infty }({\mathbb {R}}^{n})\), \(j=\overline {1,m}\), it follows that

$$ \bigl\Vert {\mathcal{H}}^{m} \bigr\Vert _{\prod _{j=1}^{m} H_{\alpha _{j}}^{\infty }( {\mathbb {R}}^{n})\to H_{\alpha }^{\infty }({\mathbb {R}}^{n})}\le \frac{1}{v_{mn}} \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} \vert y_{j} \vert ^{-\alpha _{j}}\prod _{j=1}^{m}dV(y_{j}), $$
(20)

and consequently the boundedness of the operator. From (19) and (20) the equality in (16) follows. □

Let

$$ I_{m}:=\frac{1}{v_{mn}} \int _{ \vert \vec{y} \vert < 1}\prod_{j=1}^{m} \vert y_{j} \vert ^{- \alpha _{j}}\prod _{j=1}^{m}dV(y_{j}). $$
(21)

Employing the polar coordinates \(y_{j}=\rho _{j}\zeta _{j}\), \(j=\overline {1,m}\), and Fubini’s theorem, we obtain

$$\begin{aligned} I_{m}&=\frac{1}{v_{mn}} \underbrace{ \int _{\mathbb{S}}\cdots \int _{\mathbb{S}}}_{m \text{ times}} \int _{\sum _{j=1}^{m}\rho _{j}^{2}< 1,\rho _{j}>0, j=\overline {1,m}} \prod_{j=1}^{m} \rho _{j}^{-\alpha _{j}}\prod_{j=1}^{m} \rho _{j}^{n-1}\,d\rho _{1}\cdots \,d\rho _{m}\,d\sigma (\zeta _{1})\cdots \,d\sigma (\zeta _{m}) \\ &=\frac{\sigma _{n}^{m}}{v_{mn}} \int _{\sum _{j=1}^{m}\rho _{j}^{2}< 1, \rho _{j}>0, j=\overline {1,m}}\prod_{j=1}^{m} \rho _{j}^{n-1-\alpha _{j}}\,d\rho _{1}\cdots \,d\rho _{m}. \end{aligned}$$
(22)

By using the m-dimensional spherical coordinates

$$\begin{aligned}& \rho _{1}= r\cos \varphi _{1}, \\& \rho _{2}= r\sin \varphi _{1}\cos \varphi _{2}, \\& \rho _{3}= r\sin \varphi _{1}\sin \varphi _{2}\cos \varphi _{3}, \\& \vdots \\& \rho _{m-1}= r\sin \varphi _{1}\sin \varphi _{2}\cdots \sin \varphi _{m-2}\cos \varphi _{m-1}, \\& \rho _{m}= r\sin \varphi _{1}\sin \varphi _{2}\cdots \sin \varphi _{m-2} \sin \varphi _{m-1}, \end{aligned}$$

where \(r\ge 0\) is the radial coordinate and \(\varphi _{j}\), \(j=\overline {1,m-1}\), are angular coordinates, \(\varphi _{j}\in [0,\pi ]\), \(j=\overline {1,m-2}\), \(\varphi _{m-1}\in [0,2\pi )\), and the known fact that the associated Jacobian is

$$ \vert J_{m} \vert =r^{m-1}\sin ^{m-2} \varphi _{1}\sin ^{m-3}\varphi _{2}\cdots \sin \varphi _{m-2}, $$

in (22), we have

$$\begin{aligned} I_{m}={}&\frac{\sigma _{n}^{m}}{v_{mn}} \int _{0}^{1}r^{mn-1-\sum _{j=1}^{m} \alpha _{j}}\, dr \\ &{}\times \int _{0}^{\pi /2}\cdots \int _{0}^{\pi /2} \prod _{j=1}^{m-1}( \sin \varphi _{j})^{n(m-j)-1-\sum _{i=j+1}^{m}\alpha _{i}}( \cos \varphi _{j})^{n-1-\alpha _{j}}\, d\varphi _{1}\cdots \, d\varphi _{m-1} \\ ={}&\frac{\sigma _{n}^{m}}{v_{mn}(mn-\alpha )}\prod_{j=1}^{m-1} \int _{0}^{1}t_{j}^{n(m-j)-1- \sum _{i=j+1}^{m}\alpha _{i}} \bigl(1-t_{j}^{2}\bigr)^{\frac{n-2-\alpha _{j}}{2}}\,dt_{j} \\ ={}&\frac{\sigma _{n}^{m}2^{1-m}}{v_{mn}(mn-\alpha )}\prod_{j=1}^{m-1} \int _{0}^{1}s_{j}^{\frac{n(m-j)-\sum _{i=j+1}^{m}\alpha _{i}}{2}-1}(1-s_{j})^{ \frac{n-\alpha _{j}}{2}-1}\, ds_{j} \\ ={}&\frac{\sigma _{n}^{m}2^{1-m}}{v_{mn}(mn-\alpha )}\prod_{j=1}^{m-1}B \biggl(\frac{n(m-j)-\sum_{i=j+1}^{m}\alpha _{i}}{2}, \frac{n-\alpha _{j}}{2} \biggr), \end{aligned}$$
(23)

where we have also used the fact that \(\varphi _{j}\in (0,\pi /2)\), \(j=\overline {1,m-1}\), which is a consequence of integrating over a set in the first orthant, the changes of variables \(t_{j}=\sin \varphi _{j}\), \(j=\overline {1,m-1}\), and \(s_{j}=t_{j}^{2}\), \(j=\overline {1,m-1}\), as well as the definition of the beta function (see, e.g., [36, p. 437]).

By using the following well-known relation between Euler’s beta and gamma functions:

$$ B(a,b)=\frac{\Gamma (a)\Gamma (b)}{\Gamma (a+b)} $$

(see, for example, [36]), after some simple calculations, we see that the following relations hold:

$$\begin{aligned} \prod_{j=1}^{m-1}B \biggl( \frac{n(m-j)-\sum_{i=j+1}^{m}\alpha _{i}}{2}, \frac{n-\alpha _{j}}{2} \biggr) &=\prod _{j=1}^{m-1} \frac{\Gamma (\frac{n(m-j)-\sum_{i=j+1}^{m}\alpha _{i}}{2} )\Gamma (\frac{n-\alpha _{j}}{2} )}{\Gamma (\frac{n(m-(j-1))-\sum_{i=j}^{m}\alpha _{i}}{2} )} \\ &=\frac{\prod_{j=1}^{m}\Gamma (\frac{n-\alpha _{j}}{2} )}{\Gamma (\frac{nm-\alpha }{2} )}. \end{aligned}$$
(24)

Combining relations (16), (23), and (24), it follows that the following corollary holds.

Corollary 2

Let \(m\in {\mathbb {N}}\), \(\alpha _{j}>0\), \(j=\overline {1,m}\), and \(\alpha =\sum_{j=1}^{m}\alpha _{j}\). Then the operator \({\mathcal{H}}^{m}:\prod_{j=1}^{m} H_{\alpha _{j}}^{\infty }({\mathbb {R}}^{n}) \to H_{\alpha }^{\infty }({\mathbb {R}}^{n})\) is bounded if and only if (15) holds. Moreover, if (15) holds, then the following formulas hold:

$$\begin{aligned} \bigl\Vert {\mathcal{H}}^{m} \bigr\Vert _{\prod _{j=1}^{m} H_{\alpha _{j}}^{\infty }( {\mathbb {R}}^{n})\to H_{\alpha }^{\infty }({\mathbb {R}}^{n})}&= \frac{\sigma _{n}^{m}2^{1-m}}{v_{mn}(mn-\alpha )} \frac{\prod_{j=1}^{m}\Gamma (\frac{n-\alpha _{j}}{2} )}{\Gamma (\frac{nm-\alpha }{2} )} \\ &=\frac{\sigma _{n}^{m}2^{1-m}}{v_{mn}(mn-\alpha )}\prod_{j=1}^{m-1}B \biggl(\frac{n(m-j)-\sum_{i=j+1}^{m}\alpha _{i}}{2}, \frac{n-\alpha _{j}}{2} \biggr). \end{aligned}$$

Remark 2

From the above consideration and Corollary 2 we see that when \(\alpha _{j}>0\), \(j=\overline {1,m}\), condition (15) holds if and only if \(\alpha _{j}\in (0,n)\), \(j=\overline {1,m}\).

A comment on the proof of the main result in [11]

As we have already mentioned, in [11] the norm of the operator \({\mathcal{H}}:L^{p}({\mathbb {R}}^{n})\to L^{p}({\mathbb {R}}^{n})\) was calculated in a nice way by proving formula (4). In the proof of the result the authors applied the convolution inequality \(\|g*L\|_{L^{p}}\le \|g\|_{L^{p}}\|L\|_{L^{1}}\) in the group \(({\mathbb {R}}_{+},\frac{dt}{t})\). However, the operator appearing there is not a convolution in the standard sense. On the other hand, the used inequality really holds. Hence, it needs some explanations related to the inequality, which might be useful in other similar situations. Namely, the following theorem holds and its proof is analogous to the one for the convolution operator. For some information on locally compact groups and related topics, see, e.g., [15, 37].

Theorem 3

Let \(p\in [1,\infty ]\), \(f\in L^{p}(G)\), \(g\in L^{1}(G)\), where G is a locally compact group G, and μ be a right-invariant Haar measure on G. Then

$$ (f \odot g) (x):= \int _{G} f(xy)g(y)\, d\mu (y) $$

exists μ a.e. and

$$ \Vert f \odot g \Vert _{L^{p}}\le \Vert f \Vert _{L^{p}} \Vert g \Vert _{L^{1}}. $$
(25)

Proof

If \(p=1\), then the result follows from Fubini’s theorem and the right-invariance of measure μ. If \(p=\infty \), then we have

$$\begin{aligned} \|f \odot g\|_{L^{\infty}}={}& \operatorname{ess}\sup_{x\in G}\bigg|\int_{G} f(xy)g(y)\,d\mu(y)\bigg| \le \operatorname{ess}\sup_{x\in G}\int_{G} \big|f(xy)\big|\big|g(y)\big|\,d\mu(y)\\ \le{}&\int_{G} \|f\|_{L^{\infty}}\big|g(y)\big|\,d\mu(y)=\|f\|_{L^{\infty}}\|g\|_{L^{1}}. \end{aligned}$$

If \(p\in (1,\infty )\) and if we use the notation \(p'=p/(p-1)\), then by using the Hölder inequality, Fubini’s theorem, a change of variables, and the right-invariance of measure μ, we have

$$\begin{aligned} \Vert f \odot g \Vert _{L^{p}}^{p}&= \int _{G} \biggl\vert \int _{G} f(xy)g(y)\,d\mu (y) \biggr\vert ^{p} \,d\mu (x)\le \int _{G} \biggl( \int _{G} \bigl\vert f(xy) \bigr\vert \bigl\vert g(y) \bigr\vert \,d\mu (y) \biggr)^{p}\,d\mu (x) \\ &= \int _{G} \biggl( \int _{G} \bigl\vert f(xy) \bigr\vert \bigl\vert g(y) \bigr\vert ^{1/p} \bigl\vert g(y) \bigr\vert ^{1/p'}\,d\mu (y) \biggr)^{p}\,d\mu (x) \\ &\le \int _{G} \int _{G} \bigl\vert f(xy) \bigr\vert ^{p} \bigl\vert g(y) \bigr\vert \,d\mu (y) \biggl( \int _{G} \bigl\vert g(y) \bigr\vert \,d\mu (y) \biggr)^{p/p'}\,d\mu (x) \\ &\le \Vert g \Vert _{L^{1}}^{p/p'} \int _{G} \bigl\vert g(y) \bigr\vert \int _{G} \bigl\vert f(xy) \bigr\vert ^{p}\,d\mu (x)\,d\mu (y) \\ &= \Vert g \Vert _{L^{1}}^{p-1} \int _{G} \bigl\vert g(y) \bigr\vert \int _{G} \bigl\vert f(z) \bigr\vert ^{p}\,d\mu (z)\,d\mu (y) \\ &= \Vert f \Vert _{L^{p}}^{p} \Vert g \Vert _{L^{1}}^{p}, \end{aligned}$$

from which it follows that (25) holds, and since \(f \odot g\in L^{p}(G)\) it follows that \((f \odot g)(x)\) exists μ a.e. □

Remark 3

Inequality (25) can be also obtained by using Minkowski’s inequality. Indeed, by using the inequality and the right-invariance of measure μ, we have

$$\begin{aligned} \Vert f \odot g \Vert _{L^{p}}&= \biggl( \int _{G} \biggl\vert \int _{G} f(xy)g(y)\,d\mu (y) \biggr\vert ^{p} \,d\mu (x) \biggr)^{1/p} \\ &\le \biggl( \int _{G} \biggl( \int _{G} \bigl\vert f(xy) \bigr\vert \bigl\vert g(y) \bigr\vert \,d\mu (y) \biggr)^{p}\,d\mu (x) \biggr)^{1/p} \\ &\le \int _{G} \biggl( \int _{G} \bigl\vert f(xy) \bigr\vert ^{p}\,d\mu (x) \biggr)^{1/p} \bigl\vert g(y) \bigr\vert \,d\mu (y) \\ &= \int _{G} \biggl( \int _{G} \bigl\vert f(z) \bigr\vert ^{p}\,d\mu (z) \biggr)^{1/p} \bigl\vert g(y) \bigr\vert \,d\mu (y)= \Vert f \Vert _{L^{p}} \Vert g \Vert _{L^{1}}. \end{aligned}$$

Remark 4

By using inequality (25) instead of the corresponding one for the convolution operator, the proof of the main result in [11] is clear and complete.

Availability of data and materials

Not applicable.

References

  1. 1.

    Rudin, W.: Real and Complex Analysis, 3rd edn. Higher Mathematics Series. McGraw-Hill, New York (1976)

    Google Scholar 

  2. 2.

    Avetisyan, K.: Hardy Bloch-type spaces and lacunary series on the polydisk. Glasg. Math. J. 49, 345–356 (2007)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 54(1), 70–79 (1993)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Stević, S.: Norm of weighted composition operators from Bloch space to \(H^{\infty }_{\mu }\) on the unit ball. Ars Comb. 88, 125–127 (2008)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Stević, S.: Norm and essential norm of composition followed by differentiation from α-Bloch spaces to \(H^{\infty }_{\mu }\). Appl. Math. Comput. 207, 225–229 (2009)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Stević, S.: Norm of weighted composition operators from α-Bloch spaces to weighted-type spaces. Appl. Math. Comput. 215, 818–820 (2009)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Stević, S.: Norm of some operators from logarithmic Bloch-type spaces to weighted-type spaces. Appl. Math. Comput. 218, 11163–11170 (2012)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Hardy, G.H.: Notes on some points in the integral calculus LX: an inequality between integrals. Messenger Math. 54, 150–156 (1925)

    Google Scholar 

  9. 9.

    Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1952)

    Google Scholar 

  10. 10.

    Hardy, G.H.: Notes on some points in the integral calculus, LXI: further inequalities between integrals. Messenger Math. 57, 12–16 (1927)

    Google Scholar 

  11. 11.

    Christ, M., Grafakos, L.: Best constants for two nonconvolution inequalities. Proc. Am. Math. Soc. 123, 1687–1693 (1995)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Dunford, N., Schwartz, J.T.: Linear Operators, Part I: General Theory. Interscience, New York (1958)

    Google Scholar 

  13. 13.

    Fu, Z., Grafakos, L., Lu, S., Zhao, F.: Sharp bounds for m-linear Hardy and Hilbert operators. Houst. J. Math. 38(1), 225–244 (2012)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Grafakos, L.: Best bounds for the Hilbert transform on \(L^{p}({\mathbb {R}}^{1})\). Math. Res. Lett. 4(4), 469–471 (1997)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Grafakos, L.: Classical Fourier Analysis, 3rd edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2014)

    Google Scholar 

  16. 16.

    Grafakos, L., Montgomery-Smith, S.: Best constants for uncentered maximal functions. Bull. Lond. Math. Soc. 29(1), 60–64 (1997)

    Article  Google Scholar 

  17. 17.

    Grafakos, L., Montgomery-Smith, S., Motrunich, O.: A sharp estimate for the Hardy–Littlewood maximal function. Stud. Math. 134(1), 57–67 (1999)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Grafakos, L., Savage, T.: Best bounds for the Hilbert transform on \(L^{p}({\mathbb {R}}^{1})\), a corrigendum. Math. Res. Lett. 22(5), 1333–1335 (2015)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Li, H., Li, S.: Norm of an integral operator on some analytic function spaces on the unit disk. J. Inequal. Appl. 2013, Article ID 342 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Pelczynski, A.: Norms of classical operators in function spaces. Astérisque 131, 137–162 (1985)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Stević, S.: Norms of some operators from Bergman spaces to weighted and Bloch-type space. Util. Math. 76, 59–64 (2008)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Stević, S.: Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball. Appl. Math. Comput. 215(6), 2199–2205 (2009)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Stević, S.: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 354, 426–434 (2009)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Stević, S.: Norm and essential norm of an integral-type operator from the Dirichlet space to the Bloch-type space on the unit ball. Abstr. Appl. Anal. 2010, Article ID 134969 (2010)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Stević, S.: Norm of an integral-type operator from Dirichlet to Bloch space on the unit disk. Util. Math. 83, 301–303 (2010)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Stević, S.: Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains. Appl. Math. Comput. 217, 2870–2876 (2010)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Stević, S.: Norms of some operators on bounded symmetric domains. Appl. Math. Comput. 216, 187–191 (2010)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Stević, S.: Essential norm of some extensions of the generalized composition operators between kth weighted-type spaces. J. Inequal. Appl. 2017, Article ID 220 (2017)

    Article  Google Scholar 

  29. 29.

    Benke, G., Chang, D.C.: A note on weighted Bergman spaces and the Cesáro operator. Nagoya Math. J. 159, 25–43 (2000)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Li, S.: On an integral-type operator from the Bloch space into the \(Q_{K}(p,q)\) space. Filomat 26(2), 331–339 (2012)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Pan, C.: On an integral-type operator from \(Q_{K}(p,q)\) spaces to α-Bloch spaces. Filomat 25(3), 163–173 (2011)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Yang, W.: On an integral-type operator between Bloch-type spaces. Appl. Math. Comput. 215(3), 954–960 (2009)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Zhu, X.: On an integral-type operator between \(H^{2}\) space and weighted Bergman spaces. Bull. Belg. Math. Soc. Simon Stevin 18(1), 63–71 (2011)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Zhu, X.: On an integral-type operator from Privalov spaces to Bloch-type spaces. Ann. Pol. Math. 101(2), 139–148 (2011)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Faris, W.: Weak Lebesque spaces and quantum mechanical binding. Duke Math. J. 43, 365–373 (1976)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Zorich, V.A.: Mathematical Analysis II. Springer, Berlin (2004)

    Google Scholar 

  37. 37.

    Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1962)

    Google Scholar 

Download references

Acknowledgements

Not applicable.

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

The author has contributed solely to the writing of this paper. He read and approved the manuscript.

Corresponding author

Correspondence to Stevo Stević.

Ethics declarations

Competing interests

The author declares that he has no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stević, S. Note on norm of an m-linear integral-type operator between weighted-type spaces. Adv Differ Equ 2021, 187 (2021). https://doi.org/10.1186/s13662-021-03346-4

Download citation

MSC

  • 47A30
  • 47B38

Keywords

  • Operator norm
  • m-linear integral-type operator
  • Weighted-type space
  • Locally compact group
\