Norms of some operators between weighted-type spaces and weighted Lebesgue spaces

1.   Introduction

By ${\mathbb R}$ we denote the set of real numbers, by ${\mathbb R}_+$ the interval $[0, +\infty)$, the space of continuous functions on a set $\Omega$ we denote by $C(\Omega),$ whereas the space of continuously differentiable functions on $\Omega$ we denote by $C^1(\Omega).$ A vector $(x_1, \ldots, x_n)\in {\mathbb R}^n$ we denote by $x.$ If $c\in {\mathbb R}$, then by $\vec c$ we denote the vector $(c, c, \ldots, c)$ (for example, $\vec 1 = (1, \ldots, 1)$). By $\langle x, y\rangle$ we denote the Euclidean inner product of vectors $x, y\in {\mathbb R}^n$, that is, $\langle x, y\rangle = \sum_{j = 1}^nx_jy_j.$ The Lebesgue measure on ${\mathbb R}^n$ we denote by $dV(x)$, whereas by $d\sigma(\zeta)$ we denote the surface measure on the unit sphere ${\mathbb S}\subset {\mathbb R}^n$. A function $w:\Omega\to {\mathbb R}$ is called a weight function or simply weight if it is positive and continuous. The class of all weights on $\Omega$ we denote by $W(\Omega).$

Let $w\in W(\Omega)$. The weighted-type space $C_w(\Omega)$ consists of all $f\in C(\Omega)$ such that

By using a standard argument, which is applied to the space $C(\Omega)$, it is shown that $C_w(\Omega)$ is a Banach space. Various weighted-type spaces of continuous or analytic functions and operators on them have been investigated considerably for several decades (see, e.g., ^{[1,2,12,19,25,32,33,34,35,38,42,43,48,50,51]} and the related references therein).

Let ${\mathcal L}_w^p({\mathbb R}^n) = {\mathcal L}_w^p$, where $p\ge 1$ and $w\in W({\mathbb R}^n)$, be the weighted ${\mathcal L}^p$ space consisting of all measurable functions $f$ such that

With the norm $\|\cdot\|_{{\mathcal L}_w^p}$ the space ${\mathcal L}_w^p$ is Banach.

Let $X$ and $Y$ be normed spaces, and $L:X\to Y$ be a linear operator. We say that the operator is bounded if there is $M\ge 0$, such that

for every $x\in X$ (^{[6,26,27,44,45]}).

Norm of the operator is defined by

where $B_X$ denotes the unit ball in the space $X$.

Finding norms of linear operators is one of the basic problems in operator theory. Many classical results can be found in books and surveys on functional analysis, operator theory and inequalities (see, for example, ^{[6,7,9,10,16,23,26,27,44,45]}; see also some of the original sources ^[13,14,28]). For some recent results in the topic, including some on multi-linear operators (for the definition and some examples see [52,p. 51–55]), see, for example, ^{[4,9,11,20,33,34,35,36,38,40,41,42]} and the related references therein.

Let $u$ be a function defined on $\Omega$. Then by $M_u$ we denote the multiplication operator

where $f$ is a function on $\Omega.$

There has been some interest in the multiplication operators on spaces of functions ^[35,49]. Motivated by some of our previous results on calculating and estimating norms of concrete operators and a problem in ^[45], here we present some formulas for norms of the multiplication and several integral-type operators between weighted-type spaces. We also calculate norm of an integral-type operator on some subspaces of ${\mathcal L}_w^p({\mathbb R}^n)$ space. For various integral-type operators see, e.g., ^{[3,4,5,7,8,9,10,16,19,20,21,22,24,29,30,31,32,37,39,40,41,42,43,46,47]}. Some of the formulas we have got long time ago, but have never published them. Some of the formulas could be matters of folklore, but we could not found references.

2.   Main results

This section presents our main results and some analyses.

2.1.   Multiplication operator between weighted-type spaces

The following result is a simple and basic one, and should be a matter of folklore. However, it is useful and instructive, because of which we give a proof.

Theorem 1. Let $w_1, w_2\in W(\Omega)$. Then the operator $M_u:C_{w_1}(\Omega)\to C_{w_1w_2}(\Omega)$ is bounded if and only if

Moreover, if (2.1) holds then

Proof. First, assume that condition (2.1) holds, that is, that

Then, we have

from which by taking the supremum over the ball $B_{C_{w_1}(\Omega)}$ we get

From (2.3) and (2.4) the boundedness of the operator $M_u:C_{w_1}(\Omega)\to C_{w_1w_2}(\Omega)$ follows.

Now assume that the operator $M_u:C_{w_1}(\Omega)\to C_{w_1w_2}(\Omega)$ is bounded. Since $w_1$ is a positive continuous function we see that $1/w_1$ is also such a function. Note that

Further, we have

From (2.5), (2.6) and the boundedness of the operator $M_u:C_{w_1}(\Omega)\to C_{w_1w_2}(\Omega)$ it follows that

which means that (2.1) holds.

If condition (2.1) holds, then from the inequalities in (2.4) and (2.7), we immediately obtain formula (2.2).

Remark 1. Note that the simple fact in (2.5) plays one of the decisive roles in finding the norm of the operator $M_u:C_{w_1}(\Omega)\to C_{w_1w_2}(\Omega)$. Related facts are very useful in finding norms of concrete operators acting from weighted-type spaces and will be also used further in this paper.

2.2.   Appearance of an integral-type operator in differential equations

Consider the initial value problem

where $f, {\beta}\in C({\mathbb R}_+)$.

By using the Euler multiplier $e^{\int_0^t{\beta}(\zeta)d\zeta}$ from (2.8) we have

By integrating the last relation and using condition (2.9), after some calculation, we obtain

Note that formula (2.10) presents a linear operator, say, $L$ which is defined as follows

and acts from $C({\mathbb R}_+)$ into the subspace of $C^1({\mathbb R}_+)$ consisting of all $g\in C^1({\mathbb R}_+)$ such that $g(0) = 0.$

Consider the operator from $C_{w_1}({\mathbb R}_+)$ to $C_{w_2}({\mathbb R}_+)$. Using the definitions of the spaces $C_{w_1}({\mathbb R}_+)$ and $C_{w_2}({\mathbb R}_+)$, we have

from which it follows that

From (2.5) and since

we have

From (2.11) and (2.12) we obtain

From the analysis that we have just conduced it follows that the following result holds.

Theorem 2. Let $w_1, w_2\in W({\mathbb R}_+)$, ${\beta}\in C({\mathbb R}_+)$ and

Then the operator $L:C_{w_1}({\mathbb R}_+)\to C_{w_2}({\mathbb R}_+)$ is bounded if and only if

Moreover, if the operator is bounded then

Let

where ${\delta}\in {\mathbb R}_+$, and let

The following example shows that for some functions $w_1, w_2$ and ${\beta}$ the norm of the operator $L:C_{w_1}({\mathbb R}_+)\to C_{w_2}({\mathbb R}_+)$ can be explicitly calculated ([45,Problem 7.31]).

Corollary 1. Let $w_1(t) = e^{{\alpha} t}$, ${\alpha}\ge 0$, $w_2(t) = e^{{\gamma} t}$, ${\beta}(t) = {\beta}$, and ${\beta} > {\alpha}\ge {\gamma}.$ Then the operator $L:C_{\alpha}({\mathbb R}_+)\to C_{\gamma}({\mathbb R}_+)$ is bounded and the following statements hold.

(a) If ${\alpha} = {\gamma}$, then

(b) If ${\alpha} > {\gamma}$, then

Proof. By Theorem 2, we have that formula (2.15) holds with $w_1(t) = e^{{\alpha} t}$, $w_2(t) = e^{{\gamma} t}$ and ${\beta}(t) = {\beta}$, that is,

(a) Since ${\alpha} = {\gamma}$ from (2.18) we have

(b) In this case from (2.18) we have

Let $g(t): = e^{({\gamma}-{\alpha})t}-e^{({\gamma}-{\beta})t}$, then we have $g(0) = 0$, $\lim_{t\to+\infty}g(t) = 0$ (since ${\gamma} < {\alpha} < {\beta}$), and $g(t) = e^{({\gamma}-{\beta})t}(e^{({\beta}-{\alpha})t}-1)\ge 0$, $t\in {\mathbb R}_+$. Since

we have that $g'(t) = 0$ if and only if

Hence,

from which together with (2.19) and some calculation, formula (2.17) follows.

2.3.   An extension of Theorem 2 and its corollaries

Let

where $t = (t_1, \ldots, t_n)$, $s = (s_1, \ldots, s_n)$, $s_j, t_j\in {\mathbb R}_+$, $j = \overline{1, n},$ and $g, h\in C({\mathbb R}_+^n)$.

The following theorem is an extension of Theorem 2.

Theorem 3. Let $v, w, h, g\in W({\mathbb R}^n_+)$ and operator $L$ be given in (2.20). Then the operator $L:C_w({\mathbb R}^n_+)\to C_v({\mathbb R}^n_+)$ is bounded if and only if

and if it is bounded then the norm of the operator is equal to $\widetilde M$.

Proof. Assume that (2.21) holds. Then we have

from which along with (2.21) the boundedness of the operator $L:C_w({\mathbb R}^n_+)\to C_v({\mathbb R}^n_+)$ follows. Moreover, we have

If the operator $L:C_w({\mathbb R}^n_+)\to C_v({\mathbb R}^n_+)$ is bounded, then since the function $f_0(t) = \frac1{w(t)}$ belongs to $C_w({\mathbb R}_+^n)$ and $\|f_0\|_w = 1$, we have

from which together with the boundedness of the operator $L:C_w({\mathbb R}^n_+)\to C_v({\mathbb R}^n_+)$ and positivity of functions $g$ and $w$ we obtain (2.21). From (2.22) and (2.23) we obtain

completing the proof.

The following corollary is an extension of Corollary 1.

Corollary 2. Let $v, w\in W({\mathbb R}_+^n)$, $j = \overline{1, n}$, ${\beta}_j\in C({\mathbb R}_+)$, $j = \overline{1, n},$ and

Then the operator $L:C_w({\mathbb R}_+^n)\to C_v({\mathbb R}_+^n)$ is bounded if and only if

Moreover, if the operator is bounded then

The following integral-type operator is a special case of operator (2.24)

Let $C_{\vec{\delta}}$, ${\delta}_j\ge 0$, $j = \overline{1, n}$, be the class of all $f\in C({\mathbb R}_+^n)$ such that

The following consequence of Corollary 2 is an ultimate extension of Corollary 1.

Corollary 3. Let $w_j(t) = e^{{\alpha}_j t_j}$, ${\beta}_j(t) = {\beta}_j$, and ${\beta}_j > {\alpha}_j\ge {\gamma}_j,$ $j = \overline{1, n}$. Then the operator $\widetilde L:C_{\vec {\alpha}}({\mathbb R}_+^n)\to C_{\vec {\gamma}}({\mathbb R}_+^n)$ is bounded and

Proof. By using (2.26), (2.27), Corollary 1 and Corollary 2, we have

as desired.

Remark 2. The norm in formula (2.28) is achieved for the function

Indeed, we have $f_{\vec{\alpha}}\in C({\mathbb R}^n_+),$

and

From (2.28)–(2.30) the claim follows.

2.4.   An integral-type operator between weighted-type spaces

Let $g\in C([0, 1)^n)$ and

where $x\in [0, 1)^n$. The operator on the polydisk was studied in ^[32].

From now on, for the operator in (2.31) we use the notation

By $\mathcal Q_{\vec {\gamma}}$ we denote the space of all $f\in C([0, 1)^n)$ such that

where $\vec {\gamma} = ({\gamma}_1, \ldots, {\gamma}_n)$ is such that ${\gamma}_j > 0$, $j = \overline{1, n}.$ The quantity $\|\cdot\|_{\mathcal Q_{\vec {\gamma}}}$ is a norm on the space.

In the theorem which follows we estimate norm of the operator $T_g:{\mathcal Q}_{\vec{\alpha}}\to {\mathcal Q}_{\vec{\alpha}+\vec{\beta}-\vec 1}$, under some conditions posed on the vectors $\vec{\alpha}$ and $\vec{\beta}$, and calculate it for a concrete function $g$.

Theorem 4. Let $\vec{\alpha}, \vec{\beta}\in {\mathbb R}_+^n$ be such that ${\alpha}_j+{\beta}_j > 1,$ $j = \overline{1, n}$, and

Then the operator $T_g:{\mathcal Q}_{\vec{\alpha}}\to {\mathcal Q}_{\vec{\alpha}+\vec{\beta}-\vec 1}$ is bounded and

If additionally

then

Proof. Suppose that relation (2.32) holds. Let $f$ be an arbitrary function in ${\mathcal Q_{\vec {\alpha}}}$ and $x$ be an arbitrary point in the cube $[0, 1)^n$. Then by using the definition of the spaces ${\mathcal Q}_{\vec{\alpha}}$ and ${\mathcal Q}_{\vec{\beta}}$, some known inequalities, as well as some calculations it follows that

from which it follows that

for every $x\in[0, 1)^n$ and $f\in {\mathcal Q}_{\vec{\alpha}}.$

By taking the supremum in (2.36) over the set $[0, 1)^n$, it follows that the following inequality holds

for every $f\in {\mathcal Q}_{\vec{\alpha}}.$

By taking the supremum in (2.37) over the unit ball $B_{{\mathcal Q}_{\vec {\beta}}}$ the boundedness of the operator $T_g:{\mathcal Q}_{\vec{\alpha}}\to{\mathcal Q}_{\vec{\alpha}+\vec{\beta}-\vec 1}$ follows.

Moreover, from inequality (2.37) we obtain the following estimate for the norm of the operator

Now, assume that the operator $T_g:{\mathcal Q}_{\vec{\alpha}}\to {\mathcal Q}_{\vec{\alpha}+\vec{\beta}-\vec 1}$ is bounded and that function $g$ is defined as in (2.34).

Let

then

By using (2.34), (2.39) and (2.40), as well as some standard calculations it follows that

From (2.38), (2.41), and since in this case $\|g\|_{{\mathcal Q}_{\vec{\beta}}} = 1$, we have

finishing the proof of the theorem.

Generally speaking operator (2.31) can be considered on functions defined on any set of the form

where $c_j\in[0, +\infty]$, $j = \overline{1, n},$ and where we exclude the case $\prod_{j = 1}^n[0, +\infty].$

Our next result considers the boundedness of operator (2.31) between such spaces.

Theorem 5. Let $u, v, w\in W(I),$ $g\in C_v(I)$, where the set $I$ has one of the forms in (2.42). If

then the operator $T_g:C_w(I)\to C_u(I)$ is bounded.

If additionally

then

Proof. Using the definitions of the spaces $C_w(I)$ and $C_u(I)$, and the change of variables $s_j = x_jt_j$, $j = \overline{1, n}$, we have

for every $x\in I$ and $f\in C_w(I)$.

Multiplying (2.46) by $u(x)$, then taking the supremum over the set $I$ we have

from which it follows that

Using the assumption $g\in C_v(I)$, (2.43) and (2.47) the boundedness of $T_g:C_w(I)\to C_u(I)$ follows.

If (2.44) holds, then

Now, note that for $\widetilde f_0(x) = \frac1{w(x)}$ we have

Further, we have

From (2.49) and (2.50) we obtain

Combining (2.47), (2.48) and (2.50) we get (2.45).

Remark 3. Note that in the case

we have

from which by taking the supremum in (2.52) over the set ${\mathbb R}^n_+$ it follows that

Here, as usual

the standard supremum norm.

2.5.   Another integral-type operator between weighted-type spaces

Let $g\in C([0, 1)^n)$ and

where $x\in {\mathbb R}^n.$ From now on, for the operator in (2.53) we use the notation

Let $u\in W({\mathbb R}^n)$ and

The following theorem holds.

Theorem 6. Let $g\in C([0, 1)^n)$, $g(x)\ge 0$, $x\in {\mathbb R}^n$, $u, v\in W({\mathbb R}^n)$, such that

for some ${\alpha}_j\in {\mathbb R}_+$, $j = \overline{1, n}.$

Then the operator $\widehat T_g:C_u({\mathbb R}^n)\to C_v({\mathbb R}^n)$ is bounded if and only if

Moreover, if the operator $\widehat T_g:C_u({\mathbb R}^n)\to C_v({\mathbb R}^n)$ is bounded then

Proof. Assume that (2.55) holds. Let $f\in C_u({\mathbb R}^n).$ Then by using the definition of the norm in $C_u({\mathbb R}^n)$ and (2.54) we have

from which it follows that

By taking the supremum in (2.57) over the set ${\mathbb R}^n\setminus\{\vec 0\}$, it follows that the following inequality holds

By taking the supremum in (2.58) over the unit ball $B_{C_u({\mathbb R}^n)}$ the boundedness of the operator $\widehat T_g:C_u({\mathbb R}^n)\to C_v({\mathbb R}^n)$ follows. Moreover, we have

Now assume that the operator $\widehat T_g:C_u({\mathbb R}^n)\to C_v({\mathbb R}^n)$ is bounded. Let

Then

By using (2.54), (2.60) and (2.61), as well as some standard calculations it follows that

from which (2.55) follows.

If the operator $\widehat T_g:C_u({\mathbb R}^n)\to C_v({\mathbb R}^n)$ is bounded then from (2.59) and (2.62) we get (2.56), finishing the proof of the theorem.

The following theorem is proved similar to Theorem 6, so we omit the proof.

Theorem 7. Let $g\in C[0, 1)$, $g(t)\ge 0$, $t\in {\mathbb R}$, $u, v\in W({\mathbb R}^n)$, such that

for some ${\alpha} > 0$ and every $t\in[0, 1)$ and $x\in {\mathbb R}^n,$ and

Then the operator $\widehat L_g:C_u({\mathbb R}^n)\to C_v({\mathbb R}^n)$ is bounded if and only if

Moreover, if the operator $\widehat L_g:C_u({\mathbb R}^n)\to C_v({\mathbb R}^n)$ is bounded then

Example 1. Let

where $1\le \min\{p, q\}\le \max\{p, q\} < +\infty$ and for $r\ge 1$

Since all the norms on a finite-dimensional linear space are equivalent (here the linear space is ${\mathbb R}^n$), we have that there are positive constants $C_1$ and $C_2$ such that

Hence, we have

Note also that in this case we have

Hence, to guaranty the boundedness of the operator $\widehat L_g:C_u({\mathbb R}^n)\to C_v({\mathbb R}^n)$ in this case, the corresponding condition in (2.65) holds if the function $g$ satisfies the condition

2.6.   On a Hardy integral operator

Let $\widehat {\mathcal L}_w^p({\mathbb R}^n) = \widehat {\mathcal L}_w^p$ be a linear subspace of ${\mathcal L}_w^p$ containing constant functions, and such that the integral means

are non-increasing for each $f\in \widehat {\mathcal L}_w^p$.

Example 2. An example of such a space consists of all harmonic functions on ${\mathbb R}^n$ ^[18,31], for which the integral means are nondecreasing functions (see, e.g., ^[17]; for one-dimensional case see ^[26]).

Theorem 8. Let $\mu$ be a nonnegative Borel measure on the interval $[0, 1]$, $w\in W({\mathbb R}^n)$ be a radial function such that

and

Then the operator $L_\mu:\widehat{\mathcal L}_w^p({\mathbb R}^n)\to \widehat{\mathcal L}_w^p({\mathbb R}^n)$ is bounded if and only if

Moreover, if the operator $L_\mu:\widehat{\mathcal L}_w^p({\mathbb R}^n)\to \widehat{\mathcal L}_w^p({\mathbb R}^n)$ is bounded then

Proof. First assume that (2.68) holds. By using Minkowski's integral inequality (see, e.g., ^[16,30]), polar coordinates (see, e.g., ^[18] or [26,p.150]), the assumption that $w$ is radial, i.e., $w(r\zeta) = w(r)$, $x = r\zeta\in {\mathbb R}^n$, and the monotonicity of the integral means, we have

from which it follows that

Now, assume that the operator $L_\mu:\widehat{\mathcal L}_w^p({\mathbb R}^n)\to \widehat{\mathcal L}_w^p({\mathbb R}^n)$ is bounded. Note that from (2.66) we have

On the other hand, by the definition of the space $\widehat{\mathcal L}_w^p$, we have $\widehat f_0(x)\equiv 1\in \widehat{\mathcal L}_w^p.$ From this and since

we get

If the operator $L_\mu:\widehat{\mathcal L}_w^p({\mathbb R}^n)\to \widehat{\mathcal L}_w^p({\mathbb R}^n)$ is bounded, then from (2.70) and (2.71) we get (2.69).

Remark 4. The operator in (2.67) is a Hardy integral-type operator ^[15].

3.   Conclusions

Here we calculate the norms of several concrete operators between weighted-type spaces of continuous functions on several domains, as well as the norm of an integral-type operator on some subspaces of the weighted Lebesgue spaces. Several methods, ideas and tricks, which could be used in some other settings, are presented.

Acknowledgments

The paper was made during the investigation supported by the Ministry of Education, Science and Technological Development of Serbia, contract no. 451-03-68/2022-14/200029.

Conflict of interest

The author declares that he has no competing interest.