Abstract
The complete characterization of the weighted \(L^{p}-L^{r}\) inequalities of supremum operators on the cones of monotone functions for all \(0< p,r\leq \infty\) is given.
Similar content being viewed by others
1 Introduction
Let \(\mathbb{R}_{+}:=[0,\infty)\). Denote \(\mathfrak{M}\) the set of all measurable functions on \(\mathbb{R}_{+}\), \(\mathfrak{M}^{+}\subset\mathfrak {M}\) the subset of all non-negative functions and \(\mathfrak {M}^{\downarrow}\subset\mathfrak{M}^{+}\) (\(\mathfrak{M}^{\uparrow}\subset\mathfrak{M}^{+}\)) is the cone of all non-increasing (non-decreasing) functions. Also denote by \(\mathscr {C}\subset\mathfrak{M}\) the set of all continuous functions on \(\mathbb {R}_{+}\). If \(0< p\leq\infty\) and \(v\in\mathfrak{M}^{+}\) we define
Let \(w\in\mathfrak{M}^{+}\) and \(k(x,y)\geq0\) is a Borel function on \([0,\infty)^{2}\) satisfying Oinarov’s condition: \(k(x,y)=0\) if \(x< y\), and there is a constant \(D\geq1\) independent of \(x\geq z\geq y\geq0\) such that
The mapping properties between weighted \(L^{p}\) spaces of Hardy type operators involved are very well studied. See e.g. the books [1, 2] and [3] and the references therein. We also mention the following examples of articles in this area: [4–8] and [9]. Recently, it has been discovered that it is of great interest to study also some corresponding supremum operators instead of the usual such Hardy type (arithmetic mean) operators. The interest comes both from purely mathematical point of view but also from various applications where such kernels many times are the unit impulse answers to the problem at hand and the best constants means the operator norms of the corresponding transfer of the energy of the ‘signals’ measured in weighted \(L^{p}\) spaces.
We consider supremum operators of the form
Let \(0< p, r\leq\infty\) and \(u, v\in\mathfrak{M}^{+}\). The paper is devoted to the necessary and sufficient conditions for the inequalities
where the constants \(C_{T}\) and others are taken as the least possible.
This problem was first studied for the inequality (1.3) in [10], Theorem 3.2, in a case when \(k(x,y)=1\), \(w\in \mathscr {C}\). This result was extended in [11] for the case \(k(x,y)\) satisfying (1.1) with a discrete form of a criterion for \(0< r< p<\infty\). With different supremum operators some similar problems were studied in [12–22]. This area is currently developing intensively and finds many interesting applications.
Section 2 is devoted to preliminaries. The border cases \(0< r< p=\infty\), \(0< p< r=\infty\) and \(r=p=\infty\) are solved in Section 3. In Section 4 we characterize the case \(k(x,y)= 1\), which is essentially used in Section 5 with the main results of the paper.
We use signs := and =: for determining new quantities and \(\mathbb {Z}\) for the set of all integers. For positive functionals F and G we write \(F\lesssim G\), if \(F\leq cG\) with some positive constant c, which depends only on irrelevant parameters. \(F\approx G\) means \(F\lesssim G\lesssim F\) or \(F=cG\). \(\chi_{E}\) denotes the characteristic function (indicator) of a set E. Uncertainties of the form \(0\cdot \infty\), \(\frac{\infty}{\infty}\) and \(\frac{0}{0}\) are taken to be zero. □ stands for the end of a proof.
2 Preliminaries
We denote
Let \(0< p,r<\infty\). By [11], Lemma 2.1, and the monotone convergence theorem the inequality (1.2) is equivalent to
If
then (2.1) is equivalent to
Analogously, if
then (1.3), (1.4), and (1.5) are equivalent to
and
respectively.
For the border cases \(0< p< r=\infty\), \(0< r< p=\infty\), and \(r=p=\infty\) we have the following four groups of inequalities:
for the operator T;
for the operator S;
for the operator \(\mathscr {T}\), and
for the operator \(\mathscr {S}\). We characterize the inequalities (2.6)-(2.17) in the next section.
To deal with the inequalities (2.1)-(2.5) we study first the case \(k(x,y) = 1\) and then a general case.
3 Border cases of summation parameters
For a measurable function \(v\in\mathfrak{M}^{+}\) we define monotone envelopes (see [23], Section 2) as follows:
Theorem 3.1
For the best possible constants of the inequalities (2.6)-(2.8) we have
Proof
Observe that if \(k(x,y)\) satisfies (1.1), then \([k(x,y)]^{p}\) satisfies (1.1) too with a constant \(D_{p}\geq1\). If \(x\leq t\), then
Hence,
It implies (see [11], Proposition 3.1)
and (2.6) is equivalent to
where
Thus,
Since by a well-known theorem ([24], Theorem 1.1)
we obtain (3.1).
Now, (2.7) is equivalent to the inequality
The lower bound of (3.2) follows from (3.5) with \(f=\frac {1}{v^{\downarrow}}\) and the upper bound from the estimate \(f(y)\leq\frac {\|f\|_{L^{\infty}_{v^{\downarrow}}}}{v^{\downarrow}(y)}\).
The proof of (3.3) is the same. □
Analogously, we can prove the following.
Theorem 3.2
For the best possible constants of the inequalities (2.9)-(2.17) we have
4 The case \(k(x,y)=1\)
Let \(u, v_{0}, w_{0}\in\mathfrak{M}^{+}\) be weights. We suppose for simplicity that \(0<\int_{0}^{t}u <\infty \), for all \(t>0\), \(\int _{0}^{\infty}u =\infty\) and define the functions \(\sigma: [0;\infty)\rightarrow[0;\infty)\), \(\sigma^{-1}: [0;\infty )\rightarrow[0;\infty)\), by
Let \(\sigma^{2}:=\sigma(\sigma)\). For \(0\leq c < d< \infty\) and \(h\in\mathfrak{M}^{+}\) we put
We need the following partial cases of [21], Theorems 2.1 and 2.3 (see also [19, 20]).
Theorem 4.1
Let \(0< r<\infty\). Then:
-
(a)
For validity of the inequality
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} w_{0}(y) \int _{0}^{y} h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$(4.1)it is necessary and sufficient that the inequality
$$\biggl( \int_{0}^{\infty}u(x)\bigl[w_{0}^{\downarrow}(x) \bigr]^{r} \biggl( \int_{0}^{x}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq A_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$holds and the constant
$$A_{1}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{1}{r}}\Vert H_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}}} ,&r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{1-r}} \Vert H_{[\sigma^{-1}(x), \sigma(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}}}\,dx )^{\frac {1-r}{r}}, &0< r< 1, \end{cases} $$is finite. Moreover, \(C_{0}\approx A_{0}+A_{1}\).
-
(b)
For validity of the inequality
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} w_{0}(y) \int _{y}^{\infty}h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$(4.2)it is necessary and sufficient that the inequality
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{2}(x)}w_{0}(y)\Bigr]^{r} \biggl( \int_{\sigma^{2}(x)}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac {1}{r}}\leq {B}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$holds and the constant
$${B}_{1}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{1}{r}}\Vert H^{*}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}}} ,&r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{1-r}} \Vert H^{*}_{[\sigma^{-1}(x), \sigma(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}}}\,dx )^{\frac {1-r}{r}}, &0< r< 1, \end{cases} $$is finite. Moreover, \(C_{1}\approx{B}_{0}+{B}_{1}\).
Using Theorem 4.1 we characterize (1.2) and (1.3) with \(k(x,y) = 1\).
Theorem 4.2
Let \(0< p, r<\infty\) and \(k(x,y) = 1\). Then, for the best possible constants of the inequalities (1.2) and (1.3) the following equivalences hold:
where
Proof
Since (1.2) ⇔ (2.2) and (1.3) ⇔ (2.3), the proof follows by applying Theorem 4.1 with r replaced by \(\frac{r}{p}\), \(w_{0}=w^{p}\), \(v_{0}=V_{\ast}\) in (4.1) and \(v_{0}=V\) in (4.2). Thus, \(C_{T}\approx \mathscr {A}_{0}^{\prime}+\mathscr {A}_{1}^{\prime}\), where \(\mathscr {A}_{0}^{\prime}\) is the best constant in the inequality
and
If \(k(x,y)\geq0\) is a measurable kernel on \(\mathbb{R}_{+}\times\mathbb {R}_{+}\) and
then by well-known results ([25], Chapter XI, Section 1.5, Theorem 4, see also [24], Theorem 1.1)
If \(k(x,y)=w(x)\chi_{[0,x]}(y)u(y)\) and \(0< q<1\), then ([26], Theorem 3.3)
Applying (4.5) and (4.6) to (4.4) we find that \(\mathscr {A}_{0}\approx \mathscr {A}_{0}^{\prime}\). Again, applying (4.5), when
we obtain
Similarly, using the monotonicity of \(V_{\ast}\), we find
and the estimate \(\mathscr {A}_{1}\approx \mathscr {A}_{1}^{\prime}\) follows.
For the second part we observe that \(C_{S}\approx \mathscr {B}_{0}^{\prime}+\mathscr {B}_{1}^{\prime}\), where \(\mathscr {B}_{0}^{\prime}\) is the least constant in the inequality
and
By a change of variables we see that (4.7) is equivalent to
where \(V_{\sigma^{2}}(y):=V(\sigma^{2}(t))\). By the same argument as above it follows that \(\mathscr {B}_{0}^{\prime}\approx \mathscr {B}_{0}\) and \(\mathscr {B}_{1}^{\prime}\approx \mathscr {B}_{1}\). □
Analogously, we obtain the sharp estimates for the best constants in (1.4) and (1.5).
Suppose for simplicity that \(0<\int_{t}^{\infty}u <\infty \) for all \(t>0\), \(\int_{0}^{\infty}u =\infty\) and define the functions \(\zeta: [0;\infty)\rightarrow[0;\infty)\), \(\zeta^{-1}: [0;\infty )\rightarrow[0;\infty)\), by
Let \(\zeta^{2}:=\zeta(\zeta)\). For \(0\leq c < d< \infty\) and \(h\in\mathfrak{M}^{+}\) we put
We need the following partial cases of [21], Theorems 3.1 and 3.2.
Theorem 4.3
Let \(0< r<\infty\). Then:
-
(a)
For validity of the inequality
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} w_{0}(y) \int_{y}^{\infty}h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{2}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$it is necessary and sufficient that the inequality
$$\biggl( \int_{0}^{\infty}u(x)\bigl[w_{0}^{\uparrow}(x) \bigr]^{r} \biggl( \int_{x}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq D_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$holds and the constant
$$D_{1}:= \textstyle\begin{cases} \sup_{t>0} (\int_{t}^{\infty}u )^{\frac{1}{r}}\Vert \mathscr {H}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}}} ,& r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{x}^{\infty}u )^{\frac{r}{1-r}} \Vert \mathscr {H}_{[\zeta^{-1}(x), \zeta(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}}}\,dx )^{\frac {1-r}{r}}, & 0< r< 1, \end{cases} $$is finite. Moreover, \(C_{2}\approx D_{0}+D_{1}\).
-
(b)
For validity of the inequality
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} w_{0}(y) \int_{0}^{y} h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{3}\|h\|_{L^{1}_{v_{0}}}, \quad h\in \mathfrak{M}^{+}, $$it is necessary and sufficient that the inequality
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{\zeta^{-2}(x)\leq y\leq x}w_{0}(y)\Bigr]^{r} \biggl( \int_{0}^{\zeta^{-2}(x)}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {E}_{0}\|h\|_{L^{1}_{v_{0}}}, \quad h\in \mathfrak{M}^{+}, $$holds and the constant
$${E}_{1}:= \textstyle\begin{cases} \sup_{t>0} (\int_{t}^{\infty}u )^{\frac{1}{r}}\Vert \mathscr {H}^{*}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}}} ,&r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{x}^{\infty}u )^{\frac{r}{1-r}} \Vert \mathscr {H}^{*}_{[\zeta^{-1}(x), \zeta(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}}}\,dx )^{\frac {1-r}{r}}, &0< r< 1, \end{cases} $$is finite. Moreover, \(C_{3}\approx{E}_{0}+{E}_{1}\).
Using Theorem 4.3 we characterize (1.4) and (1.5) with \(k(x,y) = 1\).
Theorem 4.4
Let \(0< p, r<\infty\) and \(k(x,y) = 1\). Then for the best possible constants of the inequalities (1.4) and (1.5) the following equivalences hold:
where
5 Main results
To deal with the kernel transformation we need the following extension of Theorem 4.1 following from [21], Theorems 4.1 and 4.3.
Theorem 5.1
Let \(0< r<\infty\), \(u, v_{0}, w_{0} \in\mathfrak{M}^{+}\) and \(k_{0}(x,y)\) satisfies Oinarov’s condition (1.1). Then:
-
(a)
For validity of the inequality
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k_{0}(y,x)w_{0}(y) \int_{0}^{y} h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq{\mathbf{C}}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$(5.1)it is necessary and sufficient that the inequalities
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k_{0}(y,x)w_{0}(y) \Bigr]^{r} \biggl( \int_{0}^{x}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{A}}_{0}\|h\|_{L^{1}_{v_{0}}}, \quad h\in\mathfrak{M}^{+}, $$and
$$\biggl( \int_{0}^{\infty}u(x)\bigl[k_{0}\bigl( \sigma^{2}(x),x\bigr)\bigr]^{r} \biggl(\mathop{ \operatorname{ess\,sup}}_{y\geq \sigma^{2}(x)} w_{0}(y) \int_{0}^{y}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{A}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$hold and the constant
$${\mathbf{A}}_{2}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{1}{r}}\Vert H_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\cdot,t)}} ,& r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{1-r}} \Vert H_{[\sigma^{-1}(x), \sigma^{2}(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\cdot,\sigma ^{-1}(x))}}\,dx )^{\frac{1-r}{r}}, &r< 1, \end{cases} $$is finite. Moreover, \({\mathbf{C}}_{0}\approx{\mathbf{A}}_{0}+{\mathbf{A}}_{1}+{\mathbf{A}}_{2}\).
-
(b)
For validity of the inequality
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k_{0}(y,x)w_{0}(y) \int_{y}^{\infty}h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq{\mathbf{C}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h \in\mathfrak{M}^{+}, $$(5.2)it is necessary and sufficient that the inequalities
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{3}(x)}k_{0}(y,x)w_{0}(y) \Bigr]^{r} \biggl( \int_{\sigma^{3}(x)}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac {1}{r}}\leq {\mathbf{B}}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$and
$$\biggl( \int_{0}^{\infty}u(x)\bigl[k_{0}\bigl( \sigma^{2}(x),x\bigr)\bigr]^{r} \biggl(\mathop{ \operatorname{ess\,sup}}_{y\geq \sigma^{2}(x)} w_{0}(y) \int_{y}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{B}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$hold and the constant
$${\mathbf{B}}_{2}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{1}{r}}\Vert H^{*}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\cdot,t)}} ,& r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{1-r}} \Vert H^{*}_{[\sigma^{-1}(x), \sigma^{2}(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\cdot,\sigma ^{-1}(x))}}\,dx )^{\frac{1-r}{r}}, & r< 1, \end{cases} $$is finite. Moreover, \({\mathbf{C}}_{0}\approx{\mathbf{B}}_{0}+{\mathbf{B}}_{1}+{\mathbf{B}}_{2}\).
Using Theorem 5.1 we obtain the characterization of (1.2) and (1.3) for \(0< p, r<\infty\). Denote
Theorem 5.2
Let \(0< p, r<\infty\). Then, for the best possible constants of the inequalities (1.2) and (1.3) the following equivalences hold:
where
Proof
We start with the inequality (1.2). Since (1.2) ⇔ (2.1), then applying Theorem 5.1 we see that
where \(\mathbb{A}_{0}^{\prime}\) and \(\mathbb{A}_{1}^{\prime}\) are the best constants in the inequalities
and
Applying (4.5) and (4.6) we see that \(\mathbb {A}_{0}^{\prime}\approx\mathbb{A}_{0}\) and \(\mathbb{A}_{2}^{\prime}\approx\mathbb {A}_{2}\). By a change of variable we find that (5.4) is equivalent to the inequality
which is governed by Theorem 4.1. Arguing analogously to the proof of Theorem 4.2 we see that
where \(\mathbb{A}_{1,0}^{\prime}\) is the best constant of the inequality
and
for \(0< r< p\). Again applying (4.5) and (4.6) we see that \(\mathbb{A}_{1,0}^{\prime}\approx\mathbb{A}_{1,0}\) and \(\mathbb {A}_{1,1}^{\prime}\approx\mathbb{A}_{1,1}\).
The proof for the inequality (1.3) is similar. □
Analogously, we obtain the sharp estimates for the best constants in (1.4) and (1.5). To this end we need the following extension of Theorem 4.3 from [21], Theorems 5.1 and 5.2.
Theorem 5.3
Let \(0< r<\infty\), \(u, v_{0}, w_{0} \in\mathfrak{M}^{+}\) and \(k_{0}(x,y)\) satisfy Oinarov’s condition (1.1). Then:
-
(a)
For validity of the inequality
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k_{0}(x,y)w_{0}(y) \int_{y}^{\infty}h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq{\mathbf{C}}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$it is necessary and sufficient that the inequalities
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k_{0}(x,y)w_{0}(y) \Bigr]^{r} \biggl( \int_{x}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{A}}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$and
$$\begin{aligned}& \biggl( \int_{0}^{\infty}u(x)\bigl[k_{0}\bigl(x, \zeta^{-2}(x)\bigr)\bigr]^{r} \biggl(\mathop{\operatorname{ess \,sup}}_{0\leq y\leq\zeta^{-2}(x)} w_{0}(y) \int_{y}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}} \\& \quad \leq {\mathbf{A}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, \end{aligned}$$hold and the constant
$${\mathbf{A}}_{2}:= \textstyle\begin{cases} \sup_{t>0} (\int_{t}^{\infty}u )^{\frac{1}{r}}\Vert \mathscr {H}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(t,\cdot)}} ,&r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{x}^{\infty}u )^{\frac{r}{1-r}} \Vert \mathscr {H}_{[\zeta^{-1}(x), \zeta^{2}(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\zeta ^{2}(x),\cdot)}}\,dx )^{\frac{1-r}{r}}, &r< 1, \end{cases} $$is finite. Moreover, \({\mathbf{C}}_{0}\approx{\mathbf{A}}_{0}+{\mathbf{A}}_{1}+{\mathbf{A}}_{2}\).
-
(b)
For validity of the inequality
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k_{0}(y,x)w_{0}(y) \int_{0}^{y} h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq{\mathbf{C}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h \in\mathfrak{M}^{+}, $$it is necessary and sufficient that the inequalities
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{\zeta^{-3}(x)\leq y\leq x}k_{0}(x,y)w_{0}(y) \Bigr]^{r} \biggl( \int_{0}^{\zeta^{-3}(x)}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{B}}_{0}\|h\|_{L^{1}_{v_{0}}}, \quad h\in\mathfrak{M}^{+}, $$and
$$\biggl( \int_{0}^{\infty}u(x)\bigl[k_{0}\bigl(x, \zeta^{-2}(x)\bigr)\bigr]^{r} \biggl(\mathop{\operatorname{ess \,sup}}_{0\leq y\leq\zeta^{-2}(x)} w_{0}(y) \int_{0}^{y} h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{B}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$hold and the constant
$${\mathbf{B}}_{2}:= \textstyle\begin{cases} \sup_{t>0} (\int_{t}^{\infty}u )^{\frac{1}{r}}\Vert \mathscr {H}^{*}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(t,\cdot)}} ,& r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{x}^{\infty}u )^{\frac{r}{1-r}} \Vert \mathscr {H}^{*}_{[\zeta^{-1}(x), \zeta^{2}(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\zeta ^{2}(x),\cdot)}}\,dx )^{\frac{1-r}{r}}, &r< 1, \end{cases} $$is finite. Moreover, \({\mathbf{C}}_{0}\approx{\mathbf{B}}_{0}+{\mathbf{B}}_{1}+{\mathbf{B}}_{2}\).
Using Theorem 5.3 we obtain the characterization of (1.4) and (1.5) for \(0< p, r<\infty\). Denote
Theorem 5.4
Let \(0< p, r<\infty\). Then for the best possible constants of the inequalities (1.4) and (1.5) the following equivalences hold:
where
References
Kufner, A, Persson, LE: Weighted Inequalities of Hardy Type. World Scientific, River Edge (2003)
Kufner, A, Maligranda, L, Persson, LE: The Hardy Inequality: About Its History and Some Related Results. Vydavatelsky Servis Publishing House, Pilsen (2007)
Kokalishvili, V, Meskhi, A, Persson, LE: Weighted Norm Inequalities for Integral Transforms with Product Kernels. Nova Science Publishers, New York (2010)
Kalybay, A, Person, LE, Temirkhanova, A: A new discrete Hardy-type inequality with kernels and monotone functions. J. Inequal. Appl. 2015, 321 (2015)
Machihara, S, Ozawa, T, Wadade, H: Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space. J. Inequal. Appl. 2015, 281 (2015)
Mejjaoli, H: Hardy-type inequalities associated with the Weinstein operator. J. Inequal. Appl. 2015, 267 (2015)
Persson, L-E, Shambilova, GE, Stepanov, VD: Hardy-type inequalities on the weighted cones of quasi-concave functions. Banach J. Math. Anal. 9(2), 21-34 (2015)
Oguntuase, J, Fabelurin, O, Adeagbu-Sheikh, A, Persson, LE: Time scale Hardy inequalities with ‘broken’ exponent p. J. Inequal. Appl. 2015, 17 (2015)
Persson, LE, Shaimardan, S: Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator. J. Inequal. Appl. 2015, 296 (2015)
Gogatishvili, A, Opic, B, Pick, L: Weighted inequalities for Hardy-type operators involving suprema. Collect. Math. 57, 227-255 (2006)
Stepanov, VD: On a supremum operator. In: Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends. Operator Theory: Advances and Applications, vol. 219, pp. 233-242 (2012)
Gogatishvili, A, Pick, L: A reduction theorem for supremum operators. J. Comput. Appl. Math. 208, 270-279 (2007)
Cwikel, M, Pustylnik, E: Weak type interpolation near ‘endpoint’ spaces. J. Funct. Anal. 171, 235-277 (2000)
Evans, WD, Opic, B: Real interpolation with logarithmic functions and reiteration. Can. J. Math. 52, 920-960 (2000)
Pick, L: Optimal Sobolev embeddings. Rudolph-Lipshitz-Vorlesungsreihe, no. 43. Rheinische Friedrich-Wilhelms-Universität Bonn (2002)
Prokhorov, DV: Inequalities for Riemann-Liouville operator involving suprema. Collect. Math. 61, 263-276 (2010)
Prokhorov, DV: Lorentz norm inequalities for the Hardy operator involving suprema. Proc. Am. Math. Soc. 140, 1585-1592 (2012)
Prokhorov, DV: Boundedness and compactness of a supremum-involving integral operator. Proc. Steklov Inst. Math. 283, 136-148 (2013)
Prokhorov, DV, Stepanov, VD: On weighted Hardy inequalities in mixed norms. Proc. Steklov Inst. Math. 283, 149-164 (2013)
Prokhorov, DV, Stepanov, VD: Estimates for a class of sublinear integral operators. Dokl. Math. 89, 372-377 (2014)
Prokhorov, DV, Stepanov, VD: Weighted inequalities for quasilinear integral operators on the semiaxis and application to the Lorentz spaces. Sb. Math. 207(8), 135-162 (2016). doi:10.1070/SM8535
Krepela, M: Integral conditions for Hardy type operators involving suprema. Collect. Math. (2016). doi:10.1007/s13348-016-0170-6
Sinnamon, G: Transferring monotonicity in weighted norm inequalities. Collect. Math. 54, 181-216 (2003)
Gogatishvili, A, Stepanov, VD: Reduction theorems for weighted integral inequalities on the cone of monotone functions. Russ. Math. Surv. 68(4), 597-664 (2013)
Kantorovich, LV, Akilov, GP: Functional Analysis. Pergamon, Oxford (1982)
Sinnamon, G, Stepanov, VD: The weighted Hardy inequality: new proofs and the case \({p=1}\). J. Lond. Math. Soc. 54, 89-101 (1996)
Acknowledgements
The research work of GE Shambilova and VD Stepanov was carried out at the Peoples’ Friendship University of Russia and financially supported by the Russian Science Foundation (Project no. 16-41-02004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Persson, LE., Shambilova, G.E. & Stepanov, V.D. Weighted Hardy type inequalities for supremum operators on the cones of monotone functions. J Inequal Appl 2016, 237 (2016). https://doi.org/10.1186/s13660-016-1168-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-016-1168-z