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
$$\begin{aligned}& L^{p}_{v}:= \biggl\{ f\in\mathfrak{M}:\|f\|_{L^{p}_{v}} := \biggl( \int_{0}^{\infty}\bigl\vert f(x)\bigr\vert ^{p}v(x)\,dx \biggr)^{\frac{1}{p}}< \infty \biggr\} , \\& L^{\infty}_{v}:= \Bigl\{ f\in\mathfrak{M}:\|f\|_{L^{\infty}_{v}} :=\mathop{\operatorname{ess\,sup}}_{x\geq0} v(x)\bigl\vert f(x)\bigr\vert < \infty \Bigr\} . \end{aligned}$$
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
$$ \frac{1}{D} \bigl(k(x,z)+k(z,y) \bigr)\leq k(x,y)\leq D \bigl(k(x,z)+k(z,y) \bigr). $$
(1.1)
Anzeige
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
$$\begin{aligned}& (Tf) (x) =\mathop{\operatorname{ess\,sup}}_{y\geq x} k(y,x)w(y)f(y),\quad f \in \mathfrak{M}^{\uparrow}, \\& (Sf) (x) =\mathop{\operatorname{ess\,sup}}_{y\geq x} k(y,x)w(y)f(y), \quad f\in \mathfrak{M}^{\downarrow}, \\& ({\mathscr {T}}f) (x) =\mathop{\operatorname{ess\,sup}}_{0\leq y\leq x} k(x,y)w(y)f(y),\quad f\in\mathfrak{M}^{\downarrow}, \\& ({\mathscr {S}}f) (x) =\mathop{\operatorname{ess\,sup}}_{0\leq y\leq x} k(x,y)w(y)f(y),\quad f\in\mathfrak{M}^{\uparrow}. \end{aligned}$$
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.
$$\begin{aligned}& \Vert Tf\Vert _{L^{r}_{u}} \leq C_{T}\Vert f \Vert _{L^{p}_{v}},\quad f\in \mathfrak{M}^{\uparrow}, \end{aligned}$$
(1.2)
$$\begin{aligned}& \Vert Sf\Vert _{L^{r}_{u}} \leq C_{S}\Vert f \Vert _{L^{p}_{v}}, \quad f\in \mathfrak{M}^{\downarrow}, \end{aligned}$$
(1.3)
$$\begin{aligned}& \Vert {\mathscr {T}}f\Vert _{L^{r}_{u}} \leq C_{{\mathscr {T}}} \Vert f\Vert _{L^{p}_{v}},\quad f\in\mathfrak{M}^{\downarrow}, \end{aligned}$$
(1.4)
$$\begin{aligned}& \Vert {\mathscr {S}}f\Vert _{L^{r}_{u}} \leq C_{{\mathscr {S}}} \Vert f\Vert _{L^{p}_{v}},\quad f\in\mathfrak{M}^{\uparrow}, \end{aligned}$$
(1.5)
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.
Anzeige
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.
$$V(t):= \int_{0}^{t} v,\qquad V_{\ast}(t):= \int_{t}^{\infty}v. $$
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} \bigl(k(y,x)w(y) \bigr)^{p} \int_{0}^{y} h \biggr]^{\frac{r}{p}}u(x)\,dx \biggr)^{\frac{p}{r}} \leq C_{T}^{p} \int_{0}^{\infty}hV_{\ast}, \quad h\in \mathfrak{M}^{+}. $$
(2.1)
$$T_{p}h(x):=\mathop{\operatorname{ess\,sup}}_{y\geq x} \bigl(k(y,x)w(y) \bigr)^{p} \int_{0}^{y} h, $$
$$ \Vert T_{p}h\Vert _{L^{\frac{r}{p}}_{u}} \leq C_{T}^{p}\Vert h\Vert _{L^{1}_{V_{\ast}}},\quad h\in \mathfrak{M}^{+}. $$
(2.2)
$$\begin{aligned}& S_{p}h(x):=\mathop{\operatorname{ess\,sup}}_{y\geq x} \bigl(k(y,x)w(y) \bigr)^{p} \int_{y}^{\infty}h, \\& \mathscr {T}_{p}h(x):=\mathop{\operatorname{ess\,sup}}_{0\leq y\leq x} \bigl(k(x,y)w(y) \bigr)^{p} \int_{y}^{\infty}h, \\& \mathscr {S}_{p}h(x):=\mathop{\operatorname{ess\,sup}}_{0\leq y\leq x} \bigl(k(x,y)w(y) \bigr)^{p} \int_{0}^{y} h, \end{aligned}$$
$$\begin{aligned}& \Vert S_{p}h\Vert _{L^{\frac{r}{p}}_{u}} \leq C_{S}^{p}\Vert h\Vert _{L^{1}_{V}},\quad h\in \mathfrak{M}^{+}, \end{aligned}$$
(2.3)
$$\begin{aligned}& \Vert \mathscr {T}_{p}h\Vert _{L^{\frac{r}{p}}_{u}} \leq C_{\mathscr {T}}^{p}\Vert h\Vert _{L^{1}_{V}},\quad h\in \mathfrak{M}^{+}, \end{aligned}$$
(2.4)
$$ \Vert \mathscr {S}_{p}h\Vert _{L^{\frac{r}{p}}_{u}} \leq C_{\mathscr {S}}^{p}\Vert h\Vert _{L^{1}_{V_{\ast}}},\quad h\in \mathfrak{M}^{+}, $$
(2.5)
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.
$$\begin{aligned}& \mathop{\operatorname{ess\,sup}}_{x\geq0} \bigl[u(x) \bigr]^{p}T_{p}h(x)\leq C_{T,p}^{p} \int _{0}^{\infty}hV_{\ast}, \quad h\in \mathfrak{M}^{+}, \end{aligned}$$
(2.6)
$$\begin{aligned}& \biggl( \int_{0}^{\infty}\Bigl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k(y,x)w(y)f(y) \Bigr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{T,r}\|f\|_{L^{\infty}_{v}},\quad f\in \mathfrak{M}^{\uparrow}, \end{aligned}$$
(2.7)
$$\begin{aligned}& \mathop{\operatorname{ess\,sup}}_{x\geq0} u(x) \Bigl[ \mathop{\operatorname{ess\,sup}}_{y\geq x} k(y,x)w(y)f(y) \Bigr]\leq C_{T,\infty} \|f\|_{L^{\infty}_{v}}, \quad f\in\mathfrak{M}^{\uparrow}, \end{aligned}$$
(2.8)
$$\begin{aligned}& \mathop{\operatorname{ess\,sup}}_{x\geq0} \bigl[u(x) \bigr]^{p}S_{p}h(x)\leq C_{S,p}^{p} \int _{0}^{\infty}hV, \quad h\in\mathfrak{M}^{+}, \end{aligned}$$
(2.9)
$$\begin{aligned}& \biggl( \int_{0}^{\infty}\Bigl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k(y,x)w(y)f(y) \Bigr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{S,r}\|f\|_{L^{\infty}_{v}},\quad f\in \mathfrak{M}^{\downarrow}, \end{aligned}$$
(2.10)
$$\begin{aligned}& \mathop{\operatorname{ess\,sup}}_{x\geq0} u(x) \Bigl[ \mathop{\operatorname{ess\,sup}}_{y\geq x} k(y,x)w(y)f(y) \Bigr]\leq C_{S,\infty} \|f\|_{L^{\infty}_{v}},\quad f\in\mathfrak{M}^{\downarrow}, \end{aligned}$$
(2.11)
$$\begin{aligned}& \mathop{\operatorname{ess\,sup}}_{x\geq0} \bigl[u(x) \bigr]^{p}\mathscr {T}_{p}h(x)\leq C_{\mathscr {T},p}^{p} \int_{0}^{\infty}hV,\quad h\in\mathfrak{M}^{+}, \end{aligned}$$
(2.12)
$$\begin{aligned}& \biggl( \int_{0}^{\infty}\Bigl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k(x,y)w(y)f(y) \Bigr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{\mathscr {T},r}\|f\|_{L^{\infty}_{v}}, \quad f\in \mathfrak{M}^{\downarrow}, \end{aligned}$$
(2.13)
$$\begin{aligned}& \mathop{\operatorname{ess\,sup}}_{x\geq0} u(x) \Bigl[ \mathop{\operatorname{ess\,sup}}_{0\leq y\leq x} k(x,y)w(y)f(y) \Bigr]\leq C_{\mathscr {T},\infty} \|f\|_{L^{\infty}_{v}}, \quad f\in\mathfrak{M}^{\downarrow}, \end{aligned}$$
(2.14)
$$\begin{aligned}& \mathop{\operatorname{ess\,sup}}_{x\geq0} \bigl[u(x) \bigr]^{p}\mathscr {S}_{p}h(x)\leq C_{\mathscr {S},p}^{p} \int_{0}^{\infty}hV_{\ast},\quad h\in \mathfrak{M}^{+}, \end{aligned}$$
(2.15)
$$\begin{aligned}& \biggl( \int_{0}^{\infty}\Bigl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k(x,y)w(y)f(y) \Bigr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{\mathscr {S},r}\|f\|_{L^{\infty}_{v}},\quad f\in \mathfrak{M}^{\uparrow}, \end{aligned}$$
(2.16)
$$\begin{aligned}& \mathop{\operatorname{ess\,sup}}_{x\geq0} u(x) \Bigl[ \mathop{\operatorname{ess\,sup}}_{0\leq y\leq x} k(x,y)w(y)f(y) \Bigr]\leq C_{\mathscr {S},\infty} \|f\|_{L^{\infty}_{v}}, \quad f\in\mathfrak{M}^{\uparrow}, \end{aligned}$$
(2.17)
3 Border cases of summation parameters
For a measurable function \(v\in\mathfrak{M}^{+}\) we define monotone envelopes (see [23], Section 2) as follows:
$$\begin{aligned}& v^{\downarrow}(x):=\mathop{\operatorname{ess\,sup}}_{y\geq x} v(y), \\ & v^{\uparrow}(x):=\mathop{\operatorname{ess\,sup}}_{0\leq y\leq x} v(y). \end{aligned}$$
Theorem 3.1
For the best possible constants of the inequalities (2.6)-(2.8) we have
$$\begin{aligned}& C_{T,p}\approx\sup_{x\geq0}u^{\uparrow}(x) \mathop{\operatorname{ess\,sup}}_{y\geq x}\frac{k(y,x)w(y)}{V^{1/p}_{\ast}(y)}, \end{aligned}$$
(3.1)
$$\begin{aligned}& C_{T,r}= \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x}\frac{k(y,x)w(y)}{v^{\downarrow}(y)} \biggr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}}, \end{aligned}$$
(3.2)
$$\begin{aligned}& C_{T,\infty}=\mathop{\operatorname{ess\,sup}}_{x\geq0} u(x) \biggl[\mathop{\operatorname{ess\,sup}}_{y\geq x}\frac{k(y,x)w(y)}{v^{\downarrow}(y)} \biggr]. \end{aligned}$$
(3.3)
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).
$$\begin{aligned} \begin{aligned} T_{p}h(t)&=\mathop{\operatorname{ess\,sup}}_{y\geq t} \bigl(k(y,t)w(y) \bigr)^{p} \int_{0}^{y} h \\ &\leq D_{p} \mathop{\operatorname{ess\,sup}}_{y\geq t} \bigl(k(y,x)w(y) \bigr)^{p} \int _{0}^{y} h \\ &\leq D_{p} \mathop{\operatorname{ess\,sup}}_{y\geq x} \bigl(k(y,x)w(y) \bigr)^{p} \int _{0}^{y} h=D_{p}T_{p}h(x). \end{aligned} \end{aligned}$$
$$T_{p}h(x)\approx\sup_{t\geq x}T_{p}h(t):= \varphi(x)\in\mathfrak {M}^{\downarrow}. $$
$$\begin{aligned} \mathop{\operatorname{ess\,sup}}_{x\geq0} \bigl[u(x)\bigr]^{p}T_{p}h(x)& \approx\mathop{\operatorname{ess\,sup}}_{x\geq0} \bigl[u(x) \bigr]^{p}\varphi(x) \\ &=\mathop{\operatorname{ess\,sup}}_{x\geq0} \bigl[u(x)\bigr]^{p} \sup_{t\geq x}\varphi (t)=\sup_{t\geq0}\varphi(t) \bigl[u^{\uparrow}(t)\bigr]^{p} \\ &\approx\sup_{x\geq0} \bigl[u^{\uparrow}(x) \bigr]^{p}T_{p}h(x), \end{aligned}$$
$$ \sup_{x\geq0} \bigl[u^{\uparrow}(x) \bigr]^{p}\|H_{x}h\|_{L^{\infty}_{(k(\cdot,x)w(\cdot ))^{p}}}\lesssim C_{T,p}^{p} \int_{0}^{\infty}hV_{\ast},\quad h\in \mathfrak{M}^{+}, $$
(3.4)
$$H_{x}h(y):=\chi_{[x,\infty)}(y) \int_{0}^{y}h. $$
$$C_{T,p}^{p}\approx\sup_{x\geq0} \bigl[u^{\uparrow}(x)\bigr]^{p}\|H_{x}\|_{L^{1}_{V_{\ast}}\to L^{\infty}_{(k(\cdot,x)w(\cdot))^{p}}}. $$
$$\|H_{x}\|_{L^{1}_{V_{\ast}}\to L^{\infty}_{(k(\cdot,x)w(\cdot))^{p}}}=\mathop{\operatorname{ess \,sup}}_{y\geq x}\frac{(k(y,x)w(y))^{p}}{V_{\ast}(y)}, $$
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 \(f(y)\leq\frac {\|f\|_{L^{\infty}_{v^{\downarrow}}}}{v^{\downarrow}(y)}\).
$$ \biggl( \int_{0}^{\infty}\Bigl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k(y,x)w(y)f(y) \Bigr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{T,r}\|f\|_{L^{\infty}_{v^{\downarrow}}},\quad f\in \mathfrak{M}^{\uparrow}. $$
(3.5)
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
$$\begin{aligned}& C_{S,p}\approx\sup_{x\geq0}u^{\uparrow}(x) \mathop{\operatorname{ess\,sup}}_{y\geq x}\frac{k(y,x)w(y)}{V^{1/p}(y)}, \end{aligned}$$
(3.6)
$$\begin{aligned}& C_{S,r}= \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x}\frac{k(y,x)w(y)}{v^{\uparrow}(y)} \biggr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}}, \end{aligned}$$
(3.7)
$$\begin{aligned}& C_{S,\infty}=\mathop{\operatorname{ess\,sup}}_{x\geq0} u(x) \biggl[\mathop{\operatorname{ess\,sup}}_{y\geq x}\frac{k(y,x)w(y)}{v^{\uparrow}(y)} \biggr]. \end{aligned}$$
(3.8)
$$\begin{aligned}& C_{\mathscr {T},p}\approx\sup_{x\geq0}u^{\downarrow}(x) \mathop{\operatorname{ess\,sup}}_{0\leq y\leq x}\frac{k(x,y)w(y)}{V^{1/p}(y)}, \end{aligned}$$
(3.9)
$$\begin{aligned}& C_{\mathscr {T},r}= \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x}\frac{k(x,y)w(y)}{v^{\uparrow}(y)} \biggr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}}, \end{aligned}$$
(3.10)
$$\begin{aligned}& C_{\mathscr {T},\infty}=\mathop{\operatorname{ess\,sup}}_{x\geq0} u(x) \biggl[\mathop{\operatorname{ess\,sup}}_{0\leq y\leq x}\frac{k(x,y)w(y)}{v^{\uparrow}(y)} \biggr], \end{aligned}$$
(3.11)
$$\begin{aligned}& C_{\mathscr {S},p}\approx\sup_{x\geq0}u^{\downarrow}(x) \mathop{\operatorname{ess\,sup}}_{0\leq y\leq x}\frac{k(y,x)w(y)}{V_{\ast}^{1/p}(y)}, \end{aligned}$$
(3.12)
$$\begin{aligned}& C_{\mathscr {S},r}= \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x}\frac{k(y,x)w(y)}{v^{\downarrow}(y)} \biggr]^{r}u(x)\,dx \biggr)^{\frac{1}{r}}, \end{aligned}$$
(3.13)
$$\begin{aligned}& C_{\mathscr {S},\infty}=\mathop{\operatorname{ess\,sup}}_{x\geq0} u(x) \biggl[\mathop{\operatorname{ess\,sup}}_{0\leq y\leq x}\frac {k(y,x)w(y)}{v^{\downarrow}(y)} \biggr]. \end{aligned}$$
(3.14)
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]).
$$\begin{aligned}& \sigma(x):= \inf \biggl\{ y>0: \int_{0}^{y}u \geq2 \int_{0}^{x}u \biggr\} , \\& \sigma^{-1}(x):= \inf \biggl\{ y>0: \int_{0}^{y}u \geq\frac{1}{2} \int_{0}^{x}u \biggr\} . \end{aligned}$$
$$\begin{aligned}& H_{c}h(x):=\chi_{[c,\infty)}(x) \int_{0}^{x}h, \\& H_{c,d}h(x):=\chi_{[c,d)}(x) \int_{\sigma^{-1}(c)}^{x}h, \\& H^{*}_{c}h(x):=\chi_{[c,\infty)}(x) \int_{x}^{\infty}h, \\& H^{*}_{c,d}h(x):=\chi_{[c,d)}(x) \int_{x}^{\sigma(d)}h. \end{aligned}$$
Theorem 4.1
Let
\(0< r<\infty\). Then:
(a)
For validity of the inequality
it is necessary and sufficient that the inequality
holds and the constant
is finite. Moreover, \(C_{0}\approx A_{0}+A_{1}\).
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} w_{0}(y) \int _{0}^{y} h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$
(4.1)
$$\biggl( \int_{0}^{\infty}u(x)\bigl[w_{0}^{\downarrow}(x) \bigr]^{r} \biggl( \int_{0}^{x}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq A_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$
$$A_{1}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{1}{r}}\Vert H_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}}} ,&r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{1-r}} \Vert H_{[\sigma^{-1}(x), \sigma(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}}}\,dx )^{\frac {1-r}{r}}, &0< r< 1, \end{cases} $$
(b)
For validity of the inequality
it is necessary and sufficient that the inequality
holds and the constant
is finite. Moreover, \(C_{1}\approx{B}_{0}+{B}_{1}\).
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} w_{0}(y) \int _{y}^{\infty}h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$
(4.2)
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{2}(x)}w_{0}(y)\Bigr]^{r} \biggl( \int_{\sigma^{2}(x)}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac {1}{r}}\leq {B}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$
$${B}_{1}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{1}{r}}\Vert H^{*}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}}} ,&r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{1-r}} \Vert H^{*}_{[\sigma^{-1}(x), \sigma(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}}}\,dx )^{\frac {1-r}{r}}, &0< r< 1, \end{cases} $$
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
$$ C_{T}\approx \mathscr {A}_{0}+ \mathscr {A}_{1},\qquad C_{S}\approx \mathscr {B}_{0}+\mathscr {B}_{1}, $$
(4.3)
$$\begin{aligned}& \mathscr {A}_{0} =\sup_{t>0}\bigl[V_{\ast}(t) \bigr]^{-\frac{1}{p}}\biggl( \int_{t}^{\infty}u\bigl[w^{\downarrow}\bigr]^{r}\biggr)^{\frac{1}{r}},\quad r\geq p, \\& \mathscr {A}_{0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V_{\ast}(x) \bigr]^{-1} \int_{x}^{\infty}u\bigl[w^{\downarrow}\bigr]^{r}\biggr)^{\frac{r}{p-r}}u(x)\bigl[w^{\downarrow}(x) \bigr]^{r}\,dx \biggr)^{\frac{p-r}{pr}}, \quad 0< r< p, \\& \mathscr {A}_{1} =\sup_{t>0}\biggl( \int_{0}^{t} u\biggr)^{\frac{1}{r}}\sup _{y\geq t}\frac{w^{\downarrow}(y)}{[V_{\ast}(y)]^{\frac{1}{p}}},\quad r\geq p, \\& \mathscr {A}_{1} =\biggl( \int_{0}^{\infty}u(x) \biggl( \int_{0}^{x} u\biggr)^{\frac{r}{p-r}} \biggl(\mathop{ \operatorname{ess\,sup}}_{\sigma^{-1}(x)\leq y\leq\sigma(x)}\frac{[w(y)]^{p}}{V_{\ast}(y)} \biggr)^{\frac{r}{p-r}} \,dx\biggr)^{\frac {p-r}{pr}}, \quad 0< r< p, \\& \mathscr {B}_{0} =\sup_{t>0}\bigl[V\bigl( \sigma^{2}(t)\bigr)\bigr]^{-\frac{1}{p}}\biggl( \int_{0}^{t} u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{2}(x)}w(y)\Bigr]^{r}\,dx\biggr)^{\frac{1}{r}},\quad r \geq p, \\& \mathscr {B}_{0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V\bigl( \sigma^{2}(z)\bigr)\bigr]^{-1} \int_{0}^{z} u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{2}(x)}w(y)\Bigr]^{r}\,dx\biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathscr {B}_{0} ={}}{}\times u(z)\Bigl[\mathop{\operatorname{ess\,sup}}_{z\leq y\leq\sigma^{2}(z)}w(y) \Bigr]^{r}\,dz\biggr)^{\frac{p-r}{pr}}, \quad 0< r< p, \\& \mathscr {B}_{1} =\sup_{t>0}\biggl( \int_{0}^{t} u\biggr)^{\frac{1}{r}}\mathop{ \operatorname{ess\,sup}}_{y\geq t}\frac{w(y)}{[V(y)]^{\frac{1}{p}}}, \quad r\geq p, \\& \mathscr {B}_{1} =\biggl( \int_{0}^{\infty}u(x) \biggl( \int_{0}^{x} u\biggr)^{\frac{r}{p-r}} \biggl(\mathop{ \operatorname{ess\,sup}}_{\sigma^{-1}(x)\leq y\leq\sigma(x)}\frac{[w(y)]^{p}}{V(y)} \biggr)^{\frac{r}{p-r}} \,dx\biggr)^{\frac {p-r}{pr}},\quad 0< r< p. \end{aligned}$$
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.
$$ \biggl( \int_{0}^{\infty}u(x)\bigl[w^{\downarrow}(x) \bigr]^{r} \biggl( \int_{0}^{x}h \biggr)^{\frac{r}{p}}\,dx \biggr)^{\frac{p}{r}}\leq \bigl[\mathscr {A}_{0}^{\prime}\bigr]^{p}\|h\|_{L^{1}_{V_{\ast}}},\quad h\in\mathfrak{M}^{+}, $$
(4.4)
$$\bigl[\mathscr {A}_{1}^{\prime}\bigr]^{p}= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{p}{r}}\Vert H_{t}\Vert _{L^{1}_{V_{\ast}} \rightarrow L^{\infty}_{w^{p}}} ,&r\geq p, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{p-r}} \Vert H_{[\sigma^{-1}(x), \sigma(x)]} \Vert ^{\frac {r}{p-r}}_{L^{1}_{V_{\ast}}\rightarrow L^{\infty}_{w^{p}}}\,dx )^{\frac {p-r}{r}}, &0< r< p. \end{cases} $$
$$Kf(x):= \int_{0}^{\infty}k(x,y)f(y)\,dy, $$
$$\begin{aligned} \Vert K\Vert _{L^{1}\to L^{q}}=\mathop{\operatorname{ess \,sup}}_{s\geq0}\bigl\Vert k(\cdot,s)\bigr\Vert _{L^{q}},\quad 1\leq q\leq\infty. \end{aligned}$$
(4.5)
$$ \Vert K\Vert _{L^{1}\to L^{q}}\approx \biggl( \int_{0}^{\infty}\Bigl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} u(y) \Bigr]^{\frac{q}{1-q}} \biggl( \int_{x}^{\infty}w^{q} \biggr)^{\frac {q}{1-q}} \bigl[w(x)\bigr]^{q}\,dx \biggr)^{\frac{1-q}{q}}. $$
(4.6)
$$k(x,y)=w^{p}(x)\chi_{[t,\infty)}(x)\frac{\chi_{[0,x]}(y)}{V_{\ast}(y)} $$
$$\begin{aligned} \begin{aligned} \Vert H_{t}\Vert _{L^{1}_{V_{\ast}} \rightarrow L^{\infty }_{w^{p}}}&=\mathop{\operatorname{ess \,sup}}_{s\geq0}\bigl\Vert k(\cdot,s)\bigr\Vert _{L^{\infty}}= \mathop{\operatorname{ess\,sup}}_{s\geq0} \frac{1}{V_{\ast}(s)}\mathop{ \operatorname{ess\,sup}}_{\{x\geq t\}\cap\{x\geq s\}}w^{p}(x) \\ &=\sup_{s\geq0}\frac{1}{V_{\ast}(s)}\bigl[w^{\downarrow}\bigl( \max(t,s)\bigr)\bigr]^{p} \\ &=\max \biggl(\sup_{0\leq s\leq t}\frac{[w^{\downarrow}(t)]^{p}}{V_{\ast}(s)}, \sup _{ s\geq t}\frac{[w^{\downarrow}(s)]^{p}}{V_{\ast}(s)} \biggr)= \sup_{ s\geq t} \frac{[w^{\downarrow}(s)]^{p}}{V_{\ast}(s)}. \end{aligned} \end{aligned}$$
$$\begin{aligned} \Vert H_{[\sigma^{-1}(x), \sigma(x)]} \Vert ^{\frac {r}{p-r}}_{L^{1}_{V_{\ast}}\rightarrow L^{\infty}_{w^{p}}}&= \mathop{ \operatorname{ess\,sup}}_{s\geq\sigma^{-2}(x)}\frac{1}{V_{\ast}(s)} \mathop{ \operatorname{ess\,sup}}_{\{\sigma^{-1}(x)\leq y\leq\sigma(x)\}\cap\{y\geq s\}}w^{p}(y) \\ &=\sup_{\sigma^{-1}(x)\leq s\leq\sigma(x)}\frac{1}{V_{\ast}(s)}\sup_{\sigma^{-1}(x)\leq y\leq\sigma(x)} \mathop{\operatorname{ess\,sup}}_{s\leq y\leq\sigma(x)}w^{p}(y) \\ &=\mathop{\operatorname{ess\,sup}}_{\sigma^{-1}(x)\leq y\leq\sigma(x)}\frac {w^{p}(y)}{V_{\ast}(y)} \end{aligned}$$
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 \(\mathscr {B}_{1}^{\prime}\approx \mathscr {B}_{1}\). □
$$ \biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{2}(x)} w(y)\Bigr]^{r} \biggl( \int_{\sigma^{2}(x)}^{\infty}h \biggr)^{\frac{r}{p}}\,dx \biggr)^{\frac {p}{r}}\leq \bigl[\mathscr {B}_{0}^{\prime}\bigr]^{p}\|h\|_{L^{1}_{V}},\quad h\in\mathfrak{M}^{+}, $$
(4.7)
$$\bigl[\mathscr {B}_{1}^{\prime}\bigr]^{p}= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{p}{r}}\Vert H_{t}^{\ast} \Vert _{L^{1}_{V} \rightarrow L^{\infty}_{w^{p}}} ,& r\geq p, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{p-r}} \Vert H_{[\sigma^{-1}(x), \sigma(x)]}^{\ast} \Vert ^{\frac {r}{p-r}}_{L^{1}_{V}\rightarrow L^{\infty}_{w^{p}}}\,dx )^{\frac {p-r}{r}}, & 0< r< p. \end{cases} $$
$$ \biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{2}(x)} w(y)\Bigr]^{r} \biggl( \int_{x}^{\infty}h \biggr)^{\frac{r}{p}}\,dx \biggr)^{\frac{p}{r}}\leq \bigl[\mathscr {B}_{0}^{\prime}\bigr]^{p}\|h\|_{L^{1}_{V_{\sigma^{2}}}},\quad h\in\mathfrak{M}^{+}, $$
(4.8)
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.
$$\begin{aligned}& \zeta(x):= \sup \biggl\{ y>0: \int_{y}^{\infty}u \geq\frac{1}{2} \int_{x}^{\infty}u \biggr\} , \\& \zeta^{-1}(x):= \sup \biggl\{ y>0: \int_{y}^{\infty}u \geq2 \int_{x}^{\infty}u \biggr\} . \end{aligned}$$
$$\begin{aligned}& \mathscr {H}_{d}h(x):=\chi_{(0,d]}(x) \int_{x}^{\infty}h, \\& \mathscr {H}_{c,d}h(x):=\chi_{(c,d]}(x) \int_{x}^{\zeta(d)}h, \\& \mathscr {H}^{*}_{c}h(x):=\chi_{(0,d]}(x) \int_{0}^{x}h, \\& \mathscr {H}^{*}_{c,d}h(x):=\chi_{(c,d]}(x) \int_{\zeta^{-1}(c)}^{x}h. \end{aligned}$$
Theorem 4.3
Let
\(0< r<\infty\). Then:
(a)
For validity of the inequality
it is necessary and sufficient that the inequality
holds and the constant
is finite. Moreover, \(C_{2}\approx D_{0}+D_{1}\).
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} w_{0}(y) \int_{y}^{\infty}h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{2}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$
$$\biggl( \int_{0}^{\infty}u(x)\bigl[w_{0}^{\uparrow}(x) \bigr]^{r} \biggl( \int_{x}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq D_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in \mathfrak{M}^{+}, $$
$$D_{1}:= \textstyle\begin{cases} \sup_{t>0} (\int_{t}^{\infty}u )^{\frac{1}{r}}\Vert \mathscr {H}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}}} ,& r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{x}^{\infty}u )^{\frac{r}{1-r}} \Vert \mathscr {H}_{[\zeta^{-1}(x), \zeta(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}}}\,dx )^{\frac {1-r}{r}}, & 0< r< 1, \end{cases} $$
(b)
For validity of the inequality
it is necessary and sufficient that the inequality
holds and the constant
is finite. Moreover, \(C_{3}\approx{E}_{0}+{E}_{1}\).
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} w_{0}(y) \int_{0}^{y} h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq C_{3}\|h\|_{L^{1}_{v_{0}}}, \quad h\in \mathfrak{M}^{+}, $$
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{\zeta^{-2}(x)\leq y\leq x}w_{0}(y)\Bigr]^{r} \biggl( \int_{0}^{\zeta^{-2}(x)}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {E}_{0}\|h\|_{L^{1}_{v_{0}}}, \quad h\in \mathfrak{M}^{+}, $$
$${E}_{1}:= \textstyle\begin{cases} \sup_{t>0} (\int_{t}^{\infty}u )^{\frac{1}{r}}\Vert \mathscr {H}^{*}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}}} ,&r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{x}^{\infty}u )^{\frac{r}{1-r}} \Vert \mathscr {H}^{*}_{[\zeta^{-1}(x), \zeta(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}}}\,dx )^{\frac {1-r}{r}}, &0< r< 1, \end{cases} $$
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
$$ C_{\mathscr {T}}\approx \mathscr {D}_{0}+\mathscr {D}_{1}, \qquad C_{\mathscr {S}}\approx \mathscr {E}_{0}+ \mathscr {E}_{1}, $$
$$\begin{aligned}& \mathscr {D}_{0} =\sup_{t>0}\bigl[V(t) \bigr]^{-\frac{1}{p}}\biggl( \int_{0}^{t} u\bigl[w^{\uparrow}\bigr]^{r}\biggr)^{\frac{1}{r}}, \quad r\geq p, \\& \mathscr {D}_{0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V(x)\bigr]^{-1} \int_{0}^{x} u\bigl[w^{\uparrow}\bigr]^{r}\biggr)^{\frac{r}{p-r}}u(x)\bigl[w^{\uparrow}(x) \bigr]^{r}\,dx\biggr)^{\frac {p-r}{pr}},\quad 0< r< p, \\& \mathscr {D}_{1} =\sup_{t>0}\biggl( \int_{t}^{\infty}u\biggr)^{\frac{1}{r}}\sup _{0< y< t}\frac{w^{\uparrow}(y)}{[V(y)]^{\frac{1}{p}}},\quad r\geq p, \\& \mathscr {D}_{1} =\biggl( \int_{0}^{\infty}u(x) \biggl( \int_{x}^{\infty}u \biggr)^{\frac{r}{p-r}} \biggl( \mathop{\operatorname{ess\,sup}}_{\zeta^{-1}(x)\leq y\leq\zeta(x)}\frac {[w(y)]^{p}}{V(y)} \biggr)^{\frac{r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}},\quad 0< r< p, \\& \mathscr {E}_{0} =\sup_{t>0}\bigl[V_{\ast} \bigl(\zeta^{-2}(t)\bigr)\bigr]^{-\frac{1}{p}} \biggl( \int_{t}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{\zeta^{-2}(x)\leq y\leq x}w(y)\Bigr]^{r}\,dx\biggr)^{\frac{1}{r}},\quad r \geq p, \\& \mathscr {E}_{0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V_{\ast}\bigl( \zeta ^{-2}(z)\bigr)\bigr]^{-1} \int_{z}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{\zeta^{-2}(x)\leq y\leq x}w(y)\Bigr]^{r}\,dx\biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathscr {E}_{0} ={}}{}\times u(z)\Bigl[\mathop{\operatorname{ess\,sup}}_{\zeta^{-2}(z)\leq y\leq z}w(y) \Bigr]^{r}\,dz\biggr)^{\frac{p-r}{pr}},\quad 0< r< p, \\& \mathscr {E}_{1} =\sup_{t>0}\biggl( \int_{t}^{\infty}u\biggr)^{\frac {1}{r}} \mathop{ \operatorname{ess\,sup}}_{0\leq y\leq t}\frac{w(y)}{[V_{\ast}(y)]^{\frac{1}{p}}},\quad r\geq p, \\& \mathscr {E}_{1} =\biggl( \int_{0}^{\infty}u(x) \biggl( \int_{x}^{\infty}u \biggr)^{\frac{r}{p-r}} \biggl( \mathop{\operatorname{ess\,sup}}_{\zeta^{-1}(x)\leq y\leq\zeta(x)}\frac {[w(y)]^{p}}{V_{\ast}(y)} \biggr)^{\frac{r}{p-r}}\,dx\biggr)^{\frac {p-r}{pr}},\quad 0< r< p. \end{aligned}$$
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
it is necessary and sufficient that the inequalities
and
hold and the constant
is finite. Moreover, \({\mathbf{C}}_{0}\approx{\mathbf{A}}_{0}+{\mathbf{A}}_{1}+{\mathbf{A}}_{2}\).
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k_{0}(y,x)w_{0}(y) \int_{0}^{y} h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq{\mathbf{C}}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$
(5.1)
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k_{0}(y,x)w_{0}(y) \Bigr]^{r} \biggl( \int_{0}^{x}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{A}}_{0}\|h\|_{L^{1}_{v_{0}}}, \quad h\in\mathfrak{M}^{+}, $$
$$\biggl( \int_{0}^{\infty}u(x)\bigl[k_{0}\bigl( \sigma^{2}(x),x\bigr)\bigr]^{r} \biggl(\mathop{ \operatorname{ess\,sup}}_{y\geq \sigma^{2}(x)} w_{0}(y) \int_{0}^{y}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{A}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$
$${\mathbf{A}}_{2}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{1}{r}}\Vert H_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\cdot,t)}} ,& r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{1-r}} \Vert H_{[\sigma^{-1}(x), \sigma^{2}(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\cdot,\sigma ^{-1}(x))}}\,dx )^{\frac{1-r}{r}}, &r< 1, \end{cases} $$
(b)
For validity of the inequality
it is necessary and sufficient that the inequalities
and
hold and the constant
is finite. Moreover, \({\mathbf{C}}_{0}\approx{\mathbf{B}}_{0}+{\mathbf{B}}_{1}+{\mathbf{B}}_{2}\).
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k_{0}(y,x)w_{0}(y) \int_{y}^{\infty}h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq{\mathbf{C}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h \in\mathfrak{M}^{+}, $$
(5.2)
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{3}(x)}k_{0}(y,x)w_{0}(y) \Bigr]^{r} \biggl( \int_{\sigma^{3}(x)}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac {1}{r}}\leq {\mathbf{B}}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$
$$\biggl( \int_{0}^{\infty}u(x)\bigl[k_{0}\bigl( \sigma^{2}(x),x\bigr)\bigr]^{r} \biggl(\mathop{ \operatorname{ess\,sup}}_{y\geq \sigma^{2}(x)} w_{0}(y) \int_{y}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{B}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$
$${\mathbf{B}}_{2}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{1}{r}}\Vert H^{*}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\cdot,t)}} ,& r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{1-r}} \Vert H^{*}_{[\sigma^{-1}(x), \sigma^{2}(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\cdot,\sigma ^{-1}(x))}}\,dx )^{\frac{1-r}{r}}, & r< 1, \end{cases} $$
Using Theorem 5.1 we obtain the characterization of (1.2) and (1.3) for \(0< p, r<\infty\). Denote
$$\begin{aligned}& W_{k}(x):=\mathop{\operatorname{ess\,sup}}_{y\geq x} k(y,x)w(y),\qquad \mathscr {W}_{k}(x):=\mathop{\operatorname{ess\,sup}}_{x\leq y\leq\sigma^{3}(x)} k(y,x)w(y), \\& \mathbf{w}_{\sigma}(w) (x):=\mathop{\operatorname{ess \,sup}}_{x\leq y\leq\sigma^{2}(x)} w(y), \\& k_{\sigma}(x):=k\bigl(\sigma^{2}(x),x\bigr),\qquad g_{\sigma^{k}}(y):=g\bigl(\sigma^{k}(y)\bigr), \\& \sigma_{0}(x):= \inf \biggl\{ y>0: \int_{0}^{y}u[k_{\sigma}]^{r} \geq2 \int_{0}^{x}u[k_{\sigma}]^{r} \biggr\} , \\& \sigma_{0}^{-1}(x):= \inf \biggl\{ y>0: \int_{0}^{y}u[k_{\sigma}]^{r} \geq\frac{1}{2} \int_{0}^{x}u[k_{\sigma}]^{r} \biggr\} . \end{aligned}$$
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
$$ C_{T}\approx\mathbb{A}_{0}+ \mathbb{A}_{1,0}+\mathbb{A}_{1,1}+\mathbb{A}_{2}, \qquad C_{S}\approx\mathbb{B}_{0}+\mathbb{B}_{1,0}+ \mathbb{B}_{1,1}+\mathbb{B}_{2}, $$
(5.3)
$$\begin{aligned}& \mathbb{A}_{0} =\sup_{t>0}\bigl[V_{\ast}(t) \bigr]^{-\frac{1}{p}}\biggl( \int_{t}^{\infty}u[W_{k}]^{r} \biggr)^{\frac{1}{r}},\quad r\geq p, \\& \mathbb{A}_{0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V_{\ast}(x) \bigr]^{-1} \int_{x}^{\infty}u[W_{k}]^{r} \biggr)^{\frac{r}{p-r}}u(x)\bigl[W_{k}(x)\bigr]^{r}\,dx \biggr)^{\frac {p-r}{pr}}, \quad 0< r< p, \\& \mathbb{A}_{1,0} =\sup_{t>0}\bigl[(V_{\ast})_{\sigma^{2}}(t) \bigr]^{-\frac{1}{p}} \biggl( \int_{t}^{\infty}u\bigl[k_{\sigma}w_{\sigma^{2}}^{\downarrow}\bigr]^{r}\biggr)^{\frac {1}{r}}, \quad r\geq p, \\& \mathbb{A}_{1,0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[(V_{\ast})_{\sigma ^{2}}(x) \bigr]^{-1} \int_{x}^{\infty}u\bigl[k_{\sigma}w_{\sigma^{2}}^{\downarrow}\bigr]^{r} \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{A}_{1,0} ={}}{}\times u(x)\bigl[k_{\sigma}(x)w_{\sigma^{2}}^{\downarrow}(x) \bigr]^{r}\,dx \biggr)^{\frac{p-r}{pr}}, \quad 0< r< p, \\& \mathbb{A}_{1,1} =\sup_{t>0}\biggl( \int_{0}^{t} u[k_{\sigma}]^{r} \biggr)^{\frac {1}{r}}\mathop{\operatorname{ess\,sup}}_{y\geq t} \frac{w_{\sigma ^{2}}(y)}{[(V_{\ast})_{\sigma^{2}}(y)]^{\frac{1}{p}}},\quad r\geq p, \\& \mathbb{A}_{1,1} =\biggl( \int_{0}^{\infty}u(x)\bigl[k_{\sigma}(x) \bigr]^{r}\biggl( \int_{0}^{x} u[k_{\sigma}]^{r} \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{A}_{1,1} ={}}{}\times \biggl(\mathop{\operatorname{ess\,sup}}_{\sigma_{0}^{-1}(x)\leq y\leq\sigma_{0}(x)} \frac{[w_{\sigma^{2}}(y)]^{p}}{(V_{\ast})_{\sigma^{2}}(y)} \biggr)^{\frac {r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}},\quad 0< r< p, \\& \mathbb{A}_{2} =\sup_{t>0}\biggl( \int_{0}^{t} u\biggr)^{\frac{1}{r}} \mathop{ \operatorname{ess\,sup}}_{y\geq t}\frac{w(y)k(y,t)}{[V_{\ast}(y)]^{\frac {1}{p}}}, \quad r\geq p, \\& \mathbb{A}_{2} =\biggl( \int_{0}^{\infty}u(x) \biggl( \int_{0}^{x} u\biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{A}_{2} ={}}{}\times \biggl(\mathop{\operatorname{ess\,sup}}_{\sigma^{-1}(x)\leq y\leq\sigma^{2}(x)} \frac{[w(y)k(y,\sigma^{-1}(x))]^{p}}{V_{\ast}(y)} \biggr)^{\frac {r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}}, \quad 0< r< p, \\& \mathbb{B}_{0} =\sup_{t>0}\bigl[V_{\sigma^{3}}(t) \bigr]^{-\frac{1}{p}}\biggl( \int_{0}^{t} u[\mathscr {W}_{k}]^{r} \biggr)^{\frac{1}{r}},\quad r\geq p, \\& \mathbb{B}_{0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V_{\sigma^{3}}(z) \bigr]^{-1} \int_{0}^{z} u[\mathscr {W}_{k}]^{r} \biggr)^{\frac{r}{p-r}}u(z)\bigl[\mathscr {W}_{k}(z)\bigr]^{r} \,dz\biggr)^{\frac{p-r}{pr}}, \quad 0< r< p, \\& \mathbb{B}_{1,0} =\sup_{t>0}\bigl[V_{\sigma^{3}} \bigl(\sigma_{0}^{2}(t)\bigr)\bigr]^{-\frac {1}{p}}\biggl( \int_{0}^{t} u\bigl[k_{\sigma}\mathbf{w}_{\sigma_{0}}({w}_{\sigma ^{3}})\bigr]^{r} \biggr)^{\frac{1}{r}},\quad r\geq p, \\& \mathbb{B}_{1,0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V_{\sigma^{2}}\bigl( \sigma _{0}^{2}(t)\bigr) (z)\bigr]^{-1} \int_{0}^{z} u[k_{\sigma}]^{r} \bigl[\mathbf{w}_{\sigma _{0}}({w}_{\sigma^{3}})\bigr]^{r} \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{B}_{1,0} ={}}{}\times u(z)\bigl[k_{\sigma}(z)\mathbf{w}_{\sigma_{0}}({w}_{\sigma ^{3}}) (z)\bigr]^{r}\,dz\biggr)^{\frac{p-r}{pr}}, \quad 0< r< p, \\& \mathbb{B}_{1,1} =\sup_{t>0}\biggl( \int_{0}^{t} u[k_{\sigma}]^{r} \biggr)^{\frac{1}{r}} \mathop{\operatorname{ess\,sup}}_{y\geq t} \frac{{w}_{\sigma^{3}}(y)}{[V_{\sigma ^{3}}(y)]^{\frac{1}{p}}}, \quad r\geq p, \\& \mathbb{B}_{1,1} =\biggl( \int_{0}^{\infty}u(x)\bigl[k_{\sigma}(x) \bigr]^{r}\biggl( \int_{0}^{x} u[k_{\sigma}]^{r} \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{B}_{1,1} ={}}{}\times \biggl(\mathop{\operatorname{ess\,sup}}_{\sigma_{0}^{-1}(x)\leq y\leq\sigma_{0}(x)} \frac{[{w}_{\sigma^{3}}(y)]^{p}}{V_{\sigma^{3}}(y)} \biggr)^{\frac {r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}},\quad 0< r< p, \\& \mathbb{B}_{2} =\sup_{t>0}\biggl( \int_{0}^{t} u\biggr)^{\frac{1}{r}}\mathop{ \operatorname{ess\,sup}}_{y\geq t}\frac{w(y)k(y,t)}{[V(y)]^{\frac{1}{p}}},\quad r\geq p, \\& \mathbb{B}_{2} =\biggl( \int_{0}^{\infty}u(x) \biggl( \int_{0}^{x} u\biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{B}_{2} ={}}{}\times \biggl(\mathop{\operatorname{ess\,sup}}_{\sigma^{-1}(x)\leq y\leq\sigma^{2}(x)} \frac{[w(y)k(y,\sigma^{-1}(x))]^{p}}{V(y)} \biggr)^{\frac {r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}},\quad 0< r< p. \end{aligned}$$
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 \(\mathbb {A}_{1,1}^{\prime}\approx\mathbb{A}_{1,1}\).
$$C_{T}\approx\mathbb{A}_{0}^{\prime}+ \mathbb{A}_{1}^{\prime}+\mathbb{A}_{2}^{\prime}, $$
$$ \begin{aligned} & \biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{y\geq x} k(y,x)w(y)\Bigr]^{r} \biggl( \int_{0}^{x}h \biggr)^{\frac{r}{p}}\,dx \biggr)^{\frac{p}{r}}\leq \bigl[\mathbb{A}_{0}^{\prime}\bigr]^{p}\|h\|_{L^{1}_{V_{\ast}}},\quad h\in\mathfrak{M}^{+}, \\ & \biggl( \int_{0}^{\infty}u(x)\bigl[k\bigl(\sigma^{2}(x),x \bigr)\bigr]^{r} \biggl(\mathop{\operatorname{ess\,sup}}_{y\geq \sigma^{2}(x)} \bigl[w(y)\bigr]^{p} \int_{0}^{y}h \biggr)^{\frac {r}{p}}\,dx \biggr)^{\frac{p}{r}}\leq \bigl[\mathbb{A}_{1}^{\prime}\bigr]^{p}\|h\|_{L^{1}_{V_{\ast}}},\quad h\in\mathfrak{M}^{+}, \end{aligned} $$
(5.4)
$$\bigl[\mathbb{A}_{2}^{\prime}\bigr]^{p}:= \textstyle\begin{cases} \sup_{t>0} (\int_{0}^{t}u )^{\frac{p}{r}}\Vert H_{t}\Vert _{L^{1}_{V_{\ast}} \rightarrow L^{\infty}_{[w(\cdot)k(\cdot,t)]^{p}}} ,& r\geq p, \\ (\int_{0}^{\infty}u(x) (\int_{0}^{x}u )^{\frac{r}{p-r}} \Vert H_{[\sigma^{-1}(x), \sigma^{2}(x)]} \Vert ^{\frac {r}{p-r}}_{L^{1}_{V_{\ast}}\rightarrow L^{\infty}_{[w(\cdot)k(\cdot,\sigma ^{-1}(x))]^{p}}}\,dx )^{\frac{p-r}{r}}, &0< r< p. \end{cases} $$
$$\begin{aligned}& \biggl( \int_{0}^{\infty}u(x)\bigl[k_{\sigma}(x) \bigr]^{r} \biggl(\mathop{\operatorname{ess\,sup}}_{y\geq x} \bigl[w_{\sigma^{2}}(y)\bigr]^{p} \int_{0}^{y}h \biggr)^{\frac {r}{p}}\,dx \biggr)^{\frac{p}{r}} \\& \quad \leq \bigl[\mathbb{A}_{1}^{\prime}\bigr]^{p}\|h\|_{L^{1}_{[V_{\ast}]_{\sigma^{2}}}},\quad h\in\mathfrak{M}^{+}, \end{aligned}$$
(5.5)
$$\mathbb{A}_{1}^{\prime}\approx\mathbb{A}_{1,0}^{\prime}+ \mathbb {A}_{1,1}^{\prime}, $$
$$ \biggl( \int_{0}^{\infty}u(x)\bigl[k_{\sigma}(x) \bigr]^{r}\bigl[w_{\sigma^{2}}^{\downarrow}(x)\bigr]^{r} \biggl( \int_{0}^{x}h \biggr)^{\frac{r}{p}}\,dx \biggr)^{\frac {p}{r}}\leq \bigl[\mathbb{A}_{1,0}^{\prime}\bigr]^{p}\|h\|_{L^{1}_{[V_{\ast}]_{\sigma^{2}}}},\quad h\in \mathfrak{M}^{+}, $$
$$\begin{aligned}& \bigl[\mathbb{A}_{1,1}^{\prime}\bigr]^{p}:=\sup _{t>0} \biggl( \int_{0}^{t}u[k_{\sigma}]^{r} \biggr)^{\frac{p}{r}}\Vert H_{t}\Vert _{L^{1}_{[V_{\ast}]_{\sigma^{2}}} \rightarrow L^{\infty}_{w_{\sigma^{2}}^{p}}},\quad r\geq p, \\& \bigl[\mathbb{A}_{1,1}^{\prime}\bigr]^{p}:= \biggl( \int_{0}^{\infty}u(x)\bigl[k_{\sigma}(x) \bigr]^{r} \biggl( \int_{0}^{x}u[k_{\sigma}]^{r} \biggr)^{\frac{r}{p-r}} \Vert H_{[\sigma_{0}^{-1}(x), \sigma_{0}(x)]} \Vert ^{\frac {r}{p-r}}_{L^{1}_{[V_{\ast}]_{\sigma^{2}}} \rightarrow L^{\infty}_{w_{\sigma ^{2}}^{p}}} \,dx \biggr)^{\frac{p-r}{r}}, \end{aligned}$$
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
it is necessary and sufficient that the inequalities
and
hold and the constant
is finite. Moreover, \({\mathbf{C}}_{0}\approx{\mathbf{A}}_{0}+{\mathbf{A}}_{1}+{\mathbf{A}}_{2}\).
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k_{0}(x,y)w_{0}(y) \int_{y}^{\infty}h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq{\mathbf{C}}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k_{0}(x,y)w_{0}(y) \Bigr]^{r} \biggl( \int_{x}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{A}}_{0}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$
$$\begin{aligned}& \biggl( \int_{0}^{\infty}u(x)\bigl[k_{0}\bigl(x, \zeta^{-2}(x)\bigr)\bigr]^{r} \biggl(\mathop{\operatorname{ess \,sup}}_{0\leq y\leq\zeta^{-2}(x)} w_{0}(y) \int_{y}^{\infty}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}} \\& \quad \leq {\mathbf{A}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, \end{aligned}$$
$${\mathbf{A}}_{2}:= \textstyle\begin{cases} \sup_{t>0} (\int_{t}^{\infty}u )^{\frac{1}{r}}\Vert \mathscr {H}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(t,\cdot)}} ,&r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{x}^{\infty}u )^{\frac{r}{1-r}} \Vert \mathscr {H}_{[\zeta^{-1}(x), \zeta^{2}(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\zeta ^{2}(x),\cdot)}}\,dx )^{\frac{1-r}{r}}, &r< 1, \end{cases} $$
(b)
For validity of the inequality
it is necessary and sufficient that the inequalities
and
hold and the constant
is finite. Moreover, \({\mathbf{C}}_{0}\approx{\mathbf{B}}_{0}+{\mathbf{B}}_{1}+{\mathbf{B}}_{2}\).
$$ \biggl( \int_{0}^{\infty}\biggl[\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k_{0}(y,x)w_{0}(y) \int_{0}^{y} h \biggr]^{r} u(x)\,dx \biggr)^{\frac{1}{r}} \leq{\mathbf{C}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h \in\mathfrak{M}^{+}, $$
$$\biggl( \int_{0}^{\infty}u(x)\Bigl[\mathop{\operatorname{ess \,sup}}_{\zeta^{-3}(x)\leq y\leq x}k_{0}(x,y)w_{0}(y) \Bigr]^{r} \biggl( \int_{0}^{\zeta^{-3}(x)}h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{B}}_{0}\|h\|_{L^{1}_{v_{0}}}, \quad h\in\mathfrak{M}^{+}, $$
$$\biggl( \int_{0}^{\infty}u(x)\bigl[k_{0}\bigl(x, \zeta^{-2}(x)\bigr)\bigr]^{r} \biggl(\mathop{\operatorname{ess \,sup}}_{0\leq y\leq\zeta^{-2}(x)} w_{0}(y) \int_{0}^{y} h \biggr)^{r}\,dx \biggr)^{\frac{1}{r}}\leq {\mathbf{B}}_{1}\|h\|_{L^{1}_{v_{0}}},\quad h\in\mathfrak{M}^{+}, $$
$${\mathbf{B}}_{2}:= \textstyle\begin{cases} \sup_{t>0} (\int_{t}^{\infty}u )^{\frac{1}{r}}\Vert \mathscr {H}^{*}_{t}\Vert _{L^{1}_{v_{0}} \rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(t,\cdot)}} ,& r\geq1, \\ (\int_{0}^{\infty}u(x) (\int_{x}^{\infty}u )^{\frac{r}{1-r}} \Vert \mathscr {H}^{*}_{[\zeta^{-1}(x), \zeta^{2}(x)]} \Vert ^{\frac {r}{1-r}}_{L^{1}_{v_{0}}\rightarrow L^{\infty}_{w_{0}(\cdot)k_{0}(\zeta ^{2}(x),\cdot)}}\,dx )^{\frac{1-r}{r}}, &r< 1, \end{cases} $$
Using Theorem 5.3 we obtain the characterization of (1.4) and (1.5) for \(0< p, r<\infty\). Denote
$$\begin{aligned}& W^{*}_{k}(x):=\mathop{\operatorname{ess \,sup}}_{0\leq y\leq x} k(x,y)w(y), \qquad \mathscr {W}^{*}_{k}(x):= \mathop{\operatorname{ess\,sup}}_{\zeta^{-3}(x)\leq y\leq x} k(x,y)w(y), \\& {\Omega}_{\zeta}(w) (x):=\mathop{\operatorname{ess\,sup}}_{\zeta^{-2}(x)\leq y\leq x} w(y), \\& k_{\zeta}(x):=k\bigl(x,\zeta^{-2}(x)\bigr),\qquad g_{\zeta^{-k}}(y):=g\bigl(\zeta^{-k}(y)\bigr), \\& \zeta_{0}(x):= \sup \biggl\{ y>0: \int_{y}^{\infty}u[k_{\zeta}]^{r} \geq \frac{1}{2} \int_{x}^{\infty}u[k_{\zeta}]^{r} \biggr\} , \\& \zeta_{0}^{-1}(x):= \sup \biggl\{ y>0: \int_{y}^{\infty}u[k_{\zeta}]^{r} \geq2 \int_{x}^{\infty}u[k_{\zeta}]^{r} \biggr\} . \end{aligned}$$
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
$$ C_{\mathscr {T}}\approx\mathbb{D}_{0}+ \mathbb{D}_{1,0}+\mathbb {D}_{1,1}+\mathbb{D}_{2}, \qquad C_{\mathscr {S}}\approx\mathbb{E}_{0}+\mathbb {E}_{1,0}+\mathbb{E}_{1,1}+\mathbb{E}_{2}, $$
(5.6)
$$\begin{aligned}& \mathbb{D}_{0} =\sup_{t>0}\bigl[V(t) \bigr]^{-\frac{1}{p}}\biggl( \int_{0}^{t} u\bigl[W^{*}_{k} \bigr]^{r}\biggr)^{\frac{1}{r}},\quad r\geq p, \\& \mathbb{D}_{0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V(x)\bigr]^{-1} \int_{0}^{x} u\bigl[W^{*}_{k} \bigr]^{r}\biggr)^{\frac{r}{p-r}}u(x)\bigl[W^{*}_{k}(x) \bigr]^{r}\,dx\biggr)^{\frac {p-r}{pr}},\quad 0< r< p, \\& \mathbb{D}_{1,0} =\sup_{t>0}\bigl[V_{\zeta^{-2}}(t) \bigr]^{-\frac{1}{p}}\biggl( \int _{0}^{t} u\bigl[k_{\zeta}w_{\zeta^{-2}}^{\uparrow}\bigr]^{r}\biggr)^{\frac{1}{r}},\quad r\geq p, \\& \mathbb{D}_{1,0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[V_{\zeta^{-2}}(x) \bigr]^{-1} \int _{0}^{x} u\bigl[k_{\zeta}w_{\zeta^{-2}}^{\uparrow}\bigr]^{r}\biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{D}_{1,0} ={}}{}\times u(x)\bigl[k_{\zeta}(x) w_{\zeta^{-2}}^{\uparrow}(x) \bigr]^{r}\,dx\biggr)^{\frac {p-r}{pr}},\quad 0< r< p, \\& \mathbb{D}_{1,1} =\sup_{t>0}\biggl( \int_{t}^{\infty}u[k_{\zeta}]^{r} \biggr)^{\frac{1}{r}} \mathop{\operatorname{ess\,sup}}_{0\leq y\leq t } \frac{w_{\zeta ^{-2}}(y)}{[V_{\zeta^{-2}}(y)]^{\frac{1}{p}}}, \quad r\geq p, \\& \mathbb{D}_{1,1} =\biggl( \int_{0}^{\infty}u(x)\bigl[k_{\zeta}(x) \bigr]^{r}\biggl( \int _{x}^{\infty}u[k_{\zeta}]^{r} \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{D}_{1,1} ={}}{}\times \biggl(\mathop{\operatorname{ess\,sup}}_{\zeta_{0}^{-1}(x)\leq y\leq\zeta_{0}(x)} \frac{[w_{\zeta^{-2}}(y)]^{p}}{V_{\zeta^{-2}}(y)} \biggr)^{\frac {r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}},\quad 0< r< p, \\& \mathbb{D}_{2} =\sup_{t>0}\biggl( \int_{t}^{\infty}u\biggr)^{\frac {1}{r}}\mathop{ \operatorname{ess\,sup}}_{0\leq y\leq t } \frac{w(y)k(t,y)}{[V(y)]^{\frac{1}{p}}}, \quad r\geq p, \\& \mathbb{D}_{2} =\biggl( \int_{0}^{\infty}u(x) \biggl( \int_{x}^{\infty}u \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{D}_{2} ={}}{}\times \biggl(\mathop{\operatorname{ess\,sup}}_{\zeta^{-1}(x)\leq y\leq\zeta^{2}(x)} \frac{[w(y)k(\zeta^{2}(x),y)]^{p}}{V(y)} \biggr)^{\frac{r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}}, \quad 0< r< p, \\& \mathbb{E}_{0} =\sup_{t>0}\bigl[(V_{\ast})_{\zeta^{-3}}(t) \bigr]^{-\frac{1}{p}} \biggl( \int_{t}^{\infty}u\bigl[\mathscr {W}^{*}_{k} \bigr]^{r}\biggr)^{\frac{1}{r}},\quad r\geq p, \\& \mathbb{E}_{0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[(V_{\ast})_{\zeta ^{-3}}(z) \bigr]^{-1} \int_{z}^{\infty}u\bigl[\mathscr {W}^{*}_{k} \bigr]^{r}\biggr)^{\frac {r}{p-r}}u(z)\bigl[\mathscr {W}^{*}_{k}(z) \bigr]^{r}\,dz\biggr)^{\frac{p-r}{pr}},\quad 0< r< p, \\& \mathbb{E}_{1,0} =\sup_{t>0}\bigl[(V_{\ast})_{\zeta^{-2}} \bigl(\zeta _{0}^{-2}(t)\bigr)\bigr]^{-\frac{1}{p}}\biggl( \int_{t}^{\infty}u[k_{\zeta}]^{r} \bigl[{\Omega }_{\zeta_{0}}(w_{\zeta^{-2}})\bigr]^{r} \biggr)^{\frac{1}{r}},\quad r\geq p, \\& \mathbb{E}_{1,0} =\biggl( \int_{0}^{\infty}\biggl(\bigl[(V_{\ast})_{\zeta^{-2}} \bigl(\zeta _{0}^{-2}(t)\bigr) (z)\bigr]^{-1} \int_{z}^{\infty}u[k_{\zeta}]^{r} \bigl[{\Omega}_{\zeta _{0}}(w_{\zeta^{-2}})\bigr]^{r} \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{E}_{1,0} ={}}{}\times u(z)\bigl[k_{\zeta}(z)\bigr]^{r}\bigl[{ \Omega}_{\zeta_{0}}(w_{\zeta ^{-2}})\bigr]^{r}\,dz \biggr)^{\frac{p-r}{pr}},\quad 0< r< p, \\& \mathbb{E}_{1,1} =\sup_{t>0}\biggl( \int_{t}^{\infty}u[k_{\zeta}]^{r} \biggr)^{\frac{1}{r}} \mathop{\operatorname{ess\,sup}}_{0\leq y\leq t } \frac{w_{\zeta ^{-2}}(y)}{[(V_{\ast})_{\zeta^{-2}}(y)]^{\frac{1}{p}}}, \quad r\geq p, \\& \mathbb{E}_{1,1} =\biggl( \int_{0}^{\infty}u(x)\bigl[k_{\zeta}(x) \bigr]^{r}\biggl( \int _{x}^{\infty}u[k_{\zeta}]^{r} \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{E}_{1,1} ={}}{}\times \biggl(\mathop{\operatorname{ess\,sup}}_{\zeta_{0}^{-1}(x)\leq y\leq\zeta_{0}(x)} \frac{[w_{\zeta^{-2}}(y)]^{p}}{(V_{\ast})_{\zeta^{-2}}(y)} \biggr)^{\frac{r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}},\quad 0< r< p, \\& \mathbb{E}_{2} =\sup_{t>0}\biggl( \int_{t}^{\infty}u\biggr)^{\frac{1}{r}} \mathop{ \operatorname{ess\,sup}}_{0\leq y\leq t }\frac {w(y)k(t,y)}{[V_{*}(y)]^{\frac{1}{p}}},\quad r\geq p, \\& \mathbb{E}_{2} =\biggl( \int_{0}^{\infty}u(x) \biggl( \int_{x}^{\infty}u \biggr)^{\frac{r}{p-r}} \\& \hphantom{\mathbb{E}_{2} ={}}{}\times \biggl(\mathop{\operatorname{ess\,sup}}_{\zeta^{-1}(x)\leq y\leq\zeta^{2}(x)} \frac{[w(y)k(\zeta^{2}(x),y)]^{p}}{V_{*}(y)} \biggr)^{\frac {r}{p-r}}\,dx\biggr)^{\frac{p-r}{pr}},\quad 0< r< p. \end{aligned}$$
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).
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.
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.