Criteria for boundedness of a class of integral operators

from $L_{p}$ to $L_{q}$ for $1<q<p<\infty$

R. Oinarova,111Corresponding author<EMAIL_ADDRESS>A. Temirkhanovaa, A.
Kalybayb

aL.N. Gumilyov Eurasian National University, 2 Satpaev Str., Astana 101008,
Kazakhstan

bKIMEP University, 4 Abay Ave., Almaty 050010, Kazakhstan

###### Abstract

In the paper, we consider integral operators with non-negative kernels
satisfying conditions, which are less restrictive than conditions studied
earlier. We establish criteria for the boundedness of these operators in
Lebesgue spaces.

2000 Mathematics Subject Classification. 26D10, 26D15, 47B34.

Key words and phrases. Integral operator, boundedness, weighted inequality,
weight function, kernel.

##  1 Introduction

Let $I=(0,\infty)$, $1<p,q<\infty$ and $p^{\prime}=\frac{p}{p-1}$. Suppose
that $u$ and $v$ are positive functions locally summable on $I$ such that
$u\in L_{q}^{loc}(I)$ and $v\in L_{p^{\prime}}^{loc}(I)$.

We consider the problem of boundedness from $L_{p}(I)$ to $L_{q}(I)$,
$1<q<p<\infty$, of the following integral operators

$\mathcal{K}^{+}f(x)=\int\limits_{0}^{x}u(x)K(x,s)v(s)f(s)ds,\leavevmode\nobreak\
x>0,$ (1.1)
$\mathcal{K}^{-}g(s)=\int\limits_{s}^{\infty}v(s)K(x,s)u(x)g(x)dx,\leavevmode\nobreak\
s>0,$ (1.2)

with a kernel $K(x,s)\geq 0$, $x\geq s>0$, i.e., the validity of the
inequalities

$\|\mathcal{K}^{+}f\|_{q}\leq C\|f\|_{p},\leavevmode\nobreak\ f\in L_{p}(I),$
(1.3) $\|\mathcal{K}^{-}f\|_{q}\leq C\|f\|_{p},\leavevmode\nobreak\ f\in
L_{p}(I),$ (1.4)

where $\|\cdot\|_{p}$ is the standard norm of the space $L_{p}(I)$.

Taking $K(\cdot,\cdot)\equiv 1$ in (1.1) and (1.2), we respectively get the
weighted Hardy operator and the dual weighted Hardy operator, for which
inequalities (1.3) and (1.4) were characterized for all possible parameters
$0<q\leq\infty$ and $1\leq p\leq\infty$ (see, e.g., [1, 6, 14, 15, 16, 17, 24,
25]).

At the end of the last century, research began on Hardy operators with kernels
of forms (1.1) and (1.2). However, since the problem of characterizing
inequalities (1.3) and (1.4) with an arbitrary measurable kernel
$K(\cdot,\cdot)\geq 0$ is open, specific kernels were considered. In papers
[26, 27, 28], necessary and sufficient conditions for the validity of
inequalities (1.3) and (1.4) were found for the kernel
$K(x,s)=(x-s)^{\alpha}$, $\alpha>0$. These results were successfully used to
establish the discreteness of the spectrum of certain higher-order ‘‘polar’’
operators (see [26]).

The problem arose of selecting a class of kernels, which would cover the
kernels of known operators. At the beginning of the 1990s, in the works [5]
and [18], a class $\mathcal{O}$ was introduced. The class $\mathcal{O}$
contains many kernels of different operators and, in particular, kernels of
fractional integration operators. A measurable function $K(x,s)\geq 0$ belongs
to the class $\mathcal{O}$ if there exists a number $h\geq 1$ and
$h^{-1}\big{(}K(x,t)+K(t,s)\big{)}\leq K(x,s)\leq
h\big{(}K(x,t)+K(t,s)\big{)}$ for $x\geq t\geq s$. Criteria for the
boundedness of operators (1.1) and (1.2) with a kernel from the class
$\mathcal{O}$ have good applications in various problems of analysis, for
example, in the oscillatory and spectral theory of higher order differential
operators (see, e.g., [4, 12, 13]). In recent years, operators (1.1) and (1.2)
with a kernel from the class $\mathcal{O}$ have become the object of many
works (see, e.g., [7, 10, 11, 22, 29, 31]). In the last decade, many papers
have discussed the weighted estimates of iterated operators, where iterations
are taken with respect to different combinations of the Hardy operator, the
dual Hardy operator, operators (1.1) and (1.2) with kernels from the class
$\mathcal{O}$ (see, e.g., [8, 9, 10, 11, 22, 29, 30]). However, the kernels of
iterations of operators (1.1) and (1.2) with a kernel from the class
$\mathcal{O}$ do not belong to the class $\mathcal{O}$.

In the paper [19], the classes $\mathcal{O}_{n}^{+}$ and
$\mathcal{O}_{n}^{-}$, $n\geq 0$, of functions $K(x,s)\geq 0$, $x\geq s>0$,
were introduced. These classes are wider than the class $\mathcal{O}$.
Necessary and sufficient conditions of boundedness of operators (1.1) and
(1.2) from $L_{p}(I)$ to $L_{q}(I)$ were established for $1<p\leq q<\infty$
when their kernels $K(\cdot,\cdot)$ belong to the class
$\mathcal{O}_{n}^{+}\bigcup\mathcal{O}_{n}^{-}$, $n\geq 1$. These classes form
extension systems $\mathcal{O}_{m}^{\pm}\subset\mathcal{O}_{n}^{\pm}$, $n\geq
m\geq 0$. The work [2] shows that the iteration kernel of two operators (1.1)
and (1.2) from the classes $\mathcal{O}_{n}^{\pm}$ and $\mathcal{O}_{m}^{\pm}$
belong to the class $\mathcal{O}_{n+m+1}^{\pm}$, i.e., the class system
$\\{\mathcal{O}_{n}^{\pm},\leavevmode\nobreak\ n\geq 0\\}$ is closed with
respect to iteration. This property makes it possible to study various
problems where iterations of integral operators are involved, in particular,
the above problem of weight estimates for quasilinear operators with iteration
of operators (1.1) and (1.2).

The boundedness from $L_{p}(I)$ to $L_{q}(I)$ of operators (1.1) and (1.2)
with kernels from $\mathcal{O}_{n}^{+}$ or $\mathcal{O}_{n}^{-}$, $n>1$, has
not been established for $1<q<p<\infty$. In the paper [3], under some
assumptions on kernels criteria of boundedness of operators (1.1) and (1.2)
from $L_{p}(I)$ to $L_{q}(I)$ for $1<q<p<\infty$ were obtained when their
kernels belong to the classes $\mathcal{O}_{1}^{+}$ or $\mathcal{O}_{1}^{-}$.

The main aim of this paper is to establish criteria for the boundedness of
operators (1.1) and (1.2) from $L_{p}(I)$ to $L_{q}(I)$ for $1<q<p<\infty$,
when their kernels belong to the classes $O_{n}^{\pm}$, $n\geq 2$. To achieve
this aim, we need to find the validity of inequalities (1.3) and (1.4) for
$1<q<p<\infty$ for more general operators than operators (1.1) and (1.2), when
their kernels belong to the classes $\mathcal{O}_{n}^{\pm}$. The specific form
of these operators and in which spaces they are considered will be given
later. Now, let us define the classes $\mathcal{O}_{n}^{\pm}$, $n\geq 0$. Let
$\Omega=\\{(x,s)\in I\times I:x\geq s>0\\}$.

######  Definition 1.1.

A measurable function $K(\cdot,\cdot)\equiv K_{0}(\cdot,\cdot)\geq 0$ defined
on the set $\Omega$ belongs to the class $\mathcal{O}_{0}^{+}$
($\mathcal{O}_{0}^{-}$) if $K_{0}(x,s)\equiv r(s)\geq 0$
($K_{0}(x,s)\equiv\bar{r}(x)\geq 0$).

######  Definition 1.2.

A measurable function $K(\cdot,\cdot)\equiv K_{n}(\cdot,\cdot)\geq 0$ defined
on the set $\Omega$ belongs to the class $\mathcal{O}_{n}^{+}$, $n\geq 1$, if
the function $K(\cdot,\cdot)\geq 0$ does not decrease in the first argument
and there exists a non-negative functions $K_{n,i}(\cdot,\cdot)$,
$i=0,1,...,n-1$, and $K_{i}(\cdot,\cdot)$, $i=0,1,...,n-1$, measurable on
$\Omega$ and a number $h_{n}\geq 1$ such that
$K_{i}(\cdot,\cdot)\in\mathcal{O}_{i}^{+}$, $i=0,1,...,n-1$, and

$h_{n}^{-1}\sum\limits_{i=0}^{n}K_{n,i}(x,t)K_{i}(t,s)\leq K_{n}(x,s)\leq
h_{n}\sum\limits_{i=0}^{n}K_{n,i}(x,t)K_{i}(t,s)$ (1.5)

for all $x,t,s:x\geq t\geq s>0$, where $K_{n,n}(\cdot,\cdot)\equiv 1$, $n\geq
1$.

######  Definition 1.3.

A measurable function $K(\cdot,\cdot)\equiv K_{n}(\cdot,\cdot)\geq 0$ defined
on the set $\Omega$ belongs to the class $\mathcal{O}_{n}^{-}$, $n\geq 1$, if
the function $K(\cdot,\cdot)\geq 0$ does not increase in the second argument
and there exists a non-negative functions $K_{i,n}(\cdot,\cdot)$,
$i=0,1,...,n-1$, and $K_{i}(\cdot,\cdot)$, $i=0,1,...,n-1$, measurable on
$\Omega$ and a number $\bar{h}_{n}\geq 1$ such that
$K_{i}(\cdot,\cdot)\in\mathcal{O}_{i}^{-}$, $i=0,1,...,n-1$, and

$\bar{h}_{n}^{-1}\sum\limits_{i=0}^{n}K_{i}(x,t)K_{i,n}(t,s)\leq
K_{n}(x,s)\leq\bar{h}_{n}\sum\limits_{i=0}^{n}K_{i}(x,t)K_{i,n}(t,s)$ (1.6)

for all $x,t,s:x\geq t\geq s>0$, where $K_{n,n}(\cdot,\cdot)\equiv 1$, $n\geq
1$.

From (1.5) for $n=1$, i.e., when $K(\cdot,\cdot)\equiv
K_{1}(\cdot,\cdot)\in\mathcal{O}^{+}_{1}$, we have

$h_{1}^{-1}\left(K_{1,0}(x,t)r_{1}(s)+K_{1}(t,s)\right)\leq K_{1}(x,s)\leq
h_{1}\left(K_{1,0}(x,t)r_{1}(s)+K_{1}(t,s)\right)$ (1.7)

for all $x,t,s:x\geq t\geq s>0$.

From (1.6) for $n=1$, i.e., when $K(\cdot,\cdot)\equiv
K_{1}(\cdot,\cdot)\in\mathcal{O}^{-}_{1}$, we have

$\bar{h}_{1}^{-1}\left(K_{1}(x,t)+\bar{r}_{1}(x)K_{0,1}(t,s)\right)\leq
K_{1}(x,s)\leq\bar{h}_{1}\left(K_{1}(x,t)+\bar{r}_{1}(x)K_{0,1}(t,s)\right)$

for all $x,t,s:x\geq t\geq s>0$.

We can similarly write the analogues of relations (1.7) and (1.6) for $n=2$.

######  Remark 1.1.

There are many examples of kernels $K(\cdot,\cdot)\geq 0$ that belong to the
classes $\mathcal{O}_{n}^{\pm}$, $n\geq 1$, some of them are given in [19] and
[20]. Moreover, in [2], it is proved that the kernels of the superpositions of
two operators (1.1) and (1.2) with the kernels
$K(\cdot,\cdot)\in\mathcal{O}_{n}^{\pm}$ and
$K(\cdot,\cdot)\in\mathcal{O}_{m}^{\pm}$ belong to the class
$\mathcal{O}_{n+m+1}^{\pm}$.

######  Remark 1.2.

As shown in [19], without loss of generality, the functions
$K_{n,i}(\cdot,\cdot)$ and $K_{i,n}(\cdot,\cdot)$, $i=0,1,...,n-1$, $n\geq 1$,
can be considered as non-decreasing in the first argument and non-increasing
in the second argument. It is also shown that for any $n,j,i:n\geq j\geq i\geq
0$ we have that $K_{n,j}(x,t)K_{j,i}(t,s)\leq K_{n,i}(x,s)$ and
$K_{i,j}(x,t)K_{j,n}(t,s)\leq K_{i,n}(x,s)$ hold for $x\geq t\geq s>0$.

######  Remark 1.3.

Let $\varphi$ be a non-decreasing function on $I$. Without loss of generality,
we will assume that the function $\varphi$ is right-continuous, replacing
$\varphi(t)$ by $\varphi(t+0)$ if necessary. Then it is known (see [23,
Chapter 12]) that there exists a Borel measure $\eta$ such that
$\varphi(t)=\int\limits_{[0,t]}d\eta(x)$. If $\varphi(0)=0$, then
$\varphi(t)=\int\limits_{(0,t]}d\eta(x)$.

######  Remark 1.4.

Let $\psi$ be a non-increasing function on $I$. Without loss of generality, we
will assume that the function $\psi$ is left-continuous, replacing $\psi(t)$
by $\psi(t-0)$ if necessary. Then it is known (see [23, Chapter 12]) that
there exists a Borel measure $\theta$ such that
$\psi(t)=\int\limits_{[t,\infty]}d\theta(x)$. If
$\lim\limits_{t\to\infty}\psi(t)=\psi(\infty)=0$, then
$\psi(t)=\int\limits_{[t,\infty)}d\theta(x)$.

Below, in the statements of Theorems, instead of $d\eta(\cdot)$ and
$d\theta(\cdot)$, we will respectively write $d\varphi(\cdot)$ and
$d(-\psi(\cdot))$, which indicates that the measures are generated by the
functions $\varphi$ and $\psi$.

The paper is organized as follows: In Section 2, we establish the boundedness
of operators (1.1) and (1.2) from $L_{p}(I)$ to $L_{q}(I)$ in the case
$1<q<p<\infty$ when their kernels belong to the classes
$\mathcal{O}_{1}^{\pm}$. In Section 3, we obtain similar results for operators
(1.1) and (1.2) when their kernels belong to the classes
$\mathcal{O}_{n}^{\pm}$, $n\geq 2$.

Throughout the paper, the symbol $A\ll B$ means that $A\leq CB$ with some
constant $C>0$. The symbol $A\approx B$ means that $A\ll B\ll A$. Moreover,
$\chi_{(a,b)}$ stands for the characteristic function of the interval
$(a,b)\subset\mathbb{R}$ and $\mathbb{Z}$ is the set of integers.

##  2 Integral operators with kernels from the class $\mathcal{O}_{1}^{\pm}$

Let $\Psi$ be the set of nonnegative Borel measures on $I$. Suppose that
$\mu\in\Psi$. We consider the integral operators

$\mathbf{K^{+}}f(x)=\int\limits_{0}^{x}K(x,s)v(s)f(s)ds,\leavevmode\nobreak\
x>0,$ (2.1)
$\mathbf{K^{-}}g(s)=\int\limits_{[s,\infty]}K(x,s)v(s)g(x)d\mu(x),\leavevmode\nobreak\
s>0.$ (2.2)

Let $L_{q,\mu}(I)$ be a space of $\mu$-measurable functions on $I$, for which

$\|g\|_{q,\mu}=\left(\int\limits_{[0,\infty]}|g(x)|^{q}d\mu(x)\right)^{\frac{1}{q}}<\infty.$

Operator (2.1) acts from $L_{p}(I)$ to $L_{q,\mu}(I)$, and operator (2.2) acts
from $L_{p,\mu}(I)$ to $L_{q}(I)$. It is easy to see that operator (2.2) is
dual to operator (2.1) from $L_{q^{\prime},\mu}(I)$ to $L_{p^{\prime}}(I)$,
and (2.1) is dual to operator (2.2) from $L_{q^{\prime}}(I)$ to
$L_{p^{\prime},\mu}(I)$ with respect to the linear form
$\int\limits_{0}^{\infty}f(x)g(x)d\mu(x)$.

######  Theorem 2.1.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{1}(\cdot,\cdot)\in\mathcal{O}_{1}^{+}$. Then operator (2.1) is bounded from
$L_{p}(I)$ to $L_{q,\mu}(I)$ if and only if
$B^{+}_{1}=\max\\{B^{+}_{1,0},B^{+}_{1,1}\\}<\infty$. Moreover,
$\|\mathbf{K}^{+}\|_{p\rightarrow q}\approx B^{+}_{1}$, where
$\|\mathbf{K}^{+}\|_{p\rightarrow q}$ is the norm of operator (2.1) from
$L_{p}(I)$ to $L_{q,\mu}(I)$, where

$B^{+}_{1,0}=\left(\int\limits_{0}^{\infty}\left(\int\limits_{[z,\infty]}K_{1,0}^{q}(x,z)d\mu(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}r_{1}^{p^{\prime}}(s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}r_{1}^{p^{\prime}}(z)v^{p^{\prime}}(z)dz\right)^{\frac{p-q}{pq}},$
$B^{+}_{1,1}=\left(\int\limits_{(0,\infty]}\left(\mu\left([x,\infty]\right)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}d\left(\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,t)v^{p^{\prime}}(t)dt\right)\right)^{\frac{p-q}{pq}}.$

###### Proof.

Necessity. Let operator (2.1) be bounded from $L_{p}(I)$ to $L_{q,\mu}(I)$,
$1<q<p<\infty$, with the norm $\|\mathbf{K}^{+}\|_{p\rightarrow q}<\infty$.
Let $0\leq f\in L_{p}(I)$ have compact support. Using the relation
$K_{1}(x,t)\gg K_{1,0}(x,s)r_{1}(t)$ for $x\geq s\geq t>0$, which follows from
(1.7), by Fubini’s theorem, we have

$J=\int\limits_{[0,\infty]}\left(\mathbf{K}^{+}f(x)\right)^{q}d\mu(x)\approx\int\limits_{[0,\infty]}\int\limits_{0}^{x}K_{1}(x,s)v(s)f(s)\left(\int\limits_{0}^{s}K_{1}(x,t)v(t)f(t)dt\right)^{q-1}dsd\mu(x)$
$\gg\int\limits_{[0,\infty]}\int\limits_{0}^{x}K_{1,0}^{q}(x,s)r_{1}(s)v(s)f(s)\left(\int\limits_{0}^{s}r_{1}(t)v(t)f(t)dt\right)^{q-1}dsd\mu(x)$
$=\int\limits_{0}^{\infty}r_{1}(s)v(s)f(s)\left(\int\limits_{0}^{s}r_{1}(t)v(t)f(t)dt\right)^{q-1}\int\limits_{[s,\infty]}K_{1,0}^{q}(x,s)d\mu(x)ds.$
(2.3)

Since the function $\int\limits_{[s,\infty]}K_{1,0}^{q}(x,s)d\mu(x)$ does not
increase on $I$, then, by Remark 1.4, there exists a Borel measure
$\theta_{1}$ such that
$\int\limits_{[s,\infty]}K_{1,0}^{q}(x,s)d\mu(x)=\int\limits_{[s,\infty]}d\theta_{1}(t)$.
Replacing this relation into (2.3) and using Fubini’s theorem, we deduce

$J=\int\limits_{0}^{\infty}r_{1}(s)v(s)f(s)\left(\int\limits_{0}^{s}r_{1}(t)v(t)f(t)dt\right)^{q-1}\int\limits_{[s,\infty]}d\theta_{1}(z)ds$
$=\int\limits_{[0,\infty]}\int\limits_{0}^{z}r_{1}(s)v(s)f(s)\left(\int\limits_{0}^{s}r_{1}(t)v(t)f(t)dt\right)^{q-1}ds\,d\theta_{1}(z)$
$\approx\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}r_{1}(t)v(t)f(t)dt\right)^{q}d\theta_{1}(z).$

Then

$\left(\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}r_{1}(t)v(t)f(t)dt\right)^{q}d\theta_{1}(z)\right)^{\frac{1}{q}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q}\left(\int\limits_{0}^{\infty}f^{p}(t)dt\right)^{\frac{1}{p}}.$

Hence, by Theorem 2 from [16, §1.3, 1.3.3], we get

$B^{+}_{1,0}\ll\|\mathbf{K^{+}}\|_{p\rightarrow q}.$ (2.4)

From the boundedness of operator (2.1) from $L_{p}(I)$ to $L_{q,\mu}(I)$ it
follows the boundedness of dual operator (2.2) from $L_{q^{\prime},\mu}(I)$ to
$L_{p^{\prime}}(I)$, i.e., we have

$\left(\int\limits_{0}^{\infty}\left(\int\limits_{[s,\infty]}K_{1}(x,s)v(s)g(x)d\mu(x)\right)^{p^{\prime}}ds\right)^{\frac{1}{p^{\prime}}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q}\left(\int\limits_{[0,\infty]}|g(t)|^{q^{\prime}}d\mu(t)\right)^{\frac{1}{q^{\prime}}}$
(2.5)

for $0\leq g\in L_{q^{\prime},\mu}(I)$ having compact support. The non-
decrease in the first argument of the function
$K_{1}(\cdot,\cdot)\in\mathcal{O}^{+}_{1}$ gives that $K_{1}(t,s)\geq
K_{1}(x,s)$ for $t\geq x\geq s>0$. This fact and Lemma 2 from [21] imply that

$J^{-}=\int\limits_{0}^{\infty}(\mathbf{K}^{-}g(s))^{p^{\prime}}ds\approx\int\limits_{0}^{\infty}\int\limits_{[s,\infty]}K_{1}(x,s)v^{p^{\prime}}(s)g(x)\left(\int\limits_{[x,\infty]}K_{1}(t,s)g(t)d\mu(t)\right)^{p^{\prime}-1}d\mu(x)ds$
$\geq\int\limits_{0}^{\infty}v^{p^{\prime}}(s)\int\limits_{[s,\infty]}K_{1}^{p^{\prime}}(x,s)g(x)\left(\int\limits_{[x,\infty]}g(t)d\mu(t)\right)^{p^{\prime}-1}d\mu(x)ds$
$=\int\limits_{[0,\infty]}g(x)\left(\int\limits_{[x,\infty]}g(t)d\mu(t)\right)^{p^{\prime}-1}\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\,d\mu(x).$
(2.6)

The function $\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds$
does not decrease on $I$ and $\lim\limits_{x\to
0}\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds=0$ due to
(2.6). Therefore, by Remark 1.3, there exists a Borel measure $\eta_{1}$ such
that
$\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds=\int\limits_{(0,x]}d\eta_{1}(t)$.
Replacing this relation into (2.6) and using Fubini’s theorem, we obtain

$J^{-}=\int\limits_{[0,\infty]}g(x)\left(\int\limits_{[x,\infty]}g(t)d\mu(t)\right)^{p^{\prime}-1}\int\limits_{(0,x]}d\eta_{1}(z)d\mu(x)\approx\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}g(t)d\mu(t)\right)^{p^{\prime}}d\eta_{1}(z).$

The latter and (2.5) give that

$\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}g(t)d\mu(t)\right)^{p^{\prime}}d\eta_{1}(z)\right)^{\frac{1}{p^{\prime}}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q}\left(\int\limits_{[0,\infty]}|g(t)|^{q^{\prime}}d\mu(t)\right)^{\frac{1}{q^{\prime}}}.$

Then, on the basis of Theorem 3 from [21], we have
$B^{+}_{1,1}\ll\|\mathbf{K^{+}}\|_{p\rightarrow q}$, which, together with
(2.4), implies that

$\|\mathbf{K^{+}}\|_{p\rightarrow q}\gg B^{+}_{1}.$ (2.7)

Sufficiency. Let $B^{+}_{1}<\infty$ and $K(\cdot,\cdot)\equiv
K_{1}(\cdot,\cdot)\in\mathcal{O}_{1}^{+}$. Let $0\leq f\in L_{p}(I)$ have
compact support. Hence, the function
$F(x)=\int\limits_{0}^{x}K_{1}(x,s)v(s)f(s)ds$ is non-negative, non-decreasing
and turns to zero at $x\rightarrow 0^{+}$. Let
$\alpha_{f}=\inf\\{x>0:F(x)>0\\}$. Then $F(x)>0$ for $x>\alpha_{f}$. Then,
without loss of generality, we assume that $\alpha_{f}=0$.

We define the function (see [19])

$k(x)=\max\\{k\in\mathbb{Z}:(h_{1}+1)^{k}\leq F(x),x\in I\\},$

where $h_{1}$ from (1.7). The function $k(x)$ is non-decreasing and

$(h_{1}+1)^{k(x)}\leq F(x)\leq(h_{1}+1)^{k(x)+1},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in I.$

Let $\\{k_{i}\\}$, $(k_{i}<k_{i+1})$, be the domain of the function
$k:I\rightarrow\mathbb{Z}$ and $I_{i}$ is the preimage of the point
$k_{i}\in\mathbb{Z}$, i.e., $I_{i}=\\{x\in I:k(x)=k_{i}\\}$. Then

$I=\bigcup\limits_{i}I_{i}.$ (2.8)

Assume that $x_{i}=\inf I_{i}$. It is obvious that $x_{i}<x_{i+1}$ and
$I_{i}=[x_{i},x_{i+1})$. Therefore,

$(h_{1}+1)^{k_{i}}\leq F(x)<(h_{1}+1)^{k_{i}+1}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
x_{i}\leq x<x_{i+1}.$ (2.9)

Since $k_{i}<k_{i+1}$, then $k_{i}-1\geq k_{i-1}\geq k_{i-2}+1$ and

$(h_{1}+1)^{k_{i}-1}=(h_{1}+1)^{k_{i}}-h_{1}(h_{1}+1)^{k_{i}-1}\leq(h_{1}+1)^{k_{i}}-h_{1}(h_{1}+1)^{k_{i-2}+1}\leq
F(x_{i})-h_{1}F(x_{i-2})$
$\leq\int\limits_{x_{i-2}}^{x_{i}}K_{1}(x_{i},s)v(s)f(s)ds+\int\limits_{0}^{x_{i-2}}[K_{1}(x_{i},s)-h_{1}K_{1}(x_{i-2},s)]v(s)f(s)ds$
$\leq\int\limits_{x_{i-2}}^{x_{i}}K_{1}(x_{i},s)v(s)f(s)ds+h_{1}K_{1,0}(x_{i},x_{i-2})\int\limits_{0}^{x_{i-2}}r_{1}(s)v(s)f(s)ds.$
(2.10)

In the last relation, we have applied the right-hand side of (1.7) to
$K_{1}(x_{i},s)-h_{1}K_{1}(x_{i-2},s)$. In view of (2.8), (2.9) and (2.10),
assuming that $\bar{I}_{i}=[x_{i},x_{i+1}]$, we get

$J=\int\limits_{[0,\infty]}(\mathbf{K^{+}}f(x))^{q}d\mu(x)\ll\sum\limits_{i}\int\limits_{\bar{I}_{i}}F^{q}(x)d\mu(x)\ll\sum\limits_{i}(h_{i}+1)^{q(k_{i}-1)}\mu(\bar{I}_{i})$
$\ll\sum\limits_{i}\mu(\bar{I}_{i})\left(\int\limits_{x_{i-2}}^{x_{i}}K_{1}(x_{i},s)v(s)f(s)ds\right)^{q}+\sum\limits_{i}\mu(\bar{I}_{i})K_{1,0}^{q}(x_{i},x_{i-2})\left(\int\limits_{0}^{x_{i-2}}r_{1}(s)v(s)f(s)ds\right)^{q}$
$\leq\sum\limits_{i}\mu(\bar{I}_{i})\left(\int\limits_{x_{i-2}}^{x_{i}}K_{1}(x_{i},s)v(s)f(s)ds\right)^{q}+\sum\limits_{i}\int\limits_{\bar{I}_{i}}K_{1,0}^{q}(x,x_{i-2})d\mu(x)\left(\int\limits_{0}^{x_{i-2}}r_{1}(s)v(s)f(s)ds\right)^{q}$
$=J_{1,1}+J_{1,0}.$ (2.11)

We separately estimate $J_{1,0}$ and $J_{1,1}$. To estimate $J_{1,1}$ we use
Hölder’s inequality in the integral, then in the sum with the parameters
$\frac{p}{q}$ and $\frac{p}{p-q}$:

$J_{1,1}\leq\sum\limits_{i}\mu(\bar{I}_{i})\left(\int\limits_{x_{i-2}}^{x_{i}}K_{1}^{p^{\prime}}(x_{i},s)v^{p^{\prime}}(s)ds\right)^{\frac{q}{p^{\prime}}}\left(\int\limits_{x_{i-2}}^{x_{i}}|f(t)|^{p}dt\right)^{\frac{q}{p}}$
$\leq\left(\sum\limits_{i}\mu^{\frac{p}{p-q}}(\bar{I}_{i})\left(\int\limits_{x_{i-2}}^{x_{i}}K_{1}^{p^{\prime}}(x_{i},s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\right)^{\frac{p-q}{p}}\left(\sum\limits_{i}\int\limits_{x_{i-2}}^{x_{i}}|f(t)|^{p}dt\right)^{\frac{q}{p}}\ll\hat{B}_{1,1}\|f\|_{p}^{q},$
(2.12)

where

$\hat{B}_{1,1}=\left(\sum\limits_{i}\mu^{\frac{p}{p-q}}(\bar{I}_{i})\left(\int\limits_{x_{i-2}}^{x_{i}}K_{1}^{p^{\prime}}(x_{i},s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\right)^{\frac{p-q}{p}}.$

Using Lemma 2 from [21] and the relation
$\mu^{\frac{p}{p-q}}(\bar{I}_{i})\approx\int\limits_{[x_{i},x_{i+1}]}\mu^{\frac{q}{p-q}}([x,x_{i+1}])d\mu(x)$,
then using as above the relation
$\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds=\int\limits_{(0,x]}d\eta_{1}(z)$
and Fubini’s theorem, we obtain

$\hat{B}_{1,1}\ll\left(\sum\limits_{i}\int\limits_{[x_{i},x_{i+1}]}\mu^{\frac{q}{p-q}}([x,\infty])\left(\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}d\mu(x)\right)^{\frac{p-q}{p}}$
$\ll\left(\int\limits_{[0,\infty]}(\mu([x,\infty]))^{\frac{q}{p-q}}\left(\int\limits_{(0,x]}d\eta_{1}(z)\right)^{\frac{q(p-1)}{p-q}}d\mu(x)\right)^{\frac{p-q}{p}}$
$\approx\left(\int\limits_{[0,\infty]}(\mu([x,\infty]))^{\frac{q}{p-q}}\int\limits_{(0,x]}\left(\int\limits_{(0,s]}d\eta_{1}(z)\right)^{\frac{p(q-1)}{p-q}}d\eta_{1}(s)d\mu(x)\right)^{\frac{p-q}{p}}$
$\approx\left(\int\limits_{(0,\infty]}(\mu([s,\infty]))^{\frac{p}{p-q}}\left(\int\limits_{(0,s]}d\eta_{1}(z)\right)^{\frac{p(q-1)}{p-q}}d\eta_{1}(s)\right)^{\frac{p-q}{p}}$
$=\left(\int\limits_{(0,\infty]}(\mu([x,\infty]))^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}d\left(\int\limits_{0}^{x}K_{1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{p}}=(B^{+}_{1,1})^{q}.$

The latter and (2.12) give that

$J_{1,1}\ll(B^{+}_{1,1})^{q}\|f\|_{p}^{q}.$ (2.13)

Now, we estimate $J_{1,0}$:

$J_{1,0}=\sum\limits_{i}\int\limits_{\bar{I}_{i}}K_{1,0}^{q}(x,x_{i-2})d\mu(x)\left(\int\limits_{0}^{x_{i-2}}r_{1}(s)v(s)f(s)ds\right)^{q}=\int\limits_{[0,\infty]}\left(\int\limits_{0}^{t}r_{1}(s)v(s)f(s)ds\right)d\eta(t),$

where
$d\eta(t)=\sum\limits_{i}\int\limits_{\bar{I}_{i}}K_{1,0}^{q}(x,x_{i-2})d\mu(x)\delta(t-x_{i-2})dt$
and $\delta(\cdot)$ is the Dirac delta function. Then, by Theorem 2 from [16,
§1.3, 1.3.3], we have

$J_{1,0}\ll\hat{B}_{1,0}\left(\int\limits_{0}^{\infty}|f(t)|^{p}dt\right)^{\frac{q}{p}},$

where

$\hat{B}_{1,0}=\left(\int\limits_{0}^{\infty}(\eta([x,\infty]))^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}r_{1}^{p^{\prime}}(s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}r_{1}^{p^{\prime}}(x)v^{p^{\prime}}(x)dx\right)^{\frac{p-q}{p}}.$

Since $\eta([x,\infty])=\sum\limits_{i:x_{i-2}\geq
x}\int\limits_{\bar{I}_{i}}K^{q}_{1,0}(t,x_{i-2})d\mu(t)\ll\int\limits_{[x,\infty]}K_{1,0}^{q}(t,x)d\mu(t)$,
then $\hat{B}_{1,0}\ll(B^{+}_{1,0})^{q}$. Therefore,
$J_{1,0}\ll(B^{+}_{1,0})^{q}\|f\|_{p}^{q}$. The last estimate, together with
(2.11) and (2.13), gives that
$J\ll\max\\{B^{+}_{1,1},B^{+}_{1,0}\\}^{q}\|f\|_{p}^{q}$, i.e., operator (2.1)
is bounded from $L_{p}(I)$ to $L_{q,\mu}(I)$ and for its norm the estimate
$\|\mathbf{K^{+}}\|_{p\rightarrow q}\ll B^{+}_{1}$ holds, which and (2.7) give
that $\|\mathbf{K^{+}}\|_{p\rightarrow q}\approx B^{+}_{1}$. The proof of
Theorem 2.1 is complete. ∎

We can similarly prove the following statement.

######  Theorem 2.2.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{1}(\cdot,\cdot)\in\mathcal{O}_{1}^{-}$. Then operator (2.2) is bounded from
$L_{p,\mu}(I)$ to $L_{q}(I)$ if and only if
$B^{-}_{1}=\max\\{B^{-}_{0,1},B^{-}_{1,1}\\}<\infty$. Moreover,
$\|\mathbf{K}^{-}\|_{p\rightarrow q}\approx B^{-}_{1}$, where
$\|\mathbf{K}^{-}\|_{p\rightarrow q}$ is the norm of operator (2.2) from
$L_{p,\mu}(I)$ to $L_{q}(I)$, where

$B^{-}_{0,1}=\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}\bar{r}^{p^{\prime}}(t)d\mu(t)\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{0}^{z}K^{q}_{0,1}(z,s)v^{q}(s)ds\right)^{\frac{q}{p-q}}d\left(\int\limits_{0}^{z}K^{q}_{0,1}(z,s)v^{q}(s)ds\right)\right)^{\frac{p-q}{pq}},$
$B^{-}_{1,1}=\left(\int\limits_{0}^{\infty}\left(\int\limits_{[z,\infty]}K^{p^{\prime}}_{1}(x,z)d\mu(x)\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{0}^{z}v^{q}(s)ds\right)^{\frac{q}{p-q}}v^{q}(z)dz\right)^{\frac{p-q}{pq}}.$

##  3 Operators (1.1) and (1.2) with kernels from the classes
$\mathcal{O}_{n}^{\pm}$

In this Section, we establish the boundedness of operators (1.1) and (1.2)
from $L_{p}(I)$ to $L_{q}(I)$, $1<q<p<\infty$, when their kernels belong to
the class $\mathcal{O}_{n}^{\pm}$. It follows from Theorems 2.1 and 2.2 that
in relation (1.7) the function $r_{1}$ can be taken equal to unity, since it
can be combined with the weight functions of operators (1.1) and (1.2).
Therefore, further we suppose that $K_{0}(\cdot,\cdot)\equiv 1$ in (1.5), and
(1.5) has the form

$h_{n}^{-1}\left(K_{n,0}(x,t)+\sum\limits_{i=1}^{n-1}K_{n,i}(x,t)K_{i}(t,s)+K_{n}(t,s)\right)\leq
K_{n}(x,s)$ $\leq
h_{n}\left(K_{n,0}(x,t)+\sum\limits_{i=1}^{n-1}K_{n,i}(x,t)K_{i}(t,s)+K_{n}(t,s)\right),\leavevmode\nobreak\
\leavevmode\nobreak\ x\geq t\geq s>0.$ (3.1)

Assume that

$B^{+}_{n,i}=\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{0}^{z}K^{p^{\prime}}_{i}(z,t)v^{p^{\prime}}(t)dt\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

######  Theorem 3.1.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$, $n\geq 1$. Then operator (2.1) is
bounded from $L_{p}(I)$ to $L_{q,\mu}(I)$ if and only if
$B^{+}_{n}=\max\limits_{0\leq i\leq n}B^{+}_{n,i}<\infty$. Moreover,
$\|\mathbf{K^{+}}\|_{p\rightarrow q}\approx B^{+}_{n}$, where
$\|\mathbf{K^{+}}\|_{p\rightarrow q}$ is the norm of operator (2.1) from
$L_{p}(I)$ to $L_{q,\mu}(I)$. In addition,

$B^{+}_{n,i}\approx\left(\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$ (3.2)

######  Remark 3.1.

Relations of form (3.2) are presented in the statements of all Theorems
because they are suitable for the applications listed in Introduction. We only
prove relation (3.2), since the other relations are proved similarly.

###### Proof.

Necessity. Let operator (2.1) be bounded from $L_{p}(I)$ to $L_{q,\mu}(I)$,
i.e., the inequality

$\left(\int\limits_{[0,\infty]}\left(\int\limits_{0}^{x}K_{n}(x,s)v(s)f(s)ds\right)^{q}d\mu(x)\right)^{\frac{1}{q}}\leq\|\mathbf{K^{+}}\|_{p\rightarrow
q}\left(\int\limits_{0}^{\infty}|f(t)|^{p}dt\right)^{\frac{1}{p}}$ (3.3)

holds for all $0\leq f\in L_{p}(I)$ having compact support.

Using the relations $K_{n}(x,t)\gg K_{n,0}(x,s)$ for $x\geq s\geq t>0$ and, in
particular, the relation $K_{n}(x,s)\gg K_{n,0}(x,s)$ for $x\geq s>0$, which
follows from (3.1), by Lemma 1 from [21] and Fubini’s theorem, we have

$J_{n}\equiv\int\limits_{[0,\infty]}\left(\int\limits_{0}^{x}K_{n}(x,s)v(s)f(s)ds\right)^{q}d\mu(x)$
$\approx\int\limits_{[0,\infty]}\int\limits_{0}^{x}K_{n}(x,s)v(s)f(s)\left(\int\limits_{0}^{s}K_{n}(x,t)v(t)f(t)dt\right)^{q-1}dsd\mu(x)$
$\gg\int\limits_{[0,\infty]}\int\limits_{0}^{x}K_{n,0}^{q}(x,s)v(s)f(s)\left(\int\limits_{0}^{s}v(t)f(t)dt\right)^{q-1}dsd\mu(x)$
$=\int\limits_{0}^{\infty}v(s)f(s)\left(\int\limits_{0}^{s}v(t)f(t)dt\right)^{q-1}\int\limits_{[s,\infty]}K_{n,0}^{q}(x,s)d\mu(x)ds.$
(3.4)

Remark 1.2 gives that the function
$\int\limits_{[s,\infty)}K_{n,0}^{q}(x,s)d\mu(x)$ does not increase on $I$.
Hence, by Remark 1.4, there exists a Borel measure $\theta_{n,0}$ such that
$\int\limits_{[s,\infty]}K_{n,0}^{q}(x,s)d\mu(x)=\int\limits_{[s,\infty]}d\theta_{n,0}(z)$.
Replacing this relation into (3.4) and using Fubini’s theorem, we get

$J_{n}\gg\int\limits_{0}^{\infty}v(s)f(s)\left(\int\limits_{0}^{s}v(t)f(t)dt\right)^{q-1}\int\limits_{[s,\infty]}d\theta_{n,0}(z)ds\approx\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}v(t)f(t)dt\right)^{q}d\theta_{n,0}(z).$

Then from (3.3) we obtain

$\left(\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}v(t)f(t)dt\right)^{q}d\theta_{n,0}(z)\right)^{\frac{1}{q}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q}\left(\int\limits_{0}^{\infty}|f(t)|^{p}dt\right)^{\frac{1}{p}}$

and, on the basis of Theorem 2 from [16, §1.3, 1.3.3], we find that

$B^{+}_{n,0}\ll\|\mathbf{K^{+}}\|_{p\rightarrow q}.$ (3.5)

From the boundedness of operator (2.1) from $L_{p}(I)$ to $L_{q,\mu}(I)$ it
follows the boundedness of dual operator (2.2) from $L_{q^{\prime},\mu}(I)$ to
$L_{p^{\prime}}(I)$, i.e., we have

$\left(\int\limits_{0}^{\infty}v^{p^{\prime}}(s)\left(\int\limits_{[s,\infty]}K_{n}(x,s)g(x)d\mu(x)\right)^{p^{\prime}}ds\right)^{\frac{1}{p^{\prime}}}\leq\|\mathbf{K^{+}}\|_{p\rightarrow
q}\left(\int\limits_{[0,\infty]}|g(t)|^{q^{\prime}}d\mu(t)\right)^{\frac{1}{q^{\prime}}}$
(3.6)

for $0\leq g\in L_{q^{\prime}}(I)$ having compact support. Since
$K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$, then, by the definition, the
function $K_{n}(\cdot,\cdot)$ does not decrease in the first argument. Using
the last fact, Lemma 2 from [21] and Fubini’s theorem, we deduce that

$J_{n}^{*}=\int\limits_{0}^{\infty}v^{p^{\prime}}(s)\left(\int\limits_{[s,\infty]}K_{n}(x,s)g(x)d\mu(x)\right)^{p^{\prime}}ds$
$\approx\int\limits_{0}^{\infty}v^{p^{\prime}}(s)\int\limits_{[s,\infty]}K_{n}(x,s)g(x)\left(\int\limits_{[x,\infty]}K_{n}(t,s)g(t)d\mu(t)\right)^{p^{\prime}-1}d\mu(x)ds$
$\geq\int\limits_{0}^{\infty}v^{p^{\prime}}(s)\int\limits_{[s,\infty]}K^{p^{\prime}}_{n}(x,s)g(x)\left(\int\limits_{[x,\infty]}g(t)d\mu(t)\right)^{p^{\prime}-1}d\mu(x)ds$
$=\int\limits_{[0,\infty]}g(x)\left(\int\limits_{[x,\infty]}g(t)d\mu(t)\right)^{p^{\prime}-1}\int\limits_{0}^{x}K_{n}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\,d\mu(x).$
(3.7)

The function $\int\limits_{0}^{x}K_{n}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds$
does not decrease on $I$ and $\lim\limits_{x\to
0}\int\limits_{0}^{x}K_{n}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds=0$ due to
(3.7). Therefore, by Remark 1.3, there exists a Borel measure $\eta_{n}$ such
that
$\int\limits_{0}^{x}K_{n}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds=\int\limits_{(0,x]}d\eta_{n}(z)$.
Using this relation in (3.7), Fubini’s theorem and Lemma 2 from [21], we have

$J_{n}^{*}\gg\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}g(t)d\mu(t)\right)^{p^{\prime}}d\eta_{n}(z).$

The latter and (3.6) give that

$\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty])}g(t)d\mu(t)\right)^{p^{\prime}}d\eta_{n}(z)\right)^{\frac{1}{p^{\prime}}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q}\left(\int\limits_{[0,\infty]}|g(t)|^{q^{\prime}}d\mu(t)\right)^{\frac{1}{q^{\prime}}}.$

Then, by Theorem 3 from [21], we get

$B^{+}_{n,n}\ll\|\mathbf{K^{+}}\|_{p\rightarrow q}.$ (3.8)

From (3.1) it follows that

$K_{n}(x,s)\gg
K_{n,i}(x,t)K_{i}(t,s)=\chi_{(s,x)}(t)K_{n,i}(x,t)K_{i}(t,s),\leavevmode\nobreak\
i=1,2,...,n-1,$ (3.9)

for $x\geq t\geq s>0$. Let $\varphi:I\to[0,\infty)$ and
$\int\limits_{0}^{\infty}\varphi(t)dt=1$. Multiplying both sides of (3.9) by
$\varphi(t)$ and integrating with respect to $t\in I$, we obtain

$K_{n}(x,s)\gg\int\limits_{s}^{x}K_{n,i}(x,t)K_{i}(t,s)\varphi(t)dt,\leavevmode\nobreak\
i=1,2,...,n-1.$

Then, using Fubini’s theorem and the fact that the function $K_{i}(\tau,t)$
does not decrease in the first argument, for an arbitrary
$i\in\\{1,2,...,n-1\\}$ we get

$\int\limits_{[0,\infty]}\left(\int\limits_{0}^{x}K_{n}(x,s)v(s)f(s)ds\right)^{q}d\mu(x)$
$=\int\limits_{[0,\infty]}\int\limits_{0}^{x}K_{n}(x,s)v(s)f(s)ds\left(\int\limits_{0}^{x}K_{n}(x,t)v(t)f(t)dt\right)^{q-1}d\mu(x)$
$\geq\int\limits_{[0,\infty]}\int\limits_{0}^{x}\left(\int\limits_{s}^{x}K_{n,i}(x,z)K_{i}(z,s)\varphi(z)dz\right)v(s)f(s)ds$
$\times\left(\int\limits_{0}^{x}\left(\int\limits_{t}^{x}K_{n,i}(x,\tau)K_{i}(\tau,t)\varphi(\tau)d\tau\right)v(t)f(t)dt\right)^{q-1}d\mu(x)$
$=\int\limits_{[0,\infty]}\int\limits_{0}^{x}K_{n,i}(x,z)\int\limits_{0}^{z}K_{i}(z,s)v(s)f(s)ds\varphi(z)dz$
$\times\left(\int\limits_{0}^{x}K_{n,i}(x,\tau)\int\limits_{0}^{\tau}K_{i}(\tau,t)v(t)f(t)dt\varphi(\tau)d\tau\right)^{q-1}d\mu(x)$
$\geq\int\limits_{0}^{\infty}\int\limits_{0}^{z}K_{i}(z,s)v(s)f(s)ds\int\limits_{[z,\infty]}K_{n,i}(x,z)$
$\times\left(\int\limits_{z}^{x}K_{n,i}(x,\tau)\int\limits_{0}^{\tau}K_{i}(\tau,t)v(t)f(t)dt\varphi(\tau)d\tau\right)^{q-1}d\mu(x)\varphi(z)dz$
$\geq\int\limits_{0}^{\infty}\left(\int\limits_{0}^{z}K_{i}(z,s)v(s)f(s)ds\right)^{q}\int\limits_{[z,\infty]}K_{n,i}(x,z)\left(\int\limits_{z}^{x}K_{n,i}(x,\tau)\varphi(\tau)d\tau\right)^{q-1}d\mu(x)\varphi(z)dz.$

Based on (3.3), the latter gives that

$\left(\int\limits_{o}^{\infty}\left(\int\limits_{0}^{z}K_{i}(z,s)v(s)f(s)ds\right)^{q}\int\limits_{[z,\infty]}K_{n,i}(x,z)\left(\int\limits_{z}^{x}K_{n,i}(x,\tau)\varphi(\tau)d\tau\right)^{q-1}d\mu(x)\varphi(z)dz\right)^{\frac{1}{q}}$
$\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q}\left(\int\limits_{0}^{\infty}f^{p}(t)dt\right)^{\frac{1}{p}},\leavevmode\nobreak\
i=1,2,...,n-1.$ (3.10)

This inequality is the same as inequality (3.3) for $n=i$. Therefore, due to
(3.8), which holds for all $n>1$, we have

$\sup\limits_{\varphi}\left(\int\limits_{(0,\infty]}\left(\int\limits_{z}^{\infty}\int\limits_{[t,\infty]}K_{n,i}(x,t)\left(\int\limits_{t}^{x}K_{n,i}(x,\tau)\varphi(\tau)d\tau\right)^{q-1}d\mu(x)\varphi(t)dt\right)^{\frac{p}{p-q}}\right.$
$\left.\times\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}d\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{pq}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q}.$ (3.11)

Since

$\int\limits_{z}^{\infty}\int\limits_{[t,\infty]}K_{n,i}(x,t)\left(\int\limits_{t}^{x}K_{n,i}(x,\tau)\varphi(\tau)d\tau\right)^{q-1}d\mu(x)\varphi(t)dt$
$=\int\limits_{[z,\infty]}\int\limits_{z}^{x}K_{n,i}(x,t)\varphi(t)\left(\int\limits_{t}^{x}K_{n,i}(x,\tau)\varphi(\tau)d\tau\right)^{q-1}dtd\mu(x)\approx\int\limits_{[z,\infty]}\left(\int\limits_{z}^{x}K_{n,i}(x,t)\varphi(t)dt\right)^{q}d\mu(x),$

then from (3.11) we have

$\sup\limits_{\varphi}\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}\left(\int\limits_{z}^{x}K_{n,i}(x,t)\varphi(t)dt\right)^{q}d\mu(x)\right)^{\frac{p}{p-q}}\right.$
$\left.\times\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}d\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{pq}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q}.$

Let $\tau\in I$. Assume that $\varphi(t)=\delta(t-\tau)$. Then

$\sup\limits_{\tau\in
I}\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}\left(\int\limits_{0}^{\infty}\chi_{(z,x)}(t)K_{n,i}(x,t)\delta(t-\tau)dt\right)^{q}d\mu(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}K_{i}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.$
$\left.\times
d\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{pq}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q},$

which, basing on the non-increase of the function $K_{n,i}(x,t)$ in the second
argument, implies that

$B^{+}_{n,i}=\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.$
$\left.\times
d\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{pq}}\ll\|\mathbf{K^{+}}\|_{p\rightarrow
q},$

i.e.,

$B^{+}_{n,i}\ll\|\mathbf{K^{+}}\|_{p\rightarrow q},\leavevmode\nobreak\
i=1,2,...,n-1,$

holds. The latter, together with (3.5) and (3.8), gives that

$B^{+}_{n}=\max\limits_{0\leq i\leq
n}B^{+}_{n,i}\ll\|\mathbf{K^{+}}\|_{p\rightarrow q}.$ (3.12)

Sufficiency. Let $B^{+}_{n}<\infty$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$. Let $0\leq f\in L_{p}(I)$ have
compact support. Next, we proceed in the same way as in the proof of the
sufficient part of Theorem 2.1. Here we take the function
$F_{n}(x)=\int\limits_{0}^{x}K_{n}(x,s)v(s)f(s)ds$ instead of the function
$F(\cdot)$ and, as in Theorem 2.1, integers $\\{k_{i}\\}\subset\mathbb{Z},$
$(k_{i}<k_{i+1})$ and points $\\{x_{i}\\}\subset I$ are such that

$I=\bigcup\limits_{i}I_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ I_{i}=[x_{i},x_{i+1}),\,\,\,\bar{I}_{i}=[x_{i},x_{i+1}],$
(3.13) $(h_{n}+1)^{k_{i}}\leq F_{n}(x)<(h_{n}+1)^{k_{i}+1}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
x_{i}\leq x<x_{i+1},$ (3.14)

where $h_{n}$ is from relation (3.1). Based on (3.1), relation (2.10) has the
form

$(h_{n}+1)^{k_{i}-1}\leq(h_{n}+1)^{k_{i}-1}-h_{n}(h_{n}+1)^{k_{i-2}+1}\leq\int\limits_{x_{i-2}}^{x_{i}}K_{n}(x_{i},s)v(s)f(s)ds$
$+\int\limits_{0}^{x_{i-2}}[K_{n}(x_{i},s)-h_{n}K_{n}(x_{i-2},s)]v(s)f(s)ds$
$\ll\int\limits_{x_{i-2}}^{x_{i}}K_{n}(x_{i},s)v(s)f(s)ds+K_{n,0}(x_{i},x_{i-2})\int\limits_{0}^{x_{i-2}}v(s)f(s)ds$
$+\sum\limits_{\gamma=1}^{n-1}K_{n,\gamma}(x_{i},x_{i-2})\int\limits_{0}^{x_{i-2}}K_{\gamma}(x_{i-2},s)v(s)f(s)ds.$
(3.15)

Due to (3.13), (3.14) and (3.15) we have

$J_{n}\equiv\int\limits_{[0,\infty]}(\mathbf{K_{n}}^{+}f(x))^{q}d\mu(x)\ll\sum\limits_{i}\int\limits_{\bar{I}_{i}}F_{n}^{q}(x)d\mu(x)\leq\sum\limits_{i}(h_{n}+1)^{q(k_{i}+1)}\mu(\bar{I}_{i})$
$\ll\sum\limits_{i}(h_{n}+1)^{q(k_{i}-1)}\mu(\bar{I}_{i})\ll\sum\limits_{i}\mu(\bar{I}_{i})\left(\int\limits_{x_{i-2}}^{x_{i}}K_{n}(x_{i},s)v(s)f(s)ds\right)^{q}$
$+\sum\limits_{i}\mu(\bar{I}_{i})K^{q}_{n,0}(x_{i},x_{i-2})\left(\int\limits_{0}^{x_{i-2}}v(s)f(s)ds\right)^{q}$
$+\sum\limits_{\gamma=1}^{n-1}\sum\limits_{i}\mu(\bar{I}_{i})K_{n,\gamma}^{q}(x_{i},x_{i-2})\left(\int\limits_{0}^{x_{i-2}}K_{\gamma}(x_{i-2},s)v(s)f(s)ds\right)^{q}=J_{n,n}+J_{n,0}+\sum\limits_{\gamma=1}^{n-1}J_{n,\gamma}.$
(3.16)

We estimate $J_{n,n}$, $J_{n,0}$ and $J_{n,\gamma}$, $\gamma=1,2,...,n-1$,
separately. To estimate $J_{n,n}$ we use Hölder’s inequality in the integral,
then in the sum with the parameters $\frac{p}{q}$ and $\frac{p}{p-q}$:

$J_{n,n}\leq\sum\limits_{i}\mu(\bar{I}_{i})\left(\int\limits_{x_{i-2}}^{x_{i}}K_{n}^{p^{\prime}}(x_{i},s)v^{p^{\prime}}(s)ds\right)^{\frac{q}{p^{\prime}}}\left(\int\limits_{x_{i-2}}^{x_{i}}f^{p}(t)dt\right)^{\frac{q}{p}}$
$\leq\left(\sum\limits_{i}\left(\mu(\bar{I}_{i})\right)^{\frac{p}{p-q}}\left(\int\limits_{x_{i-2}}^{x_{i}}K_{n}^{p^{\prime}}(x_{i},s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\right)^{\frac{p-q}{p}}$
$\times\left(\sum\limits_{i}\int\limits_{x_{i-2}}^{x_{i}}f^{p}(t)dt)\right)^{\frac{q}{p}}\ll\hat{B}_{n,n}\|f\|_{p}^{q},$
(3.17)

where

$\hat{B}_{n,n}=\left(\sum\limits_{i}\left(\mu(\bar{I}_{i})\right)^{\frac{p}{p-q}}\left(\int\limits_{x_{i-2}}^{x_{i}}K_{n}^{p^{\prime}}(x_{i},s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\right)^{\frac{p-q}{p}}.$

Then, applying Lemmas 1 and 2 from [21], relation
$\int\limits_{0}^{x}K_{n}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds=\int\limits_{(0,x]}d\eta_{n}(z)$,
which follows from the condition $B^{+}_{n}<\infty$, Remark 1.3 and Fubini’s
theorem, we get

$\hat{B}_{n,n}\ll\left(\sum\limits_{i}\int\limits_{[x_{i},x_{i+1}]}\left(\mu([x,\infty])\right)^{\frac{q}{p-q}}\left(\int\limits_{0}^{x}K_{n}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}d\mu(x)\right)^{\frac{p-q}{p}}$
$\ll\left(\int\limits_{[0,\infty]}\left(\mu([x,\infty])\right)^{\frac{q}{p-q}}\left(\int\limits_{(0,x]}d\eta_{n}(z)\right)^{\frac{q(p-1)}{p-q}}d\mu(x)\right)^{\frac{p-q}{p}}$
$\ll\left(\int\limits_{[0,\infty]}\left(\mu([x,\infty])\right)^{\frac{q}{p-q}}\int\limits_{(0,x]}\left(\int\limits_{(0,s]}d\eta_{n}(z)\right)^{\frac{p(q-1)}{p-q}}d\eta_{n}(s)d\mu(x)\right)^{\frac{p-q}{p}}$
$\ll\left(\int\limits_{(0,\infty]}\left(\mu([s,\infty])\right)^{\frac{p}{p-q}}\left(\int\limits_{(0,s]}d\eta_{n}(z)\right)^{\frac{p(q-1)}{p-q}}d\eta_{n}(s)\right)^{\frac{p-q}{p}}$
$=\left(\int\limits_{(0,\infty]}\left(\mu([x,\infty])\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}K_{n}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}d\left(\int\limits_{0}^{x}K_{n}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{p}}=(B^{+}_{n,n})^{q}.$

The latter and (3.17) give that

$J_{n,n}\ll(B^{+}_{n,n})^{q}\|f\|_{p}^{q}.$ (3.18)

Now, we estimate $J_{n,0}$:

$J_{n,0}=\sum\limits_{i}\mu(\bar{I}_{i})K^{q}_{n,0}(x_{i},x_{i-2})\left(\int\limits_{0}^{x_{i-2}}v(s)f(s)ds\right)^{q}\leq\sum\limits_{i}\int\limits_{[x_{i},x_{i+1}]}K^{q}_{n,0}(x,x_{i-2})d\mu(x)$
$\times\left(\int\limits_{0}^{x_{i-2}}v(s)f(s)ds\right)^{q}=\int\limits_{[0,\infty]}\left(\int\limits_{0}^{t}v(s)f(s)ds)\right)^{q}d\eta_{n,0}(t),$
(3.19)

where
$d\eta_{n,0}(t)=\sum\limits_{i}\int\limits_{[x_{i},x_{i+1}]}K_{n,0}^{q}(x,x_{i-2})d\mu(x)\delta(t-x_{i-2})dt$.
Then by Theorem 3 from [16, §1.3, 1.3.3] we obtain

$J_{n,0}\ll\hat{B}_{n,0}\|f\|_{p}^{q},$ (3.20)

where

$\hat{B}_{n,0}=\left(\int\limits_{0}^{\infty}(\eta_{n,0}([x,\infty]))^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}v^{p^{\prime}}(x)dx\right)^{\frac{p-q}{p}}.$

Since

$\eta_{n,0}([x,\infty])=\sum\limits_{i:x_{i-2}\geq
x}\int\limits_{[x_{i},x_{i+1}]}K_{n,0}^{q}(t,x_{i-2})d\mu(t)\ll\int\limits_{[x,\infty]}K_{n,0}^{q}(t,x)\mu(t),$

then $\hat{B}_{n,0}\ll(B^{+}_{n,0})^{q}$. Therefore, from (3.20) we find that

$J_{n,0}\ll(B^{+}_{n,0})^{q}\|f\|_{p}^{q}.$ (3.21)

Note that relations (3.18) and (3.21) hold for all $n\geq 1$.

Next, we estimate $J_{n,j}$, $j=1,2,...,n-1$:

$J_{n,j}\leq\sum\limits_{i}\int\limits_{[x_{i},x_{i+1}]}K^{q}_{n,j}(x,x_{i-2})d\mu(x)\left(\int\limits_{0}^{x_{i-2}}K_{j}(x_{i-2},s)v(s)f(s)ds\right)^{q}$
$=\int\limits_{[0,\infty]}\left(\int\limits_{0}^{t}K_{j}(t,s)v(s)f(s)ds\right)^{q}d\eta_{n,j}(t),$
(3.22)

where
$d\eta_{n,j}(t)=\sum\limits_{i}\int\limits_{[x_{i},x_{i+1}]}K_{n,j}^{q}(x,x_{i-2})d\mu(x)\delta(t-x_{i-2})dt.$
Further, by the method of mathematical induction, we show that

$J_{n,j}\ll\max\limits_{0\leq i\leq j}(B^{+}_{n,i})^{q}\|f\|_{p}^{q}$ (3.23)

for all $j=1,2,...,n-1$.

Hence, from (3.18), (3.21) and (3.23) we get

$J_{n}\ll(B^{+}_{n})^{q}\|f\|_{p}^{q},$

i.e., operator (2.1) is bounded from $L_{p}(I)$ to $L_{q,\mu}(I)$ and for its
norm $\|\mathbf{K^{+}}\|_{p\rightarrow q}$ from $L_{p}(I)$ to $L_{q,\mu}(I)$
we have the estimate

$\|\mathbf{K^{+}}\|_{p\rightarrow q}\ll B^{+}_{n}.$ (3.24)

Let $j=1$. Since the measure $\eta_{n,j}$ is generated by a non-increasing
function $\psi(z)=\sum\limits_{i:x_{i-2}\geq
z}\int\limits_{[x_{i},x_{i+1}]}K_{n,j}^{q}(x,x_{i-2})d\mu(x)$, i.e.,
$\int\limits_{[z,\infty]}d\eta_{n,j}(t)=\psi(z)$, $z\in I$, then it is obvious
that $\eta_{n,j}\in\Psi$ for all $j=1,2,...,n-1$. Therefore, applying Theorem
2.1 to $J_{n,1}$, we deduce

$J_{n,1}\ll\max\\{\hat{B}_{n,1,0},\hat{B}_{n,1,1}\\}\|f\|_{p}^{q},$ (3.25)

where

$\hat{B}_{n,1,0}=\left(\int\limits_{0}^{\infty}\left(\int\limits_{[z,\infty]}K_{1,0}^{q}(t,z)d\eta_{n,1}(t)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}v^{p^{\prime}}(z)dz\right)^{\frac{p-q}{p}},$
$\hat{B}_{n,1,1}=\left(\int\limits_{(0,\infty]}\left(\eta_{n,1}([z,\infty])\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}K_{1}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}d\left(\int\limits_{0}^{z}K_{1}^{p^{\prime}}(z,t)v^{p^{\prime}}(t)dt\right)\right)^{\frac{p-q}{p}}.$

Using Remark 1.2, we obtain

$\int\limits_{[z,\infty]}K_{1,0}^{q}(t,z)d\eta_{n,1}(t)=\sum\limits_{i:x_{i-2}\geq
z}K_{1,0}^{q}(x_{i-2},z)\int\limits_{[x_{i},x_{i+1}]}K_{n,1}^{q}(x,x_{i-2})d\mu(x)$
$\ll\sum\limits_{i:x_{i-2}\geq
z}\int\limits_{[x_{i},x_{i+1}]}K^{q}_{n,0}(x,z)d\mu(x)\ll\int\limits_{[z,\infty]}K_{n,0}^{q}(x,z)d\mu(x).$

Then

$\int\limits_{[z,\infty]}K_{1,0}^{q}(t,z)d\eta_{n,1}(t)\ll\int\limits_{[z,\infty]}K_{n,0}^{q}(x,z)d\mu(x).$

Replacing the obtained estimate in the expression $\hat{B}_{n,1,0}$, we get

$\hat{B}_{n,1,0}\ll(B^{+}_{n,0})^{q}.$ (3.26)

Since $\eta_{n,1}([z,\infty])=\sum\limits_{i:x_{i-2}\geq
z}\int\limits_{[x_{i},x_{i+1}]}K_{n,1}^{q}(x,x_{i-2})d\mu(x)\ll\int\limits_{[z,\infty]}K_{n,1}^{q}(x,z)d\mu(x)$,
then $\hat{B}_{n,1,1}\ll(B^{+}_{n,1})^{q}$. Thus, taking into account (3.25)
and (3.26), we have

$J_{n,1}\ll\max\\{B^{+}_{n,0},B^{+}_{n,1}\\}^{q}\|f\|_{p}^{q},$

i.e., relation (3.23) holds for $j=1$ for any $n>1$.

Let $1\leq k<n-1$ and relation (3.23) hold for $j=1,2,...,k$ for any $n>k$ and
$\mu\in\Psi$. We show that it holds for $j=k+1$. From (3.22) we obtain

$J_{n,k+1}\leq\int\limits_{[0,\infty]}\left(\int\limits_{0}^{t}K_{k+1}(t,s)v(s)f(s)ds\right)d\eta_{n,k+1}(t).$
(3.27)

Since $\eta_{n,k+1}\in\Psi$, then from (3.27) and (3.16) we have

$\int\limits_{[0,\infty]}\left(\int\limits_{0}^{t}K_{k+1}(t,s)v(s)f(s)ds\right)d\eta_{n,k+1}(t)\ll
J_{n,k+1,k+1}+J_{n,k+1,0}+\sum\limits_{\gamma=1}^{k}J_{n,k+1,\gamma}.$ (3.28)

From (3.18) and (3.21) we respectively get

$J_{n,k+1,k+1}\ll(B^{+}_{n,k+1,k+1})^{q}\|f\|^{q}_{p},\leavevmode\nobreak\
\leavevmode\nobreak\ J_{n,k+1,0}\ll(B^{+}_{n,k+1,0})^{q}\|f\|^{q}_{p},$ (3.29)

where

$(B^{+}_{n,k+1,k+1})^{q}=\left(\int\limits_{(0,\infty]}\left(\eta_{n,k+1}([x,\infty])\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}K_{k+1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.d\left(\int\limits_{0}^{x}K_{k+1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{p}},$
$(B^{+}_{n,k+1,0})^{q}=\left(\int\limits_{0}^{\infty}\left(\int\limits_{[x,\infty]}K_{k+1,0}^{q}(t,x)\,d\eta_{n,k+1}(t)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}v^{p^{\prime}}(x)dx\right)^{\frac{p-q}{p}}.$

Since

$\eta_{n,k+1}([x,\infty])=\sum\limits_{i:x_{i-2}\geq
x}\int\limits_{[x_{i},x_{i+1}]}K_{n,k+1}^{q}(t,x_{i-2})d\mu(t)\ll\int\limits_{[x,\infty]}K_{n,k+1}^{q}(t,x)d\mu(t),$

and

$\int\limits_{[x,\infty]}K_{k+1,0}^{q}(t,x)\,d\eta_{n,k+1}(t)=\sum\limits_{i:x_{i-2}\geq
x}K_{k+1,0}^{q}(x_{i-2},x)\int\limits_{[x_{i},x_{i+1}]}K_{n,k+1}^{q}(t,x_{i-2})d\mu(t)$
$=\sum\limits_{i:x_{i-2}\geq
x}\int\limits_{[x_{i},x_{i+1}]}K_{n,k+1}^{q}(t,x_{i-2})K_{k+1,0}^{q}(x_{i-2},x)d\mu(t)\ll\int\limits_{[x,\infty]}K_{n,0}^{q}(t,x)d\mu(t)$
(3.30)

due to Remark 1.2, then

$(B^{+}_{n,k+1,k+1})^{q}\ll\left(\int\limits_{(0,\infty]}\left(\int\limits_{[x,\infty]}K_{n,k+1}^{q}(t,x)d\mu(t)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}K_{k+1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.$
$\left.\times
d\left(\int\limits_{0}^{x}K_{k+1}^{p^{\prime}}(x,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{p}}=(B^{+}_{n,k+1})^{q},$
$(B^{+}_{n,k+1,0})^{q}\ll\left(\int\limits_{0}^{\infty}\left(\int\limits_{[x,\infty]}K_{n,0}^{q}(t,x)d\mu(t)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{x}v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}v^{p^{\prime}}(x)dx\right)^{\frac{p-q}{p}}=(B^{+}_{n,0})^{q}.$

Since $1\leq\gamma\leq k$ in (3.28), by the assumption, we have

$J_{n,k+1,\gamma}\ll\max\limits_{0\leq
i\leq\gamma}(B^{+}_{n,k+1,i})^{q}\|f\|_{p}^{q},$

where

$(B^{+}_{n,k+1,i})^{q}=\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}K_{k+1,i}^{q}(x,z)d\eta_{n,k+1}(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.$
$\left.\times
d\left(\int\limits_{0}^{z}K^{p^{\prime}}_{i}(z,t)v^{p^{\prime}}(t)dt\right)\right)^{\frac{p-q}{p}}$
$\ll\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.$
$\left.\times
d\left(\int\limits_{0}^{z}K^{p^{\prime}}_{i}(z,t)v^{p^{\prime}}(t)dt\right)\right)^{\frac{p-q}{p}}=(B^{+}_{n,i})^{q}.$

In the last relation, as for (3.30), we have used the expression for
$d\eta_{n,k+1}(x)$ and Remark 1.2. Then

$J_{n,k+1,\gamma}\ll\max\limits_{0\leq
i\leq\gamma}(B^{+}_{n,i})^{q}\|f\|_{p}^{q},\leavevmode\nobreak\
\leavevmode\nobreak\ \gamma=1,2,...,k.$ (3.31)

Now, from (3.27), (3.28), (3.29) and (3.31) we deduce that (3.23) holds for
$j=k+1$.

Thus, we have proved that (3.23) holds. Hence, (3.24) holds, which, together
with (3.12), gives that $\|\mathbf{K^{+}}\|_{p\rightarrow q}\approx
B^{+}_{n}$. Let us show that relation (3.2) holds. Let $i\in\\{0,1,...,n\\}$.

The function $\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)$ does not
increase, therefore, by Remark 1.4, there exists a Borel measure
$\theta_{n,i}$ such that
$\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)=\int\limits_{[s,\infty]}d\theta_{n,i}(z)$.
By Lemma 2 from [21], we have

$\left(\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)\right)^{\frac{p}{p-q}}=\left(\int\limits_{[s,\infty]}d\theta_{n,i}(z)\right)^{\frac{p}{p-q}}\approx\int\limits_{[s,\infty]}\left(\int\limits_{[x,\infty]}d\theta_{n,i}(z)\right)^{\frac{q}{p-q}}d\theta_{n,i}(x).$

Replacing this relation into the expression $B^{+}_{n,i}$ and using Fubini’s
theorem, we get

$(B^{+}_{n,i})^{\frac{pq}{p-q}}\approx\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)\right)^{\frac{q}{p-q}}\\\
\times d\left(-\int\limits_{[z,\infty]}K_{n,i}^{q}(x,z)d\mu(x)\right).$

Hence, relation (3.2) holds. The proof of Theorem 3.1 is complete. ∎

Now, we consider operator (2.2) from $L_{p,\mu}(I)$ to $L_{q}(I)$ when its
kernel $K(\cdot,\cdot)\in\mathcal{O}_{n}^{-}$, $n\geq 1$. In (1.6) we assume
that $K_{0}(\cdot,\cdot)\equiv 1$.

We can similarly prove the following statement.

######  Theorem 3.2.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{-}$, $n>1$. Then operator (2.2) is
bounded from $L_{p,\mu}(I)$ to $L_{q}(I)$ if and only if
$B^{-}_{n}=\max\limits_{0\leq i\leq n}B^{-}_{i,n}<\infty$. Moreover,
$\|\mathbf{K^{-}}\|_{p\rightarrow q}\approx B^{-}_{n}$, where
$\|\mathbf{K^{-}}\|_{p\rightarrow q}$ is the norm of operator (2.2) from
$L_{p,\mu}(I)$ to $L_{q}(I)$ and

$B^{-}_{i,n}=\left(\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}K_{i,n}^{q}(z,s)v^{q}(s)ds\right)^{\frac{p}{p-q}}\left(\int\limits_{[z,\infty]}K_{i}^{p^{\prime}}(x,z)d\mu(x)\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{[z,\infty]}K_{i}^{p^{\prime}}(x,z)d\mu(x)\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}K_{i}^{p^{\prime}}(x,z)d\mu(x)\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{0}^{z}K_{i,n}^{q}(z,s)v^{q}(s)ds\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{0}^{z}K_{i,n}^{q}(z,s)v^{q}(s)ds\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

Now, we consider operator (2.2) from $L_{p,\mu}(I)$ to $L_{q}(I)$ and operator
(2.1) from $L_{p}(I)$ to $L_{q,\mu}(I)$ for $1<q<p<\infty$ when their kernels
belong to the classes $\mathcal{O}_{n}^{+}$ and $\mathcal{O}_{n}^{-}$, $n>1$,
respectively. In view of mutual duality of operators (2.2) and (2.1), from
Theorems 3.2 and 3.1 we respective have the following two theorems.

######  Theorem 3.3.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{-}$, $n\geq 1$. Then operator (2.1) is
bounded from $L_{p}(I)$ to $L_{q,\mu}(I)$ if and only if
$\bar{B}^{+}_{n}=\max\limits_{0\leq i\leq n}\bar{B}^{+}_{i,n}<\infty$.
Moreover, $\|\mathbf{K}^{+}\|_{p\rightarrow q}\approx\bar{B}_{n}^{+}$, where
$\|\mathbf{K}^{+}\|_{p\rightarrow q}$ is the norm of operator (2.1) from
$L_{p}(I)$ to $L_{q,\mu}(I)$ and

$\bar{B}^{+}_{i,n}=\left(\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}K_{i,n}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{[z,\infty]}K_{i}^{q}(x,z)d\mu(x)\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{[z,\infty]}K_{i}^{q}(x,z)d\mu(x)\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}K_{i}^{q}(x,z)d\mu(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}K_{i,n}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{0}^{z}K_{i,n}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

######  Theorem 3.4.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$, $n>1$. Then operator (2.2) is
bounded from $L_{p,\mu}(I)$ to $L_{q}(I)$ if and only if
$\bar{B}^{-}_{n}=\max\limits_{0\leq i\leq n}\bar{B}^{-}_{n,i}<\infty$.
Moreover, $\|\mathbf{K}^{-}\|_{p\rightarrow q}\approx\bar{B}_{n}^{-}$, where
$\|\mathbf{K}^{-}\|_{p\rightarrow q}$ is the norm of operator (2.2) from
$L_{p,\mu}(I)$ to $L_{q}(I)$ and

$\bar{B}^{-}_{n,i}=\left(\int\limits_{(0,\infty]}\left(\int\limits_{[z,\infty]}K_{n,i}^{p^{\prime}}(x,z)d\mu(x)\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{0}^{z}K_{i}^{q}(z,s)v^{q}(s)ds\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{0}^{z}K^{q}_{i}(z,t)v^{q}(t)dt\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{[0,\infty]}\left(\int\limits_{0}^{z}K_{i}^{q}(z,s)v^{q}(s)ds\right)^{\frac{p}{p-q}}\left(\int\limits_{[z,\infty]}K_{n,i}^{p^{\prime}}(x,z)d\mu(x)\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{[z,\infty]}K_{n,i}^{p^{\prime}}(x,z)d\mu(x)\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=1,2,...,n-1.$

Let $\mu\in\Psi$. We consider the integral operator

$\mathbb{K^{-}}g(s)=\int\limits_{s}^{\infty}K(x,s)u(x)g(x)dx,\leavevmode\nobreak\
s>0,$ (3.32)

acting from $L_{p}(I)$ to $L_{q,\mu}(I)$, and the integral operator

$\mathbb{K^{+}}f(x)=\int\limits_{[0,x]}u(x)K(x,s)f(s)d\mu(s),\leavevmode\nobreak\
x>0,$ (3.33)

acting from $L_{p,\mu}(I)$ to $L_{q}(I)$.

Proceeding as above, we obtain analogues of Theorems 3.1–3.4.

######  Theorem 3.5.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{-}$, $n\geq 1$. Then operator (3.32) is
bounded from $L_{p}(I)$ to $L_{q,\mu}(I)$ if and only if
$\hat{B}^{-}_{n}=\max\limits_{0\leq i\leq n}\hat{B}^{-}_{i,n}<\infty$.
Moreover, $\|\mathbb{K^{-}}\|_{p\rightarrow q}\approx\hat{B}^{-}_{n}$, where
$\|\mathbb{K^{-}}\|_{p\rightarrow q}$ is the norm of operator (3.32) from
$L_{p}(I)$ to $L_{q,\mu}(I)$ and

$\hat{B}^{-}_{i,n}=\left(\int\limits_{[0,\infty)}\left(\int\limits_{[0,z]}K_{i,n}^{q}(z,x)d\mu(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{z}^{\infty}K_{i}^{p^{\prime}}(s,z)u^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{z}^{\infty}K^{p^{\prime}}_{i}(t,z)u^{p^{\prime}}(t)dt\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$
$\approx\left(\int\limits_{[0,\infty]}\left(\int\limits_{z}^{\infty}K_{i}^{p^{\prime}}(s,z)u^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{[0,z]}K_{i,n}^{q}(z,x)d\mu(x)\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{[0,z]}K_{i,n}^{q}(z,x)d\mu(x)\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

######  Theorem 3.6.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$, $n\geq 1$. Then operator (3.33) is
bounded from $L_{p,\mu}(I)$ to $L_{q}(I)$ if and only if
$\hat{B}^{+}_{n}=\max\limits_{0\leq i\leq n}\hat{B}^{+}_{n,i}<\infty$.
Moreover, $\|\mathbb{K^{+}}\|_{p\rightarrow q}\approx\hat{B}^{+}_{n}$, where
$\|\mathbb{K^{+}}\|_{p\rightarrow q}$ is the norm of operator (3.33) from
$L_{p,\mu}(I)$ to $L_{q}(I)$ and

$\hat{B}^{+}_{n,i}=\left(\int\limits_{[0,\infty]}\left(\int\limits_{z}^{\infty}K_{n,i}^{q}(s,z)u^{q}(s)ds\right)^{\frac{p}{p-q}}\left(\int\limits_{[0,z]}K_{i}^{p^{\prime}}(z,x)d\mu(x)\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{[0,z]}K_{i}^{p^{\prime}}(z,x)d\mu(x)\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{[0,\infty)}\left(\int\limits_{[0,z]}K_{i}^{p^{\prime}}(z,x)d\mu(x)\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{z}^{\infty}K_{n,i}^{q}(s,z)u^{q}(s)ds\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{z}^{\infty}K_{n,i}^{q}(s,z)u^{q}(s)ds\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

######  Theorem 3.7.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$, $n\geq 1$. Then operator (3.32) is
bounded from $L_{p}(I)$ to $L_{q,\mu}(I)$ if and only if
$\tilde{B}^{-}_{n}=\max\limits_{0\leq i\leq n}\tilde{B}^{-}_{n,i}<\infty$.
Moreover, $\|\mathbb{K}^{-}\|_{p\rightarrow q}\approx\tilde{B}_{n}^{-}$, where
$\|\mathbb{K}^{-}\|_{p\rightarrow q}$ is the norm of operator (3.32) from
$L_{p}(I)$ to $L_{q,\mu}(I)$ and

$\tilde{B}^{-}_{n,i}=\left(\int\limits_{[0,\infty]}\left(\int\limits_{z}^{\infty}K_{n,i}^{p^{\prime}}(s,z)u^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{[0,z]}K_{i}^{q}(z,x)d\mu(x)\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{[0,z]}K_{i}^{q}(z,x)d\mu(x)\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{[0,\infty)}\left(\int\limits_{[0,z]}K_{i}^{q}(z,x)d\mu(x)\right)^{\frac{p}{p-q}}\left(\int\limits_{z}^{\infty}K_{n,i}^{p^{\prime}}(s,z)u^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{z}^{\infty}K_{n,i}^{p^{\prime}}(s,z)u^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

######  Theorem 3.8.

Let $1<q<p<\infty$, $\mu\in\Psi$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{-}$, $n>1$. Then operator (3.33) is
bounded from $L_{p,\mu}(I)$ to $L_{q}(I)$ if and only if
$\tilde{B}^{+}_{n}=\max\limits_{0\leq i\leq n}\tilde{B}^{+}_{n,i}<\infty$.
Moreover, $\|\mathbb{K}^{+}\|_{p\rightarrow q}\approx\tilde{B}_{n}^{+}$, where
$\|\mathbb{K}^{+}\|_{p\rightarrow q}$ is the norm of operator (3.33) from
$L_{p,\mu}(I)$ to $L_{q}(I)$ and

$\tilde{B}^{+}_{i,n}=\left(\int\limits_{[0,\infty)}\left(\int\limits_{[0,z]}K_{i,n}^{p^{\prime}}(z,x)d\mu(x)\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{z}^{\infty}K_{i}^{q}(s,z)u^{q}(s)ds\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{z}^{\infty}K^{q}_{i}(t,z)u^{q}(t)dt\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{[0,\infty]}\left(\int\limits_{z}^{\infty}K_{i}^{q}(s,z)u^{q}(s)ds\right)^{\frac{p}{p-q}}\left(\int\limits_{[0,z]}K_{i,n}^{p^{\prime}}(z,x)d\mu(x)\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{[0,z]}K_{i,n}^{p^{\prime}}(z,x)d\mu(x)\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=1,2,...,n-1.$

Let the measure $\mu\in\Psi$ be absolutely continuous with respect to the
Lebesgue measure with density $u^{q}$, where $u$ is a non-negative function
such that $u\in L_{q}^{log}(I)$. Then
$\int\limits_{0}^{\infty}\left(\mathbf{K^{+}}f(x)\right)^{q}d\mu(x)=\int\limits_{0}^{\infty}\left(\mathcal{K^{+}}f(x)\right)^{q}dx$,
$f\geq 0$, and the boundedness of operator (2.1) from $L_{p}(I)$ to
$L_{q,\mu}(I)$ coincides with the boundedness of operator (1.1) from
$L_{p}(I)$ to $L_{q}(I)$ with the norm $\|\mathbf{K^{+}}\|_{p\to
q}=\|\mathcal{K^{+}}\|_{p\to q}$. Therefore, from Theorems 3.1 and 3.3 we
respectively deduce the following statements.

######  Theorem 3.9.

Let $1<q<p<\infty$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$, $n\geq 1$. Then operator (1.1) is
bounded from $L_{p}(I)$ to $L_{q}(I)$ if and only if
$\mathcal{B}^{+}_{n}=\max\limits_{0\leq i\leq n}\mathcal{B}^{+}_{n,i}<\infty$.
Moreover, $\|\mathcal{K}^{+}\|_{p\rightarrow q}\approx\mathcal{B}_{n}^{+}$,
where $\|\mathcal{K}^{+}\|_{p\rightarrow q}$ is the norm of operator (1.1)
from $L_{p}(I)$ to $L_{q}(I)$ and

$\mathcal{B}^{+}_{n,i}=\left(\int\limits_{(0,\infty]}\left(\int\limits_{z}^{\infty}K_{n,i}^{q}(x,z)u^{q}(x)dx\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{0}^{z}K^{p^{\prime}}_{i}(z,t)v^{p^{\prime}}(t)dt\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{[0,\infty)}\left(\int\limits_{0}^{z}K_{i}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{z}^{\infty}K_{n,i}^{q}(x,z)u^{q}(x)dx\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{z}^{\infty}K_{n,i}^{q}(x,z)u^{q}(x)dx\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

######  Theorem 3.10.

Let $1<q<p<\infty$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{-}$, $n\geq 1$. Then operator (1.1) is
bounded from $L_{p}(I)$ to $L_{q}(I)$ if and only if
$\mathbb{B}^{+}_{n}=\max\limits_{0\leq i\leq n}\mathbb{B}^{+}_{i,n}<\infty$.
Moreover, $\|\mathcal{K}^{+}\|_{p\rightarrow q}\approx\mathbb{B}_{n}^{+}$,
where $\|\mathcal{K}^{+}\|_{p\rightarrow q}$ is the norm of operator (1.1)
from $L_{p}(I)$ to $L_{q}(I)$ and

$\mathbb{B}^{+}_{i,n}=\left(\int\limits_{[0,\infty)}\left(\int\limits_{0}^{z}K_{i,n}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{z}^{\infty}K_{i}^{q}(x,z)u^{q}(x)dx\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{z}^{\infty}K_{i}^{q}(x,z)u^{q}(x)dx\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{(0,\infty]}\left(\int\limits_{z}^{\infty}K_{i}^{q}(x,z)u^{q}(x)dx\right)^{\frac{p}{p-q}}\left(\int\limits_{0}^{z}K_{i,n}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{0}^{z}K_{i,n}^{p^{\prime}}(z,s)v^{p^{\prime}}(s)ds\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,2,...,n.$

Now, we establish a criterion for the boundedness of operator (1.2) from
$L_{p}(I)$ to $L_{q}(I)$ when its kernel $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$. Using mutual duality of operators
(1.1) and (1.2), from Theorems 3.9 and 3.10 we respectively obtain the
following theorems.

######  Theorem 3.11.

Let $1<q<p<\infty$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{+}$, $n\geq 1$. Then operator (1.2) is
bounded from $L_{p}(I)$ to $L_{q}(I)$ if and only if
$\mathcal{B}^{-}_{n}=\max\limits_{0\leq i\leq n}\mathcal{B}^{-}_{n,i}<\infty$.
Moreover, $\|\mathcal{K}^{-}\|_{p\rightarrow q}\approx\mathcal{B}^{-}_{n}$,
where $\|\mathcal{K}^{-}\|_{p\rightarrow q}$ is the norm of operator (1.2)
from $L_{p}(I)$ to $L_{q}(I)$ and

$\mathcal{B}^{-}_{n,i}=\left(\int\limits_{(0,\infty]}\left(\int\limits_{z}^{\infty}K_{n,i}^{p^{\prime}}(x,z)u^{p^{\prime}}(x)dx\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{0}^{z}K_{i}^{q}(z,s)v^{q}(s)ds\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{0}^{z}K^{q}_{i}(z,t)v^{q}(t)dt\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{[0,\infty)}\left(\int\limits_{0}^{z}K_{i}^{q}(z,s)v^{q}(s)ds\right)^{\frac{p}{p-q}}\left(\int\limits_{z}^{\infty}K_{n,i}^{p^{\prime}}(x,z)u^{p^{\prime}}(x)dx\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{z}^{\infty}K_{n,i}^{p^{\prime}}(x,z)u^{p^{\prime}}(x)dx\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

######  Theorem 3.12.

Let $1<q<p<\infty$ and $K(\cdot,\cdot)\equiv
K_{n}(\cdot,\cdot)\in\mathcal{O}_{n}^{-}$, $n\geq 1$. Then operator (1.2) is
bounded from $L_{p}(I)$ to $L_{q}(I)$ if and only if
$\mathbb{B}^{-}_{n}=\max\limits_{0\leq i\leq n}\mathbb{B}^{-}_{i,n}<\infty$.
Moreover, $\|\mathcal{K}^{-}\|_{p\rightarrow q}\approx\mathbb{B}^{-}_{n}$,
where $\|\mathcal{K}^{-}\|_{p\rightarrow q}$ is the norm of operator (1.2)
from $L_{p}(I)$ to $L_{q}(I)$ and

$\mathbb{B}^{-}_{i,n}=\left(\int\limits_{[0,\infty)}\left(\int\limits_{0}^{z}K_{i,n}^{q}(z,s)v^{q}(s)ds\right)^{\frac{p}{p-q}}\left(\int\limits_{z}^{\infty}K_{i}^{p^{\prime}}(x,z)u^{p^{\prime}}(x)dx\right)^{\frac{p(q-1)}{p-q}}\right.\\\
\left.\times
d\left(-\int\limits_{z}^{\infty}K_{i}^{p^{\prime}}(x,z)u^{p^{\prime}}(x)dx\right)\right)^{\frac{p-q}{pq}}$
$\approx\left(\int\limits_{(0,\infty]}\left(\int\limits_{z}^{\infty}K_{i}^{p^{\prime}}(x,z)u^{p^{\prime}}(x)dx\right)^{\frac{q(p-1)}{p-q}}\left(\int\limits_{0}^{z}K_{i,n}^{q}(z,s)v^{q}(s)ds\right)^{\frac{q}{p-q}}\right.\\\
\left.\times
d\left(\int\limits_{0}^{z}K_{i,n}^{q}(z,s)v^{q}(s)ds\right)\right)^{\frac{p-q}{pq}},\leavevmode\nobreak\
i=0,1,...,n.$

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11institutetext:  1Harvard University 2Boston College 3CMU 4Google Research

# Structure-Preserving Instance Segmentation via Skeleton-Aware Distance
Transform

Zudi Lin1 Currently affiliated with Amazon Alexa. Work was done before joining
Amazon. Donglai Wei2 Aarush Gupta3 Work was done during an internship at
Harvard University. Xingyu Liu1∗ Deqing Sun4 Hanspeter Pfister1

###### Abstract

Objects with complex structures pose significant challenges to existing
instance segmentation methods that rely on boundary or affinity maps, which
are vulnerable to small errors around contacting pixels that cause noticeable
connectivity change. While the distance transform (DT) makes instance
interiors and boundaries more distinguishable, it tends to overlook the intra-
object connectivity for instances with varying width and result in over-
segmentation. To address these challenges, we propose a skeleton-aware
distance transform (SDT) that combines the merits of object skeleton in
preserving connectivity and DT in modeling geometric arrangement to represent
instances with arbitrary structures. Comprehensive experiments on
histopathology image segmentation demonstrate that SDT achieves state-of-the-
art performance.

##  1 Introduction

Figure 1:  Skeleton-aware distance transform (SDT). Given (a) instance masks,
(b) the boundary map is prone to false merge errors at object contact pixels
while (c) the distance transform (DT) struggles to preserve object
connectivity. (d) Our SDT can both separate touching instances and enforce
object connectivity.

Instances with complex shapes arise in many biomedical domains, and their
morphology carries critical information. For example, the structure of gland
tissues in microscopy images is essential in accessing the pathological stages
for cancer diagnosis and treatment. These instances, however, are usually
closely in touch with each other and have non-convex structures with parts of
varying widths (Fig. 1a), posing significant challenges for existing
segmentation methods.

In the biomedical domain, most methods [3, 14, 4, 22, 13] first learns
intermediate representations and then convert them into masks with standard
segmentation algorithms like connected-component labeling and watershed
transform. These representations are not only efficient to predict in one
model forward pass but also able to capture object geometry (i.e., precise
instance boundary), which are hard for top-down methods using low-resolution
features for mask generation. However, existing representations have several
restrictions. For example, boundary map is usually learned as a pixel-wise
binary classification task, which makes the model conduct relatively local
predictions and consequently become vulnerable to small errors that break the
connectivity between adjacent instances (Fig. 1b). To improve the boundary
map, Deep Watershed Transform (DWT) [1] predicts the Euclidean distance
transform (DT) of each pixel to the instance boundary. This representation is
more aware of the structure for convex objects, as the energy value for
centers is significantly different from pixels close to the boundary. However,
for objects with non-convex morphology, the boundary-based distance transform
produces multiple local optima in the energy landscape (Fig. 1c), which tends
to break the intra-instance connectivity when applying thresholding and
results in over-segmentation.

To preserve the connectivity of instances while keeping the precise instance
boundary, in this paper, we propose a novel representation named skeleton-
aware distance transform (SDT). Our SDT incorporate object skeleton, a concise
and connectivity-preserving representation of object structure, into the
traditional boundary-based distance transform (DT) (Fig. 1d). In quantitative
evaluations, we show that our proposed SDT achieves leading performance on
histopathology image segmentation for instances with various sizes and complex
structures. Specifically, under the Hausdorff distance for evaluating shape
similarity, our approach improves the previous state-of-the-art method by
relatively $10.6\%$.

###  1.1 Related Work

Instance Segmentation. Bottom-up instance segmentation approaches have become
de facto for many biomedical applications due to the advantage in segmenting
objects with arbitrary geometry. U-Net [14] and DCAN [4] use fully
convolutional models to predict the boundary map of instances. Since the
boundary map is not robust to small errors that can significantly change
instance structure, shape-preserving loss [22] adds a curve fitting step in
the loss function to enforce boundary connectivity. In order to further
distinguish closely touching instances, deep watershed transform (DWT) [1]
predicts the distance transform (DT) that represents each pixel as its
distance to the closest boundary. However, for complex structure with parts of
varying width, the boundary-based DT tends to produce relatively low values
for thin connections and consequently causes over-segmentation. Compared to
DWT, our SDT incorporates object skeleton (also known as medial axis) [2, 24,
8] that concisely captures the topological connectivity into standard DT to
enforce both the geometry and connectivity.

Object Skeletonization. Object skeleton [15] is a one-pixel wide
representation of object masks that can be calculated by topological thinning
[24, 8, 12] or medial axis transform [2]. The vision community has been
working on direct object skeletonization from images [16, 7, 9, 19]. Among the
works, only Shen et al. [16] shows the application of the skeleton on
segmenting single-object images. We instead focus on the more challenging
instance segmentation task with multiple objects closely touching each other.
Object skeletons are also used to correct errors in pre-computed segmentation
masks [11]. Our SDT framework instead use the skeleton in the direct
segmentation from images.

##  2 Skeleton-Aware Distance Transform

###  2.1 SDT Energy Function

Given an image, we aim to design a new representation $E$ for a model to
learn, which is later decoded into instances with simple post-processing.
Specifically, a good representation for capturing complex-structure masks
should have two desired properties: precise geometric boundary and robust
topological connectivity.

Let $\Omega$ denote an instance mask, and $\Gamma_{b}$ be the boundary of the
instance (pixels with other object indices in a small local neighborhood). The
boundary (or affinity) map is a binary representation where
$E|_{\Gamma_{b}}=0$ and $E|_{\Omega\setminus\Gamma_{b}}=1.$ Taking the merits
of DT in modeling the geometric arrangement and skeleton in preserving
connectivity, we propose a new representation $E$ that satisfies:

$0=E|_{\Gamma_{b}}<E|_{\Omega\setminus(\Gamma_{b}\cup\Gamma_{s})}<E|_{\Gamma_{s}}=1$
(1)

Here $E|_{\Omega\setminus\Gamma_{s}}<E|_{\Gamma_{s}}=1$ indicates that there
is only one global maximum for each instance, and the value is assigned to a
pixel if and only if the pixel is on the skeleton. This property avoids
ambiguity in defining the object interior and preserve connectivity. Besides,
$E|_{\Omega\setminus\Gamma_{b}}>E|_{\Gamma_{b}}=0$ ensures that boundary is
distinguishable as the standard DT, which produces precise geometric boundary.

For the realization of $E$, let $x$ be a pixel in the input image, and $d$ be
the metric, e.g., Euclidean distance. The energy function for distance
transform (DT) is defined as $E_{\text{DT}}(x)=d(x,\Gamma_{b})$, which starts
from 0 at object boundary and increases monotonically when $x$ is away from
the boundary. Similarly, we can define an energy function $d(x,\Gamma_{s})$
representing the distance from the skeleton. It vanishes to 0 when the pixel
approaches the object skeleton. Formally, we define the energy function of the
skeleton-aware distance transform (Fig. 2) as

$E_{SDT}(x)=\left(\frac{d(x,\Gamma_{b})}{d(x,\Gamma_{s})+d(x,\Gamma_{b})}\right)^{\alpha},~{}\alpha>0$
(2)

where $\alpha$ controls the curvature of the energy surface111We add
$\epsilon=10^{-6}$ to the denominator to avoid dividing by 0 for the edge case
where a pixel is both instance boundary and skeleton (i.e., a one-pixel wide
part).. When $0<\alpha<1$, the function is concave and decreases faster when
being close to the boundary, and vice versa when $\alpha>1$. In the ablation
studies, we demonstrate various patterns of the model predictions given
different $\alpha$.

Besides, since common skeletonization algorithms can be sensitive to small
perturbations on the object boundary and produce unwanted branches, we smooth
the masks before computing the object skeleton by Gaussian filtering and
thresholding to avoid complex branches.

Figure 2: Illustration of the SDT energy function. (a) Given an instance mask
$\Omega$, (b) we calculate the distances of a pixel to both the skeleton and
boundary. (c) Our energy function ensures a uniform maximum value of 1 on the
skeleton and minimum value of 0 on the boundary, with a smooth interpolation
in between.

Learning Strategy. Given the ground-truth SDT energy map, there are two ways
to learn it using a CNN model. The first way is to regress the energy using
$L_{1}$ or $L_{2}$ loss. In the regression mode, the output is a single-
channel image. The second way is to quantize the $[0,1]$ energy space into $K$
bins and rephrase the regression task into a classification task [1, 18],
which makes the model robust to small perturbations in the energy landscape.
For the classification mode, the model output has $(K+1)$ channels with one
channel representing the background region. We fix the bin size to 0.1 without
tweaking, making $K=10$. Softmax is applied before calculating the cross-
entropy loss. We test both learning strategies in the experiments to
illustrate the optimal setting for SDT.

###  2.2 SDT Network

Network Architecture. Directly learning the energy function with a fully
convolutional network (FCN) can be challenging. Previous approaches either
first regress an easier direction field representation and then use additional
layers to predict the desired target [1], or take the multi-task learning
approach to predict additional targets at the same time [21, 22, 16].

Figure 3:  Overview of the SDT framework. (a) Training Phase: target SDT is
calculated conditioned on the distance to both the boundary and skeleton. A
FCN maps the image into the energy space to minimize the loss. (b) Inference
Phase: we threshold the SDT to generate skeleton segments, which is processed
into seeds with the connected component labeling. Finally, the watershed
transform algorithm takes the seeds and the reversed SDT energy to yield the
masks.  Figure 4: Skeleton generation rule. (a) Given an instance and the
global skeleton, (b) the partial skeleton cropped from the global skeleton can
be different from (c) the local skeleton generated from the cropped mask. For
SDT, we calculate the local skeleton to prevent the model from extrapolating
the unseen parts.

Fortunately, with recent progress in FCN architectures, it becomes feasible to
learn the target energy map in an end-to-end fashion. Specifically, in all the
experiments, we use a DeepLabV3 model [5] with a ResNet [6] backbone to
directly learn the SDT energy without additional targets (Fig. 3, Training
Phase). We also add a CoordConv [10] layer before the 3rd stage in the
backbone network to introduce spatial information into the segmentation model.

Target SDT Generation. There is an inconsistency problem in object skeleton
generation: part of the complete instance skeleton can be different from the
skeleton of the instance part (Fig. 4). Some objects may touch the image
border due to either a restricted field of view (FoV) of the imaging devices
or spatial data augmentation like the random crop. If pre-computing the
skeleton, we will get local skeleton (Fig. 4c) for objects with missing masks
due to imaging restrictions, and partial skeleton (Fig. 4b) due to spatial
data augmentation, which causes ambiguity. Therefore we calculate the local
skeleton for SDT on-the-fly after all spatial transformations instead of pre-
computing to prevent the model from hallucinating the structure of parts
outside of the currently visible region. In inference, we always run
predictions on the whole images to avoid inconsistent predictions. We use the
skeletonization algorithm in Lee et al. [8], which is less sensitive to small
perturbations and produces skeletons with fewer branches.

Instance Extraction from SDT. In the SDT energy map, all boundary pixels share
the same energy value and can be processed into segments by direct
thresholding and connected component labeling, similar to DWT [1]. However,
since the prediction is never perfect, the energy values along closely
touching boundaries are usually not sharp and cause split-errors when applying
a higher threshold or merge-errors when applying a lower threshold.

Therefore we utilize a skeleton-aware instance extraction (Fig. 3, Inference
Phase) for SDT. Specifically, we set a threshold $\theta=0.7$ so that all
pixels with the predicted energy bigger than $\theta$ are labeled as skeleton
pixels. We first perform connected component labeling of the skeleton pixels
to generate seeds and run the watershed algorithm on the reversed energy map
using the seeds as basins (local optima) to generate the final segmentation.
We also follow previous works [4, 22] and refine the segmentation by hole-
filling and removing small spurious objects.

##  3 Experiments

###  3.1 Histopathology Instance Segmentation

Accurate instance segmentation of gland tissues in histopathology images is
essential for clinical analysis, especially cancer diagnosis. The diversity of
object appearance, size, and shape makes the task challenging.

Dataset and Evaluation Metric. We use the gland segmentation challenge dataset
[17] that contains colored light microscopy images of tissues with a wide
range of histological levels from benign to malignant. There are 85 and 80
images in the training and test set, respectively, with ground truth
annotations provided by pathologists. According to the challenge protocol, the
test set is further divided into two splits with 60 images of normal and 20
images of abnormal tissues for evaluation. Three evaluation criteria used in
the challenge include instance-level F1 score, Dice index, and Hausdorff
distance, which measure the performance of object detection, segmentation, and
shape similarity, respectively. For the instance-level F1 score, an IoU
threshold of 0.5 is used to decide the correctness of a prediction.

Figure 5:  Visual comparison on histopathology image segmentation. (First 2
rows) Compared with shape-preserving loss (SPL) [22], our SDT unambiguously
separates closely touching objects while preserving the structure of
complicated masks. (The 3rd row) Compared with FullNet [13], our model infers
the SDT energy of instance masks from a global structure perspective instead
of boundary that relies on relatively local predictions, which produces high-
quality masks.

Methods in Comparison. We compare SDT with previous state-of-the-art
segmentation methods, including DCAN [4], multi-channel network (MCN) [21],
shape-preserving loss (SPL) [22] and FullNet [13]. We also compare with
suggestive annotation (SA) [23], and SA with model quantization (QSA) [20],
which use multiple FCN models to select informative training samples from the
dataset. With the same training settings as our SDT, we also report the
performance of skeleton with scales (SS) and traditional distance transform
(DT).

Training and Inference. Since the training data is relatively limited due to
the challenges in collecting medical images, we apply pixel-level and spatial-
level augmentations, including random brightness, contrast, rotation, crop,
and elastic transformation, to alleviate overfitting. We set $\alpha=0.8$ for
our SDT in Eqn. 2. We use the classification learning strategy and optimize a
model with 11 output channels (10 channels for energy quantized into ten bins
and one channel for background). We train the model for 20k iterations with an
initial learning rate of $5\times 10^{-4}$ and a momentum of $0.9$. The same
settings are applied to DT. At inference time, we apply argmax to get the
corresponding bin index of each pixel and transform the energy value to the
original data range. Finally, we apply the watershed-based instance extraction
rule described in Sec. 2.2.

Specifically for SS, we set the number of output channels to two, with one
channel predicting skeleton probability and the other predicting scales. Since
the scales are non-negative, we add a ReLU activation for the second channel
and calculate the regression loss. Masks are generated by morphological
dilation. We do not quantize the scales as DT and SDT since even ground-truth
scales can yield masks unaligned with the instance boundary with quantization.

Results. Our SDT framework achieves state-of-the-art performance on 5 out of 6
evaluation metrics on the gland segmentation dataset (Table 1). With the
better distinguishability of object interior and boundary, SDT can
unambiguously separate closely touching instances (Fig. 5, first two rows),
performs better than previous methods using object boundary representations
[4, 22]. Besides, under the Hausdorff distance for evaluating shape-similarity
between ground-truth and predicted masks, our SDT reports an average score of
44.82 across two test splits, which improves the previous state-of-the-art
approach (i.e., FullNet with an average score of 50.15) by $10.6\%$. We also
notice the different sensitivities of the three evaluation metrics. Taking the
instance D (Fig. 5, 3rd row) as an example: both SDT and FullNet [13] have 1.0
F1-score (IoU threshold 0.5) for the correct detection; SDT has a slightly
higher Dice Index (0.956 vs. 0.931) for better pixel-level classification; and
our SDT has significantly lower Hausdorff distance (24.41 vs. 48.81) as SDT
yields a mask with much more accurate morphology.

###  3.2 Ablation Studies

Loss Function. We compare the regression mode using L1 and L2 losses with the
classification mode using cross-entropy loss. There is a separate channel for
background under the classification mode where the energy values are quantized
into bins. However, for regression mode, if the background value is 0, we need
to use a threshold $\tau>0$ to decide the foreground region, which results in
shrank masks. To separate the background region from the foreground objects,
we assign an energy value of $-b$ to the background pixels $(b\geq 0)$. To
facilitate the regression, given the predicted value $\hat{y}_{i}$ for pixel
$i$, we apply a sigmoid function ($\sigma$) and affine transformation so that
$\hat{y}_{i}^{\prime}=(1+b)\cdot\sigma(\hat{y}_{i})-b$ has a range of
$(-b,1)$. We set $b=0.1$ for the experiments. We show that under the same
settings, the model trained with quantized energy reports the best results
(Table 2). We also notice that the model trained with $L_{1}$ loss produces a
much sharper energy surface than the model trained with $L_{2}$ loss, which is
expected.

Method | F1 Score $\uparrow$ | Dice Index $\uparrow$ | Hausdorff $\downarrow$
---|---|---|---
Part A | Part B | Part A | Part B | Part A | Part B
DCAN [4] | 0.912 | 0.716 | 0.897 | 0.781 | 45.42 | 160.35
MCN [21] | 0.893 | 0.843 | 0.908 | 0.833 | 44.13 | 116.82
SPL [22] | 0.924 | 0.844 | 0.902 | 0.840 | 49.88 | 106.08
SA [23] | 0.921 | 0.855 | 0.904 | 0.858 | 44.74 | 96.98
FullNet [13] | 0.924 | 0.853 | 0.914 | 0.856 | 37.28 | 88.75
QSA [20] | 0.930 | 0.862 | 0.914 | 0.859 | 41.78 | 97.39
SS | 0.872 | 0.765 | 0.853 | 0.797 | 54.86 | 116.33
DT | 0.918 | 0.846 | 0.896 | 0.848 | 41.84 | 90.86
SDT | 0.931 | 0.866 | 0.919 | 0.851 | 32.29 | 82.40

Table 1: Comparison with existing methods on the gland segmentation. Our SDT
achieves better or on par F1 score and Dice Index, and significantly better
Hausdorff distance for evaluating shape similarity. DT and SS represent
distance transform and skeleton with scales.

Setting | F1 Score $\uparrow$ | Dice Index $\uparrow$ | Hausdorff $\downarrow$
---|---|---|---
Part A | Part B | Part A | Part B | Part A | Part B
Loss |  |  |  |  |  |
L1 | 0.916 | 0.842 | 0.903 | 0.850 | 39.76 | 94.83
L2 | 0.896 | 0.833 | 0.885 | 0.837 | 49.11 | 110.24
CE | 0.931 | 0.866 | 0.919 | 0.851 | 32.29 | 82.40
Curvature |  |  |  |  |  |
$\alpha=0.6$ | 0.912 | 0.845 | 0.914 | 0.855 | 36.25 | 91.24
$\alpha=0.8$ | 0.931 | 0.866 | 0.919 | 0.851 | 32.29 | 82.40
$\alpha=1.0$ | 0.926 | 0.858 | 0.907 | 0.849 | 35.73 | 86.73
Skeleton |  |  |  |  |  |
Partial | 0.899 | 0.831 | 0.896 | 0.837 | 47.50 | 105.19
Local | 0.931 | 0.866 | 0.919 | 0.851 | 32.29 | 82.40

Table 2: Ablations studies on the gland dataset. The results suggest that the
model trained with cross-entropy loss with $\alpha=0.8$ and local skeleton
generation achieves the best performance.

Curvature. We also compare different $\alpha$ in Eqn. 2 that controls the
curvature of the energy landscape. Table 2 shows that $\alpha=0.8$ achieves
the best overall performance, which is slightly better than $\alpha=1.0$.
Decreasing $\alpha$ to 0.6 introduces more merges and make the results worse.

Global/Local Skeleton. In Sec. 2.2 we show the inconsistency problem of global
and local skeletons. In this study, we set $\alpha=0.8$ and let the model
learn the pre-computed SDT energy for the training set. The results show that
pre-computed SDT significantly degrades performance (Table 2). We argue this
is because pre-computed energy not only introduces inconsistency for instances
touching the image border but also restricts the diversity of SDT energy maps.

##  4 Conclusion

In this paper, we introduce the skeleton-aware distance transform (SDT) to
capture both the geometry and topological connectivity of instance masks with
complex shapes. For multi-class problems, we can use class-aware semantic
segmentation to mask the SDT energy trained for all objects that is agnostic
to their classes. We hope this work can inspire more research on not only
better representations of object masks but also novel models that can better
predict those representations with shape encoding. We will also explore the
application of SDT in the more challenging 3D instance segmentation setting.

Acknowledgments. This work has been partially supported by NSF awards
IIS-2239688 and IIS-2124179. This work has also been partially supported by
gift funding and GCP credits from Google.

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