Interpolation Theorems for Nonlinear Operators in General Morrey-Type Spaces and Their Applications

Abstract

We prove new interpolation theorems for a sufficiently wide class of nonlinear operators in Morrey-type spaces. In particular, these theorems apply to Urysohn integral operators. We also obtain analogs of the Marcinkiewicz–Calderón and Stein–Weiss–Peetre interpolation theorems and establish a criterion of $(p,q)$ quasiweak boundedness of the Urysohn operator.

1. Introduction

Let $(U,\mu)$ be a space with measure $\mu$ and $\Omega$ be a set. By $G_\Omega=\{G_{t,y}\}_{t>0,\;y\in\Omega}$ we will denote a two-parameter family of $\mu$-measurable sets $G_{t,y}$ satisfying the condition

$$G_{t,y}\subset G_{s,y}\qquad \text{for}\quad 0<t<s\quad \text{and} \quad y\in\Omega.$$

We will call such a family a net. For $y\in\Omega$, we set $G_{\{y\}}=\{G_{t,y}\}_{t>0}$. We say that these nets are generated by the net $G_\Omega$. If the sets $G_{t,y}$, $t>0$, are independent of the parameter $y$, we denote such a net by $G=\{G_t\}_{t>0}$.

Let $0<p,q\leq \infty$ and $0<\lambda< \infty$. Define the space $M^\lambda_{p,q}(G_\Omega,\mu)$ of all $\mu$-measurable functions $f \colon\, U\to{\mathbb R}$ such that

$$\|f\|_{M^\lambda_{p,q}(G_\Omega,\mu)}= \left(\intop_0^ \infty \!\left(t^{-\lambda} \sup_{y\in\Omega}\!\left(\ \intop_{G_{\smash{t,y}}}\!\mathopen|f(x)|^p\,d\mu\right)^{\!\smash{1/p}} \,\right)^{\!q}\frac{dt}{t}\right)^{\!1/q} < \infty$$

for $q< \infty$ and

$$\|f\|_{M^\lambda_{p, \infty }(G_\Omega,\mu)}= \sup_{t>0,\; y\in\Omega}t^{-\lambda} \left(\ \intop_{G_{\smash{t,y}}}\!\mathopen|f(x)|^p\,d\mu\right)^{\!1/p}< \infty$$

for $q= \infty$.

If $U={\mathbb R}^n$, $\mu$ is the Lebesgue measure, and $G_{t,y}=B(y,t)$ (ball of radius $t$ centered at $y$), then we denote this space by $M^\lambda_{p,q,\Omega}$. In particular, for $q= \infty$ and $\Omega={\mathbb R}^n$, this is the classical Morrey space $M_p^\lambda$.

If $U={\mathbb R}^n$, $\mu$ is the Lebesgue measure, $\Omega=\{0\}$, and $G_t=G_{t,0}=B(0,t)$, then $M^\lambda_{p,q}(G_\Omega,\mu)$ is the local Morrey-type space $LM^\lambda_{p,q}$, which was introduced and used to study the properties of the maximal and fractional maximal operators by V. I. Burenkov, H. V. Guliyev, and V. S. Guliyev [4–7]. If $\Omega=\{y\}$ and $G_t=G_{t,y}=B(y,t)$, then we denote the corresponding local Morrey-type space by $LM^\lambda_{p,q,y}$.

The problem of real interpolation of Morrey-type spaces was addressed in [26, 14, 23, 25, 3, 17]. It follows from the results of [23] that the interpolation space $(M_p^{\lambda_0},M_p^{\lambda_1})_{ \theta , \infty }$, where $1\le p< \infty$ and $0<\lambda_0,\lambda_1<n/p$, satisfies the inclusion

$$\bigl(M_p^{\lambda_0}, M_p^{\lambda_1}\bigr)_{ \theta , \infty }\subset M_p^\lambda\qquad \text{with} \quad \lambda=(1- \theta )\lambda_0+ \theta \lambda_1, \quad 0< \theta <1.$$

In [25, 3], this inclusion was shown to be strict.

In [18], Lemarié-Rieusset proved that for $1\le p_0,p_1< \infty$, $0<\lambda_0<n/p_0$, and $0<\lambda_1<n/p_1$, the inclusion

$$\bigl(M_{p_0}^{\lambda_0}, M_{p_1}^{\lambda_1}\bigr)_{ \theta , \infty }\subset M_p^\lambda,\qquad \text{where} \quad \frac{1}{p}=\frac{1- \theta }{p_0}+\frac{ \theta }{p_1},\quad \lambda=(1- \theta )\lambda_0+ \theta \lambda_1, \quad 0< \theta <1,$$

holds if and only if $p_0=p_1$.

In [10, 12, 8], it was shown that the scale of local Morrey-type spaces $LM_{p,q}^\lambda$ with fixed $p$ (in contrast to the scale of Morrey spaces $M^\lambda_p$) is closed under interpolation; namely, it was proved that if $0<p,q_0,q_1,q\leq \infty$ and $\lambda_0\neq\lambda_1$, $0<\lambda_0,\lambda_1< \infty$, then

$$\bigl(LM_{p,q_0}^{\lambda_0},LM_{p,q_1}^{\lambda_1}\bigr)_{ \theta ,q}= LM_{p,q}^\lambda,\qquad \text{where} \quad \lambda=(1- \theta )\lambda_0+ \theta \lambda_1, \quad 0< \theta <1.$$

In [12], the interpolation spaces were described for much more general spaces than $LM_{p,q}^\lambda$.

In [8, 9], the following interpolation theorem for quasi-additive operators was proved.

Theorem 1.

Let $\Omega\subset{\mathbb R}^n$. Let $0<p,q,\sigma,\tau\leq \infty ,$ $0\le\alpha_0,\alpha_1< \infty$ with $\alpha_0,\alpha_1>0$ if $\sigma< \infty ,$ $\alpha_0\neq\alpha_1,$ $0\leq\beta_0,\beta_1< \infty ,$ $\beta_0\neq\beta_1,$ $0< \theta <1,$ and

$$ \alpha=(1- \theta )\alpha_0+ \theta \alpha_1, \qquad \beta=(1- \theta )\beta_0+ \theta \beta_1.$$

(1.1)

Let $T$ be a quasi-additive operator on $\bigcup_{y\in\Omega}\bigl(LM^{\alpha_0}_{p,\sigma,y}+LM^{\alpha_1}_{p,\sigma,y}\bigr)$ with a quasi-additivity constant $A$.

If for some $M_0,M_1>0$ the inequalities

$$ \|Tf\|_{LM^{\beta_i}_{q, \infty ,y}}\leq M_i \|f\|_{LM^{\alpha_i}_{p,\sigma,y}}$$

(1.2)

hold for all $y\in\Omega$ and all functions $f\in LM^{\alpha_i}_{p,\sigma,y},$ $i=0,1,$ then the inequality

$$ \|Tf\|_{M^{\beta}_{q,\tau, \Omega}}\leq cAM_0^{1- \theta }M_1^{ \theta } \|f\|_{M^\alpha_{p,\tau,\Omega}}$$

(1.3)

holds for all functions $f\in M^\alpha_{p,\tau, \Omega},$ where $c>0$ depends only on $\alpha_0,$ $\alpha_1,$ $\beta_0,$ $\beta_1,$ $q,$ $\sigma,$ $\tau,$ and $\theta$.

In this theorem, the strong estimate (1.3) is derived from the weak (in a sense) estimates (1.2).

In the present study, we significantly generalize this result: first, we consider a wider class of nonlinear operators, including the Urysohn integral operator

$$(Tf)(y)=\intop_U\! K(f(x),x,y)\,d\mu;$$

second, we consider a much more general class of Morrey-type spaces; third, we consider a still weaker version of estimates of type (1.2), which leads to a strong estimate of type (1.3).

In Section 2, we present a number of properties of the spaces $M_{p,q}^\lambda(G_\Omega,\mu)$ and give various examples of these spaces. In particular, in Example 1, we demonstrate that the space $M_{p,q}^\lambda(G_\Omega,\mu)$ coincides with the Lorentz space $L_{r,q}(U,\mu)$ for a certain choice of the net $G_\Omega$ and under certain relations between the parameters $p$, $q$, and $\lambda$. In Section 3, we describe the classes of nonlinear operators under consideration (partially quasi-additive operators) and give examples of operators from these classes, including the Urysohn integral operator. In Section 4, we prove general interpolation theorems for partially quasi-additive operators in the spaces $M_{p,q}^\lambda(G_\Omega,\mu)$. In Section 5, using these theorems, we prove an analog of Calderón’s interpolation theorem for partially quasi-additive operators. In Section 6, we prove a weak version of the Stein–Weiss–Peetre interpolation theorem for the class of operators under consideration, including the Urysohn integral operator satisfying the quasiweak boundedness condition. Finally, in Section 7, we prove a quasiweak boundedness criterion for a wide class of Urysohn integral operators. The results for Urysohn integral operators were given without proofs in the authors’ note [11].

Let $F$ and $G$ be nonnegative functions defined on a set $A$. We say that $G$ dominates $F$ on $A$ and write “$F\lesssim G$ on $A$” (or simply $F\lesssim G$ when it is clear what set $A$ is meant) if there exists a $c>0$ such that $F(x)\leq c \kern1pt G(x)$ for all $x\in A$. We also say that $F$ and $G$ are equivalent on $A$ and write “$F\asymp G$ on $A$” if $F\lesssim G$ and $G\lesssim F$ on $A$.

2. Properties of Morrey-type spaces and examples

Lemma 1.

1. Let $0<q_0<q_1\leq \infty$ and $0<p,\lambda< \infty$. Then the following continuous embedding holds:

$$ M^\lambda_{p,q_0}(G_\Omega,\mu)\hookrightarrow M^\lambda_{p,q_1}(G_\Omega,\mu).$$

(2.1)

2. Let $0<p_1<p_2< \infty ,$ $0<q\le \infty ,$ and $0<\lambda_1,d< \infty$. Suppose the net $G_\Omega=\{G_{t,y}\}_{t>0,\;y\in\Omega}$ and measure $\mu$ are such that for some $C>0$

$$\mu(G_{t,y})\leq C t^d$$

for all $y\in\Omega$ and $t>0$. Then the following continuous embedding holds:

$$M^{\lambda_2}_{p_2,q}(G_\Omega,\mu)\hookrightarrow M^{\lambda_1}_{p_1,q}(G_\Omega,\mu),\qquad \textit{where}\quad \lambda_2=\lambda_1-d\biggl( \frac{1}{p_1}-\frac{1}{p_2} \biggr)>0.$$

The proof is similar to that of [9, Lemma 2.1], which was given for the case of $G_{t,y}=B(y,t)$.

Choosing the net $G_\Omega$ and the parameter $\lambda$, we can describe the quasinorms of various spaces using the quasinorm of the space $M^\lambda_{p,q}(G_\Omega,\mu)$. Here are a few examples. Based on these examples and the interpolation theorem for linear and some nonlinear operators in the spaces $M^\lambda_{p,q}(G_\Omega,\mu)$, we will then obtain, as a corollary, interpolation theorems for the corresponding spaces.

Let $f$ be a $\mu$-measurable function defined on $U$. The function

$$f^*(t)=\inf\bigl\{\sigma\ge 0 \colon\, \, \mu(\{x\in U \colon\, \kern1pt |f(x)|>\sigma\})\leq t\bigr\}, \qquad t\ge 0,$$

is called the nonincreasing rearrangement of $f$.

We say that a measure $\mu$ satisfies the regularity condition if for any $\mu$-measurable set $e$ and any $\alpha\in (0,\mu(e)/2]$ there exists a $\mu$-measurable subset $w\subset e$ such that

$$ \alpha\leq \mu(w)\leq 2\alpha.$$

(2.2)

Lemma 2.

Let $\mu$ be a regular measure and $f\in L^{\textrm{loc}}_1(U,\mu)$. Suppose also that the function $f \colon\, U\to{\mathbb C}$ is integrable on all $\mu$-measurable subsets of $U$ of finite measure. Then there exists a net $G(f)=\{G_t(f)\}_{t>0}$ such that

$$t\leq\mu(G_t(f))\leq 2t,$$

(2.3)

$$\intop_0^t\!f^*(s)\,ds\leq \intop_{G_t(f)}\!\mathopen|f(x)|\,d\mu\leq \intop_0^{2t}\!f^*(s)\,ds,$$

(2.4)

and

$$ \bigl(f\bigl(1-\chi^{}_{G_t(f)}\bigr)\bigr)^*(s)\leq f^*(t+s),$$

(2.5)

where $\chi^{}_A$ is the characteristic function of a set $A\subset U$.

Proof.

Let $t>0$. Recall that according to the properties of rearrangements we have

$$ \mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|\geq f^*(t)\bigr\}\bigr) \geq t \qquad\text{and}\qquad \mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|> f^*(t)\bigr\}\bigr)\leq t.$$

(2.6)

Put $\alpha=t-\mu(\{x\in U \colon\, |f(x)|> f^*(t)\})$. In view of (2.6), we have $\alpha \ge 0$. Let us construct the set $G_t(f)$.

If $\alpha>\mu(\{x\in U \colon\, |f(x)|= f^*(t)\})/2$, then we set $G_t(f)= \{x\in U \colon\, |f(x)|\geq f^*(t)\}$, while if $\alpha\leq \mu(\{x\in U \colon\, |f(x)|= f^*(t)\})/2$, then we set $G_t(f)= \{x\in U \colon\, |f(x)|> f^*(t)\}\cup w$, where

$$w\subset\bigl\{x\in U \colon\, \kern1pt |f(x)|= f^*(t)\bigr\},\qquad \alpha\leq \mu(w)\leq 2\alpha \le\mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|= f^*(t)\bigr\}\bigr).$$

Such a set exists since the measure $\mu$ is regular.

Let us prove inequality (2.3). Indeed, in the first case, according to (2.6),

$$\begin{aligned} \, t&\leq \mu(G_t(f))=\mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|> f^*(t)\bigr\}\bigr)+\mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|= f^*(t)\bigr\}\bigr) \\[5pt] &\leq \mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|> f^*(t)\bigr\}\bigr) +2\alpha = 2t-\mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|> f^*(t)\bigr\}\bigr)\leq 2t. \end{aligned}$$

In the second case,

$$\begin{aligned} \, t&=\alpha+\mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|> f^*(t)\bigr\}\bigr)\leq \mu(G_t(f)) =\mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|> f^*(t)\bigr\}\bigr)+\mu(w) \\[5pt] &\leq \mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|> f^*(t)\bigr\}\bigr)+2\alpha\leq 2t. \end{aligned}$$

Moreover,

$$ \sup_{\mu(e)\leq t}\,\intop_e\mathopen|f(x)|\,d\mu\leq \intop_{G_t(f)}\!\mathopen|f(x)|\,d\mu.$$

(2.7)

Indeed,

$$\begin{aligned} \, t&=\alpha+\mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|>f^*(t)\bigr\}\bigr)\leq\mu(w)+\mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|>f^*(t)\bigr\}\bigr) = \mu(G_t(f)) \\[5pt] &\leq \mu\bigl(\bigl\{x\in U \colon\, \kern1pt |f(x)|> f^*(t)\bigr\}\bigr)+2\alpha=t+\alpha\leq 2t. \end{aligned}$$

Let $e$ be an arbitrary $\mu$-measurable set with $\mu(e)\leq t$. It is clear from the definition of the set $G_t(f)$ that $|f(x)|\geq f^*(t)$ for $x\in G_t(f)\setminus e$ and $|f(x)|\leq f^*(t)$ for $x\in e\setminus G_t(f)$. Therefore,

$$\begin{aligned} \, \intop_{G_t(f)}\!\mathopen|f(x)|\,d\mu-\intop_e\mathopen|f(x)|\,d\mu &=\intop_{G_t(f)\setminus e}\!\mathopen|f(x)|\,d\mu-\intop_{e\setminus G_t(f)}\!\mathopen|f(x)|\,d\mu \\[4pt] &\geq f^*(t)\bigl(\mu(G_t(f)\setminus e)-\mu(e\setminus G_t(f))\bigr)\geq f^*(t) \bigl(\mu(G_t(f))-\mu(e)\bigr). \end{aligned}$$

Since $\mu(G_t(f))\geq t$ and $\mu(e)\leq t$, we obtain $f^*(t) (\mu(G_t(f))-\mu(e))\geq 0$, and so

$$\intop_{G_t(f)}\!\mathopen|f(x)|\,d\mu-\intop_e\mathopen|f(x)|\,d\mu\geq 0.$$

Thus, according to the properties of rearrangements and inequalities (2.7) and (2.3),

$$\intop_0^t\!f^*(s)\,ds=\sup_{\mu(e)\leq t}\,\intop_e\mathopen|f(x)|\,d\mu \leq \intop_{G_t(f)}\!\mathopen|f(x)|\,d\mu\leq \intop_0^{\mu(G_t(f))}f^*(s)\,ds \leq \intop_0^{2t}\!f^*(s)\,ds;$$

i.e., (2.4) holds.

Next, for any $s,t>0$, we have $f^*(s+t)=\inf A_{s,t}$, where

$$\begin{aligned} \, A_{s,t}&= \bigl\{\sigma\geq 0 \colon\, \,\mu(\{x\in U \colon\, \kern1pt |f(x)|>\sigma\})\leq s+t\bigr\} \\[5pt] &=\bigl\{\sigma\geq 0 \colon\, \,\mu(\{x\in G_t(f) \colon\, \kern1pt |f(x)|>\sigma\}) +\mu(\{x\in U\setminus G_t(f) \colon\, \kern1pt |f(x)|>\sigma\})\leq s+t\bigr\} \\[5pt] &=\bigl\{\sigma\geq 0 \colon\, \,\mu(\{x\in G_t(f) \colon\, \kern1pt |f(x)|>\sigma\}) +\mu\bigl(\bigl\{x\in U \colon\, \kern1pt \bigl|\bigl(f\bigl(1-\chi^{}_{G_t(f)}\bigr)\bigr)(x)\bigr|>\sigma\bigr\}\bigr) \leq s+t\bigr\}. \end{aligned}$$

It follows from the last representation of the set $A_{s,t}$ that either $\mu(\{x\in G_t(f) \colon\, \kern1pt |f(x)|>\sigma\})\leq t$ or $\mu(\{x\in U \colon\, \kern1pt |(f(1-\chi^{}_{G_t(f)}))(x)|>\sigma\})\leq s$, i.e., $A_{s,t}\subset B_t\cup C_{s,t}$, where

$$\begin{aligned} \, B_t&=\bigl\{\sigma\geq 0 \colon\, \,\mu(\{x\in G_t(f) \colon\, \kern1pt |f(x)|>\sigma\})\leq t\bigr\},\\[5pt] C_{s,t}&= \bigl\{\sigma\geq 0 \colon\, \,\mu\bigl(\bigl\{x\in U \colon\, \kern1pt \bigl|\bigl(f\bigl(1-\chi^{}_{G_t(f)}\bigr)\bigr)(x)\bigr|>\sigma\bigr\}\bigr)\leq s\bigr\}. \end{aligned}$$

Since $|f(x)|\geq f^*(t)$ for $x\in G_t(f)$ and $\bigl|\bigl(f\bigl(1-\chi^{}_{G_t(f)}\bigr)\bigr)(x)\bigr|\leq f^*(t)$ for any $x\in U$, we have

$$\begin{aligned} \, f^*(t+s)&=\inf A_{s,t}\geq \inf(B_t\cup C_{s,t})=\min\{\inf B_t,\inf C_{s,t}\} \\[5pt] &\geq \min\bigl\{f^*(t), \bigl(f\bigl(1-\chi^{}_{G_t(f)}\bigr)\bigr)^*(t)\bigr\} =\bigl(f\bigl(1-\chi^{}_{G_t(f)}\bigr)\bigr)^*(t). \ \quad\Box\end{aligned}$$

Example 1.

Let $0<r< \infty$ and $0<q\leq \infty$. The Lorentz space $L_{r,q}(U,\mu)$ is defined as the set of all $\mu$-measurable functions $f$ such that

$$\|f\|_{L_{r,q}(U,\mu)}=\left(\intop_0^ \infty \bigl(t^{1/r}f^*(t)\bigr)^q \kern1pt \frac{dt}{t} \right)^{\!1/q}< \infty .$$

Let $\mu$ be a regular measure, $1<r< \infty$, and $\lambda=1/r'$, where $1/r+1/r'=1$. Then

$$ \|f\|_{L_{r,q}(U,\mu)}\asymp \|f\|_{M_{1,q}^{1/r'}(G,\mu)},$$

(2.8)

where the implied equivalence constants depend only on the parameters $q$ and $r$.

Indeed, it is well known that for $1<r< \infty$ the Lorentz space $L_{r,q}(U,\mu)$ can be equivalently defined in terms of the average of the nonincreasing rearrangement $f^{**}(t)=t^{-1}\intop_0^t f^*(s)\,ds$. By Lemma 2 there exists a net $G(f)=\{G_t\}_{t>0}$ for which inequalities (2.4) hold. Therefore,

$$\begin{aligned} \, \|f\|_{L_{r,q}} &\asymp \left(\intop_0^ \infty \bigl(t^{1/r}f^{**}(t)\bigr)^q \kern1pt \frac{dt}{t} \right)^{\!1/q} = \left(\intop_0^ \infty \!\left(t^{1/r-1}\intop_0^t\! f^*(s)\,ds\right)^{\!q} \frac{dt}{t} \right)^{\!1/q} \\[4pt] &\asymp \left(\intop_0^ \infty \!\left(t^{1/r-1}\intop_{G_t\smash{(f)}}\!\mathopen|f(x)|\,dx\right)^{\!q}\frac{dt}{t} \right)^{\!1/q} =\|f\|_{M_{1,q}^{1/r'}(G,\mu)}. \end{aligned}$$

Example 2.

Let $w$ be a positive $\mu$-measurable function. Put $G_t=\{x\in U \colon\, 1/\omega(x)<t\}$ for $t>0$. Then, up to a constant factor, $\|f\|_{M_{p,p}^\lambda(G,\mu)}$ coincides with

$$\|f\|_{L_p(w^\lambda)}=\left(\,\intop_U\bigl(w^\lambda(x)|f(x)|\bigr)^p\,d\mu\right)^{\!1/p}.$$

Example 3.

Let $0<p\leq \infty$, $U={\mathbb R}^n,$ and $\mu$ be the Lebesgue measure on ${\mathbb R}^n$. Let $v$ be a positive strictly increasing locally absolutely continuous function on $(0, \infty )$ such that

$$\lim_{t\to +0}v(t)=0 \qquad\text{and}\qquad \lim_{t\to + \infty }v(t)= \infty .$$

If $G=\{G_t\}_{t>0}$, where $G_t=B(0,v^ {-1} (t))$, $t>0$, then the space $M_{p,q}^1(G)$ coincides with one of the variants of the general local Morrey-type space $LM_{p,q}^{v(\cdot)}$ of all functions $f \in L_p^{\textrm{loc}}({\mathbb R}^n)$ with finite quasinorm

$$\|f\|_{LM_{p,q}^{v(\cdot)}}=\left(\intop_0^ \infty \bigl((v(r))^{-1} \|f\|_{L_p(B(0,r))}\bigr)^q \kern1pt \frac{dv(r)}{v(r)} \right)^{\!1/q}$$

(with the usual modification for $q= \infty$). Setting $v(r)=t$, we have

$$\|f\|_{LM_{p,q}^{v(\cdot)}}=\|f\|_{M^1_{p,q}(G)}.$$

Moreover, for $0<\lambda < \infty$, we have $M^\lambda_{p,q}(G)=LM_{p,q}^{v^\lambda(\cdot)}$ and

$$\|f\|_{LM_{p,q}^{v^\lambda(\cdot)}}=\lambda^{1/q}\|f\|_{M^\lambda_{p,q}(G)}.$$

Example 4.

Let $0<p,q\leq \infty$ and $w$ be a positive measurable function on $(0, \infty )$ such that $\|w\|_{L_q(t, \infty )}< \infty$ for any $t>0$,

$$\lim_{t\to +0}\mathopen\|w\|_{L_q(t, \infty )}= \infty \qquad\text{and}\qquad \lim_{t\to + \infty }\mathopen\|w\|_{L_ \infty (t, \infty )}=0\quad\text{for } \ q= \infty$$

(for $q< \infty$, the equality $\lim_{t\to+ \infty }\mathopen\|w\|_{L_q(t, \infty )}=0$ is obvious). Let us set $v(t)=q^{-1/q}\|w\|^{-1}_{L_q(t, \infty )}$ in the previous example. Then the space $M^1_{p,q}(G)$ coincides with a variant of the general local Morrey-type space $LM_{p,q,w(\cdot)}$ of all functions $f\in L^{\textrm{loc}}({\mathbb R}^n)$ with finite quasinorm

$$\|f\|_{LM_{p,q,w(\cdot)}}=\bigl\|w(r)\|f\|_{L_p(B(0,r))}\bigr\|_{L_q(0, \infty )}.$$

Moreover,

$$\|f\|_{LM_{p,q,w(\cdot)}}=\|f\|_{M^1_{p,q}(G)},$$

since $(v(r))^{-q-1}v'(r)=w(r)$ for a.e. $r>0$.

3. Some classes of nonlinear operators, examples

Let $(U,\mu)$ be a space with measure $\mu$ and $Z(U)$ be a linear space of $\mu$-measurable functions $f \colon\, U\to {\mathbb R}$ such that $f\chi^{}_w\in Z(U)$ for any function $f\in Z(U)$ and any $\mu$-measurable set $w\subset U$. Let $(V,\nu)$ be a space with measure $\nu$ and $M(V)$ be the space of all $\nu$-measurable functions $f \colon\, V\to{\mathbb R}$.

An operator $T$ is said to be quasi-additive on $Z(U)$ if $T \colon\, Z(U)\to M(V)$ and there exists an $A\geq 1$ such that

$$ |(T(f+g))(x)| \le A \bigl(|(Tf)(x)| + |(Tg)(x)|\bigr) \qquad\text{for a.e. }\, x\in V$$

(3.1)

for any functions $f,g\in Z(U)$.

We say that an operator $T$ is partially additive on $Z(U)$ if $T \colon\, Z(U)\to M(V)$ and for any function $f\in Z(U)$ and any $\mu$-measurable set $w\subset U$ one has

$$ (Tf)(x)=(T(f\chi^{}_w))(x) + (T(f\chi^{}_{U\setminus w}))(x) \qquad\text{for a.e. }\, x\in V.$$

(3.2)

We say that an operator $T$ is partially quasi-additive on $Z(U)$ if $T \colon\, Z(U)\to M(V)$ and there exists an $A\geq 1$ such that for any function $f\in Z(U)$ and any $\mu$-measurable set $w\subset U$ one has

$$ |(T f)(x)| \le A\bigl(|(T(f\chi^{}_w))(x)| + |(T(f\chi^{}_{U\setminus w}))(x)|\bigr) \qquad\text{for a.e. }\, x\in V$$

(3.3)

(i.e., inequality (3.1) holds with $f$ and $g$ changed to $f\chi^{}_w$ and $f\chi^{}_{U\setminus w}$, respectively). Here $A$ is a partial quasi-additivity constant.

If $A=1$ in (3.1), then $T$ is a subadditive operator; accordingly, if $A=1$ in (3.3), then $T$ is a partially subadditive operator.

Let $K \colon\, E\times U\times V\to {\mathbb R}$, where $E=\bigcup_{f\in Z}f(U)$, and $T \colon\, Z(U)\to M(V)$ be an operator defined as follows: for any $f\in Z(U)$,

$$ (Tf)(y)=\intop_U\! K(f(x),x,y)\,d\mu$$

(3.4)

under the assumption that this integral exists and is finite for a.e. $y\in V$. This operator is called the Urysohn integral operator. For the properties of the Urysohn operator and, in particular, the conditions on $K$ under which this integral exists and is finite for a.e. $y\in V$, we refer the reader to the book [16] and papers [21, 22].

In particular, if $f\in L_p(U,\mu)$,

$$|K(z,x,y)|\leq c |z|^\alpha|K(1,x,y)|\qquad\text{for all} \quad x\in U,\quad y\in V,\quad z\in f(U)$$

with some $0\le\alpha\leq p$ and $c>0$, and $\|K(1,\cdot \kern1pt ,y)\|_{L_r(U,\mu)}< \infty$ for a.e. $y\in V$, where $1/r=1-\alpha/p$, then the integral in (3.4) exists and is finite for a.e. $y\in V$.

Lemma 3.

For any function $f\in Z(U),$ any $\mu$ -measurable set $w\subset U,$ and any $y\in V,$

$$ (Tf)(y)=(T(f\chi^{}_w))(y)+(T(f\chi^{}_{U\setminus w}))(y)-(T(0))(y).$$

(3.5)

Proof.

Since the integral is additive with respect to the measure,

$$\begin{aligned} \, (Tf)(y)-{}&(T(f\chi^{}_w))(y) =\intop_U\! K(f(x),x,y)\,d\mu -\intop_U\! K\bigl(f(x)\chi^{}_w(x),x,y\bigr)\,d\mu \\ &=\intop_{U\setminus w}\! K\bigl(f(x)\chi^{}_{U\setminus w}(x),x,y\bigr)\,d\mu+\intop_w\! K\bigl(f(x)\chi^{}_w (x),x,y\bigr)\,d\mu \\ &\phantom{={}}{}-\intop_{U\setminus w}\! K(0,x,y)\,d\mu- \intop_w\! K\bigl(f(x)\chi^{}_w(x),x,y\bigr)\,d\mu \\ &=\intop_U\! K\bigl(f(x)\chi^{}_{U\setminus w} (x),x,y\bigr)\,d\mu -\intop_w\! K(0,x,y)\,d\mu -\intop_{U\setminus w}\! K(0,x,y)\,d\mu \\ &=\intop_U\! K\bigl(f(x)\chi^{}_{U\setminus w}(x),x,y\bigr)\,d\mu -\intop_U\! K(0,x,y)\,d\mu = (T(f\chi^{}_{U\setminus w}))(y)-(T(0))(y). \ \quad\Box\end{aligned}$$

Corollary 1.

If

$$ (T(0))(y)=\intop_U\! K(0,x,y)\,d\mu=0 \qquad\textit{for a.e. }\, y\in V,$$

(3.6)

then $T$ is a partially additive operator.

Corollary 2.

If, for any $f\in Z(U),$

$$ (Tf)(y)\geq 0 \qquad\textit{for a.e. }\, y\in V$$

(3.7)

(in particular, if the kernel $K$ is nonnegative), then $T$ is a partially subadditive operator.

Proof.

According to (3.5), we have

$$\begin{aligned} \, |(Tf)(y)|&=(Tf)(y)=(T(f\chi^{}_w))(y)+(T(f\chi^{}_{U\setminus w}))(y)-(T(0))(y) \\[4pt] &\le (T(f\chi^{}_w))(y)+(T(f\chi^{}_{U\setminus w}))(y) = |(T(f\chi^{}_w))(y)|+|(T(f\chi^{}_{U\setminus w}))(y)|. \ \quad\Box\end{aligned}$$

Corollary 3.

If there is a $B>0$ such that for any $f\in Z(U)$

$$ |(T(0))(y)|\leq B|(Tf)(y)| \qquad\textit{for a.e. }\, y\in V,$$

(3.8)

then $T$ is a partially quasi-additive operator with partial quasi-additivity constant $A=B+1$.

Proof.

According to (3.5), we have

$$\begin{aligned} \, |(Tf)(y)|&\leq|(T(f\chi^{}_w))(y)|+|(T(f\chi^{}_{U\setminus w}))(y)|+|(T(0))(y)| \\[4pt] &\leq |(T(f\chi^{}_w))(y)|+|(T(f\chi^{}_{U\setminus w}))(y)|+B|(T(f\chi^{}_w))(y)| \\[4pt] &\leq(B+1)\bigl(|(T(f\chi^{}_w))(y)|+|(T(f\chi^{}_{U\setminus w}))(y)|\bigr). \ \quad\Box\end{aligned}$$

Remark 1.

If we are interested in the classical interpolation theorems for the Urysohn integral operator and, hence, in estimates of the form

$$ \|Tf\|_Y\leq C \|f\|_X, \qquad f\in X,$$

(3.9)

where $X$ and $Y$ are some quasinormed function spaces and $C>0$ is independent of $f$, then condition (3.6) is necessary. (If we take $f=0$ in (3.9), then $(Tf)(y)=0$ for a.e. $y\in V$.)

4. Interpolation theorems for Morrey-type spaces

Lemma 4 (Hardy inequalities).

Let $\xi>0$ and $0<\sigma\leq \tau\leq \infty$. Then, for any nonnegative Lebesgue measurable function $g$ on $(0, \infty ),$

$$\left(\intop_0^ \infty \! \left(t^{-\xi}\!\left(\intop_0^t (g(r))^\sigma \kern1pt \frac{dr}{r} \right)^{\!\smash{1/\sigma}}\,\right)^{\!\tau} \frac{dt}{t} \right)^{\!1/\tau} \leq (\xi\sigma)^{-1/\sigma}\!\left(\intop_0^ \infty \bigl(t^{-\xi}g(t)\bigr)^\tau \kern1pt \frac{dt}{t} \right)^{\!1/\tau}$$

and

$$\left(\intop_0^ \infty \!\left(t^\xi\!\left(\intop_t^ \infty (g(r))^\sigma \kern1pt \frac{dr}{r} \right)^{\!\smash{1/\sigma}}\,\right)^{\!\tau} \frac{dt}{t} \right)^{\!1/\tau} \leq (\xi\sigma)^{-1/\sigma}\!\left(\intop_0^ \infty \bigl(t^\xi g(t)\bigr)^\tau \kern1pt \frac{dt}{t} \right)^{\!1/\tau}.$$

Let $(U,\mu)$ be a space with measure $\mu,$ $s>0,$ and $G_\Omega=\{G_{t,y}\}_{t>0,\; y\in\Omega}$ be a net. Define the nets $\check{G}{} _\Omega^s=\{ \check{G}{} ^s_{t,y}\}_{t>0,\; y\in\Omega}$, $\widehat{G}{} _\Omega^s=\{ \widehat{G}{} ^s_{t,y}\}_{t>0,\; y\in\Omega}$ and $\check{G}{} _{\{y\}}^s=\{ \check{G}{} ^s_{t,y}\}_{t>0}$, $\widehat{G}{} _{\{y\}}^s=\{ \widehat{G}{} ^s_{t,y}\}_{t>0}$ for $y\in\Omega$, where

$$ \check{G}{} ^s_{t,y}= \begin{cases} \displaystyle G_{t,y} & \text{ if }\, \displaystyle s\geq t,\\\displaystyle G_{s,y} & \text{ if }\, \displaystyle s<t \end{cases}\qquad\text{and}\qquad \widehat{G}{} ^s_{t,y}= \begin{cases} \displaystyle \varnothing & \text{ if }\, \displaystyle s\geq t,\\\displaystyle G_{t,y} & \text{ if }\, \displaystyle s<t. \end{cases}$$

(4.1)

Remark 2.

1. If $G=\{G_t\}_{t>0}$ is a filtration, i.e., $G$ is a system of expanding $\sigma$-algebras of measurable sets, then, in the theory of stochastic processes, the procedure defined by the first relation in (4.1) is called a stop corresponding to the moment $s$, while the procedure defined by the second relation in (4.1) defines a start corresponding to the moment $s$. These transformations play an important role in constructing interpolation methods for stochastic processes [1, 2, 19]. In this section, we present the main results of the study, in which these transformations also play a significant role.

2. Note that

$$\|f\|_{M^\alpha_{p,\sigma} \left(\smash{ \widehat{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) } =\left(\intop_s^ \infty \bigl(t^{-\alpha}\|f\|_{L_p(G_{t,y},\mu)}\bigr)^\sigma \kern1pt \frac{dt}{t} \right)^{\!1/\sigma}$$

and

$$\begin{aligned} \, \|f\|_{M^\alpha_{p,\sigma} \left(\smash{ \check{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) } &= \left(\intop_0^s\bigl(r^{-\alpha}\|f\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} +\intop_s^ \infty \bigl(r^{-\alpha}\|f\|_{L_p(G_{s,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!1/\sigma} \\[2pt] &= \left(\intop_0^s\bigl(r^{-\alpha} \|f\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} +\|f\|^\sigma_{L_p(G_{s,y},\mu)}\intop_s^ \infty r^{-\alpha\sigma} \kern1pt \frac{dr}{r} \right)^{\!1/\sigma} \\[2pt] &= \left(\intop_0^s\bigl(r^{-\alpha} \|f\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} + (\alpha\sigma)^{-1}s^{-\alpha\sigma} \|f\|^\sigma_{L_p(G_{s,y},\mu)} \right)^{\!1/\sigma} \\[2pt] &\asymp s^{-\alpha} \|f\|_{L_p(G_{s,y},\mu)}+\left(\intop_0^s\bigl(r^{-\alpha}\|f\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!1/\sigma}. \end{aligned}$$

Lemma 5.

Let $0<p,\sigma,\tau\leq \infty$ and $0\le\alpha_0<\alpha<\alpha_1$ with $\alpha_0>0$ if $\sigma< \infty ,$ and let $f\in M^\alpha_{p,\tau}(G_\Omega,\mu)$. Then, for arbitrary $y\in\Omega$ and $s>0,$

$$\|f\|_{M^{\alpha_0}_{p,\sigma} \left(\smash{ \check{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) }< \infty \qquad\textit{and}\qquad \bigl\|f\bigl(1-\chi^{}_{G_{s,y}}\bigr)\bigr\|_{M^{\alpha_1}_{p,\sigma}(G_{\{y\}},\mu)} \leq \|f\|_{M_{p,\sigma}^{\alpha_1} \left(\smash{ \widehat{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) } < \infty .$$

Proof.

Let $f\in M^\alpha_{p,\tau}(G_\Omega,\mu)$, $y\in\Omega$, and $s>0$. Then

$$\begin{aligned} \, \|f\|_{M^{\alpha_0}_{p,\sigma}( \check{G}{} _{\{y\}},\mu)} &\asymp s^{-\alpha_0}\|f\|_{L_p(G_{s,y},\mu)}+\left(\intop_0^s\bigl(r^{-\alpha_0}\|f\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!1/\sigma} \\[2pt] &\leq s^{-\alpha_0}\|f\|_{L_p(G_{s,y},\mu)}+\|f\|_{M^\alpha_{p, \infty }(G_\Omega,\mu)}\! \left(\intop_0^s r^{(\alpha-\alpha_0)\sigma} \kern1pt \frac{dr}{r} \right)^{\!1/\sigma} < \infty . \end{aligned}$$

To obtain the second relation, notice that

$$\begin{aligned} \, \bigl\|f\bigl(1-\chi^{}_{G_{s,y}}\bigr)\bigr\|_{M^{\alpha_1}_{p,\sigma}(G_{\{y\}},\mu)} &=\left(\intop_s^ \infty \bigl(r^{-\alpha_1}\bigl\|f\bigl(1-\chi^{}_{G_{s,y}}\bigr)\bigr\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!1/\sigma} \\[2pt] &\leq \left(\intop_s^ \infty (r^{-\alpha_1}\|f\|_{L_p(G_{r,y},\mu)})^\sigma \kern1pt \frac{dr}{r} \right)^{\!1/\sigma} = \|f\|_{M_{p,\sigma}^{\alpha_1} \left(\smash{ \widehat{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) } \end{aligned}$$

and

$$\|f\|_{M_{p,\sigma}^{\alpha_1} \left(\smash{ \widehat{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) } = \left(\intop_s^ \infty \bigl(r^{-\alpha_1}\|f\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!1/\sigma} \leq M^\alpha_{p, \infty }(G_\Omega,\mu)\! \left(\intop_s^ \infty r^{(\alpha-\alpha_1)\sigma} \kern1pt \frac{dr}{r} \right)^{\!1/\sigma}< \infty .\ \quad\Box$$

Theorem 2.

Let $(U,\mu)$ and $(V,\nu)$ be spaces with measures $\mu$ and $\nu,$ and let $\Omega\subset{\mathbb R}^n$.

Let $G_\Omega=\{G_{t,y}\}_{t>0,\; y\in\Omega}$ be a net in $U,$ $\check{G}{} _\Omega^s$ and $\widehat{G}{} _\Omega^s$ for $s>0$ be the nets in $U$ defined in (4.1), and $\check{G}{} _{\{y\}}^s$ and $\widehat{G}{} _{\{y\}}^s$ for $s>0$ and $y\in\Omega$ be the nets generated by $\check{G}{} _\Omega^s$ and $\widehat{G}{} _\Omega^s$ in $U$.

Let $F_\Omega=\{F_{t,y}\}_{t>0,\; y\in\Omega}$ be a net in $V$ and $F_{\{y\}}$ for $y\in\Omega$ be the nets generated by $F_\Omega$ in $V$.

Let $0<p,q,\sigma,\tau\leq \infty ,$ $0\le\alpha_0,\alpha_1< \infty$ with $\alpha_0,\alpha_1>0$ if $\sigma< \infty ,$ $\alpha_0\neq \alpha_1,$ $0\leq \beta_0,\beta_1< \infty ,$ $\beta_0\neq \beta_1,$ and $0< \theta <1,$ and let $\alpha$ and $\beta$ be as defined in (1.1).

Let $f\in M^\alpha_{p,\tau}(G_\Omega,\mu),$ $U_f=\{f\chi^{}_w \colon\, w\subset U \text{ is a } \mu\text{-measurable subset}\},$ and $T$ be a quasi-additive operator on $U_f$ with quasi-additivity constant $A$.

If for some $M_0,M_1>0$ the inequalities

$$\bigl\|T\bigl(f\chi^{}_{G_{s,y}}\bigr)\bigr\|_{M^{\beta_0}_{q, \infty }(F_{\{y\}},\nu)} \leq M_0\|f\|_{M^{\alpha_0}_{p,\sigma} \left(\smash{ \check{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) },$$

(4.2)

$$\bigl\|T\bigl(f\bigl(1-\chi^{}_{G_{s,y}}\bigr)\bigr)\bigr\|_{M^{\beta_1}_{q, \infty }(F_{\{y\}},\nu)} \leq M_1 \|f\|_{M_{p,\sigma}^{\alpha_1} \left(\smash{ \widehat{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) }$$

(4.3)

hold for all $y\in\Omega$ and $s>0,$ then the inequality

$$ \|Tf\|_{M^{\beta}_{q,\tau}(F_\Omega,\nu)}\leq cAM_0^{1- \theta }M_1^{ \theta } \|f\|_{M^\alpha_{p,\tau}(G_\Omega,\mu)}$$

(4.4)

holds with a constant $c>0$ depending only on $\alpha_0,$ $\alpha_1,$ $\beta_0,$ $\beta_1,$ $q,$ $\sigma,$ $\tau,$ and $\theta$.

Remark 3.

1. According to Lemma 5, for a fixed function $f\in M^\alpha_{p,q}(G_\Omega,\mu)$, the norms in the hypotheses of the theorem are well defined.

2. This theorem, albeit similar to the classical interpolation theorems, has a significant difference. The point is that the hypotheses and the statement of the theorem are formulated for a fixed function $f\in M^\alpha_{p,q}(G_\Omega,\mu)$. We can say that here we deal not with interpolation of operators but rather with interpolation of inequalities for a fixed function. This fact makes the statement more universal for applications (see Section 5). In particular, the sets $G_{s,y}$ can be chosen depending on $f$.

3. Consider the nonlinear integral operator

$$(Tf)(y)=\intop_U\! K(f(x),x,y)\,d\mu\qquad \text{such that}\quad (T(0))(y)=\intop_U\! K(0,x,y)\,d\mu \neq 0.$$

The classical interpolation theorems do not apply to this operator (see Section 3). At the same time, if $T$ satisfies one of the conditions (3.7) or (3.8), Theorem 2 makes sense.

Proof of Theorem 2 .

Without loss of generality, we assume that $\alpha_0<\alpha_1$. Since the embedding $M^\alpha_{p,\sigma_1}(G_{\{y\}},\mu)\hookrightarrow M^\alpha_{p,\sigma_2}(G_{\{y\}},\mu)$ holds for $\sigma_1<\sigma_2$, we can also assume without loss of generality that $0<\sigma\leq\tau\leq \infty$.

Let $f\in M^\alpha_{p,\tau}(G_\Omega,\mu)$, $y\in\Omega$, and $s>0$. Since $f\chi^{}_w\in M^\alpha_{p,\tau}(G_\Omega,\mu)$ for any $\mu$-measurable set $w\subset U$ and the operator $T$ is quasi-additive on the set $U_f$, it follows that

$$\begin{aligned} \, \|Tf\|_{L_q(F_{t,y},\nu)} &\leq A\bigl\| \kern1pt |T(f\chi^{}_{G_{s,y}})|+|T(f(1-\chi^{}_{G_{s,y}}))| \kern1pt \bigr\|_{L_q(F_{t,y},\nu)} \\[4pt] &\leq 2^{(1/q-1)_+}A\bigl(\|T(f\chi^{}_{G_{s,y}})\|_{L_q(F_{t,y},\nu)}+\|T(f(1-\chi^{}_{G_{s,y}}))\|_{L_q(F_{t,y},\nu)}\bigr). \end{aligned}$$

From condition (4.2), we have

$$\begin{aligned} \, \|T(f\chi^{}_{G_{s,y}})\|_{L_q(F_{t,y},\nu)} &=t^{\beta_0}t^{-\beta_0}\|T(f\chi^{}_{G_{s,y}})\|_{L_q(F_{t,y},\nu)} \leq t^{\beta_0}\sup_{r>0}r^{-\beta_0}\|T(f\chi^{}_{G_{s,y}})\|_{L_q(F_{r,y}\nu)} \\[4pt] &= t^{\beta_0}\|T(f\chi^{}_{G_{s,y}})\|_{M^{\beta_0}_{q, \infty ,y}(F,\nu)} \leq M_0 t^{\beta_0}\|f\|_{M^{\alpha_0}_{p,\sigma}( \check{G}{} _{\{y\}},\mu)} \\[2pt] &\asymp M_0t^{\beta_0} \left(s^{-\alpha}\|f\|_{L_p(G_{s,y},\mu)} +\left(\intop_0^s\bigl(r^{-\alpha}\|f\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!\smash{1/\sigma}}\,\right)\!. \end{aligned}$$

Hence, the following inequality holds for arbitrary $s>0$ and $y\in\Omega$:

$$\|T(f\chi^{}_{G_{s,y}})\|_{L_q(F_{t,y},\nu)} \leq c_1 M_0 t^{\beta_0} \!\left[\left(\intop_0^s\bigl(r^{-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!\smash{1/\sigma}}+s^{-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{s,y},\mu)}\right]\!,$$

where $c_1>0$ depends only on $\alpha_0$, $\alpha_1$, and $\sigma$.

Let us use condition (4.3) to estimate the second term:

$$\begin{aligned} \, &\|T(f(1-\chi^{}_{G_{s,y}}))\|_{L_q(F_{t,y},\nu)}=t^{\beta_1}t^{-\beta_1}\|T(f(1-\chi^{}_{G_{s,y}}))\|_{L_q(F_{t,y},\nu)} \leq M_1 t^{\beta_1}\|f\|_{M_{p,\sigma}^{\alpha_1} \left(\smash{ \widehat{G}{} ^s_{\{y\}},\mu}\vphantom{\widehat G}\right) } \\[2pt] &\qquad=M_1 t^{\beta_1}\!\left(\intop_s^ \infty \bigl(t^{-\alpha_1}\|f\|_{L_p(G_{t,y},\mu)}\bigr)^\sigma \kern1pt \frac{dt}{t} \right)^{\!1/\sigma} \leq M_1 t^{\beta_1}\!\left(\intop_s^ \infty \biggl(t^{-\alpha_1}\sup_{y\in\Omega} \|f\|_{L_p(G_{t,y},\mu)} \biggr)^{\!\sigma} \frac{dt}{t} \right)^{\!1/\sigma}. \end{aligned}$$

Thus, for all $t>0$, $s>0$, and $y\in\Omega$, we obtain

$$\begin{aligned} \, \|Tf\|_{L_q(F_{t,y})} &\leq c_2 A\!\left(M_0 t^{\beta_0}\!\left[ \left(\intop_0^s\biggl(r^{-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{r,y})} \biggr)^{\!\sigma} \frac{dr}{r} \right)^{\!\smash{1/\sigma}}+s^{-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{s,y},\mu)}\right]\right. \\[4pt] &\left.\!\phantom{={}}{}+ M_1t^{\beta_1}\!\left(\intop_s^ \infty \biggl(t^{-\alpha_1}\sup_{y\in\Omega} \|f\|_{L_p(G_{t,y},\mu)} \biggr)^{\!\sigma} \frac{dt}{t} \right)^{\!\smash{1/\sigma}}\,\right)\!, \end{aligned}$$

where $c_2>0$ depends only on $q$, $\alpha_0$, $\alpha_1$, and $\sigma$.

Set $s=ct^\gamma$, where $\gamma=(\beta_1-\beta_0)/(\alpha_1-\alpha_0)$ and $c>0$ is a constant to be chosen later. Then

$$\|Tf\|_{M^{\beta}_{q,\tau}(G_\Omega,\mu)} =\left(\intop_0^ \infty \biggl(t^{-\beta}\sup_{x\in\Omega} \|Tf\|_{L_q(F_{t,y})} \biggr)^{\!\tau} \frac{dt}{t} \right)^{\!1/\tau} \leq 3^{(1/\tau-1)_+} c_2 A(M_0I_1+M_0I_2+M_1I_3),$$

where

$$\begin{aligned} \, I_1&=\left(\intop_0^ \infty \!\left(t^{\beta_0-\beta}\! \left(\intop_0^{ct^\gamma}\biggl(r^{-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{r,y})} \biggr)^{\!\sigma} \frac{dr}{r} \right)^{\!\smash{1/\sigma}}\,\right)^{\!\tau} \frac{dt}{t} \right)^{\!1/\tau}, \\[3pt] I_2&=\left(\intop_0^ \infty \biggl(t^{\beta_0-\beta}(ct^\gamma)^{-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{ct^\gamma,y},\mu)} \biggr)^{\!\tau} \frac{dt}{t} \right)^{\!1/\tau}, \\[3pt] I_3&=\left(\intop_0^ \infty \!\left(t^{\beta_1-\beta}\!\left(\;\intop_{ct^\gamma}^ \infty \biggl(r^{-\alpha_1}\sup_{y\in\Omega} \|f\|_{L_p(G_{r+ct^\gamma,y})} \biggr)^{\!\sigma} \frac{dr}{r} \right)^{\!\smash{1/\sigma}}\,\right)^{\!\tau}\frac{dt}{t} \right)^{\!1/\tau}. \end{aligned}$$

Making the change $ct^\gamma\to t$, we obtain

$$I_1=\gamma^{-1/\tau}c^{ \theta (\alpha_1-\alpha_0)}J_1,\qquad I_2=\gamma^{-1/\tau}c^{ \theta (\alpha_1-\alpha_0)}J_2, \qquad I_3=\gamma^{-1/\sigma}c^{-(1- \theta )(\alpha_1-\alpha_0)}J_3,$$

where

$$\begin{aligned} \, J_1&= \left(\intop_0^ \infty \!\left(t^{- \theta (\alpha_1-\alpha_0)}\!\left(\intop_0^t\biggl(r^{-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{r,y},\mu)} \biggr)^{\!\sigma} \frac{dr}{r} \right)^{\!\smash{1/\sigma}}\,\right)^{\!\tau}\frac{dt}{t} \right)^{\!1/\tau}, \\[3pt] J_2&= \left(\intop_0^ \infty \biggl(t^{- \theta (\alpha_1-\alpha_0)-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{t,y},\mu)} \biggr)^{\!\tau}\frac{dt}{t} \right)^{\!1/\tau}=\|f\|_{M^\alpha_{p,\tau}(G_\Omega,\mu)}, \\[3pt] J_3&= \left(\intop_0^ \infty \!\left(t^{(1- \theta )(\alpha_1-\alpha_0)}\!\left(\intop_0^ \infty \biggl(r^d(r+t)^{-\alpha_1-d}\sup_{y\in\Omega} \|f\|_{L_p(G_{r+t,y})} \biggr)^{\!\sigma}\frac{dr}{r} \right)^{\!\smash{1/\sigma}} \,\right)^{\!\tau}\frac{dt}{t} \right)^{\!1/\tau}. \end{aligned}$$

To estimate $J_1$ and $J_3$, we apply the Hardy inequalities (Lemma 4), according to which we have

$$\begin{aligned} \, J_1&\leq ( \theta (\alpha_1-\alpha_0)\sigma)^{-1/\sigma}\!\left(\intop_0^ \infty \biggl(t^{-\alpha}\sup_{y\in\Omega} \|f\|_{L_p(G_{t,y})} \biggr)^{\!\tau} \frac{dt}{t} \right)^{\!1/\tau} =( \theta (\alpha_1-\alpha_0)\sigma)^{-1/\sigma}\|f\|_{M^\alpha_{p,\tau}(G_\Omega,\mu)}, \\[3pt] J_2&\leq ( \theta (\alpha_1-\alpha_0)\sigma)^{-1/\sigma}\|f\|_{M^\alpha_{p,\tau}(G_\Omega,\mu)}, \\[5pt] J_3&\leq ((1- \theta )(\alpha_1-\alpha_0)\sigma)^{-1/\sigma}\|f\|_{M^\alpha_{p,\tau}(G_\Omega,\mu)}. \end{aligned}$$

For all $c>0$, we obtain

$$\|Tf\|_{M^{\beta}_{q,\tau}(F_\Omega,\nu)} \leq c_3\bigl(M_0 c^{ \theta (\alpha_1-\alpha_0)}+M_1c^{-(1- \theta )(\alpha_1-\alpha_0)}\bigr)\|f\|_{M^\alpha_{p,\tau}(G_\Omega,\mu)},$$

where $c_3>0$ depends only on $\alpha_0$, $\alpha_1$, $\beta_0$, $\beta_1$, $q$, $\sigma$, $\tau$, and $\theta$.

Now we set $c=(M_1/M_0)^{1/(\alpha_1-\alpha_0)}$. Then

$$\|Tf\|_{M^{\beta}_{q,\tau}(F_\Omega,\nu)}\leq 2c_3M_0^{1- \theta }M_1^{ \theta }\|f\|_{M^\alpha_{p,\tau}(G_\Omega,\mu)}.\ \quad\Box$$

Remark 4.

Theorem 2 implies, in particular, Theorem 1 formulated in the Introduction. Indeed, if $U=V={\mathbb R}^n$, $\mu$ and $\nu$ are the Lebesgue measures, $G_{t,y}=B(y,t)$, $t>0$, $y\in{\mathbb R}^n$, and $\Omega={\mathbb R}^n$, then inequalities (1.2) and Theorem 1 with $f$ replaced by $f\chi^{}_{G_{s,y}}$ for $i=0$ and by $f(1-\chi^{}_{G_{s,y}})$ for $i=1$ imply inequalities (4.2) and (4.3). Hence, inequality (4.4) holds, which leads to inequality (1.3).

Let $U=V={\mathbb R}^n$, $\mu$ and $\nu$ be the Lebesgue measures, $\Omega\subset {\mathbb R}^n$, $0<p, q\leq \infty$, and $0\le\lambda< \infty$ with $\lambda>0$ if $q< \infty$. Let $v$ be a positive locally absolutely continuous strictly increasing function defined on $(0, \infty )$. Define the spaces $M_{p,q,\Omega}^\lambda(v)$ in the spirit of Example 3:

$$M_{p,q,\Omega}^\lambda(v)=\left\{f\in L_p^{\textrm{loc}}({\mathbb R}^n) \colon\, \,\|f\|_{M_{p,q,\Omega}^\lambda(v)}=\left(\intop_0^ \infty \biggl( (v(r))^{-\lambda}\sup_{y\in\Omega} \|f\|_{L_p(B(y,r))} \biggr)^{\!q} \frac{dv(r)}{v(r)} \right)^{\!\smash{1/q}}< \infty \right\}$$

for $0<q< \infty$ and

$$M_{p, \infty ,\Omega}^\lambda(v)=\biggl\{f \colon\, \, \|f\|_{M_{p, \infty ,\Omega}^\lambda(v)}=\sup_{r>0,\; y\in\Omega}(v(r))^{-\lambda} \|f\|_{L_p(B(y,r))}< \infty \biggr\}$$

for $q= \infty$.

Corollary 4.

Let $\Omega\subset{\mathbb R}^n,$ $0<p,q,\sigma,\tau\leq \infty ,$ $0\le\alpha_0,\alpha_1< \infty$ with $\alpha_0,\alpha_1>0$ if $\sigma< \infty ,$ $\alpha_0\neq \alpha_1,$ $0\leq\beta_0,\beta_1< \infty ,$ $\beta_0\neq \beta_1,$ and $0< \theta <1,$ and let $\alpha$ and $\beta$ be as defined in (1.1).

Suppose that the functions $v$ and $w$ satisfy the conditions indicated above, and let $T$ be a partially quasi-additive operator on $M^\alpha_{p,\tau,\Omega}(w)$ with partial quasi-additivity constant $A$.

If for some $M_0,M_1>0$ the inequalities

$$\|Tf\|_{M^{\beta_i}_{q, \infty ,y}(v)}\leq M_i \|f\|_{M^{\alpha_i}_{p,\sigma,y}(w)}$$

hold for all $y\in\Omega$ and all functions $f\in LM^{\alpha_i}_{p,\sigma,y}(w),$ $i=0,1,$ then the inequality

$$\|Tf\|_{M^{\beta}_{q,\tau,\Omega}(v)}\leq cAM_0^{1- \theta }M_1^{ \theta } \|f\|_{M^\alpha_{p,\tau,\Omega}(w)}$$

holds for all functions $f\in LM^\alpha_{p,\tau,\Omega}(w),$ where $c>0$ depends only on $\alpha_0,$ $\alpha_1,$ $\beta_0,$ $\beta_1,$ $q,$ $\sigma,$ $\tau,$ and $\theta$.

Proof.

Let $\mu$ be the Lebesgue measure in ${\mathbb R}^n$. Let $v^ {-1}$ and $w^ {-1}$ be the inverse functions of $v$ and $w$, respectively. Consider the nets $F=\{F_{t,y}\}_{t>0,\;y\in\Omega}$ and $G=\{G_{t,y}\}_{t>0,\;y\in\Omega}$ with

$$F_{t,y}=B(y,v^ {-1} (t)) \qquad\text{and}\qquad G_{t,y}=B(y,w^ {-1} (t)).$$

Then $M^\lambda_{p,q,\Omega}(w)$ and $M^\lambda_{p,q,y}(w)$ coincide with the spaces $M^\lambda_{p,q}(G_\Omega,\mu)$ and $M^\lambda_{p,q}(G_{\{y\}},\mu)$, respectively. Next, we apply Theorem 2. $\quad\Box$

The following interpolation theorem is purely a theorem on interpolation of inequalities (it does not contain any operator).

Theorem 3.

Let $(U,\mu)$ and $(V,\nu)$ be spaces with measures $\mu$ and $\nu,$ and let $\Omega\subset{\mathbb R}^n$.

Let $G_\Omega=\{G_{t,y}\}_{t>0,\;y\in\Omega}$ be a net in $U,$ $\check{G}{} _\Omega^s$ and $\widehat{G}{} _\Omega^s$ for $s>0$ be the nets in $U$ defined in (4.1), and $\check{G}{} _{\{y\}}^s$ and $\widehat{G}{} _{\{y\}}^s$ for $s>0$ and $y\in\Omega$ be the nets generated by $\check{G}{} _\Omega^s$ and $\widehat{G}{} _\Omega^s$ in $U$.

Let $F_\Omega=\{F_{t,y}\}_{t>0,\; y\in\Omega},$ $F^0_\Omega(s)=\{(F^0(s))_{t,y}\}_{t>0,\;y\in\Omega},$ and $F^1_\Omega(s)=\{(F^1(s))_{t,y}\}_{t>0,\;y\in\Omega}$ for $s>0$ be nets in $V,$ and let $F_{\{y\}}^0(s)$ and $F_{\{y\}}^1(s)$ for $s>0$ and $y\in\Omega$ be the nets in $V$ generated by $F^0_\Omega(s)$ and $F^1_\Omega(s)$; in addition, suppose that

$$ F_{t,y}\subset (F^0(s))_{t,y}\cup (F^1(s))_{t,y}$$

(4.5)

for all $s,t>0$ and $y\in\Omega$.

Let $0<p,q,\sigma,\tau\leq \infty ,$ $0\le\alpha_0,\alpha_1< \infty$ with $\alpha_0,\alpha_1>0$ if $\sigma< \infty ,$ $\alpha_0\neq \alpha_1,$ $0\leq\beta_0,\beta_1< \infty ,$ $\beta_0\neq \beta_1,$ and $0< \theta <1,$ and let $\alpha$ and $\beta$ be as defined in (1.1).

Let $f\in M^\alpha_{p,\tau}(G_\Omega,\mu)$ and $g$ be a $\nu$-measurable function on $V$.

If for some $M_0,M_1>0$ the inequalities

$$\|g\|_{M^{\beta_0}_{q, \infty } \left(\smash{F_{\{y\}}^0(s),\nu}\vphantom{\widehat G}\right) } \leq M_0 \|f\|_{M^{\alpha_0}_{p,\sigma} \left(\smash{ \check{G}{} _{\{y\}}^s,\mu}\vphantom{\widehat G}\right) },$$

(4.6)

$$\|g\|_{M^{\beta_1}_{q, \infty } \left(\smash{F_{\{y\}}^1(s),\nu}\vphantom{\widehat G}\right) } \leq M_1 \|f\|_{M^{\alpha_1}_{p,\sigma} \left(\smash{ \widehat{G}{} _{\{y\}}^s,\mu}\vphantom{\widehat G}\right) }$$

(4.7)

hold for all $y\in\Omega$ and $s>0,$ then the inequality

$$ \|g\|_{M^{\beta}_{q,\tau}(F_\Omega,\nu)}\leq c M_0^{1- \theta }M_1^{ \theta } \|f\|_{M^\alpha_{p,\tau}(G_\Omega,\mu)}$$

(4.8)

holds with a constant $c>0$ depending only on $q,$ $\tau,$ $\sigma,$ $\alpha_0,$ $\alpha_1,$ $\beta_0,$ $\beta_1,$ and $\theta$.

Proof.

Without loss of generality, we may assume that $\alpha_0<\alpha_1$. Since the embedding $M^\alpha_{p,\sigma_1}(G_{\{y\}},\mu)\hookrightarrow M^\alpha_{p,\sigma_2}(G_{\{y\}},\mu)$ holds for $\sigma_1<\sigma_2$, we can also assume without loss of generality that $0<\sigma\leq\tau\leq \infty$.

Let $f\in M^\alpha_{p,\tau}(G_\Omega,\mu)$, $y\in\Omega$, and $s>0$. According to (4.5),

$$\|g\|_{L_q(F_{t,y},\nu)}\leq 2^{(1/q -1)_+}\bigl(\|g\|_{L_q((F^0(s))_{t,y},\nu)}+\| g \|_{L_q((F^1(s))_{t,y},\nu)}\bigr).$$

From conditions (4.6) and (4.7), we have

$$\begin{aligned} \, \|g\|_{L_q((F^0(s))_{t,y},\nu)} &=t^{\beta_0}t^{-\beta_0}\|g\|_{L_q((F^0(s))_{t,y},\nu)} \leq t^{\beta_0}\sup_{r>0}r^{-\beta_0}\|g\|_{L_q((F^0(s))_{t,y},\nu)} \\[4pt] &=A^{\beta_0}t^{\beta_0}\|g\|_{M^{\beta_0}_{q, \infty } \left(\smash{F_{\{y\}}^0(s),\nu}\vphantom{\widehat G}\right) } \leq M_0 t^{\beta_0}\|f\|_{M^{\alpha_0}_{p,\sigma} \left(\smash{ \check{G}{} _{\{y\}}^s,\mu}\vphantom{\widehat G}\right) } \\[4pt] &\asymp M_0t^{\beta_0}\!\left(\!\left(\intop_0^s\bigl(r^{-\alpha_0} \|f\|_{L_p(G^s_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!\smash{1/\sigma}}+s^{-\alpha_0}\|f\|_{L_p(G_{s,y},\mu)}\right) \end{aligned}$$

and

$$\|f\|_{L_q((F^1(s))_{t,y},\nu)} \leq M_1 t^{\beta_1}\|f\|_{M_{p,\sigma} \left(\smash{ \widehat{G}{} ^s_{\{y\}}}\vphantom{\widehat G}\right) } = M_1t^{\beta_1}\!\left(\intop_s^ \infty \bigl(r^{-\alpha_0}\|f\|_{L_p(G^s_{r,y},\mu)}\bigr)^\sigma \kern1pt \frac{dr}{r} \right)^{\!1/\sigma}.$$

Thus, for all $t>0$, $s>0$, and $y\in\Omega$, we obtain

$$\begin{aligned} \, \|g\|_{L_q(F_{t,y})} &\leq c_2\! \left(M_0 t^{\beta_0}\!\left[\left(\intop_0^s\biggl(r^{-\alpha_0} \sup_{y\in\Omega} \|f\|_{L_p(G_{r,y})}\biggr)^{\!\sigma} \frac{dr}{r} \right)^{\!\smash{1/\sigma}}+s^{-\alpha_0}\sup_{y\in\Omega} \|f\|_{L_p(G_{s,y},\mu)}\right]\right. \\[4pt] &\left.\!\phantom{={}}{} + M_1 t^{\beta_1}\!\left(\intop_s^ \infty \biggl(t^{-\alpha_1}\sup_{y\in\Omega} \|f\|_{L_p(G_{t,y},\mu)}\biggr)^{\!\sigma} \frac{dt}{t} \right)^{\!\smash{1/\sigma}}\,\right)\!, \end{aligned}$$

where $c_2>0$ depends only on $q$, $\alpha_0$, $\alpha_1$, and $\sigma$.

The further arguments are similar to those in the proof of Theorem 2. $\quad\Box$

Remark 5.

The hypotheses of Theorem 3 contain the nets $F_\Omega$, $F_\Omega^0(s)$, and $F_\Omega^1(s)$ that satisfy condition (4.5). We now give an example of such nets, independent of $\Omega$, which will be used in Sections 5 and 6.

Let $(U,\mu)$, $Z(U)$, $(V,\nu)$, and $M(V)$ be as in Section 3. Let $T \colon\, Z(U)\to M(V)$ be a partially quasi-additive operator with partial quasi-additivity constant $A$, and let the nets $F$, $F^0(s)$, and $F^1(s)$ be defined by the following sets:

$$\begin{aligned} \, F_t&=\bigl\{x\in V \colon\, \,|(Tf)(x)|>t^{-1} h(x)\bigr\},\qquad t>0, \\[4pt] F_t^0(s)&=\bigl\{x\in V \colon\, \,|(T(f\chi^{}_{G_s}))(x)|>(2At)^{-1} h(x)\bigr\},\qquad t>0, \quad s>0, \\[4pt] F_t^1(s)&=\bigl\{x\in V \colon\, \,|(T(f(1-\chi^{}_{G_s})))(x)|>(2At)^{-1} h(x)\bigr\},\qquad t>0,\quad s>0, \end{aligned}$$

where $h$ is a positive function on $V$.

Now, let us show that the partial quasi-additivity of $T$ implies the inclusion

$$F_t\subset F_t^0(s) \cup F_t^1(s),\qquad s>0,\quad t>0.$$

Indeed, if $x\in F_t$, then $x\in V$ and

$$(T(f\chi^{}_{G_s}))(x)> \frac{h(x)}{2At}\qquad\Leftrightarrow\qquad x\in F^0_t(s)$$

or

$$(T(f(1-\chi^{}_{G_s})))(x)> \frac{h(x)}{2At} \qquad \Leftrightarrow \qquad x\in F^1_t(s),$$

since otherwise

$$(T(f\chi^{}_{G_s}))(x)\leq \frac{h(x)}{2At}, \qquad (T(f(1-\chi^{}_{G_s})))(x)\leq \frac{h(x)}{2At}$$

and

$$|(Tf)(x)| \leq A\bigl(|(T(f\chi^{}_{G_s}))(x)|+|(T(f(1-\chi^{}_{G_s})))(x)|\bigr)\leq \frac{h(x)}t,$$

which contradicts the fact that $x\in F_t$.

5. Marcinkiewicz–Calderón type interpolation theorem

Inspecting the standard proof of the Marcinkiewicz theorem, one can easily verify that the condition of quasi-additivity of the operator $T$ is applied in the proof only to sums of the form $f\chi^{}_w + f(1-\chi^{}_w)$ rather than to arbitrary sums $f_1+f_2$; i.e., it suffices to assume that the operator $T$ is partially quasi-additive. Taking this fact into account, we present the Marcinkiewicz interpolation theorem in the following form.

Theorem 4.

Let $(U,\mu)$ and $(V,\nu)$ be spaces with measures $\mu$ and $\nu$.

Let $1\leq p_0\leq p_1< \infty ,$ $1\leq q_0,q_1< \infty ,$ $q_0\neq q_1,$ $0< \theta <1,$ and

$$\frac{1}{p}=\frac{1- \theta }{p_0}+\frac{ \theta }{p_1} \geq\frac{1}{q}=\frac{1- \theta }{q_0}+\frac{ \theta }{q_1}.$$

Let $T$ be a partially quasi-additive operator on $L_{p_0}(U,\mu)+L_{p_1}(U,\mu)$ with partial quasi-additivity constant $A$.

If for some $M_0,M_1>0$ the inequalities

$$ \|Tf\|_{L_{q_i, \infty }(V,\nu)}\leq M_i \|f\|_{L_{p_i}(U,\mu)}$$

(5.1)

hold for all functions $f\in L_{p_i}(U,\mu),$ $i=0,1,$ then the inequality

$$ \|Tf\|_{L_q(V,\nu)}\leq cAM_0^{1- \theta }M_1^{ \theta } \|f\|_{L_p(U,\mu)}$$

(5.2)

holds for all $f\in L_p(U,\mu),$ where $c>0$ depends only on $p_0,$ $p_1,$ $q_0,$ $q_1,$ and $\theta$.

The well-known Calderón theorem [13] generalizes and, in a sense, strengthens the Marcinkiewicz interpolation theorem in the case when the operator is quasi-additive. There are various proofs of this theorem, but all of them employ the quasi-additivity condition in an essential way and are not generally valid in the case of just partial quasi-additivity.

We will prove an analog of the Calderón theorem for partially quasi-additive operators $T$, while requiring the regularity of the corresponding measures.

Theorem 5.

Let $(U,\mu)$ and $(V,\nu)$ be spaces with measures $\mu$ and $\nu$ satisfying the regularity condition (2.2).

Let $1\leq p_0<p_1< \infty ,$ $1\leq q_0,q_1< \infty ,$ $q_0\neq q_1,$ $0<\sigma,\tau\leq \infty ,$ $0< \theta <1,$ and

$$ \frac{1}{p}=\frac{1- \theta }{p_0}+\frac{ \theta }{p_1}, \qquad \frac{1}{q}=\frac{1- \theta }{q_0}+\frac{ \theta }{q_1}.$$

(5.3)

Let $T$ be a partially quasi-additive operator on $L_{p_0,\sigma}(U,\mu)+L_{p_1,\sigma}(U,\mu)$ with partial quasi-additivity constant $A$.

If for some $M_0,M_1>0$ the inequalities

$$ \|Tf\|_{L_{q_i, \infty }(V,\nu)}\leq M_i \|f\|_{L_{p_i,\sigma}(U,\mu)}$$

(5.4)

hold for all $f\in L_{p_i,\sigma}(U,\mu),$ $i=0,1,$ then the inequality

$$ \|Tf\|_{L_{q,\tau}(V,\nu)}\leq cAM_0^{1- \theta }M_1^{ \theta } \|f\|_{L_{p,\tau}(U,\mu)}$$

(5.5)

holds for all $f\in L_{p,\tau}(U,\mu),$ where $c>0$ depends only on $p_0,$ $p_1,$ $q_0,$ $q_1,$ $\sigma,$ $\tau,$ and $\theta$.

Proof.

Let $f\in L_{p,\tau}(U,\mu)$. It follows from Lemma 2 and Example 1 that there exist nets $G(f)=\{G_t(f)\}_{t>0}$ and $H(Tf)=\{H_t(Tf)\}_{t>0}$ that satisfy the conditions

$$\mu(G_t(f))\asymp t \qquad\text{and}\qquad \nu(H_t(Tf))\asymp t$$

and are such that

$$\begin{gathered} \, \intop_0^t(Tf)^*(s)\,ds\leq \intop_{H_t(Tf)}\!\mathopen|(Tf)(x)|\,d\nu\leq \intop_0^{2t}(Tf)^*(s)\,ds, \\ \intop_0^t\!f^*(s)\,ds\leq \intop_{G_t(f)}\!\mathopen|f(x)|\,d\mu\leq \intop_0^{2t}f^*(s)\,ds, \\[4pt] \|Tf\|_{L_{q,\tau}(V,\nu)} \asymp\|Tf\|_{M_{1,\tau}^{1/q'}(H(Tf),\nu)},\qquad\text{and}\qquad \|f\|_{L_{p,\tau}(U,\mu)}\asymp\|f\|_{M_{1,\tau}^{1/p'}(G(f),\mu)}, \end{gathered}$$

where the corresponding constants depend only on the parameter $p$, $q$, and $\tau$.

From condition (5.4), we have

$$\begin{aligned} \, &\|T(f\chi^{}_{G_s(f)})\|_{M_{1, \infty }^{1/q'_0}(H(Tf),\nu)} =\sup_{t>0}t^{-1/q'}\intop_{G_t(Tf)}\!\bigl|\bigl(T\bigl(f\chi^{}_{G_s(f)}\bigr)\bigr)(x)\bigr|\,d\mu \\ &\qquad\leq \sup_{t>0} t^{-1/q'}\sup_{\mu(e)=2t}\,\intop_e\bigl|\bigl(T\bigl(f\chi^{}_{G_s(f)}\bigr)\bigr)(x)\bigr|\,d\mu \leq \sup_{t>0} t^{-1/q'}\intop_0^{2t}\bigl(T\bigl(f\chi^{}_{G_s(f)}\bigr)\bigr)^*(t)\,dt \\[3pt] &\qquad\lesssim \bigl\|T\bigl(f\chi^{}_{G_s(f)}\bigr)\bigr\|_{L_{q_0, \infty }(D,\nu)} \leq M_0\bigl\|f\chi^{}_{G_s(f)}\bigr\|_{L_{p_0,\sigma}(U,\mu)} \\[4pt] &\qquad\lesssim M_0\bigl\|f\chi^{}_{G_s(f)}\bigr\|_{M_{1,\sigma}^{1/p_0}(G(f),\mu)} =M_0\|f\|_{M_{1,\sigma}^{1/p_0}( \check{G}{} ^s(f),\mu)}. \end{aligned}$$

According to inequalities (2.4) and (2.5), by Lemma 2 we obtain

$$\begin{aligned} \, &\bigl\|T\bigl(f\bigl(1-\chi^{}_{G_s(f)}\bigr)\bigr)\bigr\|_{M_{1, \infty }^{1/q_1}(H(Tf),\nu)} \leq \bigl\|T\bigl(f\bigl(1-\chi^{}_{G_s(f)}\bigr)\bigr)\bigr\|_{L_{q_1, \infty }(V,\nu)} \\[3pt] &\leq M_1\bigl\|f\bigl(1-\chi^{}_{G_s(f)}\bigr)\bigr\|_{L_{p_1,\sigma}(U,\mu)} \leq M_1\!\left(\intop_0^ \infty \bigl(t^{1/p_1}f^*(t+s)\bigr)^\sigma \kern1pt \frac{dt}{t} \right)^{\!1/\sigma} \\[2pt] &\leq M_1\!\left(\intop_0^ \infty \!\left(t^{1/p_1}\frac1{s+t}\intop_0^{s+t}f^*(\xi)\,d\xi\right)^{\!\sigma}\frac{dt}{t} \right)^{1/\sigma} \lesssim M_1\!\left(\intop_0^ \infty \!\left(t^{1/p_1}\frac1{t+s}\intop_{G\smash{_{t+s}(f)}}\!\mathopen|f(x)|\,d\mu\right)^{\!\sigma} \frac{dt}{t} \right)^{\!1/\sigma} \\[4pt] &\leq M_12^{(1/\sigma-1)_+} \\[2pt] &\phantom{={}}{}\times\left(\!\left(\intop_0^s\!\left(t^{1/p_1}\frac1{t+s}\intop_{G\smash{_{t+s}(f)}}\!\mathopen|f(x)|\,d\mu \right)^{\!\sigma} \frac{dt}{t} \right)^{\!\smash{1/\sigma}} +\left(\intop_s^ \infty \!\left(t^{1/p_1}\frac1{t+s}\intop_{G\smash{_{t+s}(f)}}\!\mathopen|f(x)|\,d\mu \right)^{\!\sigma} \frac{dt}{t} \right)^{\!\smash{1/\sigma}}\,\right) \\[4pt] &\leq M_12^{(1/\sigma-1)_+}\!\left(\;\intop_{G_{2s}\smash{(f)}}\! \mathopen|f(x)|\,d\mu \!\left(\intop_0^s\!\biggl( t^{1/p_1}\frac1{t+s} \biggr)^{\!\sigma}\frac{dt}{t} \right)^{\!\smash{1/\sigma}} + \left(\intop_s^ \infty \!\left( t^{1/p_1}\frac1{t}\intop_{G_{2t}\smash{(f)}}\! \mathopen|f(x)|\,d\mu\right)^{\!\sigma}\frac{dt}{t} \right)^{\!\smash{1/\sigma}}\,\right)\!. \end{aligned}$$

Taking into account that

$$\begin{aligned} \, \left(\intop_s^ \infty \!\left(t^{-1/p_1'}\intop_{G_{2t}\smash{(f)}}\!\mathopen|f(x)|\,d\mu\right)^{\!\sigma}\frac{dt}{t} \right)^{\!1/\sigma} &=2^{1/p'_1}\!\left(\,\intop_{2s}^ \infty \!\left(t^{-1/p_1'}\intop_{G_t\smash{(f)}}\!\mathopen|f(x)|\,d\mu\right)^{\!\sigma}\frac{dt}{t} \right)^{\!1/\sigma} \\[4pt] &\le 2^{1/p'_1} \|f\|_{M_{1,\sigma}^{1/p_1'}( \widehat{G}{} ^s,\mu)}, \\[4pt] \left(\intop_0^s\biggl(t^{1/p_1}\frac1{t+s} \biggr)^{\!\sigma}\frac{dt}{t} \right)^{\!1/\sigma} &= s^{-1/p_1'}\!\left(\intop_0^1\biggl(t^{1/p_1}\frac1{t+1} \biggr)^{\!\sigma}\frac{dt}{t} \right)^{\!1/\sigma} \end{aligned}$$

and

$$\begin{aligned} \, s^{-1/p_1'}\|f\|_{L_1(G_{2s}(f),\mu)} &\leq 2^{1/p_1'}\biggl( \frac\sigma{p_1'} \biggr)^{\!1/\sigma}\!\left(\,\intop_{2s}^ \infty \!\left(t^{-1/p_1'} \intop_{G_t\smash{(f)}}\!\mathopen|f(x)|\,d\mu\right)^{\!\sigma}\frac{dt}{t} \right)^{\!1/\sigma} \\[4pt] &\le 2^{1/p_1'} \biggl( \frac\sigma{p_1'} \biggr)^{\!1/\sigma}\|f\|_{M_{1,\sigma}^{1/p_1'}( \widehat{G}{} ^s,\mu)}, \end{aligned}$$

we find

$$\bigl\|T\bigl(f\bigl(1-\chi^{}_{G_s(f)}\bigr)\bigr)\bigr\|_{M_{1, \infty }^{1/q_1}(G(Tf),\nu)} \lesssim M_1\|f\|_{M_{1,\sigma}^{1/p_1'}( \widehat{G}{} ^s,\mu)}.$$

Let us apply Theorem 2:

$$\|Tf\|_{L_{q,\tau}(D,\nu)} \lesssim \|Tf\|_{M_{1,\tau}^{1/q}(G(Tf),\nu)} \lesssim M_0^{1- \theta }M_1^ \theta \|f\|_{M_{1,\tau}^{1/p}(G(f),\mu)}\lesssim M_0^{1- \theta }M_1^ \theta \|f\|_{L_{p,\tau}(U,\mu)}.$$

In the above considerations, the constants implied by the signs $\lesssim$ are independent not only of $f$ but also of $M_0$ and $M_1$. $\quad\Box$

Corollary 5.

Let $1\leq p_0<p_1< \infty ,$ $1\leq q_0,q_1< \infty ,$ $q_0\neq q_1,$ $0<\sigma,\tau\leq \infty ,$ and $0< \theta <1,$ and let $p$ and $q$ be defined by (5.3).

Let

$$(Tf)(y)=\intop_U\! K(f(x),x,y)\,d\mu,\qquad y\in V.$$

If for some $M_0,M_1>0$ the inequalities

$$ \|Tf\|_{L_{q_i, \infty }(V,\nu)}\leq M_i\|f\|_{L_{p_i,\sigma}(U,\mu)}$$

(5.6)

hold for all $f\in L_{p_i,\sigma}(U,\mu),$ $i=0,1,$ then the inequality

$$\|Tf\|_{L_{q,\tau}(V,\nu)}\leq c M_0^{1- \theta }M_1^ \theta \|f\|_{L_{p,\tau}(U,\mu)}$$

holds for all $f\in L_{p,\tau}(U,\mu),$ where $c>0$ depends only on $p_0,$ $p_1,$ $q_0,$ $q_1,$ $\sigma,$ $\theta ,$ and $\tau$.

Proof.

From condition (5.6), we have

$$\sup_{t>0}t^{1/p_0}(Tf)^*(t)\leq M_0\|f\|_{L_{p_0,1}}.$$

Setting $f=0$, we obtain

$$\intop_U\! K(0,x,y)\,d\mu =0$$

almost everywhere in $V$; in this case, as shown in Section 3, the operator $T$ is partially additive. Hence, we can apply Theorem 5. $\quad\Box$

6. Stein–Weiss–Peetre type interpolation theorems

Let $\mu$ be a measure on $U$ satisfying the regularity condition (2.2) and $w$ be a positive $\mu$-measurable (weight) function on $U$.

For $0<p\leq \infty$, denote by $L_p(U,w,\mu)$ the space of all $\mu$-measurable functions on $U$ such that

$$\|f\|_{L_p(U,w,\mu)}=\left(\,\intop_U\bigl(w(x)|f(x)|\bigr)^p\,d\mu\right)^{\!1/p}< \infty .$$

If $w\equiv1$, then $L_p(U,1,\mu)\equiv L_p(U,\mu)$; if $\mu$ is the Lebesgue measure, then $L_p(U,w,\mu)\equiv L_p(U,w)$ and $L_p(U,\mu)\equiv L_p(U)$.

Recall the Stein–Weiss–Peetre theorem.

Theorem 6.

Let $1\leq p_0\leq p_1< \infty ,$ $0< \theta <1,$ and

$$ \frac{1}{p}=\frac{1- \theta }{p_0}+\frac{ \theta }{p_1}.$$

(6.1)

Let $w_0$ and $w_1$ be positive $\mu$-measurable functions on $U,$ and let $T$ be a subadditive operator on $L_{p_0}(U,w_0,\mu)+L_{p_1}(U,w_1,\mu)$.

If for some $M_0,M_1>0$ the inequalities

$$ \|Tf\|_{L_{p_i}(U,w_i,\mu)}\leq M_i \|f\|_{L_{p_i}(U,w_i,\mu)}$$

(6.2)

hold for all $f\in L_{p_i}(U,w_i,\mu),$ $i=0,1,$ then the inequality

$$ \|Tf\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}\leq c M_0^{1- \theta }M_1^{ \theta }\|f\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}$$

(6.3)

holds for all $f\in L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu),$ where $c>0$ depends only on $p_0,$ $p_1,$ and $\theta$.

Remark 6.

In the case when $T$ is a linear operator and $p_0=p_1=p$, Theorem 6 was proved by Stein and Weiss [27]. In this case, the constant $c$ in inequality (6.3) is equal to $1$. In the present version, the theorem was proved by Peetre [24].

Theorem 7.

Let $0<p\leq q< \infty$ and $0< \theta <1$. Let $w_0$ and $w_1$ be positive $\mu$-measurable functions on $U,$ and let $T$ be a partially quasi-additive operator on $L_p(U,w_0,\mu)+L_p(U,w_1,\mu)$ with partial quasi-additivity constant $A$.

If for some $M_0,M_1>0$ the inequalities

$$\|Tf\|_{L_q(U,w_i,\mu)}\leq M_i \|f\|_{L_p(U,w_i,\mu)}$$

hold for all $f\in L_p(U,w_i,\mu),$ $i=0,1,$ then the inequality

$$\|Tf\|_{L_q(U,w_0^{1- \theta }w_1^ \theta ,\mu)}\leq cA M_0^{1- \theta }M_1^{ \theta }\|f\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}$$

holds for all $f\in L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu),$ where $c>0$ depends only on $q,$ $\alpha_0,$ $\alpha_1,$ $\lambda_0,$ $\lambda_1,$ and $\theta$.

Proof.

Let $0<\lambda_0,\lambda_1< \infty$ and $0<\alpha_0<\alpha_1< \infty$. We define nets $G_{\lambda_0,\lambda_1}=\{G_{t,\lambda_0,\lambda_1}\}_{t>0}$ and $F_{\alpha_0,\alpha_1}=\{F_{t,\alpha_0,\alpha_1}\}_{t>0}$ by setting

$$\begin{alignedat}{2} G_{t,\lambda_0,\lambda_1}&=\bigl\{x\in U \colon\, \, w_0^{1/(\lambda_1-\lambda_0)}(x)w_1^{1/(\lambda_0-\lambda_1)}(x)<t\bigr\},\qquad&& t>0,\\[4pt] F_{t,\alpha_0,\alpha_1}&=\bigl\{x\in V \colon\, \, v_0^{1/(\alpha_1-\alpha_0)}(x)v_1^{1/(\alpha_0-\alpha_1)}(x)<t\bigr\},\qquad&& t>0, \end{alignedat}$$

and let

$$d\mu_{\lambda_0,\lambda_1}=\bigl(w_0^{\lambda_1/(\lambda_1-\lambda_0)}(x)w_1^{\lambda_0/(\lambda_0-\lambda_1)}(x)\bigr)^p\,d\mu,\qquad d\nu_{\alpha_0,\alpha_1}=\bigl(v_0^{\alpha_1/(\alpha_1-\alpha_0)}(x)v_1^{\alpha_0/(\alpha_0-\alpha_1)}(x)\bigr)^q\,d\nu.$$

Then, according to Example 8 and Theorem 7 from [12],

$$\begin{gathered} \, \begin{aligned} \, M^{\lambda_0}_{p,1}(G_{\lambda_0,\lambda_1},\mu_{\lambda_0,\lambda_1}) &\hookrightarrow M^{\lambda_0}_{p,p}(G_{\lambda_0,\lambda_1},\mu_{\lambda_0,\lambda_1})=L_p(U,w_0,\mu),\\[4pt] M^{\lambda_1}_{p,1}(G_{\lambda_0,\lambda_1},\mu_{\lambda_0,\lambda_1}) &\hookrightarrow M^{\lambda_1}_{p,p}(G_{\lambda_0,\lambda_1},\mu_{\lambda_0,\lambda_1})=L_p(U,w_1,\mu),\\[4pt] M^{\alpha_0}_{q, \infty }(F_{\alpha_0,\alpha_1},\nu_{\alpha_0,\alpha_1}) &\hookleftarrow M^{\alpha_0}_{q,q}(F_{\alpha_0,\alpha_1},\nu_{\alpha_0,\alpha_1})=L_q(U,w_0,\nu),\\[4pt] M^{\alpha_1}_{q, \infty }(F_{\alpha_0,\alpha_1},\nu_{\alpha_0,\alpha_1}) &\hookleftarrow M^{\alpha_1}_{q,q}(F_{\alpha_0,\alpha_1},\nu_{\alpha_0,\alpha_1})=L_q(U,w_1,\nu), \end{aligned}\\[4pt] \begin{aligned} \, M^\lambda_{p,p}(G_{\lambda_0,\lambda_1},\mu_{\lambda_0,\lambda_1})&=L_p(U,w_0^{1- \theta }w_1^{ \theta },\mu),\qquad \lambda=({1- \theta })\lambda_0+ \theta \lambda_0,\\[4pt] M^\alpha_{q,q}(F_{\alpha_0,\alpha_1},\nu_{\alpha_0,\alpha_1})&=L_q(U,w_0^{1- \theta }w_1^{ \theta },\nu),\qquad \alpha=({1- \theta })\alpha_0+ \theta \alpha_0. \end{aligned} \end{gathered}$$

From the hypotheses of the theorem, we obtain

$$\|Tf\|_{M^{\alpha_i}_{q, \infty }(F_{\alpha_0,\alpha_1},\nu_{\alpha_0,\alpha_1})}\leq c_1 M_i\|f\|_{M^{\lambda_i}_{p,1}(G_{\lambda_0,\lambda_1},\mu_{\lambda_0,\lambda_1})},$$

where $c_1>0$ depends only on the parameters $\alpha_0$, $\alpha_1$, $\lambda_0$, and $\lambda_1$.

To complete the proof, we apply Theorem 2. $\quad\Box$

Lemma 6.

Let $0<p_0,p_1< \infty ,$ $p_0\neq p_1,$ and $0< \theta <1,$ and let $p$ be defined by (6.1).

Let $w_0$ and $w_1$ be positive $\mu$ -measurable functions on $U$ and

$$h_1(x)=\biggl( \frac{w_0(x)}{w_1(x)} \biggr)^{\!p_0p_1/(p_1-p_0)}, \qquad h_2(x)=\biggl( \frac{w_0^{p_0}(x)}{w_1^{p_1}(x)} \biggr)^{\!1/(p_1-p_0)}, \qquad x\in U,\qquad d \widetilde{\mu}{} =h_1\,d\mu.$$

Let $f \colon\, U\to {\mathbb R}$ be a $\mu$ -measurable function and $G(f)=\{G_t(f)\}_{t>0}$ with

$$G_t(f)=\biggl\{x\in U \colon\, \,|f(x)|>\frac{h_2(x)}{t}\biggr\}.$$

Then the following equalities hold for an arbitrary $\mu$-measurable set $e\subset U$:

$$\|\chi^{}_e\|_{M^{p_0}_{1,1}(G(f), \widetilde{\mu}{} )} =p_0^{-1}\|f\chi^{}_e\|_{L_{p_0}(U,w_0,\mu)}^{p_0},$$

(6.4)

$$\|1-\chi^{}_e\|_{M^{p_1}_{1,1}(G(f), \widetilde{\mu}{} )} =p_1^{-1}\|f(1-\chi^{}_e)\|_{L_{p_1}(U,w_1,\mu)}^{p_1},$$

(6.5)

$$\|1\|_{M^p_{1,1}(G(f), \widetilde{\mu}{} )} =p^{-1}\|f\|^p_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}.$$

(6.6)

Remark 7.

The functions $h_1$ and $h_2$ and the sets $G_t$ were used in [12] (see Example 9, Lemmas 3 and 4, and Theorem 8 there).

Proof of Lemma 6 .

Let $U_0(f)=\{x\in U \colon\, f(x)\neq 0\}$. Then $\bigcup_{t>0}G_t(f)=U_0(f)$ and

$$\begin{aligned} \, \|f\chi^{}_e&\|^{p_0}_{L_{p_0}(U,w_0,\mu)} =\intop_{U_0(f)}\!\bigl(|f(x)\chi^{}_e(x)|w_0(x)\bigr)^{p_0}\,d\mu \\[2pt] &=\intop_{U_0(f)}\!\biggl( \frac{w_0(x)}{w_1(x)} \biggr)^{\!p_0p_1/(p_1-p_0)} \biggl(\chi^{}_e(x)|f(x)|\biggl( \frac{w_0^{p_0}(x)}{w_1^{p_1}(x)} \biggr)^{\!1/(p_0-p_1)}\biggr)^{\!p_0} \kern1pt d\mu \\[3pt] &=\intop_{U_0(f)}\!\chi^{}_e(x) h_1(x) \bigl(|f(x)|h^{-1}_2(x)\bigr)^{p_0}\,d\mu =p_0\intop_{U_0(f)}\!\chi^{}_e(x) h_1(x)\! \left(\,\,\intop_{h_2\smash{(x)/|f(x)|}}^ \infty \! t^{-p_0-1}\,dt\right) d\mu \\[3pt] &=p_0\intop_0^ \infty t^{-p_0}\!\left(\,\,\intop_{h_2\smash{(x)/|f(x)|}<t}\!\chi^{}_e(x)h_1(x)\,d\mu\right)\!\frac{dt}{t} =p_0\intop_0^ \infty t^{-p_0}\!\left(\;\intop_{G_t\smash{(f)}}\!\chi^{}_e(x)h_1(x)\,d\mu\right)\!\frac{dt}{t} \\[3pt] &=\|\chi^{}_e\|_{M^{p_0}_{1,1}(G(f), \widetilde{\mu}{} )}, \end{aligned}$$

which implies equality (6.4). Next,

$$\begin{aligned} \, &\|f(1-\chi^{}_e)\|^{p_1}_{L_{p_1}(U,w_1,\mu)} =\intop_{U_0(f)}\! \bigl(|f(x)(1-\chi^{}_e(x))|w_1(x)\bigr)^{p_0}\,d\mu \\[2pt] &=\intop_{U_0(f)}\!\biggl( \frac{w_0(x)}{w_1(x)} \biggr)^{\!p_0p_1/(p_1-p_0)} \biggl( (1-\chi^{}_e(x))|f(x)|\biggl( \frac{w_0^{p_0}(x)}{w_1^{p_1}(x)} \biggr)^{\!1/(p_0-p_1)}\biggr)^{\!p_1} \kern1pt d\mu \\[1pt] &=\intop_{U_0(f)}\! (1-\chi^{}_e(x)) h_1(x) \bigl(|f(x)|h^{-1}_2(x)\bigr)^{p_1}\,d\mu =p_1\intop_{U_0(f)}\! (1-\chi^{}_e(x)) h_1(x)\! \left(\,\,\intop_{h_2\smash{(x)/|f(x)|}}^ \infty \! t^{-p_1-1}\,dt\right) d\mu \\[2pt] &=p_1\intop_0^ \infty t^{-p_1}\!\left(\,\,\intop_{h_2\smash{(x)/|f(x)|}<t}\!(1-\chi^{}_e(x))h_1(x)\,d\mu\right)\!\frac{dt}{t} =p_1\intop_0^ \infty t^{-p_1}\!\left(\;\intop_{G_t\smash{(f)}}\!(1-\chi^{}_e(x))h_1(x)\,d\mu\right)\!\frac{dt}{t} \\[3pt] &=\|\chi^{}_e\|_{M^{p_1}_{1,1}(G(f), \widetilde{\mu}{} )} \end{aligned}$$

and

$$\begin{aligned} \, \|1\|_{M^p_{1,1}(G(f), \widetilde{\mu}{} )} &=\intop_0^ \infty t^{-p}\intop_{G_t(f)}\!h_1(x)\,d\mu \kern1pt \frac{dt}{t} =\intop_{U_0(f)}\!h_1(x)\!\intop_0^{|f(x)|/h_2(x)}t^{-p}\,\frac{dt}{t}\,d\mu \\[3pt] &=p^{-1} \intop_U\bigl(|f(x)|w_0^{1- \theta }w_1^ \theta \bigr)^p\,d\mu. \ \quad\Box\end{aligned}$$

Theorem 8.

Let $0<p_0<p_1< \infty$ and $0< \theta <1,$ and let $p$ be defined by (6.1).

Let the functions $w_0,$ $w_1,$ $h_1,$ and $h_2$ and the measure $\widetilde{\mu}{}$ be the same as in Lemma 6, and let $T$ be a partially quasi-additive operator on $L_{p_0}(U,w_0,\mu)+L_{p_1}(U,w_1,\mu)$ with partial quasi-additivity constant $A$.

If for some $M_0,M_1>0$ the inequalities

$$ \|h_2^{-1}Tf\|_{L_{p_i, \infty } (U, \widetilde{\mu}{} )}\leq M_i \|f\|_{L_{p_i}(U,w_i,\mu)}$$

(6.7)

hold for all $f\in L_{p_i}(U,w_i,\mu),$ $i=0,1,$ then the inequality

$$ \|Tf\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}\leq c A M_0^{1- \theta }M_1^ \theta \|f\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}$$

(6.8)

holds for all $f\in L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu),$ where $c>0$ depends only on $p_0,$ $p_1,$ and $\theta$.

Remark 8.

Note that conditions (6.7) are weaker than conditions (6.2).

Indeed, for example, for $i=0$,

$$\begin{aligned} \, \|h_2^{-1}Tf\|_{L_{p_i, \infty } (U, \widetilde{\mu}{} )} &=\sup_{t>0}t\bigl( \widetilde{\mu}{} \bigl(\{x\in U \colon\, \kern1pt |(Tf)(x)|>t h_2(x)\}\bigr)\bigr)^{1/p_0} \\[3pt] &=\sup_{t>0}t\!\left(\,\,\smash{\intop_{\{x\in U \colon\, \kern1pt |(Tf)(x)|>t h_2(x)\}}}\vphantom{\intop_0}\! h_1(x)\,d\mu\right)^{\!1/p_0} \\[3pt] &\leq\sup_{t>0}\!\left(\,\,\smash{\intop_{\{x\in U \colon\, \kern1pt |(Tf)(x)|>t h_2(x)\}}}\vphantom{\intop_0} \!\biggl|\frac{(Tf)(x)}{h_2(x)}\biggr|^{p_0}h_1(x)\,d\mu\right)^{\!1/p_0} \leq \|Tf\|_{L_p(U,w_0,\mu)}. \end{aligned}$$

Proof of Theorem 8 .

Let $s>0$ and $f\in L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)$. We will use the terminology of Lemma 6. Let $G$, $F$, $F^0(s)$, and $F^1(s)$ be the nets defined by the following sets:

$$\begin{gathered} \, G_t=\bigl\{x\in U \colon\, \,|f(x)|>t h_2(x)\bigr\},\qquad F_t=\bigl\{x\in U \colon\, \,|(Tf)(x)|>t h_2(x)\bigr\},\qquad t>0,\\[4pt] \begin{alignedat}{2} F_t^0(s)&=\bigl\{x\in U \colon\, \,|(T(f\chi^{}_{G_s}))(x)|>t h_2(x)\bigr\},\qquad&& t>0, \quad s>0,\\[4pt] F_t^1(s)&=\bigl\{x\in U \colon\, \,|(T(f(1-\chi^{}_{G_s})))(x)|>t h_2(x)\bigr\},\qquad&& t>0,\quad s>0. \end{alignedat} \end{gathered}$$

These nets coincide with the nets considered in Remark 5, where $h(x)=h_2(x)$. Therefore, the condition of partial quasi-additivity of $T$ for these nets implies inclusion (4.5).

Using conditions (6.7), we obtain

$$\begin{aligned} \, &\|1\|_{M^{p_0}_{1, \infty }(F^0(s), \widetilde{\mu}{} )} =\|1\|_{M^{p_0}_{1, \infty }(G(T(f\chi^{}_{G_s})), \widetilde{\mu}{} )} =\sup_{t>0}t^{-p_0}\|1\|_{L_1(G_t(T(f\chi^{}_{G_s})), \widetilde{\mu}{} )} \\[4pt] &\qquad=\sup_{t>0}t^{-p_0} \widetilde{\mu}{} \bigl(\bigl\{x\in U \colon\, \kern1pt \bigl|\bigl(T\bigl(f\chi^{}_{G_s(f)}\bigr)\bigr)(x)\bigr|>t^{-1} h_2(x)\bigr\}\bigr) \\[4pt] &\qquad=\biggl(\sup_{t>0}t\bigl( \widetilde{\mu}{} \bigl(\bigl\{x\in U \colon\, \kern1pt \bigl|\bigl(T\bigl(f\chi^{}_{G_s(f)}\bigr)\bigr)(x)\bigr|>t h_2(x)\bigr\} \bigr)\bigr)^{1/p_0}\biggr)^{\!p_0} =\|h_2^{-1}Tf\|_{L_{p_0, \infty }(U, \widetilde{\mu}{} )} \\[4pt] &\qquad\le M_0^{p_0}\bigl\|f\chi^{}_{G_s(f)}\bigr\|^{p_0}_{L_{p_0}(U,w_0,\mu)} =p_0 M_0^{p_0}\bigl\|\chi^{}_{G_s(f)}\bigr\|_{M^{p_0}_{1,1}(G, \widetilde{\mu}{} )} =p_0 M_0^{p_0}\|1\|_{M^{p_0}_{1,1}( \check{G}{} ^s, \widetilde{\mu}{} )} \end{aligned}$$

and

$$\begin{aligned} \, &\|1\|_{M^{p_1}_{1, \infty }(F^1(s), \widetilde{\mu}{} )} =\sup_{t>0}t^{-p_1} \widetilde{\mu}{} \bigl(\bigl\{x\in U \colon\, \kern1pt \bigl|\bigl(T\bigl(1-f\chi^{}_{G_s(f)}\bigr)\bigr)(x)\bigr|>t^{-1} h_2(x)\bigr\}\bigr) \\[4pt] &\qquad=\biggl(\sup_{t>0}t\bigl( \widetilde{\mu}{} \bigl(\bigl\{x\in U \colon\, \kern1pt \bigl|\bigl(T\bigl(1-f\chi^{}_{G_s(f)}\bigr)\bigr)(x)\bigr|>t h_2(x)\bigr\} \bigr)\bigr)^{1/p_1}\biggr)^{p_1} = \|h_2^{-1}Tf\|_{L_{p_1, \infty }(U, \widetilde{\mu}{} )} \\[4pt] &\qquad\leq M_1^{p_1}\bigl\|f\bigl(1-\chi^{}_{G_s(f)}\bigr)\bigr\|^{p_1}_{L_{p_1}(U,w_1,\mu)} =M_1^{p_1}\bigl\|1-\chi^{}_{G_s(f)}\bigr\|_{M^{p_1}_{1,1}(G(f), \widetilde{\mu}{} )} \\[4pt] &\qquad= p_1M^{p_1}_1\intop_0^ \infty t^{-p_1}\bigl\|1-\chi^{}_{G_s(f)}\bigr\|_{L_1(G_t(f), \widetilde{\mu}{} )}\frac{dt}{t} \leq p_1M^{p_1}_1\intop_s^ \infty t^{-p_1}\|1\|_{L_1(G_t(f), \widetilde{\mu}{} )}\frac{dt}{t} \\[4pt] &\qquad=p_1M^{p_1}_1\|1\|_{M_{1,1}^{p_1}( \widehat{G}{} ^s, \widetilde{\mu}{} )}. \end{aligned}$$

Thus, all the hypotheses of Theorem 3 are satisfied. Hence, applying Theorem 3 combined with Lemma 6, we obtain

$$\|Tf\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}\leq c A M_0^{(1-\eta)p_0/p}M_1^{\eta p_1/p} \|f\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)},$$

where $\eta$ is such that $p=(1-\eta)p_0+\eta p_1$. In view of (6.1), we have the equalities $\eta p_1/p= \theta$ and $(1-\eta)p_0/p=1- \theta$, so we arrive at inequality (6.8). $\quad\Box$

Corollary 6.

Let the parameters $p_0,$ $p_1,$ and $\theta ,$ the functions $w_0,$ $w_1,$ $h_1,$ and $h_2,$ and the measure $\widetilde{\mu}{}$ be the same as in Theorem 8.

Let $T$ be the Urysohn integral operator defined as

$$(Tf)(y)=\intop_U\! K(f(x),x,y)\,d\mu.$$

If for some $M_0,M_1>0$ the inequalities

$$ \|h_2^{-1}Tf\|_{L_{p_i, \infty } (U, \widetilde{\mu}{} )}\leq M_i \|f\|_{L_{p_i}(U,w_i,\mu)}$$

(6.9)

hold for all $f\in L_{p_i}(U,w_i,\mu),$ $i=0,1,$ then the inequality

$$ \|Tf\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}\leq c M_0^{1- \theta }M_1^ \theta \|f\|_{L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu)}$$

(6.10)

holds for all $f\in L_p(U,w_0^{1- \theta }w_1^ \theta ,\mu),$ where $c>0$ depends only on $p_0,$ $p_1,$ and $\theta$.

Proof.

From condition (6.9) we have

$$\intop_U\! K(0,x,y)\,d\mu =0$$

almost everywhere in $V$. Hence, the operator $T$ is partially additive, which allows us to apply Theorem 8. $\quad\Box$

7. Criterion of quasiweak boundedness of the Urysohn integral operator

Let $(U,\mu)$ and $(V,\nu)$ be spaces with measures $\mu$ and $\nu$. We say that an operator $T$ is of $(p,q)$ quasiweak type if it is bounded from $L_{p,1}(U,\mu)$ to $L_{q, \infty }(V,\nu)$. If the operator $T$ is bounded from $L_{p,\tau}(U,\mu)$ to $L_{q,\tau}(V,\nu)$, then we say that it is of $(p,q)$ strong type.

Let

$$ (Tf)(y)=\intop_U\! K(f(x),x,y)\,d\mu,\qquad y\in V,$$

(7.1)

be an Urysohn integral operator.

It follows from Corollary 5 that if the operator (7.1) is of $(p_0,q_0)$ and $(p_1,q_1)$ quasiweak type, then it is of $(p,q)$ strong type, where $p$ and $q$ are defined by (5.3) with $\theta \in(0,1)$. Thus, deriving weak-type estimates for these operators is of particular interest. In this section, we give necessary and sufficient conditions for the operator (7.1) to be of $(p,q)$ quasiweak type provided that certain a priori assumptions on the kernel $K$ hold. We need some lemmas.

Lemma 7.

Let $f \colon\, U\to{\mathbb R}$ be a $\mu$-measurable function taking finitely many values,

$$f(x)=\sum_{k=1}^n\lambda_k\chi^{}_{w_k}(x),$$

where $w_k\subset U$ are $\mu$-measurable sets of finite measure and $\lambda_k\in{\mathbb R}$ with $|\lambda_1|>|\lambda_2|>\dots>|\lambda_n|>0$. Then

$$\|f\|_{L_{p,1}(U,\mu)}=p\sum_{k=1}^n (|\lambda_k|-|\lambda_{k+1}|)\Biggl(\mu\Biggl(\,\bigcup_{i=1}^kw_i\Biggr)\!\Biggr)^{\!1/p},$$

where $\lambda_{n+1}=0$.

Proof.

Notice that the nonincreasing rearrangement $f^*$ of $f$ has the form

$$f^*(t)=\sum_{k=1}^n|\lambda_k| \kern1pt \chi^{}_{[t_{k-1},t_k)}(t),$$

where $t_0=0$ and $t_k-t_{k-1}=\mu(w_k)$. Then, applying the Abel transformation, we obtain

$$\begin{aligned} \, \|f\|_{L_{p,1}(U,\mu)} &=\sum_{k=1}^n|\lambda_k|\intop_{t_{k-1}}^{t_k}\!t^{1/p-1}\,dt =\sum_{k=1}^n (|\lambda_k|-|\lambda_{k+1}|)\intop_0^{t_k}t^{1/p-1}\,dt \\[2pt] &=p\sum_{k=1}^n(|\lambda_k|-|\lambda_{k+1}|)\Biggl(\mu\Biggl(\,\bigcup_{i=1}^kw_i\Biggr)\!\Biggr)^{\!1/p}. \ \quad\Box\end{aligned}$$

Lemma 8.

Suppose that the measure $\mu$ satisfies the regularity condition (2.2) and the function $K \colon\, U\to{\mathbb R}$ is integrable on all $\mu$-measurable subsets of $U$ of finite measure. Then, for any set $e$ of positive finite measure, there exists a set $w\subset e$ such that $\mu(w)\geq\mu(e)/3$ and

$$ \intop_e\mathopen|K(x)|\,d\mu \le 4\left|\intop_w\! K(x)\,d\mu\right|.$$

(7.2)

Proof.

For an arbitrary set $e$ such that $0<\mu(e)< \infty$, define the sets

$$e_+ := \bigl\{x\in e \colon\, \,K(x) \ge 0\bigr\} \qquad\text{and}\qquad e_- := \bigl\{x\in e \colon\, \, K(x)<0\bigr\}.$$

Then

$$\intop_e\mathopen|K(x)|\,d\mu = \intop_{e_+}\! K(x)\,d\mu - \intop_{e_-}\! K(x)\,d\mu \leq 2 \max\!\left\{\,\intop_{e_+}\!K(x)\,d\mu, \left|\,\intop_{e_-}\!K(x)\,d\mu\right|\right\}.$$

For definiteness, assume that

$$\intop_{e_+}\! K(x)\,d\mu \ge \left|\,\intop_{e_-}\! K(x)\,d\mu\right|.$$

Two cases are possible: either $\mu(e_+)\ge \mu(e)/2$ or $\mu(e_+)<\mu(e)/2$. In the first case, $w=e_+$.

In the second case, there exists an $\eta\subset e_-$ such that $\mu(\eta)\geq\mu(e_-)/3$ and

$$\left|\,\intop_{\eta}\! K(x)\,d\mu\right| \le \frac12 \left|\,\intop_{e_-}\! K(x)\,d\mu\right|.$$

Indeed, according to the regularity condition (2.2) with $\alpha=\mu(e_-)/3$, there exists a set $\eta'\subset e_-$ such that $\mu(e_-)/3\le\mu(\eta')\le2\mu(e_-)/3$ and, hence, $\mu(e_-)/3\le\mu(e_-\setminus\eta')\le 2\mu(e_-)/3$. Since

$$\left|\,\intop_{e_-}\!K(x)\,d\mu\right|=\left|\,\intop_{\eta'}\!K(x)\,d\mu\right|+\left|\,\intop_{\eta''}\!K(x)\,d\mu\right|$$

for $\eta''=e_-\setminus\eta'$, at least one of the terms on the right-hand side does not exceed half the left-hand side. If this is the first term, then $\eta=\eta'$, and if the second, then $\eta=\eta''$.

Set $w = \eta\cup e_+$; then $\mu(w)=\mu(\eta)+\mu(e)\leq(\mu(e_-)+\mu(e_+))/3=\mu(e)/3$ and

$$\left|\intop_w\!K(x)\,d\mu\right| = \left|\,\intop_{e_+}\! K(x)\,d\mu + \intop_{\eta}\!K(x)\,d\mu\right| \ge \left|\,\intop_{e_+}\! K(x)\,d\mu\right| - \left|\,\intop_{\eta} K(x)\,d\mu\right| \ge \frac 12 \left|\,\intop_{e_+}\! K(x)\,d\mu\right|.$$

Thus, we have

$$\intop_e\mathopen|K(x)|\,d\mu \leq 2\left|\,\intop_{e_+}\! K(x)\,d\mu\right| \leq 4\left|\intop_w\!K(x)\,d\mu\right|.\ \quad\Box$$

Theorem 9.

Let $1< p,q< \infty ,$ $(U,\mu)$ and $(V,\nu)$ be spaces with measures $\mu$ and $\nu$ satisfying the regularity condition (2.2), and $K \colon\, {\mathbb R}\times U\times V \to{\mathbb R}$.

Suppose that the Urysohn operator (7.1) is continuous from $L_{p,\tau}(U,\mu)$ to $L_{q, \infty }(V,\nu)$ and, for some $B>0$ and for an arbitrary $\nu$-measurable set $e\subset V,$

$$ \left|\intop_e\!K(z,x,y)\,d\nu\right|\leq B|z|\left|\intop_e\!K(1,x,y)\,d\nu\right|$$

(7.3)

for a.e. $x\in U$ and any $z\in{\mathbb R}$.

Then, for any $0<\tau<1,$

$$\begin{aligned} \, \|T\|_{L_{p,\tau}(U,\mu)\to L_{q, \infty }(V,\nu)} &\asymp \|T\|_{L_{p,1}(U,\mu)\to L_{q, \infty }(V,\nu)} \\ &\asymp \sup_{\nu(e)>0,\;\mu(w)>0}\frac1{(\nu(e))^{1/q'}(\mu(w))^{1/p}} \left|\intop_e\!\intop_w\! K(1,x,y)\,d\mu\,d\nu\right|\equiv F_{p,q}(K). \end{aligned}$$

Proof.

Since $0<\tau<1$ and so $L_{p,\tau}(U,\mu)\hookrightarrow L_{p,1}(U,\mu)$, we have

$$\|T\|_{L_{p,\tau}(U,\mu)\to L_{q, \infty }(V,\nu)}\lesssim \|T\|_{L_{p,1}(U,\mu)\to L_{q, \infty }(V,\nu)}.$$

Suppose that the operator $T \colon\, L_{p,\tau}(U,\mu)\to L_{q, \infty }(V,\nu)$ is bounded. Let $e\subset V$ and $w\subset U$ be, respectively, $\nu$- and $\mu$-measurable sets such that $0<\nu(e),\mu(w)< \infty$. Then

$$\begin{aligned} \, \frac1{(\nu(e))^{1/q'}}\!\left|\intop_e (Tf)(y)\,d\nu\right| &\leq \frac1{(\nu(e))^{1/q'}}\!\intop_0^{\nu(e)}(Tf)^*(t)\,dt \leq q' \|Tf\|_{L_{q, \infty }} \\[4pt] &\leq q'\|T\|_{L_{p,\tau}(U,\mu)\to L_{q, \infty }(V,\nu)}\|f\|_{L_{p,\tau}}. \end{aligned}$$

Let $f(x)=\chi^{}_w(x)$. Taking into account that $\|\chi^{}_w(x)\|_{L_{p,\tau}}\asymp (\mu(w))^{1/p}$, we have

$$\|T\|_{L_{p,\tau}(U,\mu)\to L_{q, \infty }(V,\nu)} \geq \frac1{(\nu(e))^{1/q'}(\mu(w))^{1/p}} \left|\intop_e\!\intop_U\! K(\chi^{}_w(x),x,y)\,d\mu\,d\nu\right|.$$

It follows from (7.3) that

$$\intop_U\!\intop_e\!K(\chi^{}_w(x),x,y)\,d\nu\,d\mu = \intop_w\!\intop_e\!K(1,x,y)\,d\nu\,d\mu+\intop_{U\setminus w}\!\intop_e\!K(0,x,y)\,d\nu\,d\mu = \intop_w\! \intop_e\!K(1,x,y)\,d\nu\,d\mu.$$

Thus, since the choice of the sets $e$ and $w$ is arbitrary, we obtain

$$F_{p,q}(K)\lesssim \|T\|_{L_{p,\tau}(U,\mu)\to L_{q, \infty }(V,\nu)}\lesssim \|T\|_{L_{p,1}(U,\mu)\to L_{q, \infty }(V,\nu)}.$$

It remains to show that

$$\|T\|_{L_{p,1}(U,\mu)\to L_{q, \infty }(V,\nu)}\lesssim F_{p,q}(K).$$

Let $f$ be an arbitrary function satisfying the hypotheses of Lemma 7:

$$f(x)=\sum_{k=1}^n\lambda_k\chi^{}_{w_k}(x).$$

In view of condition (7.3), for an arbitrary $\nu$-measurable set $e$ we have

$$\left|\intop_e(Tf)(y)\,d\nu\right|\leq B\sum_{k=1}^n |\lambda_k|\intop_{w_k}\left|\intop_e\!K(\chi^{}_{w_k},x,y)\,d\nu\right|d\mu.$$

Let us apply Lemma 7:

$$\begin{aligned} \, \nonumber &\left|\intop_e(Tf)(y)\,d\nu\right| \lesssim \sum_{k=1}^n(|\lambda_k|-|\lambda_{k+1}|)\intop_{\bigcup_{i=1}^kw_i}\!\left|\intop_e\!K(1,x,y)\,d\nu\right|d\mu \\ &\qquad\leq \widetilde{F}{} _{p,q}(K)(\nu(e))^{1/q'} \sum_{k=1}^n(|\lambda_k|-|\lambda_{k+1}|)\Biggl(\mu\Biggl(\,\bigcup_{i=1}^kw_i\Biggr)\!\Biggr)^{\!1/p} =p^{-1} \widetilde{F}{} _{p,q}(K)(\nu(e))^{1/q'}\|f\|_{L_{p,1}}, \end{aligned}$$

(7.4)

where $\lambda_{n+1}=0$ and

$$\widetilde{F}{} _{p,q}(K)=\sup_{\nu(e)>0,\;\mu(w)>0}\frac1{(\nu(e))^{1/q'}(\mu(w))^{1/p}} \intop_w\left|\intop_e\!K(1,x,y)\,d\nu\right|d\mu.$$

It follows from Lemma 8 that $\widetilde{F}{} _{p,q}(K)\asymp F_{p,q}(K)$. According to [15, Lemma 1] (see also [20]), we have

$$\|Tf\|_{L_{q, \infty }}\asymp\sup_{\nu(e)>0}\frac1{(\nu(e))^{1/q'}}\left|\intop_e(Tf)(y)\,d\nu\right|.$$

Thus, from (7.4) we obtain

$$\|Tf\|_{L_{q, \infty }}\leq F_{p,q}(K)\|f\|_{L_{p,1}}.$$

The functions satisfying the hypotheses of Lemma 7 form a dense set in the space $L_{p,1}(U,\mu)$. Due to the continuity of the operator $T \colon\, L_{p,1}(U,\mu)\to L_{q, \infty }(V,\nu)$, inequality (7.4) remains valid for any function $f\in L_{p,1}(U,\mu)$; i.e.,

$$\|T\|_{L_{p,1}\to L_{q, \infty }}\lesssim F_{p,q}(K).\ \quad\Box$$

Remark 9.

In the case when $T$ is a linear integral operator

$$(Tf)(y)=\intop_U\! K(x,y)f(x)\,d\mu,$$

the statement of Theorem 9 follows from the results of [15, 20], which, in particular, include necessary and sufficient conditions for the quasiweak boundedness of linear integral operators in net spaces.