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.
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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
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
for \(q< \infty \) and
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
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
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
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
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
hold for all \(y\in\Omega\) and all functions \(f\in LM^{\alpha_i}_{p,\sigma,y},\) \(i=0,1,\) then the inequality
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
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:
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 \(C>0\)
for all \(y\in\Omega\) and \(t>0\). Then the following continuous embedding holds:
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
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
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
and
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
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
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),
In the second case,
Moreover,
Indeed,
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,
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
Thus, according to the properties of rearrangements and inequalities (2.7) and (2.3),
i.e., (2.4) holds.
Next, for any \(s,t>0\), we have \(f^*(s+t)=\inf A_{s,t}\), where
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
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
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
Let \(\mu\) be a regular measure, \(1<r< \infty \), and \(\lambda=1/r'\), where \(1/r+1/r'=1\). Then
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,
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
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
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
(with the usual modification for \(q= \infty \)). Setting \(v(r)=t\), we have
Moreover, for \(0<\lambda < \infty \), we have \(M^\lambda_{p,q}(G)=LM_{p,q}^{v^\lambda(\cdot)}\) and
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 \(t>0\),
(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
Moreover,
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
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
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
(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 \(f\in Z(U)\),
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 \(f\in L_p(U,\mu)\),
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 \(y\in V,\)
Proof.
Since the integral is additive with respect to the measure,
Corollary 1.
If
then \(T\) is a partially additive operator.
Corollary 2.
If, for any \(f\in Z(U),\)
(in particular, if the kernel \(K\) is nonnegative), then \(T\) is a partially subadditive operator.
Proof.
According to (3.5), we have
Corollary 3.
If there is a \(B>0\) such that for any \(f\in Z(U)\)
then \(T\) is a partially quasi-additive operator with partial quasi-additivity constant \(A=B+1\).
Proof.
According to (3.5), we have
Remark 1.
If we are interested in the classical interpolation theorems for the Urysohn integral operator and, hence, in estimates of the form
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 \((0, \infty ),\)
and
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
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
and
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 \(s>0,\)
Proof.
Let \(f\in M^\alpha_{p,\tau}(G_\Omega,\mu)\), \(y\in\Omega\), and \(s>0\). Then
To obtain the second relation, notice that
and
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
hold for all \(y\in\Omega\) and \(s>0,\) then the inequality
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
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
From condition (4.2), we have
Hence, the following inequality holds for arbitrary \(s>0\) and \(y\in\Omega\):
where \(c_1>0\) depends only on \(\alpha_0\), \(\alpha_1\), and \(\sigma\).
Let us use condition (4.3) to estimate the second term:
Thus, for all \(t>0\), \(s>0\), and \(y\in\Omega\), we obtain
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
where
Making the change \(ct^\gamma\to t\), we obtain
where
To estimate \(J_1\) and \(J_3\), we apply the Hardy inequalities (Lemma 4), according to which we have
For all \(c>0\), we obtain
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
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:
for \(0<q< \infty \) and
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
hold for all \(y\in\Omega\) and all functions \(f\in LM^{\alpha_i}_{p,\sigma,y}(w),\) \(i=0,1,\) then the inequality
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
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
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
hold for all \(y\in\Omega\) and \(s>0,\) then the inequality
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),
From conditions (4.6) and (4.7), we have
and
Thus, for all \(t>0\), \(s>0\), and \(y\in\Omega\), we obtain
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:
where \(h\) is a positive function on \(V\).
Now, let us show that the partial quasi-additivity of \(T\) implies the inclusion
Indeed, if \(x\in F_t\), then \(x\in V\) and
or
since otherwise
and
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
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
hold for all functions \(f\in L_{p_i}(U,\mu),\) \(i=0,1,\) then the inequality
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
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
hold for all \(f\in L_{p_i,\sigma}(U,\mu),\) \(i=0,1,\) then the inequality
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
and are such that
where the corresponding constants depend only on the parameter \(p\), \(q\), and \(\tau\).
From condition (5.4), we have
According to inequalities (2.4) and (2.5), by Lemma 2 we obtain
Taking into account that
and
we find
Let us apply Theorem 2:
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
If for some \(M_0,M_1>0\) the inequalities
hold for all \(f\in L_{p_i,\sigma}(U,\mu),\) \(i=0,1,\) then the inequality
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
Setting \(f=0\), we obtain
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
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
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
hold for all \(f\in L_{p_i}(U,w_i,\mu),\) \(i=0,1,\) then the inequality
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
hold for all \(f\in L_p(U,w_i,\mu),\) \(i=0,1,\) then the inequality
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
and let
Then, according to Example 8 and Theorem 7 from [12],
From the hypotheses of the theorem, we obtain
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
Let \(f \colon\, U\to {\mathbb R}\) be a \(\mu\) -measurable function and \(G(f)=\{G_t(f)\}_{t>0}\) with
Then the following equalities hold for an arbitrary \(\mu\)-measurable set \(e\subset U\):
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
which implies equality (6.4). Next,
and
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
hold for all \(f\in L_{p_i}(U,w_i,\mu),\) \(i=0,1,\) then the inequality
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 \(i=0\),
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:
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
and
Thus, all the hypotheses of Theorem 3 are satisfied. Hence, applying Theorem 3 combined with Lemma 6, we obtain
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
If for some \(M_0,M_1>0\) the inequalities
hold for all \(f\in L_{p_i}(U,w_i,\mu),\) \(i=0,1,\) then the inequality
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
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
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,
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
where \(\lambda_{n+1}=0\).
Proof.
Notice that the nonincreasing rearrangement \(f^*\) of \(f\) has the form
where \(t_0=0\) and \(t_k-t_{k-1}=\mu(w_k)\). Then, applying the Abel transformation, we obtain
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
Proof.
For an arbitrary set \(e\) such that \(0<\mu(e)< \infty \), define the sets
Then
For definiteness, assume that
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
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
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
Thus, we have
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 \(e\subset V,\)
for a.e. \(x\in U\) and any \(z\in{\mathbb R}\).
Then, for any \(0<\tau<1,\)
Proof.
Since \(0<\tau<1\) and so \(L_{p,\tau}(U,\mu)\hookrightarrow L_{p,1}(U,\mu)\), we have
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
Let \(f(x)=\chi^{}_w(x)\). Taking into account that \(\|\chi^{}_w(x)\|_{L_{p,\tau}}\asymp (\mu(w))^{1/p}\), we have
It follows from (7.3) that
Thus, since the choice of the sets \(e\) and \(w\) is arbitrary, we obtain
It remains to show that
Let \(f\) be an arbitrary function satisfying the hypotheses of Lemma 7:
In view of condition (7.3), for an arbitrary \(\nu\)-measurable set \(e\) we have
Let us apply Lemma 7:
where \(\lambda_{n+1}=0\) and
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
Thus, from (7.4) we obtain
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.,
Remark 9.
In the case when \(T\) is a linear integral operator
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.
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Funding
The research of E. D. Nursultanov (Sections 1–4) was supported by the Ministry of Education and Science of the Republic of Kazakhstan, project nos. AP08856479 and AP08956157. The research of V. I. Burenkov presented in Sections 1–4 was supported by the Peoples’ Friendship University of Russia and performed in the S. M. Nikol’skii Mathematical Institute within the Russian Academic Excellence Project. The research of V. I. Burenkov presented in Sections 5–7 was supported by the Russian Science Foundation under grant 19-11-00087 and performed in Steklov Mathematical Institute of Russian Academy of Sciences.
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Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 312, pp. 131–157 https://doi.org/10.4213/tm4129.
Translated by I. Nikitin
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Burenkov, V.I., Nursultanov, E.D. Interpolation Theorems for Nonlinear Operators in General Morrey-Type Spaces and Their Applications. Proc. Steklov Inst. Math. 312, 124–149 (2021). https://doi.org/10.1134/S0081543821010077
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DOI: https://doi.org/10.1134/S0081543821010077