Abstract
For all admissible values of the numerical parameters sharp sufficient conditions on the functional parameters are obtained ensuring the boundedness of the generalized Riesz potential from one general local Morrey-type space to another one, which, for a certain range of the numerical parameters, coincide with the necessary ones.
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Introduction
In this paper, we study the boundedness from one general local Morrey-type space to another one of the generalized Riesz potential
under certain assumptions on the kernel \(\rho\). Our aim is to generalize the results obtained in [6] for the case of the classical Riesz potential \(I_{\alpha },\) in which \(\rho (t)=t^{\alpha -n}\), \(t>0\), \(0<\alpha <n\). Some of the presented results were stated without proofs in [7]. Let \(F,G : A\times B \longrightarrow [0,\infty ] .\) Throughout this paper we say that F is dominated by G uniformly in \(x\in A\) and write
if there exists \(c(B)>0\) such that
(So c(B) is independent of \(x \in A,\) but may depend on \(y \in B\).) Respectively, we say that F dominates G uniformly in \(x\in A\) and write
if there exists \(c(B)>0\) such that
We also say that F is equivalent to G uniformly in \(x\in A\) and write
if F and G dominate each other uniformly in \(x\in A.\)
Definitions and basic properties of general Morrey-type spaces
In this section we recall basic facts of the theory of general Morrey-type spaces.
Let, for a Lebesgue measurable set \(\Omega \subset \mathbb {R}^{n},\) \(\mathfrak {M}(\Omega )\) denote the space of all functions \(f: \Omega \longrightarrow \mathbb {C}\) Lebesgue measurable on \(\Omega ,\) and \(\mathfrak {M^{+}}(\Omega )\) denote the subset of \(\mathfrak {M}(\Omega )\) of all non-negative functions.
Definition 2.1
Let \(0<p,\theta \le \infty\) and let \(w \in \mathfrak {M^{+}}\big ((0,\infty )\big )\) be not equivalent to 0. We denote by \(LM_{p\theta ,w (\cdot )}\) the local Morrey-type space, the space of all functions \(f\in L_{p}^{ \text {loc}}(\mathbb {R}^{n})\) with finite quasinorm
Definition 2.2
Let \(0<p,\theta \le \infty\). We denote by \(\Omega _{\theta }\) the set of all functions \(w \in \mathfrak {M^{+}}((0,\infty ))\) which are not equivalent to 0 and such that
for some \(t>0\) .
Lemma 2.1
[3,4,5]. Let \(0<p\), \(\theta \le \infty\) and let w be a non-negative Lebesgue measurable function on \((0, \infty )\), which is not equivalent to 0.
Then the space \(LM_{p\theta , w(\cdot )}\) is nontrivial if and only if \(w \in \Omega _{\theta }\).
In the sequel, keeping in mind Lemma 2.1, we always assume that \(w \in \Omega _{\theta }\) for the local Morrey-type spaces \(LM_{p\theta ,w(\cdot )}\).
It is well known that the spaces \(LM_{p \theta , w(\cdot )}\) are Banach spaces if \(1\le p\), \(\theta \le \infty\) and are quasi-Banach spaces if \(0< p <1\), or \(0<\theta < 1\), or both \(0<p\), \(\theta <1\).
For further properties of general Morrey-type spaces, operators acting in such spaces and applications see, for example, survey papers [1, 2, 12,13,14,15,16] and references therein.
\(L_{p}\) and \(WL_{p}\) estimates of generalized Riesz potentials over balls
Let \(n \in \mathbb {N}, 0<p< \infty , \Omega \subset \mathbb {R}^{n}\) be a Lebesgue measurable set. Recall that the weak \(L_{p}\)-space \(WL_{p}(\Omega )\) is the space of all Lebesgue measurable functions \(f: \Omega \rightarrow \mathbb {C}\) for which
Here \(\vert G \vert\) denotes the Lebesgue measure of a set \(G \subset \mathbb {R}^{n}.\)
Remark 3.1
Let \(M > 0.\) It follows directly from the above definition that, if f is equivalent to M on \(\Omega\), then
and if \(f : \Omega \rightarrow \mathbb {R}\) is such that \(f \geqslant M\) almost everywhere on \(\Omega\), then
Definition 3.1
Let \(n \in \mathbb {N}.\) We say that \(\rho \in S_{n}\) if \(\rho \in \mathfrak {M^{+}}((0,\infty ))\) and
1) \({\int _{0}^{r}}\rho (t)t^{n-1}dt < \infty\) for all \(r > 0,\)
2) for some \(c_{1}, c_{2}> 0\),
for all \(s,t>0\) satisfying the inequality \(\frac{t}{2}\le s\le 2t.\)
Remark 3.2
Let \(x \in \mathbb {R}^{n}\), \(r>0\), \(y \in B(x,r)\), \(z \in\) \(^{c}B(x,2r).\) Then
and
Let \(s= \vert y-z\vert\) and \(t= \vert x-z\vert .\) Then by Condition 2) of Definition 3.1 we have
\(\text {3)} \ \ c_{1}\rho (\vert x-z \vert )\le \rho (\vert y-z \vert )\le c_{2}\rho (\vert x-z \vert )\)
for all \(x \in \mathbb {R}^{n}\), \(r>0\), \(y \in B(x,r)\), \(z \in\) \(^{c}B(x,2r).\)
Remark 3.3
If functions \(\rho _{1},\rho _{2} : (0,\infty )\rightarrow (0,\infty )\) satisfy Condition 2) of Definition 3.1 with \(c_{11}\), \(c_{12}>0\), \(c_{21}\), \(c_{22}>0\), respectively, then the product \(\rho _{1}\rho _{2}\) satisfies Condition 2) with \(c_{1}=c_{11} c_{21}\) and \(c_{2}=c_{12} c_{22}\). Indeed, it suffices to multiply the inequalities
where \(s, t>0\), \(\frac{t}{2}\le s \le 2t\) and \(i=1,2\).
Remark 3.4
If a function \(\rho _{1} : (0,1]\rightarrow (0,\infty )\) satisfies Condition 2) of Definition 3.1 with \(c_{11}\), \(c_{12}>0\) and a function \(\rho _{2} : (1,\infty ) \rightarrow (0,\infty )\) satisfies Condition 2) of Definition 3.1 with \(c_{21}\), \(c_{22}>0\), then the function
satisfies Condition 2) of Definition 3.1 with certain \(c_{1}, c_{2}>0\).
Indeed, let \(s,t>0\), \(\frac{t}{2}\le s\le 2t\). If \(t\le \frac{1}{2}\), then \(s\le 1\), \(\rho (s)=\rho _{1}(s)\), \(\rho (t)=\rho _{1}(t)\) and Condition 2) of Definition 3.1 is satisfied with \(c_{11}, c_{12}>0\). Respectively, if \(t>2\), then \(s>1\), \(\rho (s)=\rho _{2}(s)\), \(\rho (t)=\rho _{2}(t)\) and Condition 2) of Definition 3.1 is satisfied with \(c_{21}, c_{22}>0\). Let \(\frac{1}{2}<t\le 2\), then \(\frac{1}{4}<s\le 4\),
hence
So, for \(\frac{1}{2}<t\le 2\), \(\frac{t}{2}\le s\le 2t\), Condition 2) of Definition 3.1 is satisfied with
The above inequality is satisfied for all \(s,t>0\), \(\frac{t}{2}\le s\le 2t\), because \(c_{11}\le 1\le c_{12}\), \(c_{21}\le 1\le c_{22}\), hence \(c_{1} \le c_{11}\), \(c_{1} \le c_{21}\), \(c_{12} \le c_{2}\), \(c_{22} \le c_{2}\).
Remark 3.5
If a function \(\rho : (0,\infty ) \rightarrow (0,\infty )\) satisfies Condition 2) of Definition 3.1, then for any \(\gamma >0\) \(c_{1\gamma }, c_{2\gamma } >0\) such that for any \(t>0\)
Indeed, if \(\frac{1}{2}\le \gamma \le 2\), then this inequality follows by Condition 2) with \(s=\gamma t\). Let \(2 \le \gamma < 4\), then by Condition 2) with \(t=2\tau\), \(\tau >0\) and \(s=\gamma \tau\) and the above inequality with \(t=\tau\) and \(\gamma =2\)
because \(\frac{1}{2} (2\tau ) \le \gamma \tau \le 2 (2\tau )\).
Next, let \(4 \le \gamma < 8\) by Condition 2) with \(t = 4\tau , \tau >0, s= \gamma \tau ,\)
then
Furthermore, if \(2^{k-1}\le \gamma <2^{k}\), \(k\in \mathbb {N}\), \(k>3 \Leftrightarrow k=\left[ \frac{\ln \gamma }{\ln 2}\right] +1\), then
The argument for the case \(0<\gamma <\frac{1}{2}\) is similar.
Example 1
Let \(\rho (t)=t^{\alpha -n}, t>0, 0<\alpha < n.\) Then \(\rho \in S_{n}\) because
and, for any \(s,t >0\) for which \(\frac{t}{2} \le s\le 2t\),
Example 2
Let \(0<\alpha <n\), \(-\infty<\beta _{1},\beta _{2}<\infty\),
and \(\rho (t) =t^{\alpha -n}\Psi _{\beta _{1}\beta _{2}}(t)\), \(t>0\). Then \(\rho \in S_{n}\).
Since the function \(\rho\) is continuous on \((0,\infty )\) it suffices to prove Condition 1) of Definition 3.1 for \(r=1.\) Recall that for any \(\varepsilon >0\) there exists \(C_{\varepsilon }\ge 1\) such that
for all \(t \in (0,1].\) If \(\beta _{1}\le 0,\) then
If \(\beta _{1}>0,\) then
if we choose \(\varepsilon >0\) to be such that \(\varepsilon \beta _{1}<\alpha .\)
As for Condition 2) of Definition 3.1, by Remark 3.3 and Example 1, it suffices to prove that the function \(\varphi _{\beta _{1},\beta _{2}}\) satisfies this condition.
Let \(s,t>0\) and \(\frac{t}{2}\le s\le 2t.\) If \(s>1,\) hence \(t>\frac{1}{2},\) then
and
hence
For similar reasons this inequality also holds if \(s\le 1,\) hence \(t\le 2.\) Therefore, for \(s\le 1\)
If \(t \le 1,\) this means that
If \(1<t\le 2,\) then
hence
which means that
So, this inequality holds for all \(s\le 1\) and all t satisfying the inequality \(\frac{t}{2} \le s\le 2t.\)
For similar reasons, for all \(s>1\) and all t satisfying the inequality \(\frac{t}{2} \le s\le 2t\)
Example 3
Let \(0<\alpha <n\), \(-\infty<\beta <\infty\) and
Then \(\rho \in S_{n}\). Indeed, Condition 1) is clearly satisfied. As for Condition 2), it is also satisfied by Remark 3.4, because \(\rho _{1}(t)=t^{\alpha -n}\) satisfies Condition 2) on (0, 1] by Example 1 and \(\rho _{2}(t) =t^{\beta }\) satisfies Condition 2) on \((1,\infty )\) by an argument similar to that of the proof of Example 1.
Note that \(t^{\beta }\) can be replaced by any function \(\rho (t)\) satisfying Condition 2) on \((1,\infty )\).
Definition 3.2
For \(\rho \in S_{n}\) and \(f \in \mathfrak {M}(\mathbb {R}^{n})\)
and
Lemma 3.1
Let \(n \in \mathbb {N},\rho \in S_{n},\) \(0 < p\le \infty , x \in \mathbb {R}^{n}, r>0, f \in \mathfrak {M}(\mathbb {R}^{n}).\) ThenFootnote 1\(G= r^{\frac{n}{p}} \left( \overline{I}_{\rho (\cdot ), 2r} \vert f \vert \right) (x).\)
uniformly in \(x \in \mathbb {R}^{n}\), \(r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n}).\)
Proof
By Property 3) of Remark 3.2 for any \(y \in B(x,r)\)
hence, by Remark 3.1, we get
for all \(x \in \mathbb {R}^{n}, r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n}).\) Here \(v_{n}\) is the volume of the unit ball in \(\mathbb {R}^{n}\).
Lemma 3.2
Let \(n \in \mathbb {N},\rho \in S_{n},\) \(0 < p\le \infty , x \in \mathbb {R}^{n}, r>0\), \(f \in \mathfrak {M}(\mathbb {R}^{n}).\) Then
and
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n}).\)
Proof
According to the propertiesFootnote 2 of the spaces \(L_{p}(\Omega )\) and \(WL_{p}(\Omega )\)
and
Next,
By Condition 3) of Remark 3.2
and
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n}).\)
Also, by Condition 3) of Remark 3.2
Hence, by (3.6)
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n}).\)
Finally, by adding inequalities (3.1) and (3.7), we get that
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n}).\)
Lemma 3.3
Let, for \(n \in \mathbb {N},\rho \in S_{n}\), \(1\le p_{1}<p_{2}<\infty\), and
where \(p_{1}^{\prime }\) is the conjugate number to \(p_{1}.\)
1. Assume that \(\rho\) is such that the function \(\varphi _{n, \rho , p_{1}, p_{2}}\) is almost non-decreasing on \((0,\infty )\), that is for some \(c>0\)
If \(1<p_{1}<p_{2}<\infty\), then
uniformly in \(x\in \mathbb {R}^{n}\), \(r>0\) and \(f\in L_{p_{1}}(B(x,2r))\).
Also, if \(p_{1}=1\), \(1< p_{2} < \infty\), then
uniformly in \(x\in \mathbb {R}^{n}\), \(r>0\) and \(f\in L_{1}(B(x,2r))\).
2. Inequality (3.10) also holds for \(p_{1}=1\) and \(1\le p_{2}\le \infty\) under the assumption
uniformly in \(r >0,\) replacing assumption (3.9).
3. Inequality (3.10) and hence inequality (3.11) also holds for \(1\le p_{1}< \infty\) and \(0 <p_{2}\le p_{1}\) under the assumption
uniformly in \(r >0,\) replacing assumption (3.9).
4. Condition (3.13) is necessary for the validity of inequalities (3.10) and (3.11) for all \(0<p_{1},p_{2} \le \infty\).
5. Condition (3.13) is necessary and sufficient for the validity of inequalities (3.10) and (3.11) for all \(1\le p_{1}< \infty\) and \(0<p_{2}\le p_{1}\).
Proof
1. Let \(1\le p_{1}<p_{2}<\infty\), \(x\in \mathbb {R}^{n}\), \(r>0\), \(z\in B(x,r)\) and \(f\in L_{p_{1}}(B(x,2r))\). Then
Since the function \(\varphi _{n,\rho , p_{1},p_{2}}\) is almost non-decreasing on \((0,\infty )\) and, for \(z\in B(x,r)\), \(y\in B(x,2r)\),
we have by (3.9)
By Remark 3.5 with \(\gamma =3\) it follows that
Hence,
and
uniformly in \(x\in \mathbb {R}^{n}\), \(r>0\), \(z\in B(x,r)\) and \(f\in L_{p_{1}}(B(x,2r))\).
Next, we apply the well-known inequalities for the Riesz potential. If \(1<p_{1}<p_{2}<\infty\), then
uniformly in \(f\in L_{p_{1}}(\mathbb {R}^{n})\). Also, if \(1<p_{2}<\infty\), then
uniformly in \(L_{1}(\mathbb {R}^{n})\).
If \(1<p_{1}<p_{2}<\infty\), then, by (3.14) and (3.15), it follows that
uniformly in \(x\in \mathbb {R}^{n}\), \(r>0\) and \(f\in L_{p_{1}}(B(x,2r))\), which is inequality (3.10).
Respectively, if \(1<p_{2}<\infty\), then by (3.14) and (3.16) there follows inequality (3.11).
2. If \(p_{1}=1\) and \(1\le p_{2}\le \infty ,\) then, by applying Young’s inequality for truncated convolutions, we get
where \(\sigma _{n}=nv_{n}\) is the surface area of the unit ball in \(\mathbb {R}^{n}.\)
By inequality (3.12) and Remark 3.5 with \(\gamma = 3\) it follows that
which implies inequality (3.10).
3. If \(1\le p_{2}=p_{1}< \infty ,\) then similarly to Step 2
By inequality (3.13) and Remark 3.5 with \(\gamma =3\) it follows that
which implies inequality (3.10).
If \(0<p_{2}< p_{1}, 1\le p_{1}<\infty ,\) then, by applying Hölder’s inequality and inequality (3.10) with \(p_{2}=p_{1}\) we get
which is inequality (3.10).
4. Assume that for some \(0<p_{1},p_{2}\le \infty\) inequality (3.10) and (3.11) is satisfied. If inequality (3.10) is satified, then inequality (3.11) is also satisfied. Take in this inequality \(x=0\) and \(f\equiv 1\). Then for any \(y\in B(0,r)\) we have \(B(y,2r) \supset B(0,r)\) and
By Remark 3.1
and \(\left\| \chi _{_{B(0,2r)}}\right\| _{L_{p_{2}}(B(0,r))}= (\sigma _{n} r^{n})^{\frac{1}{p_{1}}}\), hence by (3.11)
uniformly in \(r>0\), which implies inequality (3.13).
5. The last statement of Lemma 3.3 follows by Steps 3 and 4.
Remark 3.6
If \(1 \le p_{1}<p_{2}< \infty\) and
then inequality (3.10) for \(1<p_{1}<p_{2}<\infty\) and inequality (3.11) for \(1<p_{2}<\infty\) cannot hold for any \(f \in L_{p_{1}}{(\mathbb {R}^{n})}\), \(f \in L_{1}(\mathbb {R}^{n})\), respectively, not equivalent to 0 on \(\mathbb {R}^{n}\). Indeed, by passing to the limit as \(r \rightarrow \infty ,\) in (3.10) and (3.11), it follows that \(\parallel I_{\rho (.)}|f| \parallel _{L_{p_{2}}(\mathbb {R}^{n})}= 0\), \(\parallel I_{\rho (.)}|f| \parallel _{WL_{p_{2}}(\mathbb {R}^{n})}= 0,\) hence, f is equivalent to 0 on \(\mathbb {R}^{n}.\)
If \(\rho (t)= t^{\alpha - n}\), \(0<\alpha <n\) and \(1\le p_{1}<p_{2},\) then Condition (3.9) reduces to the condition \(\alpha \ge n(\frac{1}{p_{1}}-\frac{1}{p_{2}}),\) Condition (3.12) reduces to the condition \(\alpha > n(1-1/{p_{2}})\) and Condition (3.13) is satisfied since \(\alpha >0.\)
Corollary 3.1
Let the assumptions of Lemma 3.3 for \(n, \rho , p_{1}, p_{2}\) and the function \(\varphi _{n,\rho , p_{1}, p_{2}}\) be satisfied. Then, for \(1<p_{1}<p_{2}< \infty ,\) for \(p_{1}=1, 1<p_{2}\le \infty ,\) and for \(1\le p_{1}< \infty , 0<p_{2}\le p_{1}\)
uniformly in \(x \in \mathbb {R}^{n}\), \(r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n})\bigcap L_{p_{1}}(B(x,2r))\) and, for \(1<p_{2}<\infty\)
uniformly in \(x \in \mathbb {R}^{n}\), \(r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n})\bigcap L_{1}(B(x,2r)).\)
Lemma 3.4
Let \(f \in \mathfrak {M}(\mathbb {R}^{n})\) be a non-negative function and \(\rho\) be a non-negative, non-increasing, continuously differentiable function on \((0,\infty )\) such that \(\lim _{t\rightarrow +\infty }\rho (t)=0.\) Then for any \(x \in \mathbb {R}^{n}\) and \(r>0\)
For \(\rho (t)=t^{- \beta }\), \(t>0\), \(\beta >0\) this lemma was proved in [3] (Lemma 3).
Proof
By applying the Fubini theorem we get
Definition 3.3
Let \(n \in \mathbb {N}, 1 \le p_{1}< \infty , 0<p_{2}\le \infty .\) Then \(\rho \in S_{n, p_{1}, p_{2}}\) if \(\rho \in \mathfrak {M^{+}}((0,\infty ))\) and there exists a positive non-increasing continuously differentiable function \(\tilde{\rho }: (0,\infty ) \rightarrow (0,\infty )\) such that
1) \(\tilde{\rho }(t)\approx \rho (t)\) uniformly in \(t>0,\)
\(2)\ \lim \limits _{t \rightarrow \infty } \tilde{\rho }(t)=0,\)
\(3)\ \tilde{\rho } \in S_{n},\)
\(4)\ {\int \limits _{0}^{1}} \tilde{\rho }(t)t^{\frac{n}{p_{1}^{\prime }}-1}dt= \infty ,\) \(\int \limits _{1}^{\infty } \tilde{\rho }(t)t^{\frac{n}{p_{1}^{\prime }}-1}dt< \infty ,\)
\(5)\ \vert \tilde{\rho }^{\prime }(t)\vert t \gtrsim \tilde{\rho }(t)\) uniformly in \(t>0,\)
6) if \(0< p_{2}\le p_{1}\) and \(1\le p_{1}< \infty\), then
uniformly in \(r>0\),
7) if \(1\le p_{1}<p_{2}<\infty\), then the function \(\phi _{n,\tilde{\rho },p_{1}, p_{2}}(t)= \tilde{\rho }(t)t^{n\big (\frac{1}{p_{1}'}+\frac{1}{p_{2}}\big )}\) is almost non-decreasing on \((0,\infty )\).
Moreover, if \(p_{1}=1\) and \(1<p_{2}<\infty\), then \(\rho \in \tilde{S}_{n,1,p_{2}}\) if there exists a positive non-increasing continuously differentiable function \(\tilde{\rho }:(0,\infty )\rightarrow (0,\infty )\) such that Conditions 1)–3) and 5) are satisfied, Conditions 4) and 6) are satisfied for \(p_{1}=1\) and instead of Condition 7) the following condition is satisfied
8)
uniformly in \(r>0.\)
Remark 3.7
For the Riesz potential \(\rho (t) =t^{\alpha -n}\), \(0<\alpha <n\), \(\tilde{\rho }=\rho\). Condition 2) is satisfied because \(\alpha <n\), Condition 3) is satisfied because \(\alpha >0\), Condition 4) is satisfied if \(p_{1}<\infty\) and \(\alpha <\frac{n}{p_{1}}\), Condition 5) is obviously satisfied, Condition 6) is satisfied because \(\alpha > 0,\) Condition 7) is satisfied if \(\alpha \ge n(1/{p_{1}}-1/{p_{2}},\) Condition 8) is satisfied if \(\alpha > n(1-1/{p_{2}}).\)
Remark 3.8
Clearly, due to Condition 1), Conditions 2)–4), 6)–8) are equivalent to the conditions, obtained from them by replacing \(\tilde{\rho }\) by \({\rho }\)
Theorem 3.1
Let \(n \in \mathbb {N}, 1 \le p_{1}<\infty , 0<p_{2} \le \infty\).
1. If \(1<p_{1}<p_{2}<\infty\) or \(1 \le p_{1}< \infty\) and \(0< p_{2}\le p_{1}\), and \(\rho \in S_{n,p_{1}.p_{2}},\) then
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in L_{p_{1}}^{\text {loc}}(\mathbb {R}^{n}).\)
2. If \(p_{1}=1\) and \(0<p_{2}<\infty ,\) and \(\rho \in \tilde{S}_{n,1,p_{2}},\) then
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in L_{1}^{\text {loc}}(\mathbb {R}^{n}).\)
3. If \(p_{1}=1\) and \(1<p_{2}<\infty ,\) and \(\rho \in S_{n,1,p_{2}},\) then
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in L_{1}^{\text {loc}}(\mathbb {R}^{n}).\)
For \(\rho (t) = t^{\alpha -n}\), \(0<\alpha <n,\) hence for the Riesz potential \(I_{\alpha }\), this theorem is proved in [6] (Lemma 3.6 and Theorem 3.9).
Remark 3.9
If \(p_{1}=1\) and \(0<p_{2}<\infty\) and \(\rho \in \tilde{S}_{n,1 ,p_{2}},\) equivalence (3.22) holds if Condition 6) of Definition 3.3 for \(0 < p_{2} \le 1\) and Condition 7) of Definition 3.3 for \(1< p_{2} < \infty\) are satisfied. If \(\rho (t)=t^{\alpha -n}, 0<\alpha <n,\) then this means that \(n(1-\frac{1}{p_{2}})_{+}<\alpha <n.\)
If \(p_{1}=1\) and \(1<p_{2}<\infty\) and \(\rho \in S_{n,1 ,p_{2}}\), equivalence (3.23) holds if Condition 8) of Definition 3.3 is satisfied. If \(\rho (t)=t^{\alpha -n}\), \(0<\alpha <n,\) then this means that \(n(1-\frac{1}{p_{2}})\le \alpha <n.\)
In particular, for \(\alpha =n(1-1/{p_{2}})\) Statement 3 of Theorem 3.1 holds, while, as proved in Lemma 3.3 of [6], Statement 2 does not hold.
Proof
Step 1 (proof of (3.21)). We apply equivalence (3.2).
1.1a. To the first summand in the right-hand side of (3.2) we apply Condition 2) of Definition 3.1 with \(\rho\) replaced by \(\tilde{\rho }, s\) by r, t by 2r and inequality (3.10) and get that
uniformly in \(x \in \mathbb {R}^{n},\) \(r>0\) and \(f \in L^{\text {loc}}_{1}( \mathbb {R}^{n}).\)
1.1b (second proof of inequality (3.24)). To the first summand in the right-hand side of (3.2) we apply inequality (3.10) and Condition 2) of Definition 3.1 with \(\rho\) replaced by \(\tilde{\rho }\), firstly, s replaced by r and t by 2r, secondly, s replaced by \(\frac{\tau }{2}\) and t by \(\tau ,\) and we get that
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in L_{p_{1}}^{\text {loc}}(\mathbb {R}^{n}).\)
1.2a. In order to estimate the second summand in the right-hand side of (3.2) we obtain an estimate for \((\overline{I}_{\rho (.),r}|f|)(x).\) First we note that
Since by Condition 2) of Definition 3.1 \(\tilde{\rho }(2^{k}r)\ge c_{1}\tilde{\rho }(2^{k-1}r)\) and \(\vert x-y \vert \ge 2^{k-1}r\) for all \(y \in B(x,2^{k}r)\setminus B(x,2^{k-1}r)\), it follows that
So,
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in L_{1}^{loc}(\mathbb {R}^{n}).\)
Hence, by applying Hölder’s inequality, we get that
uniformly in \(x \in \mathbb {R}^{n}, r>0\) and \(f \in L_{1}^{loc}(\mathbb {R}^{n}).\)
Step 2 (proof of equivalences (3.22) and (3.23) for \(p_{1}=1\)). We apply equivalences (3.2) and (3.3).
2.1a. Under the assumption \(\tilde{\rho } \in \tilde{S}_{n,1,p_{2}}\) inequality (3.12) holds for \(p_{1}=1\) and \(1\le p_{2}\le \infty\) and inequality (3.13) holds for \(p_{1}=1\) and \(0<p_{2}\le 1,\) therefore, Lemma 3.3 ensures the validity of inequality (3.10) with \(p_{1}=1\) and we have
uniformly in \(x\in \mathbb {R}^{n}, \ r>0\) and \(f \in L^{\text {loc}}_{1}( \mathbb {R}^{n})\) and we get the estimate above for the first summand in (3.22).
2.1b. If \(\rho \in S_{n,1,p_{2}},\) then condition (3.9) holds for \(p_{1}=1\) and \(1<p_{2}< \infty ,\) therefore, Lemma 3.3 ensures the validity of inequality (3.11) and we, respectively, have
uniformly in \(x\in \mathbb {R}^{n}, \ r>0\) and \(f \in L^{\text {loc}}_{1}( \mathbb {R}^{n})\) and we get the estimate above for the first summand in (3.23).
2.2a. If \(\tilde{\rho }\in \tilde{S}_{n,1,p_{2}}\) and \(1\le p_{2}\le \infty\), to the second summand in the right-hand side of (3.2) we apply inequality (3.25). Equivalence (3.2) and inequalities (3.25) and (3.27) imply that
uniformly in \(x \in \mathbb {R}^{n},\) \(r>0\) and \(f \in L^{\text {loc}}_{1}(\mathbb {R}^{n}).\)
2.2b. Accordingly, if \(\tilde{\rho } \in S_{n,1,p_{2}},\) \(p_{1}=1\) and \(1<p_{2}<\infty ,\) then equivalence (3.3) and inequalities (3.25) and (3.28) imply that
uniformly in \(x \in \mathbb {R}^{n},\) \(r>0\) and \(f \in L^{\text {loc}}_{1}(\mathbb {R}^{n}).\)
2.3. For all \(y \in B(x,r)\) we have \(\vert y-z \vert \le 2r\) if \(z \in B(x,r)\) and \(\vert y-z\vert \le 2 \vert x-z \vert\) if \(z \in ^{c}B(x,r)\) (since \(\vert y-z \vert \le \vert y-x \vert + \vert x-z\vert \le r+ \vert x-z\vert \le 2 \vert x-z \vert ).\) Therefore, since \(\tilde{\rho }\) is non-increasing
By Condition 2) of Definition 3.1 with \(s=2t\) we have \(\tilde{\rho }(2t)\ge c_{1}\tilde{\rho }(z)\) for any \(t>0,\) hence, by equality (3.18) uniformly in \(x \in \mathbb {R}^{n},\) \(r>0, \ y \in B(x,r)\) and \(f \in L^{\text {loc}}_{1}(\mathbb {R}^{n})\)
Consequently, by Remark 3.1 and by Condition 5) of Definition 3.3
uniformly in \(x \in \mathbb {R}^{n},\) \(r>0\) and \(f \in L^{\text {loc}}_{1}(\mathbb {R}^{n}).\)
Step 3. Statement 1 of Theorem 3.1 follows by inequalities (3.24) and (3.26). Statement 2 follows by inequalities (3.25), (3.27) and (3.31). Statement 3 follows by inequalities (3.25), (3.28) and (3.31).
Generalized Riesz potential and Hardy operator
Let \(\mathfrak {M}^{+}((0,\infty ))\) be the subset of \(\mathfrak {M}((0,\infty ))\) consisting of all non-negative functions on \((0,\infty ).\) We denote by \(\mathfrak {M}^{+}((0,\infty );\downarrow )\) the cone of all functions in \(\mathfrak {M}^{+}((0,\infty ))\) which are non-increasing on \((0,\infty )\) and we set
Let H be the Hardy operator
Moreover, let, for \(0<p\le \infty\) and a Lebesgue measurable function \(v:(0,\infty )\rightarrow [0,\infty )\), \(L_{p_{1}v(\cdot )}(0,\infty )\) denote the space of all Lebesgue measurable functions \(f:(0,\infty )\rightarrow \mathbb {C}\) for which
Theorem 4.1
Let \(n \in \mathbb {N}\), \(1\le p_{1}< \infty ,\) \(0<p_{2}\le \infty ,\) \(0<\theta _{2}\le \infty\), \(w_{2} \in \Omega _{\theta _{2}}\), \(\rho\) be a positive continuous function on \((0,\infty )\) and
1. If \(1<p_{1}<p_{2}<\infty\) or \(1\le p_{1} <\infty\) and \(0< p_{2}\le p_{1}\), and \(\rho \in S_{n,p_{1}.p_{2}}\), then
uniformly in \(f\in \mathfrak {M} (\mathbb {R}^{n})\).
2. If \(p_{1}= 1\) and \(0<p_{2}<\infty\), and \(\rho \in \tilde{S}_{n,1,p_{2}}\), then
uniformly in all non-negative functions \(f\in \mathfrak {M} (\mathbb {R}^{n})\).
3. If \(p_{1}=1\) and \(1<p_{2}<\infty\), and \(\rho \in S_{n,1,p_{2}}\), then
uniformly in all non-negative functions \(f\in \mathfrak {M} (\mathbb {R}^{n})\).
Remark 4.1
If \(\rho (t)=t^{\alpha -n}\), \(1 \le p_{1}<\infty ,\) \(0< p_{2}<\infty ,\) \(0<\alpha <\frac{n}{p_{1}},\) then \(\mu _{n,\rho , p_{1}}(r)=r^{-\sigma },\) where \(\sigma =\frac{n}{p_{1}}-\alpha ,\) \(\mu _{n,\rho , p_{1}}^{(-1)}(r)=r^{- \frac{1}{\sigma }},\) \(v_{2}(r)= \sigma ^{-\frac{1}{\theta _{2}}}w_{2}(r^{-\frac{1}{\sigma }})r^{-\frac{n}{\sigma p_{2}}-\frac{1}{\sigma \theta _{2}}-\frac{1}{\theta _{2}}},\) \(g_{n,\rho , p_{1}}(t)\) \(= \parallel f\parallel _{L_{p_{1}}(B(0,t^{-\frac{1}{\sigma }}))}\) and Lemma 4.1 takes the form of Lemma 4.1 in [6].
Remark 4.2
Due to Condition 4) of Definition 3.3
and \(\mu _{n,\rho , p_{1}}\) is a strictly decreasing continuously differentiable function on \((0,\infty ).\) Moreover,
Proof
1. Let \(1< p_{1}< p_{2} < \infty\) or \(1 \le p_{1} \le \infty\) and \(0 < p_{2} \le p_{1}\), and \(\rho \in S_{n,p_{1},p_{2}}\). By inequality (3.21) we have \(\Big (c=\left( \int _{1}^{\infty } \rho (t)t^{\frac{n}{p'_{1}}-1}dt \right) ^{-1}\Big )\)
uniformly in \(f\in \mathfrak {M} (\mathbb {R}^{n})\).
2. Let \(p_{1} =1\) and \(0 < p_{2} \le 1\), and \(\rho \in \tilde{S}_{n,p_{1},p_{2}}\). By inequality (3.22) we have similarly to Step 1
uniformly in all non-negative functions \(f\in \mathfrak {M} (\mathbb {R}^{n})\).
3. Let \(p_{1} = 1\) and \(1< p_{2} < \infty\), and \(\rho \in S_{n,1,p_{2}}\). Equivalence (4.6) follows similarly to Step 2 from equivalence (3.23). \(\square\)
Theorem 4.2
Assume that \(n \in \mathbb {N}\), \(1\le p_{1}< \infty\), \(0<p_{2}\le \infty\), \(0<\theta _{1}\), \(\theta _{2}\le \infty\), \(w_{1}\in \Omega _{\theta _{1}}\), \(w_{2}\in \Omega _{\theta _{2}}\), \(\mu _{n,\rho , p_{1}}\) is defined by formula (4.1),
and \(v_{2}\) is defined by formula (4.2).
1. Let \(1<p_{1}<p_{2}< \infty\) or \(1\le p_{1}< \infty\) and \(0<p_{2}\le p_{1}\), and \(\rho \in S_{n,p_{1}, p_{2}}\). If the operator H is bounded from \(L_{\theta _{1}, v_{1}(\cdot )}(0,\infty )\) to \(L_{\theta _{2}, v_{2}(\cdot )}(0,\infty )\) on the cone \(\mathbb {A}\), that is
uniformly in \(g\in \mathbb {A}\), then the operator \(I_{\rho (\cdot )}\) is bounded from \(LM_{p_{1} \theta _{1},w_{1}(\cdot )}\) to \(LM_{p_{2} \theta _{2},w_{2}(\cdot )}\).
2. Let \(p_{1}=1\), \(0<p_{2}<\infty\) and \(\rho \in \tilde{S}_{n,1,p_{2}}\). Then the operator \(I_{\rho (\cdot )}\) is bounded from \(LM_{1 \theta _{1},w_{1}(\cdot )}\) to \(LM_{p_{2} \theta _{2},w_{2}(\cdot )}\) if and only if the operator H is bounded from \(L_{ \theta _{1},v_{1}(\cdot )}(0, \infty )\) to \(L_{ \theta _{2},v_{2}(\cdot )}(0, \infty )\) on the cone \(\mathbb {A}\).
3. Let \(p_{1}=1\), \(1<p_{2}<\infty\) and \(\rho \in S_{n,1,p_{2}}\). Then the operator \(I_{\rho (\cdot )}\) is bounded from \(LM_{p_{1} \theta _{1},w_{1}(\cdot )}\) to \(WLM_{p_{2} \theta _{2},w_{2}(\cdot )}\) if and only if the operator H is bounded from \(L_{ \theta _{1},v_{1}(\cdot )}(0, \infty )\) to \(L_{ \theta _{2},v_{2}(\cdot )}(0, \infty )\) on the cone \(\mathbb {A}\).
Remark 4.3
If we put \(r=\mu _{n_{1},\rho , p_{1}}(t)\) in (4.9), then, taking into account that \(\big ( \mu ^{(-1)}_{n_{1},\rho , p_{1}}(r) \big )'= \Big (\mu '_{n_{1},\rho , p_{1}}\big (\mu ^{(-1)}_{n_{1},\rho , p_{1}}(r)\big ) \Big )^{-1}\), \(r>0\), we get
and
Lemma 4.1
Assume that \(0<p_{1}, p_{2}, \theta _{1}, \theta _{2} \le \infty\), \(w_{1}\in \Omega _{\theta _{1}}\), \(\rho\) is a positive continuous function on \((0, \infty )\) such that
and the functions \(\mu _{n,\rho , p_{1}}\), \(g_{n,\rho , p_{1}}\), \(v_{1}\), \(v_{2}\) are defined by formulas (4.1), (4.3), (4.9), (4.2), respectively. Then
for all \(f \in LM_{p_{1}\theta _{1},w_{1}(\cdot )}\), for any measurable function \(\phi _{1}:(0,\infty )\rightarrow (0, \infty )\) and \(t>0\)
and for any measurable function \(\phi _{2}:(0,\infty )\rightarrow (0, \infty )\) and \(t>0\)
Proof
1. Indeed,
(We have used equality (4.11)).
2. Note that, by (3.10) and (4.7), for \(0<\theta _{1}<\infty\), \(t>0\)
and, similarly
These equalities also hold for \(\theta _{1}=\infty\), because, for example, by (4.9)
Similarly, by (4.2), (4.7) and (4.12) for \(0<\theta _{2}\le \infty\), \(t>0\)
and
The proved equalities imply equalities (4.17) and (4.20) by passing to the limit as \(t\rightarrow 0^{+}\) or \(t\rightarrow +\infty\). \(\square\)
Proof of Theorem 4.2
1. Let \(1< p_{1}< p_{2} < \infty\) or \(1 \le p_{1} < \infty\) and \(0 < p_{2} \le p_{1}\), and \(\rho \in S_{n,p_{1},p_{2}}\). Assume that the operator H is bounded from \(L_{\theta _{1},v_{1}(\cdot )}(0,\infty )\) to \(L_{\theta _{2},v_{2}(\cdot ) }(0,\infty )\) on the cone \(\mathbb {A}\). Since \(g_{n,\rho ,p_{1}}\in \mathbb {A}\), by Statement 1 of Theorem 4.1 and formula (4.14) we have
uniformly in \(f\in LM_{p_{1}\theta _{1},w_{1}(\cdot )}\), hence the operator \(I_{\rho (\cdot )}\) is bounded from \(LM_{p_{1}\theta _{1},w_{1}(\cdot )}\) to \(LM_{p_{2}\theta _{2},w_{2}(\cdot )}\).
2.1. Let \(p_{1}=1\), \(0<p_{2}<\infty\) and \(\rho \in \tilde{S}_{n,1,p_{2}}\). If the operator H is bounded from \(L_{\theta _{1},v_{1}(\cdot )(0,\infty )}\) to \(L_{\theta _{2} ,v_{2}(\cdot )}(0,\infty )\) on the cone \(\mathbb {A}\), then by Statement 2 of Theorem 4.1 as in Step 1 it follows that the operator \(I_{\rho (\cdot )}\) is bounded from \(LM_{p_{1}\theta _{1},w_{1}(\cdot )}\) to \(LM_{p_{2}\theta _{2},w_{2}(\cdot )}\).
2.2. Assume that \(I_{\rho (\cdot )}\) is bounded from \(LM_{1\theta _{1},w_{1}(\cdot )}\) to \(LM_{p_{2}\theta _{2},w_{2}(\cdot )}\). Then by Statement 2 of Theorem 4.1 and formula (4.14) with \(p_{1}=1\)
uniformly in all non-negative functions \(f \in L_{1}^{\text {loc}} (\mathbb {R}^{n})\).
Let \(g\in \mathbb {A}\) be continuously differentiable on \((0,\infty )\). Consider the positive Lebesgue measurable function h on \((0,\infty )\) defined uniquely up to equivalence by the equality
If we take in (4.22) \(f(x)=h(|x|)\), then by (4.3) \(\Vert f\Vert _{L_{1}(B(0,\mu _{n,\rho ,1}^{(-1)}(t)))}=g(t)\) and by (4.21)
uniformly in all \(g\in \mathbb {A}\) which are continuously differentiable on \((0,\infty )\).
Finally, if g is an arbitrary function in \(\mathbb {A}\), then there exist functions \(g_{k}\in \mathbb {A}\), \(k\in \mathbb {N}\), which are continuously differentiable on \((0,\infty )\) and such \(g_{k}\le g_{k+1}\), \(k\in \mathbb {N}\), and \(\lim \limits _{k\rightarrow \infty }g_{k}=g\) on \((0,\infty )\). Therefore, by passing to the limit in (4.23), with g replaced by \(g_{k}\), as \(k \rightarrow \infty\), it follows that (4.23) holds for all \(g\in \mathbb {A}\).
3. Let \(p_{1}=1\), \(1<p_{2}<\infty\) and \(\rho \in S_{n,1,p_{2}}\). In this case the argument is similar to that of Step 2, only Statement 3 of Theorem 4.1 should be used and the space \(LM_{p_{2}\theta _{2},w_{2}(\cdot )}\) should be replaced by the space \(WLM_{p_{2}\theta _{2},w_{2}(\cdot )}\). \(\square\)
Hardy’s inequality on the cone of monotonic functions
Necessary and sufficient conditions for the boundedness of the operator H from one weighted Lebesgue space \(L_{\theta _{1},v_{1}(\cdot )}(0,\infty )\) to another one \(L_{\theta _{2},v_{2}(\cdot )}(0,\infty )\) on the cone \(\mathbb {A}\) are knows for all values of \(0<\theta _{1}\), \(\theta _{2}\le \infty\). We present below these results which are corollaries of more general statements contained in the survey by A. Gogatishvili and V.D. Stepanov [10] (Theorems 2.5, 3.15 and 3.16). See also the survey by M.L. Goldman [11].
Theorem 5.1
Let \(0<\theta _{1}\), \(\theta _{2}\le \infty\), \(v_{1}\), \(v_{2}\) be non-negative Lebesgue measurable functions on \((0,\infty )\). Then the operator H is bounded from \(L_{\theta _{1},v_{1}(\cdot )}(0,\infty )\) to \(L_{\theta _{2},v_{2}(\cdot )}(0,\infty )\) on the cone \(\mathbb {A}\) if and only if
(a) if \(1<\theta _{1}\le \theta _{2}<\infty\), then
and
(b) if \(0<\theta _{1}\le 1\), \(\theta _{1} \le \theta _{2} <\infty\), then
(c) if \(1<\theta _{1}<\infty\), \(0<\theta _{2}<\theta _{1}<\infty\), then
and
where \(\frac{1}{r}=\frac{1}{\theta _{2}}-\frac{1}{\theta _{1}}\),
(d) if \(0<\theta _{2}<\theta _{1}\le 1\), then \(A_{41} := A_{31}<\infty\) and
(e) if \(0<\theta _{1}\le 1\), \(\theta _{2}=\infty\), then
(f) if \(1<\theta _{1}<\infty\), \(\theta _{2}=\infty\), then
(g) if \(\theta _{1}=\infty\), \(0<\theta _{2}<\infty\), then
(h) if \(\theta _{1}=\theta _{2}=\infty\), then
Conditions, ensuring boundedness of the generalized Riesz potential
In order to obtain sufficient conditions and necessary and sufficient conditions for \(p_{1}=1\) on the weight functions \(w_{1}\), \(w_{2}\) ensuring the boundedness of \(I_{\rho (.)}\) from \(LM_{p_{1},\theta _{1},w_{1}(.)}\) to \(LM_{p_{2},\theta _{2},w_{2}(.)}\) for \(1<p_{1}<p_{2}<\infty\) or \(1\le p_{1}<\infty\), it, clearly, suffices to apply Theorems 4.2 and 5.1.
In order to make these conditions have a simpler form we shall carry out certain changes of variables and apply equalities (4.15)–(4.20) proved in Lemma 4.1.
Theorem 6.1
Assume that \(0<\theta _{1},\theta _{2}\le \infty , w_{1} \in \Omega _{\theta _{1}}, w_{2} \in \Omega _{ \theta _{2}}\).
1. Let \(1< p_{1}< p_{2}<\infty\) or \(1 \le p_{1}< \infty\) and \(0<p_{2}\le \infty ,\) and \(\rho \in S_{n, p_{1},p_{2}}.\) Then the operator \(I_{\rho (.)}\) is bounded from \(LM_{p_{1},{ \theta _{1}}, w_{1}(\cdot ) }\) to \(LM_{p_{2},{\theta _{2}},w_{2}(\cdot ) }\) if the following conditions are satistied.
(a) If \(1<\theta _{1}\le \theta _{2}<\infty ,\) then
and
where \(w_{2,n,p_{2} }(x) =w_{2}(x)x^{\frac{n}{p_{2}}}\).
(b) If \(0<\theta _{1}\le 1\), \(\theta _{1}\le \theta _{2}<\infty ,\) then
(c) If \(1<\theta _{1}< \infty , 0<\theta _{2}< \theta _{1}< \infty ,\) then
and
where \(\frac{1}{r}=\frac{1}{\theta _{2}}-\frac{1}{\theta _{1}}\).
(d) If \(0<\theta _{2}< \theta _{1}\le 1,\) then \(B_{41}: =B_{31}<\infty\) and
(e) If \(0<\theta _{1}\le 1,\theta _{2}=\infty ,\) then
(f) If \(1<\theta _{1}< \infty ,\theta _{2}=\infty ,\) then
(g) If \(\theta _{1}=\infty\), \(0<\theta _{2}<\infty\), then
(h) If \(\theta _{1}=\theta _{2}=\infty ,\) then
2. Let \(p_{1}=1, 0<p_{2}<\infty\) and \(\rho \in \tilde{S}_{n,1,p_{2}}.\) Then the operator \(I_{\rho (.)}\) is bounded from \(LM_{1,\theta _{1},w_{1}(.)}\) to \(LM_{p_{2},\theta _{2},w_{2}(.)}\) if and only if conditions (a) - (h) with \(p_{1}=1\) are satisfied.
3. Let \(p_{1}=1, 1<p_{2}<\infty\) and \(\rho \in {S}_{n,1,p_{2}}.\) Then the operator \(I_{\rho (.)}\) is bounded from \(LM_{1,\theta _{1},w_{1}(.)}\) to \(WLM_{p_{2},\theta _{2},w_{2}(.)}\) if and only if conditions (a) - (h) with \(p_{1}=1\) are satisfied.
Proof
It sufficies to prove by using Lemma 4.1 and the appropriate changes of variables that, for \(\mu _{n,\rho ,p_{1}}\), \(v_{1}\) and \(v_{2}\) defined by formulas (4.1), (4.9), (4.2), respectively, \(A_{11}=B_{11}\). \(A_{12}=B_{12}\), \(A_{2}=B_{2}\), \(A_{31}=B_{31}\), \(A_{32}=B_{32}\), \(A_{41}=B_{41}\), \(A_{5}=B_{5}\) and \(A_{6}= c B_{6}\), \(A_{7}= c B_{7}\), \(A_{8}= c B_{8}\), where \(c=\Big (\int \limits _{1}^{\infty } \rho (t)t^{\frac{n}{p'_{1}}-1}dt \Big )^{-1}\).
(a) If \(1<\theta _{1}\le \theta _{2}<\infty\), then according to Theorem 5.1 and by (4.15) with \(\phi _{1}\equiv 1\) and (4.18) with \(\phi _{2}(s)\equiv s\), we get
and
Next, by (4.16) with
we get
(b) If \(0<\theta _{1}\le 1\), \(\theta _{1}\le \theta _{2}<\infty\), then by (4.15) with \(\phi _{1} \equiv 1\) and by (4.19′) with \(\phi _{2}(s)=\min \{t,s\}\), \(s>0\), we get
(c) If \(1<\theta _{1}<\infty\), \(0<\theta _{2}<\theta _{1}<\infty\), then by (4.15) with \(\phi _{1}\equiv 1\), (4.18) with \(\phi _{s}=s\) and by (4.19’) we get
and
(d) If \(0<\theta _{2}<\theta _{1}\le 1\), then by (4.15) with \(\phi _{1}\equiv 1\), by (4.19) with \(\phi _{2}\equiv 1\), we get
(e) If \(0<\theta _{1} \le 1\), \(\theta _{2}=\infty\), then by (4.15) with \(\phi _{1}\equiv 1\) and (4.19’) with \(\phi (y) =\min \{t,y\}\), \(y>0\), we get
(f) If \(1<\theta _{1}<\infty\), \(\theta _{2}=\infty\), then by (4.15) with \(\phi _{1}\equiv 1\) and by the same formula with
and, finally, by (4.19’) with \(\theta _{2}=\infty\) and
we get
where
(g) If \(\theta _{1}=\infty\), \(0<\theta _{2}<\infty\), then by using formula (4.15) with \(\phi _{1}\equiv 1\) and formula (4.19’) with \(\phi _{2}(x)={\int \limits _{0}^{x}} \ \ \ \frac{dy}{ \text {ess} \sup \limits _{r> \mu ^{(-1)}_{n,\rho ,p_{1}}(y) \ \ \ \ \ } w_{1}(r) }\) we get
(h) If \(\theta _{1}=\theta _{2}=\infty\), then in \(A_{8}\) according to (4.10) and (4.2)
By using formula (4.15) with \(\psi _{1}\equiv 1\) and by carrying out the following changes of variables: \(y=\mu _{n,\rho ,p_{1}}(z)\), \(z>0\), and, finally, \(t=\mu _{n,\rho ,p_{1}}(\tau )\), \(\tau >0\), we get
\(\square\)
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Notes
According to the definitions introduced in Introduction, this means that for any \(n \in \mathbb {N},\rho \in S_{n}, 0<p\le \infty\) there exists \(c(n,\rho , p)>0\) such that
$$\begin{aligned} \Vert I_{\rho (.)}\vert f\vert \Vert _{WL_{p}(B(x,r))}\gtrsim c(n,\rho , p)r^{\frac{n}{p}} \left( \overline{I}_{\rho (\cdot ), 2r} \vert f\vert \right) (x) \end{aligned}$$for all \(x \in \mathbb {R}^{n}, r>0\) and \(f \in \mathfrak {M}(\mathbb {R}^{n}).\)
In this case \(A = \left\{ \mathbb {R}^{n}, (0,\infty ),\mathfrak {M}(\mathbb {R}^{n})\right\}\), \(B= \left\{ (n,\rho ) : n \in \mathbb {N}, \rho \in S_{n}; [0,\infty \right] , F= \Vert I_{\rho (\cdot )}\vert f \vert \Vert _{WL_{p}(B(x,r))}\),
Below \(a_{+}=a\) if \(a \ge 0\) and \(a_{+}=0\) if \(a<0\).
References
V.I. Burenkov, Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I, Eurasian Math. J., 3 (2012), no. 3, 11–32.
V.I. Burenkov, Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. II, Eurasian Math. J., 4 (2013), no. 1, 21–45
V.I. Burenkov, H.V. Guliyev, Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces, Studia Math, 163 (2004), no. 2, 157-176.
V.I. Burenkov, H.V. Guliyev, V.S. Guliyev, Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces, J. Comput. Appl. Math., 208 (2007), 280-301.
V.I. Burenkov, P. Jain, T.V. Tararykova, On boundedness of the Hardy operator in Morrey-type spaces, Eurasian Math. J., 2 (2011), no. 1, 52-80.
V.I. Burenkov, A. Gogatishvili, V.S. Guliyev, R.Ch. Mustafayev, Boundedness of the Riesz potential in local Morrey-type spaces, Potential Analysis, 35 (2011), no. 1, 67-87.
V.I. Burenkov, M.A. Senouci, Boundedness of the generalized Riesz potential in local Morrey type spaces, Eurasian Math. J., 12 (2021), no. 4, 92–98.
M. Carro, L. Pick, J. Soria, V.D. Stepanov, On embeddings between classical Lorentz spaces, Math. Ineq. Appl., 4 (2001), no. 3, 397-428.
M. Carro, A. Gogatishvili, J. Martin, L. Pick, Weighted inequalities involving two Hardy operators with applications to embeddings of function spaces, J. Oper. Theory, 59 (2008), no. 2, 309-332.
A. Gogatishvili, V.D. Stepanov, Reduction theorems for weighted integral inequalities on the cone of monotone functions, Uspekhi Mat. Nauk 68 (2013), no. 4 (412), 3–68 (in Russian); English transl., Russian Math. Surveys 68 (2013), no. 4, 597–664. MR3154814
M.L. Goldman, Hardy type inequalities on the cone of quasimonotone functions, Research Report No: 98/31. Khabarovsk: Computer Center, Far Eastern Branch of the Russian Academy of Sciences, 1998, 1-69.
V.S. Guliyev, Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J., 3 (2012), no. 3, 33–61.
P.-G. Lemarié-Rieusset, The role of Morrey spaces in the study of Navier–Stokes and Euler equations, Eurasian Math. J., 3 (2012), no. 3, 62–93.
M.A. Ragusa, Operators in Morrey type spaces and applications, Eurasian Math. J., 3 (2012), no. 3, 94–109.
W. Sickel, Smoothness spaces related to Morrey spaces – a survey. I, Eurasian Math. J., 3 (2012), no. 3, 110–149.
W. Sickel, Smoothness spaces related to Morrey spaces — a survey. II, Eurasian Math. J., 4 (2013), no. 1, 82–124.
Acknowledgements
The authors are very grateful to the referees for careful reading of the paper and comments which allowed to improve the quality of the text.
Funding
Sections 1–3 of this work were performed by V.I. Burenkov and M.A. Senouci at the RUDN University and supported by the Regional Mathematical Center of the Southern Federal University with the Agreement No. 075-02-2022-893 of the Ministry of Science and Higher Education of Russia. Sections 4–6 of this work were performed by them at the V.A. Steklov Mathematical Institute and supported by the Russian Science Foundation (project no. 19-11-00087, https://rscf.ru/project/19-11-00087/).
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Burenkov, V.I., Senouci, M.A. ON BOUNDEDNESS OF THE GENERALIZED RIESZ POTENTIAL IN LOCAL MORREY-TYPE SPACES. J Math Sci 266, 765–793 (2022). https://doi.org/10.1007/s10958-022-06134-x
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DOI: https://doi.org/10.1007/s10958-022-06134-x