Abstract
We characterize the good weights for some weighted weak-type iterated and bilinear modified Hardy inequalities to hold.
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1 Introduction and results
The initial problem in the theory of weighted Hardy inequalities was the one of characterizing the positive functions w, v, the weights, such that
holds for all positive measurable function f with a positive constant C independent of f, which means that the Hardy operator \(Tf(x)=\int _a^xf\) is bounded from \(L^p(v)\) to \(L^q(w)\).
This problem was solved by Talenti [31], Muckenhoupt [23] and Bradley [4] in the case \(p\le q\), by Mazja [22] when \(1\le q<p\), Sinnamon [27, 28] for \(0<q<1<p\) and Sinnamon and Stepanov [29] for \(0<q<1=p\). Their results are the following ones.
Theorem A
([4, 22, 23, 29, 31]) Let \(1<q<\infty \), \(1\le p<\infty \) and let w, v be positive measurable functions on (a, b), where \(-\infty \le a<b\le \infty \). Then there exists a positive constant C such that inequality (1.1) holds for all nonnegative functions f if and only if
-
(i)
in the case \(p\le q\),
$$\begin{aligned} B_1\equiv \sup _{s\in (a,b)}\left( \int \limits _s^bw\right) ^{\frac{1}{q}}\Vert \chi _{(a,s)}v^{-\frac{1}{p}}\Vert _{p'}<\infty , \end{aligned}$$and the best constant C in inequality (1.1) verifies \(B_1\le C\le K(q,p)B_1\), where \(K(q,p)=\left( 1+\frac{q}{p'}\right) ^{\frac{1}{q}}\left( 1+\frac{p'}{q}\right) ^{\frac{1}{p'}}\) if \(p>1\) and \(K(q,1)=1\);
-
(ii)
in the case \(q<p\),
$$\begin{aligned} B_2\equiv \left( \int \limits _a^b\left( \int \limits _t^bw\right) ^{\frac{r}{q}}\Vert \chi _{(a,t)}v^{-\frac{1}{p}}\Vert _{p'}^{\frac{rp'}{q'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{r}}<\infty , \end{aligned}$$where \(\frac{1}{r}=\frac{1}{q}-\frac{1}{p}\), and the best constant C in inequality (1.1) verifies \(q\left( \frac{p'}{r}\right) ^{\frac{1}{q'}}B_2\le C\le q^{\frac{1}{q}}(p')^{\frac{1}{q'}}B_2\).
Weighted weak-type inequalities for T were also studied. By a weighted weak-type (p, q) inequality for T we mean the boundedness of T from \(L^p(v)\) to \(L^{q,\infty }(w)\), where
Really, weighted weak-type inequalities have been studied for the modified Hardy operators \(T_{\beta }f(x)=\beta (x)\int _a^xf\). This kind of inequalities are technically more difficult than the strong-type ones. In fact, the problem of characterizing the boundedness of \(T_{\beta }\) from \(L^p(v)\) to \(L^{q,\infty }(w)\) in the case \(q<p\) is not completely solved yet.
The first results on weighted weak-type inequalities for modified Hardy operators are due to Andersen and Muckenhoupt [2], who worked with \(\beta (x)=x^{\alpha }\), \(\alpha \in {\mathbb {R}}\), on \((0,\infty )\). The weighted weak-type inequalities with more general functions \(\beta \) were characterized in [6, 20, 21]. The following two theorems contain such characterizations.
Theorem B
([6, 21]) Let \(1\le p\le q<\infty \) and \(\beta \), v and w be positive measurable functions on (a, b), where \(-\infty \le a<b\le \infty \). Then there exists a positive constant C such that inequality
holds for all nonnegative functions f if and only if
and the best constant C in inequality (1.2) verifies \(B_3\le C\le 4B_3\).
Theorem C
([20]) Let \(0<q<p<\infty \) with \(p\ge 1\) and \(\beta \), v and w be positive measurable functions on (a, b), where \(-\infty \le a<b\le \infty \) and \(\beta \) is a monotone function. Then there exists a positive constant C such that inequality (1.2) holds for all nonnegative functions f if and only if the function \(\Psi \) defined on (a, b) by
belongs to \(L^{r,\infty }(w)\), where \(\frac{1}{r}=\frac{1}{q}-\frac{1}{p}\). In this case, the best constant C in inequality (1.2) verifies \(2^{-\frac{1}{p}}\Vert \Psi \Vert _{r,\infty ;w}\le C\le (1+4^p)^{\frac{1}{q}}\Vert \Psi \Vert _{r,\infty ;w}\).
It is worth noting that weighted weak-type inequalities for modified linear or sublinear operators are included in the topic of weighted mixed weak-type inequalities, which goes back to the work of Andersen and Muckenhoupt [2] and have been studied by several authors (see [5, 16,17,18,19, 21, 26]).
Two new kinds of Hardy inequalities are the weighted iterated and bilinear Hardy inequalities. On one hand, weighted iterated Hardy inequalities are of the form
or
and have been studied by many authors [3, 8,9,10,11, 24, 25, 30].
On the other hand, weighted strong-type bilinear Hardy inequalities
were characterized in [1] and some of their generalizations and variants have also been studied later (see, for instance, [12,13,14, 30]).
Recently, the authors have characterized in [7] the weights \(w, v_1, v_2\) for which the weighted weak-type bilinear modified Hardy inequality
holds in the cases \(0<q<\infty \), \(1\le p_1, p_2<\infty \), \(q<p_1\), \(q<p_2\) and \(\frac{1}{q}\le \frac{1}{p_1}+\frac{1}{p_2}\). In this paper, we will complete the characterization of inequality (1.7) solving the problem for the case \(\frac{1}{q}>\frac{1}{p_1}+\frac{1}{p_2}\).
As we showed in [7], inequality (1.7) is equivalent to two weighted weak-type iterated modified Hardy inequalities of the form
where \(q<p\). Therefore, we will solve the problem of the characterization of (1.8) in the case \(q<p\) and then we will get immediately the characterization of (1.7). It is worth noting that the good weights for (1.8) to hold in the case \(p\le q\) were characterized by the authors in [7].
In order to state the results for the iterated inequality (1.8), we define two functions \(\Phi , \Psi \) on (a, b) by
and
where \(\frac{1}{\theta }=\frac{1}{r}-\frac{1}{p}\).
The results are the following ones.
Theorem 1
Let p, q, r with \(0<q<p\), \(1\le p<\infty \) and \(p\le r\le \infty \). Let \(\alpha \) be a positive function in (a, b) such that
for all \(\rho , \nu \) with \(a<\rho<\nu < b\). Let us suppose that for all \(e\in (a,b)\) and all measurable sets \(\Omega \subset (e,b)\), the function \(\alpha (t) \Vert \chi _{(e,t)}u\Vert _r\) verifies
where \(\rho _1=\inf \Omega \) and \(\rho _2=\sup \Omega \). Then, (1.8) holds for all nonnegative functions f if and only if \(\Phi \in L^{\eta , \infty }(w)\), where \(\frac{1}{\eta }=\frac{1}{q} - \frac{1}{p}\). Moreover, the best constant C in inequality (1.8) verifies
if \(r<\infty \) and
if \(r=\infty \).
Theorem 2
Let p, q, r with \(0<q<p\) and \(1<r<p<\infty \). Let \(\alpha \) be a positive monotone function in (a, b) and let us suppose that (1.10) holds. Then, the weighted iterated weak-type modified Hardy inequality (1.8) holds for all nonnegative functions f if and only if \(\Phi ,\Psi \in L^{\eta , \infty }(w)\), where \(\frac{1}{\eta }=\frac{1}{q} - \frac{1}{p}\). Moreover, the best constant C in (1.8) verifies
where \(C_{r,p}=r^{\frac{1}{r}}(p')^{\frac{1}{r'}}\).
Observe that condition (1.10) holds if the function \(\alpha (t) \Vert \chi _{(e,t)}u\Vert _r\) is monotone or increases in an interval \((e,x_0)\) and decreases in \((x_0,b)\). In the same way, condition (1.9) holds, for instance, if \(\alpha \) is a positive monotone function.
As consequences of Theorems 1 and 2 we get the results for the weighted weak-type bilinear modified Hardy inequalities. In order to state them, we define the next functions on (a, b):
where \(\frac{1}{\theta }=\frac{1}{p_2'}-\frac{1}{p_1}=\frac{1}{p_1'}-\frac{1}{p_2}=1-\frac{1}{p_2}-\frac{1}{p_1}\).
The theorems read as follows.
Theorem 3
Let \(0<q<p_1,p_2<\infty \), \(p_1,p_2\ge 1\), \(\frac{1}{p_1}+\frac{1}{p_2}<\frac{1}{q}\) and \(p_2\le p_1'\). Let \(w, v_1, v_2, \beta \) be positive measurable functions on (a, b) with \(\beta \) monotone. Assume that the functions \(\alpha _i\) verify (1.9) and that for all \(e\in (a,b)\) and all measurable sets \(\Omega \subset (e,b)\), the functions \(\alpha _i(t) \Vert \chi _{(e,t)}v_i^{\frac{-1}{p_i}}\Vert _{p_i'}\), i=1,2, verify (1.10). Then the weighted weak-type bilinear modified Hardy inequality (1.7) holds if and only if \(\Phi _1, \Phi _2\in L^{\eta ,\infty }(w)\), where \(\frac{1}{\eta }=\frac{1}{q}-\frac{1}{p_1}-\frac{1}{p_2}\).
Theorem 4
Let \(0<q<p_1,p_2<\infty \), \(p_1,p_2\ge 1\), \(\frac{1}{p_1}+\frac{1}{p_2}<\frac{1}{q}\) and \(p_2> p_1'\). Let \(w, v_1, v_2, \beta \) be positive measurable functions on (a, b) with \(\beta \) monotone. Assume that the functions \(\alpha _i\) are monotone and that for all \(e\in (a,b)\) and all measurable sets \(\Omega \subset (e,b)\), the functions \(\alpha _i(t) \Vert \chi _{(e,t)}v_i^{\frac{-1}{p_i}}\Vert _{p_i'}\), i=1,2, verify (1.10). Then the weighted weak-type bilinear modified Hardy inequality (1.7) holds if and only if \(\Phi _1, \Phi _2, \Psi _1, \Psi _2\in L^{\eta ,\infty }(w)\), where \(\frac{1}{\eta }=\frac{1}{q}-\frac{1}{p_1}-\frac{1}{p_2}\).
The proofs of Theorems 1 and 2 are included in Sects. 2 and 3, respectively, while Sect. 4 contains the proofs of Theorems 2 and 4.
2 Proof of Theorem 1
First of all, let us prove the necessity of the condition. Assume that the weak-type inequality (1.8) holds and let us see that \(\Phi \in L^{\eta ,\infty }(w)\). Let \(\lambda >0\) and \(S_\lambda =\{x\in (a,b):\Phi (x)>\lambda \}\). We will prove that
where C is the constant in (1.8). Let K be a compact subset of \(S_\lambda \). For all \(z\in K\), \(z\in S_\lambda \) and then \(\Phi (z)>\lambda \). This implies the existence of \(c_z, d_z, e_z\) with \(e_z<c_z<d_z\) such that \(z\in (c_z,d_z)\) and
Then, \(K\subset \bigcup _{z\in K} (c_z, d_z)\). Since K is compact, there are \((c_{z_1},d_{z_1}), (c_{z_2},d_{z_2})\dots \) \((c_{z_N}, d_{z_N})\) such that \(K\subset \bigcup _{j=1}^N (c_{z_j},d_{z_j})\). We can also suppose that
Let, for all \(j\in \{1, 2, \ldots , N\}\) and \(x\in (a,b)\),
and
Let \(\varepsilon \) with \(0<\varepsilon <1\). Let us see that
Indeed, if \(z\in (c_{z_j},d_{z_j})\), then
This proves (2.2). Applying the weak-type inequality,
Since the last inequality holds for all \(\varepsilon \) with \(0<\varepsilon <1\), letting \(\varepsilon \rightarrow 1^-\) we get
Let \(\gamma _j= \inf _{y\in (c_{z_j},d_{z_j})} [\alpha (y)\Vert \chi _{(e_{z_j},y)}u\Vert _r]\). Then the inequality above and (2.1) yield
Then, we have
i.e.,
The last inequality implies
Since the inequality above holds for all compact \(K\subset S_{\lambda }\), the regularity of the measure \(w(x)\text {d}x\) gives
what proves that \(\Vert \Phi \Vert _{\eta , \infty ; w}\le 2^{\frac{1}{p}}C\), as we wished to show.
Now, let us prove the sufficiency of the condition. Let f be a positive function such that \(\int _a^b f^p v=1\). Let \(\lambda >0\) and \(O_{\lambda }=\{x \in (a,b): \alpha (x)\Vert \chi _{(a,x)}(t)u(t)\int _a^t f \Vert _r>\lambda \}\). Then,
The estimation of I is as follows:
Now, we will estimate II. Assume first that \(r<\infty \). Let us suppose that \(\int _a^b f <\infty \) and \(\int _a^b \left( \int _a^t f \right) ^ru^r(t)\text {d}t < \infty \), too. Let \(\{ x_k\}\) be the sequence defined by \(x_0=b\) and
The sequence \(\{ x_k\}\) decreases to a and verifies
for all k. Let \(E_k=O_\lambda \cap \{ x\in (a,b):\Phi (x)\le \lambda ^{\frac{q}{\eta }}\}\cap (x_{k+1},x_k)\). If \(x\in E_k\), we have
It is clear that, for each k, \(E_k= E_{k,1}\cup E_{k,2}\), where
and
Since \(p\le r\), by Theorem A (i), we have that
Let us see that the supremum in (2.5) is finite. Let \(\gamma \in (x_{k+2},x_{k+1})\). As \(\Phi \in L^{\eta , \infty }(w)\), \(\Phi \) is finite almost everywhere. Let \(\rho , \nu \) with \(x_{k+1}<\rho<\nu <b\) and let \(t\in (\rho ,\nu )\) such that \(\Phi (t)<\infty \). Then,
Thus, there is \({\tilde{x}}\in (\rho , \nu )\), which depends on \(\gamma \), such that
Then, applying (1.9), we get
Therefore, the supremum in (2.5) is finite. Then, for all \(x\in E_{k,1}\) we have
Let \(\varepsilon >1\). For every k, there is \(\gamma _k \in (x_{k+2},x_{k+1})\) such that
Therefore, for all \(x\in E_{k,1}\) the following inequality holds:
Since \(x_{k+1}<x\), we get
The last inequality holds for all \(x\in E_{k,1}\). Then,
Now, if we multiply both sides of this inequality by \(\displaystyle \left( \int _{E_{k,1}} w \right) ^{\frac{1}{p}}\) and apply condition (1.10), we have
where \(\rho _k^1=\inf E_{k,1}\), \(\rho _k^2=\sup E_{k,1}\) and the last inequality holds since
for all \(t\in E_{k,1}\). Thus,
Since this inequality holds for every \(\varepsilon >1\), letting \(\varepsilon \rightarrow 1^+\) and raising to p we get
Now, summing up in k, we have
what finishes the estimation of \(\int _{\cup _k E_{k,1}}w\). In order to estimate \(\int _{\cup _kE_{k,2}}w\), we will use the technique, due to Lai [15], that we have already used in [7]. Let us define the sequence \(\{ y_m'\}\) as \(y_0'=b\) and \(\int _a^{y_{m+1}'}f=\int _{y'_{m+1}}^{y_m'}f\). Let \(\{ y_n\}\) be the subsequence of \(\{ y_m'\}\) defined by \(y_0=y_0'\) and by deleting \(y'_{m+1}\) if \([y'_{m+1}, y'_m)\cap \{ x_k\}=\emptyset \). In this way, if \(y'_{m+1}=y_{n+1}\le x_{k+2}<y_n\), then \(x_{k+2}\le y'_m\) and \(y_{n+2}\le y'_{m+2}\), which yields
Let \(E_2^n=\bigcup _{\{k: y_{n+1}\le x_{k+2}<y_n\}}E_{k,2}\). If \(x\in E_2^n\), there exists k with \(y_{n+1}\le x_{k+2}<y_n\) such that \(x\in E_{k,2}\) and then, by (2.6),
Since (2.7) holds for all \(x\in E_2^n\), we have
Multiplying both sides of (2.8) by \(\left( \int _{E_2^n}w\right) ^{\frac{1}{p}}\), applying Holder’s inequality and (1.10), we get
where we have used that
for all \(t\in E_2^n\).
Then, raising to the p in (2.9) and summing, we get
which implies
This finishes the proof of the sufficiency in the case \(r<\infty \). Now, we will deal with the case \(r=\infty \). Let us consider two sequences \(\{a_n\}\) and \(\{b_n\}\), with \(\{ a_n\}\) decreasing to a and \(\{ b_n\}\) increasing to b. Then, \(\Phi \in L^{\eta , \infty }(w)\) implies that \(\Phi _n \in L^{\eta , \infty }(w, (a_n,b_n))\), where
For fixed n, there is \(r_0>p\) such that \((b_n-a_n)^{\frac{1}{r}} \le 2\) for all \(r\ge r_0\). Then, if \(r\ge r_0\) and \(a_n<e<x<b_n\), we have
Therefore, if we define \(\Phi _{n,r}(x)\) as
we have that \(\Phi _{n,r}\in L^{\eta , \infty }(w, (a_n,b_n))\) for all \(r\ge r_0\) and their norms are bounded by \(2\Vert \Phi \Vert _{\eta ,\infty ,w}\). Now, applying the Theorem in the case which we have already proved, we have that the weak-type inequality
holds, where \(C_{r,p,q}=(2^{\eta }\Vert \Phi \Vert _{\eta ,\infty ,w}^{\eta }+2^p4^{\frac{p}{r}}(4^p+K(r,p)^p))^{\frac{1}{q}} \). Since
for every x, by Fatou’s lemma we have
Now, from (2.11) and (2.12) we get
where \(C_{p,q}=(2^{\eta }\Vert \Phi \Vert _{\eta ,\infty ,w}^{\eta }+8^p+2^p)^{\frac{1}{q}}\). Finally, since (2.13) holds for all n with a constant independent of n, letting n tend to infinity and applying the monotone convergence theorem, we get (1.8) in the case \(r=\infty \).
3 Proof of Theorem 2
The necessity of condition \(\Phi \in L^{\eta ,\infty }(w)\) follows as in the proof of Theorem 1. Therefore, the best constant C in (1.8) verifies \(C\ge 2^{\frac{-1}{p}}\Vert \Phi \Vert _{\eta ,\infty , w}\). Let us prove now that (1.8) implies \(\Psi \in L^{\eta , \infty }(w)\). Let \(\lambda >0\) and \(S_{\lambda }=\{ x\in (a,b): \Psi (x)>\lambda \}\). Let K be a compact subset of \(S_{\lambda }\). If \(z\in K\), there exist \(c_z, d_z\) with \(c_z<z<d_z\) such that
Since K is compact, there exist \(z_1, z_2,\ldots ,z_N\in K\) such that \(K\subset \displaystyle \bigcup _{j=1}^N(c_{z_j},d_{z_j})\) and
Let, for each \(j\in \{ 1,2,\ldots ,N\}\),
and
If \(z\in (c_{z_j},d_{z_j})\) and \(\gamma _j=\inf _{(c_{z_j},d_{z_j})}\alpha \), we have
If \(h(x)=\displaystyle \left( \int _x^{c_{z_j}}u^r\right) ^{\frac{\theta }{rp}} \left( \int _a^xv^{1-p'}\right) ^{\frac{\theta }{r'p}}v^{1-p'}(x)\), the last factor in (3.3) can be written as follows
Let us estimate now \(\displaystyle \int _a^sh\):
Taking this estimate to (3.4), we get
Going back to (3.3) we get that for all \(z\in (c_{z_j}, d_{z_j})\),
Therefore, by (1.8), we have
Let us estimate \(\Vert f \Vert _{p,v}^q\). By definition of f, (3.1) and (3.2),
Finally, from (3.5) and (3.6) we get
which implies
Since the inequality above holds for all compact set \(K\subset S_{\lambda }\), we have that \(\Psi \in L^{\eta ,\infty }(w)\) and \(C\ge 2^{\frac{-1}{p}}r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}\Vert \Psi \Vert _{\eta ,\infty ;w}\).
Let us prove now the sufficiency. Let f be a nonnegative function with \(f\in L^1\) and \(\int _a^bf^pv=1\). Let \(\lambda >0\) and
Then, as in the proof of Theorem 1,
The estimation of I can be done as in the case \(p\le r\). For the estimation of II, we work as follows:
Firstly,
and then
Now, we will work on IV. Let \(\{ x_k\}\) be the sequence defined as in the proof of Theorem 1 and
If \(x\in E_k\),
It is clear that, for each k, \(E_k= E_{k,1}\cup E_{k,2}\), where
and
Since \(r<p\), by Theorem A (ii) we have
where \(C_{r,p}=r^{\frac{1}{r}}(p')^{\frac{1}{r'}}\). The first integral in the right-hand side of (3.7) is finite due to the monotonicity of \(\alpha \) and the proof of this fact follows the pattern of the one in Theorem 1.
If \(x\in E_{k,1}\),
which implies, due to the monotonicity of \(\alpha \),
If we multiply both terms of the last inequality by \(\displaystyle \left( \int _{E_{k,1}} w \right) ^{\frac{1}{p}}\), we get
where the last inequality holds since
\(\le \Psi (t)\) for all \(t\in E_{k,1}\). Raising to p, we have that
Now, summing up in k,
The estimation of \(\int _{\cup _k E_{k,2}} w\) is the same as the one in Theorem 1, because the relationship between r and p is not taken into account. Therefore, the proof is complete.
4 Proofs of Theorems 3 and 4
Working as in ([7], proof of Theorem 3), we have that (1.8) is equivalent to the two weighted weak-type bilinear inequalities
and
Inequality (4.1) is equivalent to
where \(\tilde{v}_1^g(x)=v_1(x)\left( \int _a^x\frac{g}{\Vert g\Vert _{p_2,v_2}}\right) ^{-p_1}\) and the constant C does not depend on g.
Since \(q<p_1\) and \(\beta \) is a monotone function, by Theorem C inequality (4.3) holds if and only if there exists \(C>0\) such that
for all g, where \(\frac{1}{r_1}=\frac{1}{q}-\frac{1}{p_1}\) and
Then (4.4) can be written as
Therefore, inequality (4.1) holds if and only if inequality (4.5) holds. Since \(p_2> r_1\), by Theorems 1 and 2, (4.5) holds if and only \(\Phi _1\in L^{\eta ,\infty }(w)\) in the case \(p_2\le p_1'\) and \(\Phi _1, \Psi _1\in L^{\eta ,\infty }(w)\) in the case \(p_1'<p_2\).
In the same way, we see that (4.2) holds if and only if \(\Phi _2\in L^{\eta ,\infty }(w)\) in the case \(p_2\le p_1'\) and \(\Phi _2, \Psi _2\in L^{\eta ,\infty }(w)\) in the case \(p_1'<p_2\).
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This research has been supported in part by Ministerio de Economía y Competitividad (Grant PGC2018-096166-B-100) and Junta de Andalucía (Grants FQM354 and UMA18-FEDERJA-002).
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Communicated by Feng Dai.
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García García, V., Ortega Salvador, P. Some new weighted weak-type iterated and bilinear modified Hardy inequalities. Ann. Funct. Anal. 15, 26 (2024). https://doi.org/10.1007/s43034-024-00327-y
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DOI: https://doi.org/10.1007/s43034-024-00327-y
Keywords
- Bilinear modified Hardy inequalities
- Iterated modified Hardy inequalities
- Weighted weak-type inequalities
- Weighted inequalities