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Boyd indices for quasi-normed function spaces with some bounds
Journal of Inequalities and Applications volume 2015, Article number: 235 (2015)
Abstract
We calculate the Boyd indices for quasi-normed rearrangement invariant function spaces with some bounds. An application to Lorentz type spaces is also given.
1 Introduction
Let \(L_{\mathrm{loc}}\) be the space of all locally integrable functions f on \({\mathbf{R}}^{n}\) and \(M^{+}\) be the cone of all locally integrable functions \(g\geq0\) on \((0,1)\) with the Lebesgue measure.
Let \(f^{\ast}\) be the decreasing rearrangement of f given by
and \(\mu_{f}\) be the distribution function of f defined by
\(\vert \cdot \vert _{n}\) denoting Lebesgue n-measure.
Also,
We use the notations \(a_{1}\lesssim a_{2}\) or \(a_{2}\gtrsim a_{1}\) for nonnegative functions or functionals to mean that the quotient \(a_{1}/a_{2}\) is bounded; also, \(a_{1}\approx a_{2}\) means that \(a_{1}\lesssim a_{2}\) and \(a_{1}\gtrsim a_{2}\). We say that \(a_{1}\) is equivalent to \(a_{2}\) if \(a_{1}\approx a_{2}\).
We consider rearrangement invariant quasi-normed spaces \(E \hookrightarrow L^{1}(\Omega)\) such that \(\|f\|_{E}=\rho_{E}(f^{\ast})<\infty \), where \(\rho_{E}\) is a quasi-norm rearrangement invariant defined on \(M^{+}\).
For simplicity, we assume that Ω is a bounded Lebesgue measurable subset of \({\mathbf{R}}^{n}\) with Lebesgue measure equal to 1 and origin lies in Ω.
There is an equivalent quasi-norm \(\rho_{p}\approx\rho_{E}\) that satisfies the triangle inequality \(\rho_{p}^{p}(g_{1}+g_{2})\leq\rho_{p}^{p}(g_{1})+\rho _{p}^{p}(g_{2})\) for some \(p\in(0,1]\) that depends only on the space E (see [1]). We say that the quasi-norm \(\rho_{E}\) satisfies Minkowski’s inequality if for the equivalent quasi-norm \(\rho_{p}\),
Usually we apply this inequality for functions \(g\in M^{+}\) with some kind of monotonicity.
Recall the definition of the lower and upper Boyd indices \(\alpha_{E}\) and \(\beta_{E}\). Let \(g_{u}(t)=g(t/u)\) if \(t<\min(1,u)\) and \(g_{u}(t)=0\) if \(\min(1,u)< t<1\), where \(g\in M^{+}\), and let
be the dilation function generated by \(\rho_{E}\). Suppose that it is finite. Then
The function \(h_{E}\) is sub-multiplicative, increasing, \(h_{E}(1)=1\), \(h_{E}(u) h_{E}(1/u)\geq1\) and hence \(0\leq\alpha _{E}\leq\beta_{E}\). We suppose that \(0<\alpha_{E}=\beta_{E}\leq1\).
If \(\beta_{E}<1\), we have by using Minkowski’s inequality that \(\rho_{E}(f^{\ast})\approx\rho_{E}(f^{\ast\ast})\). In particular, \(\|f\|_{E}\approx\rho_{E}(f^{\ast\ast})\) if \(\beta_{E}<1\). For example, consider the gamma spaces \(E=\Gamma^{q}(w)\), \(0< q\leq\infty \), w-positive weight, that is, a positive function from \(M^{+}\), with a quasi-norm \(\|f\|_{\Gamma^{q}(w)}:=\rho_{E}(f^{\ast})\), \(\rho_{E}(g):=\rho_{w,q}(\int_{0}^{1} g(tu)\,du)\), where
and
Then \(L^{\infty}(\Omega)\hookrightarrow\Gamma^{q}(w)\hookrightarrow L^{1}(\Omega)\). If \(w(t)=t^{1/p}\), \(1< p<\infty\), we write as usual \(L^{p,q}\) instead of \(\Gamma^{q}(t^{1/p})\). Consider also the classical Lorentz spaces \(\Lambda ^{q} (w)\), \(0< q\leq\infty\); \(f\in\Lambda^{q}(w)\) if \(\|f\|_{\Lambda ^{q}_{w}}:=\rho_{w,q}(f^{\ast})<\infty\), \(w(2t)\approx w(t)\). We suppose that \(L^{\infty}(\Omega)\hookrightarrow\Lambda ^{q}(w)\hookrightarrow L^{1}(\Omega)\).
The Boyd indices are useful in various problems concerning continuity of operators acting in rearrangement invariant spaces [2] or in optimal couples of rearrangement invariant spaces [3–5], and in the problems of optimal embeddings [6–8]. The main goal of this paper is to provide formulas for the Boyd indices with some bounds of rearrangement invariant quasi-normed spaces and to apply these results to the case of Lorentz type spaces.
2 Boyd indices for quasi-normed function spaces
Let \(\rho_{E}\) be a monotone quasi-norm on \(M^{+}\) and let E be the corresponding rearrangement invariant quasi-normed space consisting of all \(f\in L^{1}(\Omega)\) such that \(\|f\|_{E}=\rho_{E}(f^{\ast})<\infty\).
Theorem 2.1
Let
where \(g\in M^{+}\), and let
be the dilation function generated by \(\rho_{E}\). Suppose that it is finite. Then the Boyd indices are well defined
and they satisfy
In particular, \(0\leq\alpha_{E}\leq\beta_{E}\leq\frac{\log h_{E}(2)}{\log2}\).
Proof
We have
Indeed, since \(\min(1,uv)\leq\min(1,v)\) for \(u< v\), we find \((g_{u})_{v}(t)=g_{u}(t/(uv))\) if \(0< t<\min(1,uv)\) and \((g_{u})_{v}(t)=0\) if \(\min (1,uv)\leq t<1\). Thus (2.3) is proved. This implies that the function \(h_{E}\) is sub-multiplicative.
Further, the function \(\omega(x)=\log h_{E}(e^{x})\) is sub-additive increasing on \((-\infty,\infty)\) and \(\omega(0)=0\). Hence, by [2], Lemma 5.8, (2.2) is satisfied and evidently \(\beta _{E}\leq \frac{\log h_{E}(2)}{\log2}\).
Since \(h_{E}(1)=1\) and \(h_{E}\) is sub-multiplicative, therefore
Replacing \(u_{2}\) by \(\frac{1}{u_{1}}\), we get
which implies that
it follows that \(1\leq h_{E}(u) h_{E}(1/u)\).
We have
Indeed
if \(u>1\), then
which implies that
Since \(\beta_{E}\) is finite, therefore \(\alpha_{E}\) is also finite. Since \(h_{E}(1)=1\) and we know that \(h_{E}\) is increasing function, so
which implies that
which implies that
which implies that
and hence
 □
Let \(\rho_{H}\) be a monotone quasi-norm on \(M^{+}\) and let H be the corresponding quasi-normed space, consisting of all locally integrable functions on \((0,1)\) with a finite quasi-norm \(\|g\|_{H}=\rho_{H}(|g|)\).
Theorem 2.2
Let
where \(g\in M^{+}\), and let
be the dilation function generated by \(\rho_{H}\). Suppose that it is finite, where
Then the Boyd indices are well defined
and they satisfy
In particular, \(\frac{\log h_{H}(1/2)}{\log1/2}\leq\alpha_{H}\leq\beta _{H}\leq a/n\).
Proof
We have
Indeed, since \(\min(1,1/(uv))\leq\min(1,1/u)\) for \(u< v\), we find \(\Psi_{u}(\Psi_{v} g)(t)= g(t/(uv))\) if \(0< t<\min(1,1/(uv))\) and \(\Psi_{u}(\Psi _{v} g)(t)=g(1)\) if \(\min(1,1/(uv))\leq t<1\). Thus (2.6) is proved. This implies that the function \(h_{H}\) is sub-multiplicative. Since the function \(u^{-a/n}h_{H}(u)\) is decreasing, it follows that the function \(u^{a/n} h_{H}(1/u)\) is increasing and sub-multiplicative. Hence we can apply the results from Theorem 2.1. This establishes Theorem 2.2. □
Example 2.3
If \(E=\Lambda^{q} (t^{a} w)\), \(0\leq a\leq1\), \(0< q\leq\infty\), where w is slowly varying, then \(\alpha_{E}=\beta_{E}=a\).
Proof
We give a proof for \(0< q<\infty\), the case \(q=\infty\) is analogous. We have, for \(g\in M^{+}\),
and by a change of variables,
From the definition of a slowly varying function it follows that for every \(\varepsilon>0\), \(t^{-\varepsilon}w(t)\approx d(t)\), where d is a decreasing function. Then, for \(u>1\), we have \(d(tu)\leq d(t)\), thus
which implies that
Inserting this estimate in (2.7), we arrive at
which yields \(h_{E}(u)\lesssim u^{a+\varepsilon}\), \(u>1\). Then it follows that \(\beta_{E}\leq a+\varepsilon\). Analogously, \(\alpha_{E}\geq a-\varepsilon \). Since \(\varepsilon>0\) is arbitrary and \(\alpha_{E}\leq\beta_{E}\), we obtain \(\alpha_{E}=\beta_{E}=a\). □
Example 2.4
If \(H=L^{q}_{\ast}(w(t)t^{-\alpha})\), \(0\leq\alpha< a/n\), \(0< q\leq\infty \), where w is slowly varying, then \(\alpha_{H}=\beta_{H}=\alpha\).
Proof
We give a proof for \(0< q<\infty\), the case \(q=\infty\) is analogous. We have, for \(g\in G_{a}\),
where \(I(u)= (\int_{\min(1,1/u)}^{1} [t^{-\alpha} w(t)]^{q} \,dt/t)^{1/q}g(1)\). Note that \(I(u)=0\) for \(0< u<1\). Since for every \(\varepsilon>0\) we have \(w(t)\lesssim t^{\varepsilon}\), it follows that \(I(u)\lesssim u^{\alpha+\varepsilon} g(1)\), \(u>1\). Also, \(g(1)\rho _{H}(t^{a/n})\leq\rho_{H}(g)\) and \(\rho_{H}(t^{a/n})<\infty\) due to \(\alpha< a/n\).
On the other hand, by a change of variables,
As in the proof of the previous example, we have
therefore
Hence \(h_{H}(u)\lesssim u^{\alpha+\varepsilon}\), \(u>1\). Then it follows that \(\beta_{H}\leq\alpha+\varepsilon\). Analogously, \(\alpha_{H}\geq\alpha -\varepsilon \). Since \(\varepsilon>0\) is arbitrary and \(\alpha_{H}\leq\beta_{H}\), we obtain \(\alpha_{H}=\beta_{H}=\alpha\). □
3 Basic inequalities
Here we prove a few inequalities, which are of independent interest.
Theorem 3.1
If \(\alpha<\alpha_{H}\), then
and if \(\beta_{H}<\beta\), then
Proof
We are going to use Minkowski’s inequality for the equivalent p-norm of \(\rho_{H}\). To this end, first we replace the integrals by sums using monotonicity properties of \(g\in G_{a}\).
Thus
Applying Minkowski’s inequality, we get
The above series is convergent if we choose \(\varepsilon>0\) such that \(\varepsilon< \alpha_{H}-\alpha\), so we have
On the other hand, for \(g\in G_{a}\),
Again applying Minkowski’s inequality, we get
The above series is finite if we choose a suitable \(\varepsilon>0\) such that \(\varepsilon< \beta-\beta_{H}\). The proof is finished. □
Theorem 3.2
If \(\beta_{E}< a\), then
where \(D_{0}:=\{g \in M ^{+} : g(t) \textit{ is decreasing and } g(t)=0 \textit{ for } t \geq1\}\).
Proof
We are going to use Minkowski’s inequality for the equivalent p-norm of \(\rho_{E}\). To this end, first we replace the integral by sums using monotonicity properties of \(g\in D_{0}\).
Thus
Applying Minkowski’s inequality, we get
The above series is finite if we choose \(\varepsilon>0\) such that \(\varepsilon< a-\beta_{E}\), and this concludes the proof. □
Theorem 3.3
If \(\alpha_{E}>0\), then
Proof
We are going to use Minkowski’s inequality for the equivalent p-norm of \(\rho_{E}\). To this end, first we replace the integral by sums using monotonicity properties of \(g\in D_{0}\).
Thus
Applying Minkowski’s inequality, we get
Choosing \(\varepsilon>0\) such that \(\alpha_{E}>\varepsilon\), we conclude the proof. □
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The authors are thankful to the editor and the referees for their valuable suggestions in improving the final version of the article.
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Nazeer, W., Mehmood, Q., Nizami, A.R. et al. Boyd indices for quasi-normed function spaces with some bounds. J Inequal Appl 2015, 235 (2015). https://doi.org/10.1186/s13660-015-0754-9
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DOI: https://doi.org/10.1186/s13660-015-0754-9