Open Access

Boyd indices for quasi-normed function spaces with some bounds

  • Waqas Nazeer1,
  • Qaisar Mehmood2,
  • Abdul Rauf Nizami1 and
  • Shin Min Kang3Email author
Journal of Inequalities and Applications20152015:235

https://doi.org/10.1186/s13660-015-0754-9

Received: 12 January 2015

Accepted: 10 July 2015

Published: 29 July 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.

Keywords

rearrangement invariant function spaces Boyd indices quasi-normed function spaces

MSC

46E30 46E35

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
$$f^{\ast}(t)=\inf \bigl\{ \lambda>0:\mu_{f}(\lambda)\leq t \bigr\} ,\quad t>0, $$
and \(\mu_{f}\) be the distribution function of f defined by
$$\mu_{f}(\lambda)= \bigl\vert \bigl\{ x\in{\mathbf{R}}^{n}: \bigl\vert f(x) \bigr\vert >\lambda \bigr\} \bigr\vert _{n}, $$
\(\vert \cdot \vert _{n}\) denoting Lebesgue n-measure.
Also,
$$f^{\ast\ast}(t):=\frac{1}{t}\int_{0}^{t} f^{\ast}(s)\,ds. $$

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 \(\rho_{p}\),
$$ \rho_{p}^{p} \Bigl(\sum g_{j} \Bigr) \lesssim\sum\rho_{p}^{p}(g_{j}),\quad g_{j}\in M^{+}. $$
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
$$h_{E}(u)=\sup \biggl\{ \frac{\rho_{E}(g^{\ast}_{u})}{\rho_{E}(g^{\ast})}: g\in M^{+} \biggr\} , \quad u>0 $$
be the dilation function generated by \(\rho_{E}\). Suppose that it is finite. Then
$$\alpha_{E}:=\sup_{0< t< 1} \frac{\log h_{E}(t)}{\log t} \quad \mbox{and}\quad \beta_{E}:=\inf_{1< t< \infty} \frac{\log h_{E}(t)}{\log t}. $$
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
$$ \rho_{w,q}(g):= \biggl(\int_{0}^{1} \bigl[ g(t)w(t) \bigr]^{q} \,dt/t \biggr)^{1/q},\quad g\in M^{+} $$
(1.1)
and
$$\biggl(\int_{0}^{1} w^{q}(t)\,dt/t \biggr)^{1/q}< \infty. $$
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 [35], and in the problems of optimal embeddings [68]. 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
$$g_{u}(t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} g(t/u) & \textit{if } 0< t< \min(1,u), \\ 0 & \textit{if } \min(1,u)\leq t< 1, \end{array}\displaystyle \displaystyle \displaystyle \displaystyle \right . $$
where \(g\in M^{+}\), and let
$$h_{E}(u)=\sup \biggl\{ \frac{\rho_{E}(g^{\ast}_{u})}{\rho_{E}(g^{\ast})}: g\in M^{+} \biggr\} ,\quad u>0, $$
be the dilation function generated by \(\rho_{E}\). Suppose that it is finite. Then the Boyd indices are well defined
$$\alpha_{E}:=\sup_{0< t< 1} \frac{\log h_{E}(t)}{\log t} \quad \textit{and}\quad \beta_{E}:=\inf_{1< t< \infty} \frac{\log h_{E}(t)}{\log t} $$
and they satisfy
$$\begin{aligned}& \alpha_{E}=\lim_{t\to0} \frac{\log h_{E}(t)}{\log t}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \beta_{E}=\lim_{t\to\infty} \frac{\log h_{E}(t)}{\log t}. \end{aligned}$$
(2.2)
In particular, \(0\leq\alpha_{E}\leq\beta_{E}\leq\frac{\log h_{E}(2)}{\log2}\).

Proof

We have
$$ g_{uv}=(g_{u})_{v} \quad\mbox{if } u< v. $$
(2.3)
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
$$h_{E}(u_{1} u_{2})\leq h_{E}(u_{1})h_{E}(u_{2}). $$
Replacing \(u_{2}\) by \(\frac{1}{u_{1}}\), we get
$$h_{E}(1)\leq h_{E}(u_{1})h_{E} \biggl(\frac{1}{u_{1}} \biggr), $$
which implies that
$$1\leq h_{E}(u_{1})h_{E} \biggl(\frac{1}{u_{1}} \biggr); \quad\mbox{because } h_{E}(1)=1, $$
it follows that \(1\leq h_{E}(u) h_{E}(1/u)\).
We have
$$\alpha_{E}\leq\beta_{E}. $$
Indeed
$$\log \bigl(h_{E}(u) \bigr)\geq\log \biggl(\frac{1}{h_{E}(\frac{1}{u})} \biggr), $$
if \(u>1\), then
$$\frac{\log(h_{E}(u))}{\log u}\geq\frac{\log (\frac{1}{h_{E}(\frac {1}{u})} )}{\log u}=\frac{\log (h_{E}(\frac{1}{u}) )}{\log\frac{1}{u}}, $$
which implies that
$$\lim_{u\rightarrow\infty}\frac{\log(h_{E}(u))}{\log u}\geq\lim_{u\rightarrow\infty} \frac{\log (h_{E}(\frac{1}{u}) )}{\log \frac{1}{u}}. $$
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
$$h_{E}(u)\leq1 \quad\mbox{for } 0< u< 1, $$
which implies that
$$\log \bigl(h_{E}(u) \bigr)\leq0, $$
which implies that
$$\frac{\log(h_{E}(u))}{\log u}\geq0, $$
which implies that
$$\alpha_{E}=\sup_{0< u< 1}\frac{\log(h_{E}(u))}{\log u}\geq0, $$
and hence
$$0\leq\alpha_{E}\leq\beta_{E}. $$
 □

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
$$(\Psi_{u}g) (t)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} g(ut), & \textit{if } 0< t< \min(1,\frac{1}{u}), \\ g(1), & \textit{if } \min(1,\frac{1}{u})\leq t< 1, \end{array}\displaystyle \displaystyle \displaystyle \displaystyle \right . $$
where \(g\in M^{+}\), and let
$$h_{H}(u)=\sup \biggl\{ \frac{\rho_{H}(\Psi_{u} g)}{\rho_{H}(g)}: g \in G_{a} \biggr\} ,\quad u>0, $$
be the dilation function generated by \(\rho_{H}\). Suppose that it is finite, where
$$G_{a}:= \bigl\{ g\in M^{+}: t^{-a/n}g(t) \textit{ is decreasing} \bigr\} ,\quad a>0. $$
Then the Boyd indices are well defined
$$\alpha_{H}:=\sup_{0< t< 1} \frac{\log h_{H}(t)}{\log t} \quad \textit{and}\quad \beta_{H}:=\inf_{1< t< \infty} \frac{\log h_{H}(t)}{\log t} $$
and they satisfy
$$\begin{aligned}& \alpha_{H}=\lim_{t\to0} \frac{\log h_{H}(t)}{\log t}, \end{aligned}$$
(2.4)
$$\begin{aligned}& \beta_{H}=\lim_{t\to\infty} \frac{\log h_{H}(t)}{\log t}. \end{aligned}$$
(2.5)
In particular, \(\frac{\log h_{H}(1/2)}{\log1/2}\leq\alpha_{H}\leq\beta _{H}\leq a/n\).

Proof

We have
$$ \Psi_{uv}g=\Psi_{u}(\Psi_{v} g) \quad\mbox{if } u< v. $$
(2.6)
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 \(g\in M^{+}\),
$$\rho_{E} \bigl(g^{\ast}_{u} \bigr)= \biggl(\int _{0}^{1} \bigl[g^{\ast}_{u}(t)t^{a} w(t) \bigr]^{q} \,dt/t \biggr)^{1/q}= \biggl(\int _{0}^{\min(1,u)} \bigl[g^{\ast}(t/u)t^{a} w(t) \bigr]^{q} \,dt/t \biggr)^{1/q} $$
and by a change of variables,
$$ \rho_{E} \bigl(g^{\ast}_{u} \bigr) \leq \biggl(\int_{0}^{1} \bigl[g^{\ast}(t) (tu)^{a} w(tu) \bigr]^{q} \,dt/t \biggr)^{1/q}. $$
(2.7)
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
$$(tu)^{-\varepsilon}w(tu)\lesssim d(t u) \lesssim t^{-\varepsilon}w(t), $$
which implies that
$$ w(tu)\lesssim u^{\varepsilon}w(t),\quad u>1. $$
(2.8)
Inserting this estimate in (2.7), we arrive at
$$\rho_{E} \bigl(g^{\ast}_{u} \bigr)\lesssim u^{a+\varepsilon}\rho_{E} \bigl(g^{\ast}\bigr),\quad u>1, $$
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 \(g\in G_{a}\),
$$\begin{aligned} \rho_{H}(\Psi_{u} g)&= \biggl(\int_{0}^{1} \bigl[\Psi_{u} g(t)t^{-\alpha} w(t) \bigr]^{q} \,dt/t \biggr)^{1/q} \\ &= \biggl(\int_{0}^{\min(1,1/u)} \bigl[g(tu)t^{-\alpha} w(t) \bigr]^{q} \,dt/t \biggr)^{1/q}+I(u), \end{aligned}$$
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,
$$ \rho_{H}(\Psi_{u} g)\lesssim \biggl(\int _{0}^{1} \bigl[g(t) (t/u)^{-\alpha} w(t/u) \bigr]^{q} \,dt/t \biggr)^{1/q}+u^{\alpha+\varepsilon} \rho_{H}(g). $$
As in the proof of the previous example, we have
$$w(t/u)\lesssim u^{\varepsilon}w(t),\quad u>1, $$
therefore
$$\rho_{H}(\Psi_{u} g)\lesssim u^{\alpha+\varepsilon} \rho_{H}(g),\quad u>1, g\in G_{a}. $$
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
$$\rho_{H} \biggl(t^{\alpha}\int_{0}^{t} s^{-\alpha}g(s)\frac{ds}{s} \biggr)\lesssim\rho_{H}(g), \quad g \in G_{a} $$
and if \(\beta_{H}<\beta\), then
$$\rho_{H} \biggl(t^{\beta}\int_{t}^{1} s^{-\beta}g(s)\frac{ds}{s} \biggr)\lesssim \rho_{H}(g), \quad g \in G_{a}. $$

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
$$\begin{aligned} t^{\alpha}\int_{0}^{t} s^{-\alpha}g(s) \frac{ds}{s}&=\int_{0}^{1} v^{-\alpha }g(tv)\frac{dv}{v} \\ &=\sum_{l=-\infty}^{0}\int_{2^{l-1}}^{2^{l}} v^{-\alpha}g(tv)\frac{dv}{v} \\ &\lesssim\sum_{l=-\infty}^{0} 2^{-l\alpha}g \bigl(t2^{l} \bigr). \end{aligned}$$
Applying Minkowski’s inequality, we get
$$\begin{aligned} \rho^{p}_{H} \biggl(t^{\alpha}\int _{0}^{t} s^{-\alpha}g(s)\frac{ds}{s} \biggr) &\lesssim\sum_{l=-\infty}^{0} 2^{-lp\alpha}\rho^{p}_{H} \bigl(g \bigl(t2^{l} \bigr) \bigr) \\ &\lesssim\rho^{p}_{H}(g)\sum_{l=-\infty}^{0} 2^{-p\alpha l} h^{p}_{H} \bigl(2^{l} \bigr) \\ &\lesssim\rho^{p}_{H}(g)\sum_{l=-\infty}^{0} 2^{-p\alpha l} 2^{{lp(\alpha_{H}-\varepsilon)}} \\ &\lesssim\rho^{p}_{H}(g)\sum_{l=-\infty}^{0} 2^{{lp(\alpha _{H}-\varepsilon-\alpha)}}. \end{aligned}$$
The above series is convergent if we choose \(\varepsilon>0\) such that \(\varepsilon< \alpha_{H}-\alpha\), so we have
$$\rho_{H} \biggl(t^{\alpha}\int_{0}^{t} s^{-\alpha}g(s)\frac{ds}{s} \biggr)\lesssim\rho_{H}(g). $$
On the other hand, for \(g\in G_{a}\),
$$\begin{aligned} t^{\beta}\int_{t}^{1} s^{-\beta}g(s) \frac{ds}{s}&=\int_{1}^{\infty}\chi _{(0,1)}(tv) v^{-\beta}g(tv)\frac{dv}{v} \\ &=\sum_{l=0}^{\infty}\int_{2^{l}}^{2^{l+1}} \chi_{(0,1)}(tv) v^{-\beta }g(tv)\frac{dv}{v} \\ &\lesssim\sum_{l=0}^{\infty}2^{-l\beta}g \bigl(t2^{l} \bigr)\chi _{(0,1)} \bigl(t2^{l} \bigr). \end{aligned}$$
Again applying Minkowski’s inequality, we get
$$\begin{aligned} \rho^{p}_{H} \biggl( t^{\beta}\int _{t}^{1} s^{-\beta}g(s)\frac{ds}{s} \biggr) &\lesssim\sum_{l=0}^{\infty}2^{-l\beta p} \rho^{p}_{H} \bigl(g \bigl(t2^{l} \bigr)\chi _{(0,1)} \bigl(t2^{l} \bigr) \bigr) \\ &\lesssim\rho^{p}_{H}(g)\sum_{l=0}^{\infty}2^{-l\beta p} h^{p}_{H} \bigl(2^{l} \bigr) \\ &\lesssim\rho^{p}_{H}(g)\sum_{l=0}^{\infty}2^{-l\beta p} 2^{pl(\beta _{H}+\varepsilon)} \\ &\lesssim\rho^{p}_{H}(g)\sum_{l=0}^{\infty}2^{{lp(\beta _{H}+\varepsilon-\beta)}}. \end{aligned}$$
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
$$\rho_{E} \biggl(t^{-a}\int_{0}^{t} s^{a}g(s)\frac{ds}{s} \biggr)\lesssim\rho _{E}(g), \quad g \in D_{0}, $$
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
$$\begin{aligned} t^{-a}\int_{0}^{t} s^{a}g(s) \frac{ds}{s}&=\int_{0}^{1} v^{a}g(tv)\frac{dv}{v} \\ &=\sum_{l=-\infty}^{0}\int_{2^{l}}^{2^{l+1}} v^{a}g(tv)\frac{dv}{v} \\ &\lesssim\sum_{l=-\infty}^{0} 2^{{al}}g \bigl(t2^{l} \bigr). \end{aligned}$$
Applying Minkowski’s inequality, we get
$$\begin{aligned} \rho^{p}_{E} \biggl(t^{-a}\int _{0}^{t} s^{a}g(s)\frac{ds}{s} \biggr) &\lesssim\sum_{l=-\infty}^{0} 2^{{pal}}\rho^{p}_{E} \bigl(g \bigl(t2^{l} \bigr) \bigr) \\ &\lesssim\rho^{p}_{E}(g)\sum_{l=-\infty}^{0} 2^{{pal}}\ h^{p}_{E} \bigl(2^{1} \bigr) \\ &\lesssim\rho^{p}_{E}(g)\sum_{l=-\infty}^{0} 2^{{pal}} \ 2^{{-1p(\beta_{E}+\varepsilon)}} \\ &\lesssim\rho^{p}_{E}(g)\sum_{l=-\infty}^{0} 2^{{lp(a-\beta _{E}-\varepsilon)}}. \end{aligned}$$
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
$$\rho_{E} \biggl(\int_{t}^{1} g(u) \frac{du}{u} \biggr)\lesssim\rho _{E}(g),\quad g\in D_{0}. $$

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
$$\begin{aligned} \int_{t}^{1} g(u)\frac{du}{u} & \lesssim\int _{1}^{\infty}\chi_{(0,1)}(tv)g(tv) \frac{dv}{v} \\ &= \sum_{l=0}^{\infty}\int_{2^{l}}^{2^{l+1}} \chi_{(0,1)}(tv)g(tv)\frac {dv}{v} \\ &\lesssim\sum_{l=0}^{\infty}\chi_{(0,1)} \bigl(t2^{l} \bigr)g \bigl(t2^{l} \bigr). \end{aligned}$$
Applying Minkowski’s inequality, we get
$$\begin{aligned} \rho^{p}_{E} \biggl(\int_{t}^{1} g(u)\frac{du}{u} \biggr) &\lesssim\sum_{l=0}^{\infty}\rho^{p}_{E} \bigl(\chi_{(0,1)} \bigl(t2^{l} \bigr)g \bigl(t2^{l} \bigr) \bigr) \\ &\lesssim\rho^{p}_{E}(g)\sum_{l=0}^{\infty}h^{p}_{E} \bigl(2^{-l} \bigr) \\ &\lesssim\rho^{p}_{E}(g)\sum_{l=0}^{\infty}2^{-l(\alpha _{E}-\varepsilon)}. \end{aligned}$$
Choosing \(\varepsilon>0\) such that \(\alpha_{E}>\varepsilon\), we conclude the proof. □

Declarations

Acknowledgements

The authors are thankful to the editor and the referees for their valuable suggestions in improving the final version of the article.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Division of Science and Technology, University of Education
(2)
Department of Mathematics, Lahore Leads University
(3)
Department of Mathematics and RINS, Gyeongsang National University

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