On generalized H\"{o}lder's inequality in weak Morrey Spaces

TL;DR

This paper reestablishes a generalized H"{o}lder's inequality within weak Morrey spaces, providing sharper bounds by leveraging the relationship between weak Morrey and weak Lebesgue spaces.

Contribution

It offers a new proof of the inequality with improved bounds, connecting weak Morrey and weak Lebesgue spaces.

Findings

Sharper bounds for H"{o}lder's inequality in weak Morrey spaces

Enhanced understanding of the relation between weak Morrey and weak Lebesgue spaces

Reproves and extends previous results with improved estimates

Abstract

In this note we reprove generalized H\"{o}lder's inequality in weak Morrey spaces. In particular, we get sharper bounds than those in \cite{gunawan2}. The bounds are obtained through the relation of weak Morrey spaces and weak Lebesgue spaces.

Equations

$$\Vert f{\Vert }_{w{\mathcal{M}}_q^p}=\underset{B=B(a,r),\gamma >0}{\sup }{\left|B\right|}^{\frac{1}{q}-\frac{1}{p}}\gamma \left|\{x\in B:|f(x)|>\gamma \}\right|<\infty .$$

$$\Vert \cdot {\Vert }_{w{\mathcal{M}}_q^p}=\underset{B=B(a,r)}{\sup }{\left|B\right|}^{\frac{1}{q}-\frac{1}{p}}\Vert \cdot {\Vert }_{L^{p,\infty }(B)},$$

$$\Vert f{\Vert }_{w{\mathcal{M}}_q^{p_1}}\le \Vert f{\Vert }_{w{\mathcal{M}}_q^{p_2}}$$

$$\biggl{\|}\prod_{i=1}^{m}f_{i}\biggr{\|}_{w\mathcal{M}^{p}_{q}}\leq\prod_{i=1}^{m}\left(\frac{p_{i}}{p^{*}}\right)^{\frac{1}{p_{i}}}\|f_{i}\|_{w\mathcal{M}^{p_{i}}_{q_{i}}}.$$

$$\left|\{x\in B:\left|f_1{f}_2\cdots f_m(x)\right|>\theta \}\right|$$

$$\le \sum _{i=1}^m\Vert f_i{\Vert }_{L^{p_i,\infty }(B)}^{p_i}{\left(\frac{s_i}{s_{i-1}}\right)}^{p_i}$$

$$=\sum _{i=1}^m{a}_i{y}_i^{p_i}$$

$$a_i=\Vert f_i{\Vert }_{L^{p_i,\infty }(B)}^{p_i}\text{and}{y}_i=\frac{s_i}{s_{i-1}}.$$

$$\left|\{x\in B:\left|f_1{f}_2\cdots f_m(x)\right|>\theta \}\right|\le \sum _{i=1}^m{a}_i{y}_i^{p_i}.$$

$$y_i={\left(m^{-1}{\theta }^{-p^{\ast }}{a}_i^{-1}\prod _{j=1}^m{\left(\frac{p_j}{p^{\ast }}\right)}^{\frac{p^{\ast }}{p_j}}{a}_j^{\frac{p^{\ast }}{p_j}}\right)}^{\frac{1}{p_i}}$$

$$\left|\{x\in B:\left|f_1{f}_2\cdots f_m(x)\right|>\theta \}\right|$$

$$=\sum _{i=1}^m{a}_i\left(m^{-1}{\theta }^{-p^{\ast }}{a}_i^{-1}\prod _{j=1}^m{\left(\frac{p_j}{p^{\ast }}\right)}^{\frac{p^{\ast }}{p_j}}{a}_j^{\frac{p^{\ast }}{p_j}}\right)$$

$$={\theta }^{-p^{\ast }}\prod _{j=1}^m{\left(\frac{p_j}{p^{\ast }}\right)}^{\frac{p^{\ast }}{p_j}}\Vert f_j{\Vert }_{L^{p_j,\infty }(B)}^{p^{\ast }}.$$

$${\left|B\right|}^{\frac{1}{q}-\frac{1}{p\ast }}\theta {\left|\{x\in B:\left|f_1{f}_2\cdots f_m(x)\right|>\theta \}\right|}^{\frac{1}{p^{\ast }}}$$

$$=\prod _{i=1}^m{\left(\frac{p_i}{p^{\ast }}\right)}^{\frac{1}{p_i}}{\left|B\right|}^{\frac{1}{q_i}-\frac{1}{p_i}}\Vert f_i{\Vert }_{L^{p_i,\infty }(B)}$$

$$\le \prod _{i=1}^m{\left(\frac{p_i}{p^{\ast }}\right)}^{\frac{1}{p_i}}\Vert f_i{\Vert }_{w{\mathcal{M}}_{q_i}^{p_i}}.$$

$$\biggl{\|}\prod_{i=1}^{m}f_{i}\biggr{\|}_{w\mathcal{M}^{p*}_{q}}\leq\prod_{i=1}^{m}\left(\frac{p_{i}}{p^{*}}\right)^{\frac{1}{p_{i}}}\|f_{i}\|_{w\mathcal{M}^{p_{i}}_{q_{i}}}.$$

$$\biggl{\|}\prod_{i=1}^{m}f_{i}\biggr{\|}_{w\mathcal{M}^{p}_{q}}\leq\biggl{\|}\prod_{i=1}^{m}f_{i}\biggr{\|}_{w\mathcal{M}^{p*}_{q}}\leq\prod_{i=1}^{m}\left(\frac{p_{i}}{p^{*}}\right)^{\frac{1}{p_{i}}}\|f_{i}\|_{w\mathcal{M}^{p_{i}}_{q_{i}}},$$

$${\left(\prod _{i=1}^m{x}_i^{w_i}\right)}^{\frac{1}{w}}\le \sum _{i=1}^m\frac{w_i{x}_i}{w}.$$

$$\ln {\left(\prod _{i=1}^m{x}_i^{w_i}\right)}^{\frac{1}{w}}=\ln \left(\prod _{i=1}^m{x}_i^{\frac{w_i}{w}}\right)=\sum _{i=1}^m\frac{w_i}{w}\ln \left(x_i\right)<\ln \left(\sum _{i=1}^m\frac{w_i{x}_i}{w}\right).$$

$${\left(\prod _{i=1}^m{x}_i^{w_i}\right)}^{\frac{1}{w}}<\sum _{i=1}^m\frac{w_i{x}_i}{w},$$

$$\prod _{i=1}^m{\left(\frac{p_i}{p^{\ast }}\right)}^{\frac{1}{p_i}}\le m^{\frac{1}{p^{\ast }}}\le m.$$

$${\left(\prod _{i=1}^m{\left(\frac{p_i}{p^{\ast }}\right)}^{\frac{1}{p_i}}\right)}^{p^{\ast }}\le \sum _{i=1}^m\frac{1}{p_i}\frac{p_i}{p^{\ast }}{p}^{\ast }=m.$$

$$\prod _{i=1}^m{\left(\frac{p_i}{p^{\ast }}\right)}^{\frac{1}{p_i}}\le m^{\frac{1}{p^{\ast }}}\le m,$$

$$\biggl{\|}\prod_{i=1}^{m}f_{i}\biggr{\|}_{w\mathcal{M}^{p}_{q}}=\prod_{i=1}^{m}\left(\frac{p_{i}}{p^{*}}\right)^{\frac{1}{p_{i}}}\|f_{i}\|_{w\mathcal{M}^{p_{i}}_{q_{i}}}.$$

Full text

**ON GENERALIZED HÖLDER’S INEQUALITY IN WEAK MORREY SPACES

** ASYRAF WAJIH1 and HENDRA GUNAWAN 2

Faculty of Mathematics and Natural Sciences,

Institut Teknologi Bandung, Bandung 40132, Indonesia

1asyraf.wajih@hotmail.com, 2hgunawan@math.itb.ac.id

††footnotetext: *2000 Mathematics Subject Classification: 26D15, 46B25, 46E30

Abstract. In this note we reprove generalized Hölder’s inequality in weak Morrey spaces. In particular, we get sharper bounds than those in [2]. The bounds are obtained through the relation of weak Morrey spaces and weak Lebesgue spaces.

Key words and phrases: generalized Hölder’s inequality, weak Lebesgue spaces, weak Morrey spaces.

1. INTRODUCTION

Hölder’s inequality is one of the classic inequality which is proved by L.J. Rogers in 1888 and reproved by O. Hölder in 1889. Many researchers have obtained new results related to Hölder’s inequality. Recently, Ifronika et al. [2] obtained sufficient and necessary conditions for generalized Hölder’s inequality in generalized Morrey spaces. Furthermore, they also get similar results for weak Morrey spaces and generalized weak Morrey spaces. In this paper, we present new bounds for generalized Hölder’s inequality in weak Morrey spaces which are sharper than those in [2].

For $1\leq p\leq q<\infty$, the weak Morrey space $w\mathcal{M}^{p}_{q}(\mathbb{R}^{n})$ is the set of all measurable functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that

[TABLE]

Here $|B|$ denotes the Lebesgue measure of open ball $B=B(a,r)$ centered at $a$ with radius $r$. Note that $\|\cdot\|_{w\mathcal{M}^{p}_{q}}$ defines a quasinorm on $w\mathcal{M}^{p}_{q}(\mathbb{R}^{n})$. Furthermore, if $p=q$, then $w\mathcal{M}^{p}_{q}(\mathbb{R}^{n})=L^{p,\infty}(\mathbb{R}^{n})$. Thus, $w\mathcal{M}^{p}_{q}(\mathbb{R}^{n})$ can be considered as generalization of weak Lebesgue space $L^{p,\infty}(\mathbb{R}^{n})$. Consequently, the quasinorm $\|\cdot\|_{w\mathcal{M}^{p}_{q}}$ can be rewritten as

[TABLE]

where $\|\cdot\|_{L^{p,\infty}(B)}$ is a quasinorm on weak Lebesgue spaces $L^{p,\infty}(B)$.

From [1], we have the inclusion relation between weak Morrey spaces $w\mathcal{M}^{p_{1}}_{q}(\mathbb{R}^{n})$ and $w\mathcal{M}^{p_{2}}_{q}(\mathbb{R}^{n})$ for $1\leq p_{1}\leq p_{2}\leq q<\infty$, as stated in the following theorem.

Theorem 1.1**.**

[1]** If $1\leq p_{1}\leq p_{2}\leq q<\infty$, then $w\mathcal{M}^{p_{2}}_{q}(\mathbb{R}^{n})\subseteq w\mathcal{M}^{p_{1}}_{q}(\mathbb{R}^{n})$ with

[TABLE]

for every $f\in w\mathcal{M}^{p_{2}}_{q}(\mathbb{R}^{n})$.

2. MAIN RESULTS

Let us first state sufficient and necessary conditions for generalized Hölder’s inequality in weak Morrey spaces [2].

Theorem 2.1**.**

[2]** Let $m\geq 2$. If $1\leq p\leq q<\infty$ and $1\leq p_{i}\leq q_{i}<\infty$, $i=1,2,\dots,m$, then the following statements are equivalent:

(i)* $\sum_{i=1}^{m}\frac{1}{p_{i}}\leq\frac{1}{p}$ and $\sum_{i=1}^{m}\frac{1}{q_{i}}=\frac{1}{q}$.*

(ii)* \biggl{\|}\prod_{i=1}^{m}f_{i}\biggr{\|}_{w\mathcal{M}^{p}_{q}}\leq m\prod_{i=1}^{m}\|f_{i}\|_{w\mathcal{M}^{p_{i}}_{q_{i}}}, for every $f_{i}\in w\mathcal{M}^{p_{i}}_{q_{i}}(\mathbb{R}^{n})$, $i=1,2,\dots,m$.*

Now we will refine the statement of Theorem 2.1, particularly the part which states that (i) implies (ii). Precisely, we will replace the bound $m$ with a smaller constant.

Theorem 2.2**.**

Let $m\geq 2$, $1\leq p\leq q<\infty$, and $1\leq p_{i}\leq q_{i}<\infty$, $i=1,2,\dots,m$. If $\frac{1}{p^{*}}=\sum_{i=1}^{m}\frac{1}{p_{i}}\leq\frac{1}{p}$ and $\sum_{i=1}^{m}\frac{1}{q_{i}}=\frac{1}{q}$, then for every $f_{i}\in w\mathcal{M}^{p_{i}}_{q_{i}}(\mathbb{R}^{n})$ we have

[TABLE]

Proof. Let $f_{i}\in w\mathcal{M}^{p_{i}}_{q_{i}}$ for $i\in\{1,2,\dots,m\}$, $B=B(a,r)\subseteq\mathbb{R}^{n}$, and $\theta>0$. Suppose that $s_{0}=\theta$ and $s_{m}=1$. Note that for every $s_{1},s_{2},\dots,s_{m-1}>0$ we have

[TABLE]

with

[TABLE]

Consequently, for every $y_{1},\dots,y_{m}>0$, we have

[TABLE]

By choosing

[TABLE]

we get

[TABLE]

Moreover,

[TABLE]

By taking the supremum over all open balls $B$ and $\theta>0$, we have

[TABLE]

Hence, by using Theorem 1.1, we get

[TABLE]

which completes the proof. ∎

Next, we prove that the bound in Theorem 2.2 is in general sharper than that in Theorem 2.1. To do so, we need the following theorem.

Theorem 2.3**.**

[AM-GM Weighted Inequality] Let $m\in\mathbb{N}$. If $x_{i},w_{i}>0$ for $i=1,2,\dots,m$, and $w=\sum_{i=1}^{m}w_{i}$. Then

[TABLE]

The equality is attained when $x_{i}=x_{j}$ for every $i,j\in\{1,\dots,m\}$.

Proof. If $x_{i}=x_{j}$ for every $i,j\in\{1,2,\dots,m\}$, then equality is clearly attained. Now suppose that $x_{i}\neq x_{j}$ for some $i,j\in\{1,\dots,m\}$. Since the natural logarithm is a strictly concave function, we can use Jensen inequality to get

[TABLE]

Hence, we have

[TABLE]

as desired. ∎

Now we are ready to prove that our bound is sharper than those in [2].

Theorem 2.4**.**

Let $m\in\mathbb{N}$, $1\leq p^{*}<\infty$, and $1<p_{i}<\infty$ for every $i\in\{1,\dots,m\}$. If $\sum_{i=1}^{m}\frac{1}{p_{i}}=\frac{1}{p^{*}}$, then

[TABLE]

Proof. Let $\frac{1}{p^{*}}=\sum_{i=1}^{m}\frac{1}{p_{i}}$. By using Theorem 2.3 where $x_{i}=\frac{p_{i}}{p^{*}}$, $w_{i}=\frac{1}{p_{i}}$, and $w=\frac{1}{p^{*}}$, we get

[TABLE]

Hence,

[TABLE]

as stated. ∎

3. CONCLUDING REMARKS

In this note we already proved that generalized Hölder’s inequality in weak Morrey spaces with bound $\prod_{i=1}^{m}\left(\frac{p_{i}}{p^{*}}\right)^{\frac{1}{p_{i}}}$ and this bound is in general sharper than the one obtained in [2].

However, we still do not know whether the bound is really sharp, that is, we still do not know whether there is a function $f_{i}\in w\mathcal{M}^{p_{i}}_{q_{i}}(\mathbb{R}^{n})$, $i=1,\dots,m$ for some $m\geq 2$, such that

[TABLE]

Acknowledgement. The results in this note have been presented at Mid Year School on Analysis, Geometry, and Applications (MYSAGA) 2018. We would like to thank Ms. Ifronika for her comments on the earlier version of this note.