On generalized H\"{o}lder's inequality in weak Morrey Spaces
Asyraf Wajih, Hendra Gunawan

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
**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
1[email protected], 2[email protected]
††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 , the weak Morrey space is the set of all measurable functions such that
[TABLE]
Here denotes the Lebesgue measure of open ball centered at with radius . Note that defines a quasinorm on . Furthermore, if , then . Thus, can be considered as generalization of weak Lebesgue space . Consequently, the quasinorm can be rewritten as
[TABLE]
where is a quasinorm on weak Lebesgue spaces .
From [1], we have the inclusion relation between weak Morrey spaces and for , as stated in the following theorem.
Theorem 1.1**.**
[1]** If , then with
[TABLE]
for every .
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 . If and , , then the following statements are equivalent:
(i)* and .*
(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 , .*
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 with a smaller constant.
Theorem 2.2**.**
Let , , and , . If and , then for every we have
[TABLE]
Proof. Let for , , and . Suppose that and . Note that for every we have
[TABLE]
with
[TABLE]
Consequently, for every , we have
[TABLE]
By choosing
[TABLE]
we get
[TABLE]
Moreover,
[TABLE]
By taking the supremum over all open balls and , 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 . If for , and . Then
[TABLE]
The equality is attained when for every .
Proof. If for every , then equality is clearly attained. Now suppose that for some . 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 , , and for every . If , then
[TABLE]
Proof. Let . By using Theorem 2.3 where , , and , 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 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 , for some , 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.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] H. Gunawan, D.I. Hakim, K.L. Limanta, and A.A. Masta, “Inclusion properties of generalized Morrey spaces”, Math. Nachr. , 290 :2-3 (2017), 332-340.
- 2[2] Ifronika, M. Idris, A.A. Masta, and H. Gunawan, “Generalized Hölder’s inequality in Morrey spaces”, Mat. Vesnik. , 70 :4 (2018), 326-337.
