Search

Only show content I have access to (0)

Articles (2)

Over 3 years (2)

From year:

To year:

Apply

Mathematics (2)

muckenhoupt condition

Proceedings of the Edinburgh Mathematical Society (1)

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

View selected items
Save to my bookmarks
Export citations
Download PDF (zip)
Save to Kindle
Save to Dropbox
Save to Google Drive
Save content to
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to .

To save content items to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Please be advised that item(s) you selected are not available.
You are about to save
Your Kindle email address

Please provide your Kindle email.

@free.kindle.com @kindle.com (service fees apply)

By using this service, you agree that you will only keep content for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services Please confirm that you accept the terms of use.

×

Muckenhoupt Class Weight Decomposition and BMO Distance to Bounded Functions
Part of
- Harmonic analysis in several variables
Morten Nielsen, Hrvoje Šikić
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 62 / Issue 4 / November 2019

Published online by Cambridge University Press:

25 March 2019, pp. 1017-1031
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
We study the connection between the Muckenhoupt Ap weights and bounded mean oscillation (BMO) for general bases for ℝd. New classes of bases are introduced that allow for several deep results on the Muckenhoupt weights–BMO connection to hold in a very general form. The John–Nirenberg type inequality and its consequences are valid for the new class of Calderón–Zygmund bases which includes cubes in ℝd, but also the basis of rectangles in ℝd. Of particular interest to us is the Garnett–Jones theorem on the BMO distance, which is valid for cubes. We prove that the theorem is equivalent to the newly introduced A2-decomposition property of bases. Several sufficient conditions for the theorem to hold are analysed as well. However, the question whether the theorem fully holds for rectangles remains open.

In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality:

\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),

$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$

for some constant C > 0, where Ω is an open set in ℝN with N ⩾ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of

{\cal D}_0^{2,2} ${\cal D}_0^{2,2}$ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

Search Results

Refine search

Refine search

Actions for selected content:

2 results

Muckenhoupt Class Weight Decomposition and BMO Distance to Bounded Functions

On the generalized Hardy-Rellich inequalities

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

2 results

Muckenhoupt Class Weight Decomposition and BMO Distance to Bounded Functions

On the generalized Hardy-Rellich inequalities