Some new function spaces

01/30/2012 - 3:00pm
01/30/2012 - 4:00pm
Speaker: 
Katsuo Matsuoka, Nihon University, Japan
Abstract: 

Recently, we introduced some new function spaces, i.e. $ B_{\sigma} $-function spaces denoted by $ B_{\sigma}(E)(\mathbb{R}^n) $ and $ \dot{B}_{\sigma}(E)(\mathbb{R}^n) $.

These function spaces are defined as follows. For $ \sigma\in[0,\infty) $, let
$ B_{\sigma}(E)(\mathbb{R}^n) $ and $ \dot{B}_{\sigma}(E)(\mathbb{R}^n) $ be the sets of all functions $ f $ on $ \mathbb{R}^n $ such that $ \|f\|_{B_{\sigma}(E)}<\infty $ and $ \|f\|_{\dot{B}_{\sigma}(E)}<\infty $, respectively, where  $ \|f\|_{B_{\sigma}(E)}=\sup_{r\ge1}\dfrac{1}{r^{\sigma}}\|f\|_{E(Q_r)} $ and $ \|f\|_{\dB_{\sigma}(E)}=\sup_{r>0}\dfrac{1}{r^{\sigma}}\|f\|_{E(Q_r)}. $

Here, for each $ r>0 $$ Q_r=\{y=(y_1,y_2,\cdots,y_n)\in\mathbb{R}^n:\max_{1\le i\le n} |y_i|<r\} $ or
$ Q_r=\{y\in\mathbb{R}^n:|y|<r\} $, and $ E(Q_r) $ is a function space on $ Q_r $ with semi norm $ \|\cdot\|_{E(Q_r)} $.  For example, $ E=L^p $, $ \mathrm{Lip}_{\alpha} $, $ \mathrm{BMO} $, etc.

If $ E=L^p $ and $ \sigma=n/p $, then $ B_{\sigma}(L^p)(\mathbb{R}^n)=B^p(\mathbb{R}^n) $ which was introduced by Beurling (1964) together with its predual $ A^p(\mathbb{R}^n) $, the so-called the Beurling algebra.

Using the $ B_{\sigma} $-function spaces, we can unify a series of results on the boundedness of operators on several classical function spaces.

The talk is based on a joint work with Y. Komori-Furuya (Tokai U), E. Nakai (Ibaraki U) and Y. Sawano (Kyoto U).

Where: 
Davidson Lecture Hall, Claremont McKenna College
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