Author: Oliver Dragičević
Publisher:
Published: 2003
Total Pages: 164
ISBN-13:
DOWNLOAD EBOOKIn the first part we use the technique of averaging to give a sharp weighted estimate for the operator of convolution with $z^{-2}$ (the Ahlfors-Beurling operator) for an arbitrary $A_2$ weight. As an application we touch upon the theory of quasiconformal mappings on $\hat{\Cc}$. The Ahlfors-Beurling operator can be realized as a combination of second-order Riesz transforms. We prove a new Littlewood-Paley-type inequality which is the key to our result in the second part. As some of its consequences, we show that the scalar Riesz transforms and their vector analogues admit dimension free upper estimates of the norms when acting on $L^p({\mathbb R}^n)$ for arbitrary dimension $n$ and $p>1$. The essence of the proof is the utilization of the method of Bellman functions, which, requiring but few assumptions, appears to allow its application to other kinds of Riesz transforms. We present the proof of one such example - dimensionless boundedness of Riesz transforms on Gaussian spaces.
Book Synopsis Riesz Transforms and the Bellman Function Technique by : Oliver Dragičević
Download or read book Riesz Transforms and the Bellman Function Technique written by Oliver Dragičević and published by . This book was released on 2003 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the first part we use the technique of averaging to give a sharp weighted estimate for the operator of convolution with $z^{-2}$ (the Ahlfors-Beurling operator) for an arbitrary $A_2$ weight. As an application we touch upon the theory of quasiconformal mappings on $\hat{\Cc}$. The Ahlfors-Beurling operator can be realized as a combination of second-order Riesz transforms. We prove a new Littlewood-Paley-type inequality which is the key to our result in the second part. As some of its consequences, we show that the scalar Riesz transforms and their vector analogues admit dimension free upper estimates of the norms when acting on $L^p({\mathbb R}^n)$ for arbitrary dimension $n$ and $p>1$. The essence of the proof is the utilization of the method of Bellman functions, which, requiring but few assumptions, appears to allow its application to other kinds of Riesz transforms. We present the proof of one such example - dimensionless boundedness of Riesz transforms on Gaussian spaces.
