Singular Integral Operators on Spaces of Vanishing Mean Oscillations

Abstract: "A crowning achievement of the classical Calder\'on-Zygmund theory emergent in the 1950's was the $L^p$-boundedness, $1<p<\infty$, of Singular Integral Operators (SIO) in ${\mathbb{R}}^n$ with kernels satisfying suitable smoothness, parity, and homogeneity conditions. In particular, these include the Riesz transforms $R_j$, $1\leq j\leq n$, which are principal value SIO of convolution type with kernels, $k_j(x):=\frac{x_j}{|x|^{n+1}}$, $x\in{\mathbb{R}}^n\setminus\{0\}$, for each $j\in\{1,\dots, n\}$.\\ While such boundedness results fail for the end-point case $p=\infty$, the correct substitute is to employ the John-Nirenberg space ${\rm BMO}({\mathbb{R}}^n)$ of functions with bounded mean oscillations in ${\mathbb{R}}^n$. Pioneering work in this regard us due to C. Fefferman and E. Stein who in a paper in Acta Mathematica, 1982, have shown that some modified version $R_j^m$ of the Riesz transforms satisfy $R_j^m:{\rm BMO}({\mathbb{R}}^n)\to{\rm BMO}({\mathbb{R}}^n)$.\\ A companion space to ${\rm BMO}({\mathbb{R}}^n)$, is the Sarason space ${\rm VMO}({\mathbb{R}}^n)$ of functions with vanishing mean oscillations in ${\mathbb{R}}^n$, and the issue arises whether the (modified) Riesz transforms $R_j^m$ map ${\rm VMO}({\mathbb{R}}^n)$ boundedly into itself. In this talk I will outline a program which achieves just that and comment on further applications."

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