# Douglas Lectures

Date Time |
Location | Speaker | Title – click for abstract | |
---|---|---|---|---|

10/224:00pm |
Blocker 117 | Stefanie Petermichl University of Toulouse |
A workout program using weights: IThe Hilbert transform is a singular integral operator that gives access to harmonic conjugate functions via a convolution of boundary values. This operator is trivially a bounded linear operator in the space of square integrable functions. This is no longer obvious if we introduce a positive weight with respect to which we integrate the square. In this case, conditions on the weight need to be imposed, the so-called characteristic of the weight, both necessary and sufficient for boundedness. It is a delicate question to find the exact way in which the operator norm grows with this characteristic. Interest was spiked by a classical question surrounding quasiconformality and the Beltrame equation. Through an optimal weighted estimate of the Beurling-Ahlfors operator, the sibling of the Hilbert transform, a deep borderline regularity result of the Beltrami equation was solved. We give a historic perspective of the developments in this area of weights that spans about twenty years and that has changed our understanding of these important classical operators. We will see that the Hilbert transform is an average of dyadic shift operators. These can be seen as a coefficient shift and multiplier in a Haar wavelet expansion or as a time shifted operator on simple martingale differences. Such connections between martingale transforms and operators similar to the Hilbert transform had been understood for some time. It allows us to develop powerful tools with a probabilistic flavour to obtain deep results central to harmonic analysis. The central conjecture in sharp weighted theory was on the norm of the Hilbert transform. Its first solution involved the precise model of the dyadic shift. Since then, weighted theory evolved through many outstanding contributions, giving deep insight into the nature of such singular operators via so-called sparse domination. We highlight a probabilistic and geometric perspective on these new ideas, giving dimensionless estimates of Riesz transforms on Riemannian manifolds wi | |

10/234:00pm |
Blocker 117 | Stefanie Petermichl University of Toulouse |
A workout program using weights: IIThe Hilbert transform is a singular integral operator that gives access to harmonic conjugate functions via a convolution of boundary values. This operator is trivially a bounded linear operator in the space of square integrable functions. This is no longer obvious if we introduce a positive weight with respect to which we integrate the square. In this case, conditions on the weight need to be imposed, the so-called characteristic of the weight, both necessary and sufficient for boundedness. It is a delicate question to find the exact way in which the operator norm grows with this characteristic. Interest was spiked by a classical question surrounding quasiconformality and the Beltrame equation. Through an optimal weighted estimate of the Beurling-Ahlfors operator, the sibling of the Hilbert transform, a deep borderline regularity result of the Beltrami equation was solved. We give a historic perspective of the developments in this area of weights that spans about twenty years and that has changed our understanding of these important classical operators. We will see that the Hilbert transform is an average of dyadic shift operators. These can be seen as a coefficient shift and multiplier in a Haar wavelet expansion or as a time shifted operator on simple martingale differences. Such connections between martingale transforms and operators similar to the Hilbert transform had been understood for some time. It allows us to develop powerful tools with a probabilistic flavour to obtain deep results central to harmonic analysis. The central conjecture in sharp weighted theory was on the norm of the Hilbert transform. Its first solution involved the precise model of the dyadic shift. Since then, weighted theory evolved through many outstanding contributions, giving deep insight into the nature of such singular operators via so-called sparse domination. We highlight a probabilistic and geometric perspective on these new ideas, giving dimensionless estimates of Riesz transforms on Riemannian manifolds wi | |

10/254:00pm |
Blocker 117 | Stefanie Petermichl University of Toulouse |
A workout program using weights: IIIThe Hilbert transform is a singular integral operator that gives access to harmonic conjugate functions via a convolution of boundary values. This operator is trivially a bounded linear operator in the space of square integrable functions. This is no longer obvious if we introduce a positive weight with respect to which we integrate the square. In this case, conditions on the weight need to be imposed, the so-called characteristic of the weight, both necessary and sufficient for boundedness. It is a delicate question to find the exact way in which the operator norm grows with this characteristic. Interest was spiked by a classical question surrounding quasiconformality and the Beltrame equation. Through an optimal weighted estimate of the Beurling-Ahlfors operator, the sibling of the Hilbert transform, a deep borderline regularity result of the Beltrami equation was solved. We give a historic perspective of the developments in this area of weights that spans about twenty years and that has changed our understanding of these important classical operators. We will see that the Hilbert transform is an average of dyadic shift operators. These can be seen as a coefficient shift and multiplier in a Haar wavelet expansion or as a time shifted operator on simple martingale differences. Such connections between martingale transforms and operators similar to the Hilbert transform had been understood for some time. It allows us to develop powerful tools with a probabilistic flavour to obtain deep results central to harmonic analysis. The central conjecture in sharp weighted theory was on the norm of the Hilbert transform. Its first solution involved the precise model of the dyadic shift. Since then, weighted theory evolved through many outstanding contributions, giving deep insight into the nature of such singular operators via so-called sparse domination. We highlight a probabilistic and geometric perspective on these new ideas, giving dimensionless estimates of Riesz transforms on Riemannian manifolds wi |