Averages along Polynomial Sequences in Discrete Nilpotent Lie Groups Singular Radon Transforms

• Published in: Advances in Analysis Vol. 50; pp. 146 - 188
• Main Authors: Phong, D. H, Wainger, Stephen, Fefferman, Charles, Ionescu, Alexandru D
• Format: Book Chapter
• Language: English
• Published: United States Princeton University Press 05.01.2014
• Subjects: Abstract spaces; Algebra; Applied mathematics; Applied sciences; Applied statistics; Calculus & mathematical analysis; Chemical elements; Chemistry; Complex analysis; Differentiable manifolds; Dimensional analysis; Dimensionality; Dynamical systems; Ergodic theory; Fourier transformations; Integral transformations; Lie groups; Mathematical expressions; Mathematical functions; Mathematical manifolds; Mathematical transformations; Mathematics; Noble gases; Physical sciences; Polynomials; Pure mathematics; Radon; Real analysis; Statistical physics; Statistics; Systems science; Systems theory; Topological spaces
• ISBN: 0691159416; 9780691159416
• DOI: 10.1515/9781400848935-008

A class of interesting problems arises in studying averages of functions along polynomial sequences in discrete nilpotent groups. More precisely, assume$\mathbb{G}$is a discrete nilpotent group of step$d\ge 1$and$A:\mathbb{Z}\to \mathbb{G}$is a polynomial sequence (see Definition 1.1 below), and consider the following problems:¹ Problem 1.(L²boundedness of maximal Radon transforms) Assume $f:\mathbb{G}\to \mathbb{C}$ is a function and let \[{\mathcal M}f(g)=\underset{N\ge 0}{\mathop{\sup }}\,\frac{1}{2N+1}\sum\limits_{|n|\le N}^{{}}{|f({{A}^{-1}}(n)\cdot g)|,\quad g\in \mathbb{G}.}\] Then \[\parallel\mathcal Mf{{\parallel }_{{{L}^{2}}(\mathbb{G})}}\lesssim \parallel f{{\parallel }_{{{L}^{2}}(\mathbb{G})}}.\] Problem 2.(L²pointwise ergodic theorems) Assume $\mathbb{G}$ acts by measure-preserving transformations on a probability space X, $f\in {{L}^{2}}(X)$ , and let \[{{A}_{N}}f(x)=\frac{1}{2N+1}\sum\limits_{|n|\le N}^{{}}{f({{A}^{-1}}(n)\cdot x),\quad x\in X.}\] Then the sequence ANf converges almost everywhere in X as $N\to \infty $. Problem 3.(L²boundedness of singular