This is a reference request to some computations which I hope can be found in the literature somewhere. Let G⊂GLn be a semisimple linear algebraic group over Q. The Tamagawa measure μ on the group of adelic points G(A) is uniquely determined by the product formula. Nevertheless, it can be written as a product of … Read more
We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the following generalization: For three given points $a,b,c \in \mathbb{R}^{2}$ define $$A_{\lambda}=\{z\in \mathbb{R}^{2} \;\text{with}\;\; |z-a|+|z-b|+|z-c|=\lambda\}$$ How is the geometric description of … Read more
In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and associated theories of the Laplace operator on noncompact Lie groups. My expectation is that the theory should be reduced to coincide with that … Read more
Let T be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel K∈CZKα of order α>0 and b∈BMO(Rn). Then for f∈C∞0(Rn) define[b,T](f)=b⋅Tf−T(bf). Question. Is [b,T] is well-defined on C∞0(Rn) for all b∈BMO(Rn)? Answer AttributionSource : Link , Question Author : Timothy , Answer Author : Community
This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com. Let G be a discrete group. Let λ:G→B(ℓ2(G)) be the left regular representation of G and A(G) the Fourier algebra of G. Given f,g∈A(G), and writing f(s)=⟨λ(s)x,y⟩,g(s)=⟨λ(s)w,z⟩ for some choice x,y,w,z∈ℓ2(G), is there a way to choose (in … Read more
If I is a bounded interval in R, let 2I denote an interval with the same center point but double the length. A doubling measure on R is a (non-trivial, locally finite, Borel) measure μ such that μ(2I)≤Cμ(I) for all intervals I and some fixed constant C. A doubling weight is an L1loc function w:R→R≥0 … Read more
I am trying to show an estimate of the following form: Given any p(x) such that 1<p−≤p(x)≤p+<∞ and p(⋅) is log-Holder continuous, does there exists an R0 (depending only on p(⋅), n and log-Holder continuity of p(x)) such that ∫BRM<2R(|f|)p(x)(x) dx≤C∫B2R|f(x)|p(x) dx+1 holds for every function f∈Lp(⋅)(B2R) and every R<R0. Here M<T(|f|)(x):=supr<T1|B(x,r)|∫B(x,r)|f(y)| dy EDIT: I have been informed … Read more
Who first used the expression “dyadic model” in the sense of this blog post by Terence Tao? Say you are a harmonic analyst trying to prove a result, e.g., something like the Carleson-Hunt Theorem, but it’s just too hard. So you consider instead a simplified dyadic model, for instance using Walsh series, you work out … Read more
I am investigating the following integral \begin{equation} I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y – x|^{\mu}} \, dy \end{equation} where $f \in L_p(\mathbb{R})$, $ 1 < p < q < \infty $, and $\mu = 1 + \frac{1}{q} – \frac{1}{p}$. For the integral \begin{equation} I(x) = \int_{\mathbb{R}} \frac{f(y) }{|y – x|^{\mu}} \, dy \end{equation} there … Read more
I have some problems reading Pointwise convergence of Fourier series by Fefferman https://www.jstor.org/stable/1970917 I got stuck in Chapter 6, Lemma 5. In the proof he split the P′ into three subcollections P′k, Pk″, \mathcal P”’_k. The first and third subcollections were estimated. However, for the second, he imposed a new assumption: \varphi_k(\xi_0′)=0, where \xi’_0 is … Read more
