Local factors of Tamagawa measure

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

A generalized ellipse

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

Sobolev spaces defined on non-compact Lie groups

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

Well-definedness on C∞0(Rn)C_{0}^{\infty}(\mathbb{R}^{n})

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

Coming up with a represenation for sum of functions in the Fourier algebra

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

Continuous doubling weight vanishing on set of positive measure?

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

Integral form of maximal function estimate on variable exponent spaces

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

Question about the history of dyadic models in harmonic analysis

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

Fractional integral inequality (Hardy-Littlewood-Sobolev)

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

Fefferman’s article: Pointwise convergence of Fourier series, II

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