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1 answer
119 views

Nonstationary phase method for oscillatory integral

I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth. The stationary phase method says that if $t_0\in [a,b]$ is such that ...
charlie_beck's user avatar
0 votes
0 answers
34 views

Locally compact groupoid with range map restricted to isotropy groupoid is open

Suppose the action groupoid 𝐺=𝐻⋉𝑋, where 𝐻 is a locally compact group and 𝑋 a locally compact space is such that isotropy subgroups of H are isomorphic to each other. Can this be an example of a ...
K N SRIDHARAN NAMBOODIRI's user avatar
2 votes
1 answer
98 views

Locally compact groupoid with a Haar system such that the range map restricted to isotropy groupoid is open

Can somebody provide an example of a locally compact groupoid $G$ with a Haar system such that the range map restricted to isotropy groupoid of $G$ is open? I could not find any specific example for ...
K N Sridharan's user avatar
0 votes
0 answers
89 views

Maximal function on mixed $L^{p}$

Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is $$ \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
User091099's user avatar
1 vote
0 answers
53 views

Reference for Density question

Let $ B $ be a reflexive, separable Banach space and $ p \in (1,\infty)$. Then denote by $L^{p}(B)$ the space of all functions $$ f : \mathbb{R}^{n} \to B $$ with $$ \int_{\mathbb{R}^{n}} \vert f \...
User091099's user avatar
1 vote
0 answers
82 views

For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$

For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
Akira's user avatar
  • 835
9 votes
0 answers
240 views

What is known about when $vN(G)$ is a factor, for a locally compact group $G$?

When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group. What is known ...
Jared White's user avatar
4 votes
0 answers
68 views

Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$

For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that $$ |f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
user298455's user avatar
1 vote
0 answers
148 views

BMO estimates of singular integral operators on torus

I have the following elliptic problem: $$ \Delta u = \operatorname{div}\operatorname{div}S, $$ where $S=(S_{i,j})\colon \mathbb{T}^n\to \mathbb{R}^{n\times n} $ is bounded and $\mathbb{T}^n$ is the $n$...
M_S's user avatar
  • 123
10 votes
2 answers
594 views

Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?

Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
Hua Wang's user avatar
  • 960
4 votes
1 answer
221 views

Fourier multipliers and transference on cyclic groups

It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\...
Anton Tselishchev's user avatar
2 votes
0 answers
169 views

Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?

Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
Riku's user avatar
  • 839
1 vote
0 answers
62 views

Properties of the Fourier Transform of Countably Supported Functions on $[0,1)$

Identifying $\mathbb{R}/\mathbb{Z}$ with the interval $\left[0,1\right)$, let $C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right)$ denote the set of all functions $f:\mathbb{R}/\mathbb{Z}\rightarrow\...
MCS's user avatar
  • 1,284
1 vote
0 answers
74 views

Dimension dependence: boundedness result of the fractional Riesz integral

I am looking for the best known constant in the boundedness result of the fractional Riesz integral. In particular, I am interested in the dependence on the dimension $d$ and on the parameter $\alpha&...
user69642's user avatar
  • 778
1 vote
0 answers
122 views

Metric transforms that preserve $\ell^1$ embeddability

Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances. Work by Schoenberg and ...
Timothy Chu's user avatar
3 votes
1 answer
409 views

Riesz transform of fractional operators

I am interested in Riesz transforms linked to the fractional Laplacian and other fractional operators. I have been hunting down in the literature to find related results but I have not been able to ...
user69642's user avatar
  • 778
1 vote
0 answers
109 views

Is this a positive definite kernel?

Under which conditions on the function : \begin{array}{l|rcl} K : & \mathbb R^+ & \longrightarrow & (0, 1)\\ &t & \longmapsto & K(t) \end{array} is the symmetric ...
Abdeslam KOUBAA's user avatar
2 votes
1 answer
178 views

References for Neumann eigenfunctions

I am looking for references on eigenfunctions with Neumann boundary condition. In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...
sharpe's user avatar
  • 721
2 votes
0 answers
214 views

Why is the study of homology important? [closed]

In some fields of studies, for example, Amenability of Banach algebras and $L^2$-Betti numbers, some chain complexes are studied, why is the study of these creatures important? When and why do these ...
MSMalekan's user avatar
  • 2,118
6 votes
1 answer
134 views

Multi-parameter stationary phase asymptotic expansion

I am looking for an asymptotic expansion of the oscillatory integral of the form $$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$ as $\lambda_i\to \infty$ ...
Subhajit Jana's user avatar
4 votes
0 answers
264 views

Is the Gelfand transform strictly continuous?

Let $M$ be the Banach algebra of measures on the circle with $L_1$ naturally sitting as a closed ideal of $M$. Then $M$ carries the strict topology implemented by the family of seminorms $\|\mu\|_f = \...
Jan_Ch.'s user avatar
  • 113
11 votes
1 answer
691 views

Reference request: Fourier transform on the multiplicative group of real numbers

Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...
Jochen Glueck's user avatar
7 votes
1 answer
489 views

When the value of a function in a point is equal to its integral average over the point's neighborhood?

It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
Grove's user avatar
  • 91
3 votes
0 answers
317 views

Best constant for maximal function for locally compact groups

Maximal functions for locally compact groups have been studied. In particular papers of K. Phillips and Phillips-Taibleson (here and here) contain nice results in this direction. It might be too much ...
BigM's user avatar
  • 1,583
2 votes
1 answer
183 views

is this weighted-maximal function unbounded?

The Hardy-Littlewood maximal operator $$Mf(x)=\sup_{x\in B}\frac1{\vert B\vert}\int_B\vert f(y)\vert dy$$ where the supremum is taken over all balls $B\subset\mathbb{R}^n$ which contain $x$. It is ...
T. Amdeberhan's user avatar
5 votes
0 answers
215 views

Explicit description of the Plancherel measure for $GL_n(\mathbb{R})$

Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation $$f(g)...
Subhajit Jana's user avatar
2 votes
1 answer
481 views

Ideals of $L^1(G)$

I want to study the closed ideal structure of $L^1(G)$. Is there a good paper or book which characterizes closed ideals and maximal ideals of $L^1(G)$?
Albert harold's user avatar
10 votes
3 answers
1k views

Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in technical detail) the historical development of the various subareas of analysis and partial differential equations? I'...
2 votes
1 answer
313 views

Spherical decreasing rearrangement on the sphere

On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert f^...
guest's user avatar
  • 21
1 vote
0 answers
62 views

Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$. ...
JumpJump's user avatar
  • 679
7 votes
2 answers
385 views

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
Dustin G. Mixon's user avatar
2 votes
1 answer
333 views

Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
SMS's user avatar
  • 1,407
3 votes
1 answer
191 views

Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$ Now, by ...
anonymous's user avatar
3 votes
0 answers
140 views

convergence of $e^{it\Delta}f$

I heard of a conjecture that $e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$ but couldn't find a proper reference.
user57563's user avatar
2 votes
2 answers
741 views

Bochner's theorem for measures of positive type

Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying $\...
Evan DeCorte's user avatar
1 vote
0 answers
269 views

$L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?

Let $\mathbb T$ be a circle group. In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, $L^...
Inquisitive's user avatar
  • 1,051
25 votes
3 answers
13k views

Fourier transform of the unit sphere

The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
Francois Ziegler's user avatar
7 votes
2 answers
1k views

For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?

(This is essentially a continuation of my previous question, here.) Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
user avatar
6 votes
2 answers
469 views

Orthonormal basis for $L^2(G/H)$.

Let $G$ be a locally compact group and $H$ be a closed subgroup of $G$. Is there any way to define a reasonable orthonormal basis for $L^2(G/H)$? By "reasonable" I mean elements of the orthonormal ...
user avatar
4 votes
1 answer
471 views

Ask for theory about the weighted L^2(R^d) space.

Dear MOs, I am now considering the following norm: $$ ||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:. $$ where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
Anand's user avatar
  • 1,649
1 vote
1 answer
295 views

A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics

With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining $$\psi_{...
Tobias Kienzler's user avatar
59 votes
7 answers
29k views

Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
12 votes
1 answer
1k views

Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$

Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
András Bátkai's user avatar