All Questions
451 questions
5
votes
0
answers
196
views
Distributions and functions on the Jacquet module $C_c^\infty(X)_{H,\chi}$
Let $X$ be an $\ell$ space (in the sense of Bernstein-Zelevinski), $H$ be an $\ell$ group which acts on $X$ and $\chi$ be a character of $H$. Denote $C^\infty(X)^{H,\chi}$ the space of locally ...
- 960
1
vote
1
answer
400
views
The reproducing kernel for harmonics on compact manifolds
Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
- 617
1
vote
0
answers
192
views
The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
- 617
4
votes
1
answer
495
views
Weil's Haar measure construction from below
Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...
user1688
0
votes
0
answers
194
views
Johnson's Theorem - Proof (Runde) Clarification
I am reading Runde's Lectures on Amenability. In the proof of Johnson's theorem where he proves "$L^{1}(G)$ is amenable Banach algebra implies $G$ is amenable" : he defines a $L^1(G)$ bimodule action ...
- 185
2
votes
1
answer
285
views
Does Fourier Algebra of locally compact group separate compact sets of the group?
Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of $\...
- 185
4
votes
1
answer
952
views
Two questions on Elias Stein paper (1976)
I am working on some results related with a paper of Elias Stein (on the almost every where convergence of wavelet summation methods), and I have the following questions:
The maximal function ...
2
votes
1
answer
382
views
Is the translation/dilation of an $L^p$-multiplier again an $L^{p}$-multiplier?
Suppose that $m:\mathbb R \to \mathbb C$ satisfies: there exists $C > 0$ such that
$$\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}.$$
That is, $m$ is an $L^{p}$-multiplier. Let $M(L^{p}...
- 1,051
3
votes
0
answers
262
views
Non-compact analogue of Peter-Weyl
I have the following situation: $G$ is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation $H:=L^2(G,\mu_H)$ decomposes as
\begin{equation}
\int^{\...
- 1,813
4
votes
1
answer
257
views
condition for $f \in H^{1/2} \cap C^0$
I need to show that for $f \in H^{1/2}(S^1) \cap C^0(S^1)$ we have :
$$ \iint\limits_{S^1 \times S^1}{ \frac{|f(x)-f(y)|^2}{\sin^2(\pi(x-y))}dx dy}< + \infty $$
and that, conversely, if $f$ is a ...
- 41
5
votes
0
answers
120
views
L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension
For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator:
$M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$
(...
- 4,010
2
votes
0
answers
134
views
Restricted weak type bound at the endpoint
We know that if we have an operator that is (restricted) weak type $(p,p)$ and (strong) type $(\infty,\infty)$ with norm 1, then it's also of strong type $(q,q)$ for all $p<q<\infty$ by the real ...
- 319
7
votes
1
answer
747
views
Application of Factorization Theory to Oscillatory Integral Estimates
In the article "Some New Estimates on Oscillatory Integrals" by Bourgain in the book Essays in Honor of Elias M. Stein, Bourgain considers operators of the form
$$S_{N}g(x):=\int_{\mathbb{R}^{n}}g(y)e^...
- 873
4
votes
2
answers
1k
views
Hardy-Littlewood-Sobolev inequality in Lorentz spaces
Hardy-Littlewood-Sobolev inequality states that if $1<p<q<\infty$, $1/r=1-1/p+1/q$, then we have
$$\left\|\frac{1}{|x|^{n/r}}\ast f\right\|_{L^q(\mathbb R^n)}\le\|f\|_{L^p(\mathbb R^n).}$$
...
- 319
9
votes
1
answer
912
views
The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?
I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a ...
- 5,169
8
votes
2
answers
1k
views
What does the unique mean on weakly almost periodic functions look like?
There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
- 4,432
2
votes
0
answers
93
views
Multiplier operators on anisotropic weighted $L^2$ spaces
Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$,
$\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$
where the scalar complex function ...
- 21
2
votes
1
answer
816
views
Decoupling in mixed norm spaces
Bourgain and Demeter's proof of the $L^2$ decoupling conjecture decouples $\|f\|_{L^p}$ into an $L^2$ sum of $\|f_\theta\|_{L^p}$, where $\hat f$ is supported on a curved hypersurface $S$, where $\...
- 5,169
3
votes
3
answers
768
views
Pointwise convergence of Fourier series, Fefferman's article
This is the first time I ask a question here, so sorry if I make any mistake in the way I ask it. I'm studing Fefferman's article Pointwise Convergence of Fourier Series, and I have two questions:
1)...
user85801
0
votes
1
answer
193
views
$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$
Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
4
votes
1
answer
277
views
Is the Fourier-Stieltjes algebra of a locally compact group semi-simple?
Let $G$ be a locally compact group. Is the Fourier-Stieltjes algebra $B(G)$ semi-simple?
- 4,058
0
votes
1
answer
128
views
On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$
There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...
- 175
13
votes
3
answers
710
views
Completeness of nonharmonic Fourier Series
I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system $\Phi:...
- 131
0
votes
0
answers
54
views
Left introversion operators associated to function spaces on semigroups
I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...
- 1
20
votes
2
answers
922
views
A functional inequality about log-concave functions
Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle \...
- 3,946
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'...
3
votes
2
answers
226
views
Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$
The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as
$$
L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}),
$$
where
$$
L^2_k (\mathbb{R}^2; \...
- 2,834
2
votes
1
answer
313
views
Spherical decreasing rearrangement on the sphere
On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert f^...
- 21
13
votes
1
answer
675
views
Wavelet-like Schauder basis for standard spaces of test functions?
Edit: A more precise formulation of my question follows the separation line.
The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of ...
2
votes
1
answer
383
views
Hardy space, Lebesgue space for $p<1$,
We denote $\mathcal D'(\mathbb R^n)$ the space of distributions, and $\mathcal D(\mathbb R^n)$ the space of smooth, compactly supported functions.
Let $\rho\in \mathcal D'(\mathbb R^n)$ such that ...
- 630
2
votes
2
answers
210
views
The convolution between weighted $L^1$ space and normal $L^1$ space
Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$,
$$
\frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x)
$$
...
- 679
1
vote
0
answers
62
views
Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...
- 679
1
vote
1
answer
319
views
The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator
Consider a general bilinear multiplier operator:
$$
T(f,g)(n)=\int_{\Pi}\int_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta,
$$
where $\Pi$ is the torus, $n\in\mathbb{Z}$, $m$...
- 463
0
votes
0
answers
407
views
What does the Plancherel theorem say about positive-definite distributions?
I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem.
The ...
- 3,799
0
votes
1
answer
731
views
Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together
Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, ...
- 3
6
votes
2
answers
2k
views
Equivalent Norms on Sobolev Spaces
When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ^{\...
- 233
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 ...
- 7,633
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 ...
- 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 ...
- 31
2
votes
1
answer
396
views
$BMO$-property via a John-Nirenberg type estimate?
Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also
$$
f_B:= \frac1{|B|}\int_B f \, dx.
$$
Suppose $f \in L_{\rm loc}^p(\Omega)$ for all $1<p&...
- 632
5
votes
1
answer
421
views
application of factorization theorem
Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality
$$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$
and of course this ...
- 1,051
3
votes
0
answers
211
views
Arveson spectrum for a unitary representation of a group on a Hilbert space
Although this is not research, I think the question is a little bit too specific for math.stackexchange
Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a ...
- 189
2
votes
0
answers
119
views
Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)
Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve
$$\int_0^\infty\int_\Omega \nabla v\nabla ...
- 49
4
votes
0
answers
107
views
Is Wiener's Tauberian theorem true in Wiener space?
Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$.
Is the following true?
...
- 4,010
0
votes
0
answers
256
views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
- 325
1
vote
0
answers
133
views
Condition for boundedness in stationary phase theorem
I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and $j, ...
- 93
1
vote
0
answers
102
views
How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?
We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$
By Helson-Kahane-Katznelson-Rudin Theorem, it follows that,
"Let $F$ be a function on $\mathbb C$ and if $F(f)...
- 1,051
1
vote
1
answer
432
views
Does Trudinger inequality implies this critical Sobolev embedding?
Let $1<p<\infty$, $\mathrm{L}^p_s(\mathbb{R}^n)=J_s(\mathrm{L}^p(\mathbb{R}^n))$, where $J_s=(I-\Delta)^{-\frac{s}{2}}$, or $\mathscr{F}(J_sf)(\xi)=(1+|\xi|^2)^{-\frac{s}{2}}\hat{f}(\xi)$. And ...
- 1,616
1
vote
0
answers
206
views
What is the spectrum of $L^1(G:H)$?
Let $H$ be a compact subgroup of a locally compact topological group $G$ and
$$ L^1(G:H)=\{f\in L^1(G): R_h f=f\;\text{ a.e. }\; \forall h \in H\}$$ and $\widehat{(G:H)}=\{\xi\in \hat{G}:\xi|_H=1\}$($\...
- 91
7
votes
1
answer
450
views
Can we extend a multiplicative linear functional of a closed left ideal on whole of the algebra?
Let B be a closed left ideal of a Banach algebra A. Also, B has a right approximate identity (in B).
If g is a nonzero multiplicative linear functional on B, can we always extend g to a ...
- 91