All Questions
Tagged with fa.functional-analysis harmonic-analysis
451 questions
1
vote
0
answers
202
views
Space of analytic function and sequence space $l^p$
Let $\mathbb{D} = \{z:|z|<1\}$ be open unit disc in complex plane. Define space of analytic function:
$N^p=\{f:\mathbb{D} \to \mathbb{C} | f(z)=\sum_{n=0}^{\infty} a_n z^n, \sum_{n=0}^{\infty}|a_n|...
- 18.7k
3
votes
1
answer
480
views
Is there a uniform upper bound for this oscillatory integral?
I am wondering whether the following uniform upper bound holds:
$|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le Cab^2,$
where $0<a<b<1$, $N>N_0(a,b)\gg1$, and $C$ is a constant ...
- 31.5k
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
$$S_{N}g(x):=\int_{\mathbb{R}^{n}}g(y)e^...
CommunityBot
- 1
3
votes
1
answer
195
views
Boundedness of different Fourier transforms
Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$
Does this imply that we can take common norms in the sense that we can estimate ...
- 33
1
vote
1
answer
124
views
On a weaker condition of summability for Fourier series
The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$...
- 109k
1
vote
0
answers
112
views
Notations - Hardy and Sobolev Spaces [duplicate]
After some confusion on my part, I wanted to know is there a profound mathematical reason why both Hardy spaces and Sobolev spaces are denoted by $H^p$(1). Is it just coincidence? Does it have any ...
- 11
2
votes
1
answer
183
views
is this weighted-maximal function unbounded?
The Hardy-Littlewood maximal operator
$$Mf(x)=\sup_{x\in B}\frac1{\vert B\vert}\int_B\vert f(y)\vert dy$$
where the supremum is taken over all balls $B\subset\mathbb{R}^n$ which contain $x$.
It is ...
- 1,583
3
votes
2
answers
457
views
Integrality of complex infinite series
Let $(a_n)$ be a (double-sided) sequence of complex numbers satisfying
$$\sum_{\mathbb{Z}}\vert n\vert\,\vert a_n\vert^2<\infty, \tag1$$
$$\sum_{\mathbb{Z}}a_n\bar{a}_{n+k}=\delta_0(k), \qquad \...
- 43.2k
1
vote
0
answers
194
views
Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
CommunityBot
- 1
2
votes
0
answers
186
views
Is this simple oscillatory integral operator uniformly bounded on $L^2$?
Let $\phi(t,s)$ be a real-valued function smooth away from the diagonal, and equal to 0 on the diagonal. Assume that $0\le \phi(t,s)\le |t-s|$ for $t,s\in \mathbb{R}$. Let
$$T_\lambda f(t)=\int \frac{\...
- 171
2
votes
0
answers
125
views
Imbedding Theorems between Besov Spaces and space of continuos functions on the unit circle
I'll try to be brief.
Let us consider the Besov Space $B^{1/p}_{p, p}(\mathbb{T})$, where $1\leq p<\infty $ and $\mathbb{T}$ is the unit circle in the complex plane. I would like to know for which ...
- 141
5
votes
2
answers
2k
views
Which functions are continuous with respect to the weak topology?
Let me first introduce the restricted setting in which this question has a nice answer. I came up with this when messing around with a homework problem in a PDE class a couple years back.
Let $\phi \...
- 60.5k
2
votes
0
answers
183
views
Are there any improvements on the estimate of oscillatory integral with one-side folds?
Suppose $X$ and $Z$ are open sets in $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$, respectively. Define $T_\lambda f:L^2(Z)\to L^2(X)$ by $$T_\lambda f(x)=\int e^{i\lambda\Phi(x,z)}a(x,z)f(z)dz,$$where the ...
- 171
3
votes
1
answer
652
views
When does an inverse PDE operator have a kernel (i.e. a fundamental solution?)
Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator
$$
Lu:=-\Delta u+V(x)u
$$
where $V\geq0$ is the electric potential, ...
- 16.2k
4
votes
1
answer
140
views
Finding a special Banach algebra and a net of homomorphisms
If $A$ is a Banach agebra and $M$ is a Banach $A$-bimodule then a linear map $T:A\to M$ is called an $A$-module homomorphism if $$T(ab)=aT(b),\quad T(ab)=T(a)b,\qquad a,b\in A.$$ Also $A\hat{\otimes} ...
- 25.8k
2
votes
1
answer
3k
views
A simple question about the Hardy-Littlewood maximal function
Let $f\in L^1(\mathbb{R}^n)$. It is well known that the Hardy-Littlewood maximal function $Mf\notin L^1(\mathbb{R}^n)$ (if $f \ne 0$ a.e.), though there is a weak-type (1,1) bound for this maximal ...
CommunityBot
- 1
2
votes
0
answers
85
views
There is an analogous to the inerpolation theorem of Aronszajn-Gagliardo for multilinear operators in Banach spaces?
The Aronszajn-Gagliardo theorem states that on the category $\mathcal B$ of Banach spaces, given $A$ an interpolation space with respect to the couple $\overline A$ there exists an exact interpolation ...
- 71
1
vote
1
answer
238
views
Does the bounded extension of the Fourier multiplier operator agrees with its original explicit definition?
We consider the Fourier multiplier operator $T_0$ defined by the explicit expression
$$(T_0f)(x)=\int_{\mathbb{R}^n}{e^{ix\cdot \xi}m(\xi)\hat{f}(\xi)d\xi}, \ f\in S(\mathbb{R}^n),$$ where $S(\mathbb{...
- 171
0
votes
1
answer
697
views
How much do we know about this "local" Hardy-Littlewood maximal function?
The "local" Hardy-Littlewood maximal function is given by $$(M_\phi f)(x)= \sup_{0<\epsilon<1}|\phi_\epsilon \ast f|(x),$$ which is similar to the classical Hardy-Littlewood maximal function : $$...
- 39k
6
votes
2
answers
318
views
Natural map $C^*(G) \to M(A\rtimes G)$
Let $A$ be a C*-algebra, let $G$ be a locally compact group, and let $\alpha\colon G\to\mathrm{Aut}(A)$ be a (strongly) continuous action. It is well known that there is a natural map $\iota_G\colon C^...
- 1,702
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
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)...
- 1,692
2
votes
1
answer
172
views
Ideals of $L^1(G)$ and normal subgroups of $G$
Let $G$ be a locally compact group. Is there any correspondence between closed two-sided ideals of $L^1(G)$ and closed normal subgroups of $G$? (Especially, is there any correspondence between finite ...
- 25.8k
4
votes
0
answers
114
views
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 $\lambda:G\to B(\ell^{2}(G))$ be the left ...
- 25.8k
2
votes
0
answers
110
views
If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$
Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
CommunityBot
- 1
10
votes
1
answer
439
views
Interpolation between $L_1^0$ and $L_2^0$
Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
- 11.6k
1
vote
1
answer
1k
views
Does the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$?
I am struggling to know whether the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$.
$fg$ has compact support but I can't figure out how I can try ...
- 3,574
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") ...
- 23k
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
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 ...
- 11.6k
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
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 $\...
CommunityBot
- 1
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 ...
- 13k
4
votes
0
answers
433
views
Scattering for rapidly decaying solutions of NLS
Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...
- 3,574
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
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
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
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 ...
- 114k
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:...
- 24.2k
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 ...
- 3,984
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 $\...
- 114k
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 ...
- 1,018
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?
- 25.8k
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 ...
- 2,361
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
2
votes
2
answers
3k
views
Description of Bessel potential spaces
Hi, let $1 < p <\infty $, $ 0 < \alpha < 1$, and $ \mathscr{L}^p_\alpha(R^n) $ be the usual Bessel potential space defined by
$$
\mathscr{L}^p_\alpha = (1-\triangle)^{-\alpha/2}L^p(R^n).
$$...
- 41
10
votes
3
answers
1k
views
Historical developement of analysis and partial differential equations (especially in the 20th century)
Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
I'...
CommunityBot
- 1
2
votes
1
answer
313
views
Spherical decreasing rearrangement on the sphere
On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert f^...
CommunityBot
- 1
3
votes
2
answers
226
views
Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$
The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as
$$
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; \...
- 25.8k
6
votes
3
answers
2k
views
Generalized Hardy-Littlewood-Sobolev Inequality
The Hardy-Littlewood-Sobolev Inequality says that
$$\text{for $p,q,r\in (1,+\infty)$ such that }\quad
1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$}
$$
$$
\exists C, \forall u\in L^p(\mathbb R^n),\...
- 687