All Questions
Tagged with fa.functional-analysis harmonic-analysis
190 questions with no upvoted or accepted answers
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Laplacian on squashed spheres
Is anything known about the Laplacian on squashed spheres $S^{2n-1}_\omega$, where the ambient $C^n$ coordinates satisfy
$$ 1= \sum_{i=1}^n \omega_i |z_i|^2 $$
for fixed real numbers $\omega_i$? for ...
- 533
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124
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Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
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85
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A kernel on the d-dimensional flat torus with smoothing properties in the $L^{\infty}$-norm
Let $\rho: \mathbb{R}^d\rightarrow \mathbb{R}_+$ be smooth, symmetric, of compact support, and satisfy $\int_{\mathbb{R}^d}\rho(x)dx=1$. For each $\epsilon>0$, set $\rho_{\epsilon}(x)=\epsilon^{-d}\...
- 11
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113
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Hardy $Hp$ norm of similar function
Let $f(z)=\sum_{n=0}^{\infty} \frac{c_n}{n+1}z^n$, where sequence $c_n \in S^1=\{z:|z|=1\}.$ We observe $H^p$ norm $\|f\|_{H_p}$, where $H^p$ is Hardy space, $1 \leq p < \infty$.
Question: For the ...
- 259
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82
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Topology of the algebra $\mathbb{C}\{A\}$ for a LCA group $A$
Let $\mathcal{A}$ be a complex associative Hausdorff topological algebra, and let $A\subset\mathcal{A}$ be a locally compact Abelian (LCA) subgroup (multiplicative). The linear span $\mathbb{C}\{A\}$ ...
- 1,959
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202
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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|...
- 259
1
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194
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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 ...
- 171
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192
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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
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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 $X_1$.
...
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133
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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
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102
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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
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206
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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
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153
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On sequence of functions $(h_n)$ satisfying $\Vert\sum_{n=1}^\infty f * h_n\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert$ for all $f\in L_1(G)$
Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property
$$
\left\Vert\sum_{n=1}^\infty f * h_n\right\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert
...
- 1,697
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269
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$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^...
- 1,051
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164
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How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
- 1,051
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129
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Differentiability of $f*g$ on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f
If $f\in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$ where $\mathbb{T}$ is the circle, such that $\hat{f}\in L^{p}(\mathbb{Z})$ for some $1\leq p<\infty$, do we have that $f*g$ is ...
- 647
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154
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A problem concerning measures on locally compact spaces
I am stuck on a question for quite sometime now, although in the text it is said to be "apparent". The problem goes as the following :
Let $X$ and $Y$ be locally compact Hausdorff spaces. Then $M(X)$ ...
- 111
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134
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Tauberian measures on a locally compact abelian group
Given a locally compact abelian group $G$ with Haar measure $m$, it is well-known that there exist measures $\mu\in M(G)$ which are singular (with respect to $m$), but the convolution product $\mu\ast\...
- 4,461
1
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187
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Injective modules over Fourier algebra
Is there any article on injective modules over Fourier Algebras?
Do we have anything about injectivity of $A(G)$ as a $A(G)$-bimodule?
- 71
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1
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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 ...
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57
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Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
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Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?
Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$.
If ...
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84
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Question on approximation of norms
Suppose that $E\in Int[L_{p},L_{q}]$ for some $1<p<q<\infty$ and $E$ is $w$-concave with $1<w<\infty$. It is well-known that for each $r\geq w$, we have $E=L_{r}\odot F_{r}$ for some ...
- 45
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34
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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 ...
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36
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Sufficient condition for interpolation
If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying:
$Z_0=X$, $...
- 101
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89
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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}...
- 31
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The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 852
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145
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Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
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$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 159
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Fourier coefficient of close functions
Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as
$$ f(x) = \...
- 1
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75
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Extracting the point mass measure of some type of positive measures
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals.
Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
- 4,058
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84
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Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
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An amenable operator algebra has the total reduction property
This is from
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CB20539885C03522D141C34024707702/S1446788700014026a.pdf/div-class-title-operator-algebras-with-a-reduction-property-...
- 101
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152
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Predual of $BMO(\mathbb{T}^d) $
In 1971, Fefferman characterized the predual of $BMO(\mathbb{R}^d)$ as the Hardy space $H^1(\mathbb{R}^d)$.
Is there a characterization of the predual of $BMO(\mathbb{T}^d$)?
- 11
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194
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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
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54
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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
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405
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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
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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 $$\sum_{\alpha}\frac{z^{\...
- 325
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146
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How to bound Haar coefficients in terms of total variation?
I'm trying to get the basic idea behind the proof of Theorem 8.1 of this paper, but I'm having difficulty. Specifically, it says:
We shall show that there is a set $\Lambda_n\subset\mathcal{D}$ such ...
- 7,633
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298
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High dimensional beta integral (question following the previous post)
Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |...
- 1,649