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About weak integrals: Appendix of Folland's book "A course in abstract harmonic analysis"

Consider the following fragment from Folland's book "A course in abstract harmonic analysis": All integrals are here to interpreted in the weak sense (see p285 in Folland's book). Why is ...
Andromeda's user avatar
  • 175
4 votes
0 answers
161 views

Hodge theory in higher eigen-spaces?

Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology $$\mathcal{H}(E) \simeq H(E).$$ A classical example with differential forms ($E = (\Omega,d)$) ...
Student's user avatar
  • 5,230
3 votes
1 answer
194 views

Analytic approximation of the step function in $L^p$ norm

Motivation: Euler-Maclaurin formula uses calculus to estimate discrete sums. I wonder what one can do by reverse engineering. A concrete problem I ran into is the following. Question: Let $\chi: \...
Student's user avatar
  • 5,230
2 votes
0 answers
149 views

A closed ideal in $L^1(T)$

Let $\mathbb{T}$ be the unit circle and consider the convolution group algebra $L^1(\mathbb{T})$. Let $I_n$ be the closed ideal generated by the polynomial $p_n(z)=z^n-1$ in $L^1(\mathbb{T})$. Let $I=...
ABB's user avatar
  • 4,058
2 votes
1 answer
547 views

Shift-invariant spaces

We can define a shift-invariant space as $$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$ where convergence of the series is taken to be in $L^2(\...
AKG's user avatar
  • 49
3 votes
0 answers
77 views

Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$

My research needs help in finding examples of unitary matrices $U$ which have entries \begin{align} U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
moji's user avatar
  • 41
1 vote
0 answers
57 views

How to show the solution map of NLS is not smooth?

Let $u(\delta, t)$ satisfy $$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$ Note that the mapping: $$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$ ...
 Analyst 's user avatar
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
3 votes
1 answer
667 views

Equivalent action of convolution of directional derivative

I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented. Let $k*f(x)...
Mirar's user avatar
  • 350
9 votes
2 answers
1k views

Density of restrictions of harmonic functions inside a ball

Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let $$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
Ali's user avatar
  • 4,143
1 vote
0 answers
79 views

A problem arising from Wiener-Levy theorem on the real line

Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
ABB's user avatar
  • 4,058
1 vote
0 answers
53 views

Stability of Hajłasz-Sobolev class under post-composition

Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev? Assumptions/Setup Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
75 views

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 ...
ABB's user avatar
  • 4,058
0 votes
1 answer
88 views

An equation in the convolution measure algebra on reals

Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals. Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta_0$ be the point mass measure concentrated on ...
ABB's user avatar
  • 4,058
2 votes
0 answers
117 views

Is the cone of positive elements in $L^1(G)$ norm closed?

Let's consider $L^1(G)$, the Banach $*$- algebra of all Lebesgue integrable functions on the locally compact group $G$. Put $L^1(G)_+$ by the cone of positive elements given by $\{\sum_1^n f_i^**f_i: ...
ABB's user avatar
  • 4,058
5 votes
1 answer
496 views

Unifying two definitions of $L^\infty$

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$. Definitions: A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...
Andromeda's user avatar
  • 175
2 votes
0 answers
190 views

Inequality on the dual space of $H^s$

Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ? For instance, assume ...
Niser's user avatar
  • 93
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
1 vote
0 answers
260 views

Closure of smooth functions in Besov spaces

For real numbers $\alpha > \beta$, we know there is a continuous embedding of Besov spaces $B^\alpha_{\infty,\infty}\subset B^\beta_{\infty,\infty}$. We take the closure of the intersection $C^{\...
Inuyasha's user avatar
  • 253
2 votes
1 answer
223 views

Show that $V_G: L^2(G\times G, \mu \times \mu)\to L^2(G\times G, \mu \times \mu)$ defined by $V_G(f)(x,y) = f(xy,y)$ is well-defined

Let $G$ be a locally compact Hausdorff group and let $\mu$ be a right Haar measure on $G$. Then $\mu\times \mu$ (the Radon product of measures) is a right Haar measure on $G \times G$ and we can ...
Andromeda's user avatar
  • 175
0 votes
0 answers
84 views

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 ...
user13322's user avatar
2 votes
0 answers
120 views

Does a holomorphic function with logarithmic growth at the boundary have $L^2$ boundary values?

Let $f(z)$ be a holomorphic function on the unit disc, with logarithmic growth at the boundary: $$ |f(z)| = \mathcal O\bigg(\log\Big(\frac{1}{1-|z|}\Big)\bigg). $$ Does it follow that the (...
André Henriques's user avatar
11 votes
1 answer
413 views

Estimating the growth of the Taylor coefficients given the growth of the function at the boundary

Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies $$ |f(z)|\le \frac{1}{(1-|z|)^{k}} $$ for some fixed $k>0$. Question: What can I deduce about the growth of the ...
André Henriques's user avatar
3 votes
1 answer
171 views

Discrete singular integrals

Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties: $\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
A beginner mathmatician's user avatar
-1 votes
1 answer
78 views

Fundamental of a signal

Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$ or any reasonable Euclidean norm such that bounded ...
Arthur B's user avatar
  • 1,902
1 vote
1 answer
518 views

On level sets of smooth functions in a bounded domain

Let $\Omega$ be a bounded domain in $\mathbb R^n$, $n\geq 2$, with a smooth boundary and let $f$ be a smooth function on $\bar\Omega$. Is there a natural condition that one can impose on $f$ ( say in ...
alz812's user avatar
  • 11
1 vote
0 answers
244 views

On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)

Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
Dionel Jaime's user avatar
4 votes
0 answers
104 views

Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials? In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
Pierre's user avatar
  • 41
4 votes
0 answers
127 views

Algebra properties regarding Gevrey spaces: closed under multiplication

In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
Feng's user avatar
  • 517
3 votes
1 answer
164 views

Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?

Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
BigbearZzz's user avatar
  • 1,245
0 votes
0 answers
85 views

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-...
Korn's user avatar
  • 101
2 votes
0 answers
164 views

(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis

It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
BigbearZzz's user avatar
  • 1,245
12 votes
1 answer
727 views

A generalization of Rubio de Francia's inequality

Suppose that $\{I_m\}$ is a sequence of pairwise disjoint intervals in $\mathbb{Z}$. The well known Rubio de Francia's inequality says that for any function $f\in L^p(\mathbb{T})$, $2\le p<\infty$, ...
Anton Tselishchev's user avatar
0 votes
1 answer
212 views

The Quotient exponential operator

I have a question if you don't mind. I have the following quotient operator: $$\frac{1}{e^{d/dx}(f(x))}$$ Where $f$ is a smooth function on $R$. I would like to get rid of the denominator. IS there ...
Adam Hammam's user avatar
4 votes
0 answers
149 views

Cyclic vectors for the translation operator

Let $b\in \mathbb{R}\neq 0$, and consider the translation operators: $$ \begin{align} T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\ f &\mapsto f(\cdot + b). \end{align} $$ *Are there known ...
ABIM's user avatar
  • 5,405
5 votes
0 answers
118 views

Good (Sidon) Approximation of "Bumps"

Given a rational point $p\in S^1$ and a continuous function $f:S^1\rightarrow \mathbb C$, we say that $f$ is an $\epsilon$-bump around $p$ (for some $\epsilon>0$) if $f(p)=1,|f|_{\infty}\leq 1+\...
user3293260's user avatar
2 votes
0 answers
191 views

Convergence in $S'(\mathbb R^d)$ of the paraproduct $\dot{T}_uv$

Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$. For a fixed Littlewood-Paley decomposition $\chi \in \...
Desura's user avatar
  • 233
23 votes
9 answers
2k views

Nonseparable counterexamples in analysis

When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
3 votes
1 answer
191 views

A convolution type singular integral operator with log

Define a convolution type operator $T_m$ by $$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer. Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...
Jacob Lu's user avatar
  • 903
3 votes
2 answers
203 views

What is the distribution of the following limit?

Assume $x \in \mathbb{R}$. We already know that $$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$ Here $\delta_x$ denotes the Dirac distribution. If we ...
Jacob Lu's user avatar
  • 903
1 vote
1 answer
279 views

Understanding the regular representation of an LCA group as a 'direct integral'

The reference for what I'm asking is page $107$ from Folland's harmonic analysis. $G$ is a locally compact abelian group with dual $\hat{G}$. Let $H$ denote the Hilbert space $L^2(G)$. I'm trying to ...
Calamardo's user avatar
  • 675
6 votes
1 answer
365 views

Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?

Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm $$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$ ...
Matt Rosenzweig's user avatar
5 votes
2 answers
775 views

On inverting characteristic functions

Let $X$ be a random variable in $\mathbb{R}^n$ with distribution $\mu$ and characteristic function $\varphi$ (i.e. $\varphi(t)=\mathbb{E} e^{i\langle t,X\rangle}$). The standard inversion formula ...
TOM's user avatar
  • 2,288
6 votes
1 answer
378 views

Wiener Corollary in "An introduction to harmonic analysis" by Yitzhak Katznelson

I can't understand a lemma in "An introduction to harmonic analysis" by Yitzhak Katznelson which is stated as follows: Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\...
Christoff_ferland's user avatar
7 votes
0 answers
132 views

Smoothing property of a certain singular integral operator of non-convolution type

For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by $$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
Matt Rosenzweig's user avatar
3 votes
1 answer
203 views

Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$

Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
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
1 answer
135 views

Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$

Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that $$ \|w\|_{L^\infty(BMO)} := \sup_{t\in[0,T]}\|w(t,\cdot)\|_{BMO} \leq C $$ and $\int_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $...
M_S's user avatar
  • 123
0 votes
0 answers
152 views

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 $BMO(\mathbb{T}^d$)?
Jules Pitcho's user avatar
7 votes
2 answers
304 views

Existence of $f \in L^2(\Bbb R^n)$ with $f=g_1$ on $E$ and $\mathscr{F}(f)=g_2$ on $F$

The question has been posted here but had no response. Question: Suppose $E,F$ subsets of $\Bbb R^n$ have finite measure. Show that for any $g_1,g_2 \in L^2(\Bbb R^n)$ there exists $f \in L^2(\Bbb R^...
mathdogcmf's user avatar

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