# Tagged Questions

**2**

votes

**1**answer

220 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&...

**1**

vote

**0**answers

82 views

### Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$

Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly,
Let $f$ be a periodic ...

**5**

votes

**1**answer

131 views

### The (Hecke) double coset von Neumann algebra

It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on $l^2(\...

**2**

votes

**1**answer

195 views

### an analogue of Littlewood-Paley-Rubio de Francia theory

For any function $f$ defined on the set of integer $\mathbb{Z}$, we define its Fourier transform as the following periodic function: $$
\mathbb{F}f(\xi)=\sum_{n\in\mathbb{Z}}f(n)e^{-2\pi i n\xi}
$$
...

**1**

vote

**0**answers

119 views

### Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function:
$$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$
or to know where I can find some techniques to ...

**0**

votes

**0**answers

48 views

### Non interacting complex unit

How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...

**7**

votes

**1**answer

375 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
$...

**7**

votes

**1**answer

215 views

### Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...

**5**

votes

**1**answer

173 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**

vote

**1**answer

71 views

### Limiting absorption principle

I would like to know if there is a book (or a paper) which can give me an introduction to LAP. I tried to read some papers by myself, but I don't feel comfortable. I think that I need the basic ideas ...

**7**

votes

**2**answers

306 views

### Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...

**0**

votes

**0**answers

39 views

### Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces.
In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$
Take $f(x)= \chi_{\...

**7**

votes

**3**answers

509 views

### Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$,
and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$.
Prove or disprove the following.
Given any ...

**1**

vote

**0**answers

44 views

### Estimates for derivatives of a positive discrete harmonic function

There is the following estimation (Duffin, Discrete potential theory, Theorem 5):
Let $f$ be a discrete harmonic function in a sphere of radius $R$ with the center $p$, all in $\mathbb Z^3$. Then, if ...

**1**

vote

**0**answers

69 views

### Proof of the chain rule for fractional derivatives in F.M.Christ and M.I.Weinsten's paper

In the paper: Dispersion of small amplitude Solutions of the generalized Korteweg-de Vries equation, JOURNAL OF FUNCTIONAL ANALYSIS 100,87-109 (1991).
I find a mistake in the proof of proposition 3.1 ...

**1**

vote

**0**answers

60 views

### Stationary averages of harmonic measures

Let $A$ be a countable subset of the open unit disk $\mathbb D$ centered at 0. For a point $x\in\mathbb D$ denote by $\nu_x$ the associated harmonic measure on the boundary circle $\partial\mathbb D$.
...

**2**

votes

**2**answers

54 views

### extension of the projectivized gradient of a harmonic function

Let $(M,g)$ be a riemannian manifold, $\Delta$ the associated Laplacian, and $\{ f_i \}$ the real-valued eigenfunctions of $\Delta$. Then, $\nabla f_i \in \Gamma ^{\infty } (\mathrm{T} M) $ is defined ...

**0**

votes

**0**answers

91 views

### The Lebesgue measure of the low level sets of the two-dimmension Fourier transform of a compactly supported function

Let $f\in {{L}^{1}}\left( {{\mathbb{R}}^{2}} \right)$ be a density function with the support $\operatorname{supp}\left( f \right)\subset \left[ a,b \right]\times \left[ c,d \right]$. Denoted by $\hat{...

**5**

votes

**0**answers

145 views

### Local version of the Hardy-Littlewood-Sobolev theorem for Riesz potentials: $\|I_\alpha(f)\|_{L^q} \le C \|f\|_{L^p}$?

Recently, I have been studying the properties of the Riesz potential
$$
I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.
$$
The classical Hardy-Littlewood-Sobolev ...

**4**

votes

**0**answers

314 views

### How is $ \sum_{x \in X(\mathbb{F}_q)} \dots $ a generalization of cardinality?

This quarter Maxim Kontsevich is offering a course on exponential integral. There is not much in the way of notes, but is one page with mysterious comments.
Let $X$ be an algebraic variety over $\...

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votes

**1**answer

231 views

### Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?

Let us denote the Riesz potential in $\mathbb R^d$ by
$$
I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
By the classical Hardy-Littlewood-Sobolev theorem ...

**3**

votes

**0**answers

106 views

### Nonlinear Schrödinger blow-up for non radial solutions

I am studying a paper of Frank Merle and Pierre Raphaël,
http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf.
The equations are
$$
i\partial_tu+\Delta u=-|u|^{p-1}u
$$
on $\mathbb{R}...

**5**

votes

**3**answers

159 views

### Reference request : Besov spaces on ubounded domains

As I am relatively new to these matters, I would like to know if you could provide me a reference for Besov spaces on unbounded domains, because when I checked the first tome of Triebel's Theory of ...

**8**

votes

**1**answer

345 views

### For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?

I don't expect to find an explicit counterexample to my question, because any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the ...

**3**

votes

**0**answers

117 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 ...

**8**

votes

**1**answer

154 views

### Can the Cesaro limit of a positive definite function be negative?

Let $G$ be a countable amenable group and $\gamma:G\to\mathbb{C}$ a positive (semi)definite function (i.e. such that $\gamma(g^{-1})=\overline{\gamma(g)}$ and
$$\sum_{g,h\in G}f(g)\overline{f(h)}\...

**9**

votes

**1**answer

144 views

### Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...

**2**

votes

**1**answer

113 views

### How to prove that the convolution operator associated to a discrete measure on a LCA group has natural spectrum?

Let $\mu$ be a Borel measure with finite variation on a locally compact abelian group $G$, let $\Gamma$ denote the dual group of $G$, and let $\hat \mu: \Gamma \to \mathbb{C}$ be the Fourier-Stieltjes ...

**5**

votes

**1**answer

125 views

### Existence of doubling non-Polish metric measure spaces

Let $(X,d,\mu)$ be a metric measure space (i.e. $(X,d)$ is a metric space and $\mu$ is a Borel measure on $X$). Let's say that $X$ is doubling if there exists a constant $C \geq 1$ such that $0 < \...

**2**

votes

**0**answers

94 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 ...

**0**

votes

**1**answer

179 views

### A $C^{*}$ algebra associated to a group [closed]

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...

**9**

votes

**6**answers

1k views

### number theory which is close to analysis

I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis.
...

**4**

votes

**1**answer

175 views

### Generalisation of Lebesgue differentiation theorem to Orlicz spaces

If $f\in L_{p}^{\rm loc}(\mathbb{R}^{n})$ and $1\leq p<\infty$, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\...

**2**

votes

**2**answers

175 views

### Growth of a harmonic function on the disc

Here is my question : I have a harmonic function $h$ on the open unit disc in $D \subset \mathbb{C}$, such that $\iint_D e^{2h} d\lambda(z) \leq A < \infty$ ($d\lambda$ is the Lebesgue measure on $\...

**4**

votes

**0**answers

86 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?
...

**6**

votes

**1**answer

863 views

### Cricket and the Hardy-Littlewod maximal function

I'v read somewhere that one motivation for Hardy to define his maximal function is the game of cricket. But I can't see how they are related. Could anyone provide some more information on their ...

**3**

votes

**1**answer

178 views

### Commutator with Hilbert transform

Let $H$ be Hilbert transform on $L^p({R})$. It is well known that commutator of $H$ and function $b\in BMO$ is a bounded operator in $L^p({R})$:
$$\|[H,b] u\|_p\leq c_p\|b\|_{BMO}\|u\|_p, 1<p<\...

**4**

votes

**2**answers

176 views

### Root of positive function in Fourier algebra

Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$.
Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...

**0**

votes

**1**answer

112 views

### Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by
\begin{align}
Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t},
\end{align}
that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast f....

**0**

votes

**0**answers

147 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^{\...

**0**

votes

**0**answers

103 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 ...

**1**

vote

**0**answers

92 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)...

**2**

votes

**1**answer

136 views

### Approximation of subharmonic functions

Let $u$ be an (upper semi-continuous) locally bounded subharmonic function in a domain in $\mathbb{R}^n$. Let $\chi_\epsilon$ be a standard smoothing kernel, namely
$$\chi_\epsilon(x)=\frac{c_n}{\...

**3**

votes

**1**answer

231 views

### Which group algebras in analysis are “true group algebras”?

Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that
$$
\pi(1)=1,\qquad \pi(a\cdot b)=\pi(a)\...

**4**

votes

**2**answers

801 views

### Fourier transform of compactly supported distribution is smooth

My advisor made the comment that if $u\in \mathcal{E}'$ is a compactly supported distribution, then $\hat{u}(\xi)\in C^{\infty}(\mathbb{R}^n)$ is actually a smooth function (not merely a distribution ...

**0**

votes

**1**answer

164 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 ...

**0**

votes

**1**answer

132 views

### Uniform $L^p-L^{p'}$ bound of a Fourier multiplier

Let $(\tau,\xi)\in\mathbb{R}\times \mathbb{R}^n$, and consider the function
$$
m_{\epsilon}(\tau,\xi)=\frac{1}{\tau+|\xi|^4+\epsilon|\xi|^2+i}
$$
in $\mathbb{R}^{n+1}$. My first question is that does ...

**0**

votes

**0**answers

93 views

### Is each multiplicative linear functional on $L^1(SL(2,R):SO(2,R))$ is triviall?

We know that if $G=SL(2,R)$ and $H=SO(2,R)$ as a compact subgroup of $G$, then $\{\xi \in \hat{G}; \xi|_H=1\}$is triviall. Can we conclude that each multiplicative linear functional on $\{f\in L^1(G);...

**1**

vote

**0**answers

188 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 $\widehat{(G:H)}=\{\xi\in \hat{G}:\xi|_H=1\}$($\...

**7**

votes

**1**answer

256 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 ...