All Questions
Tagged with fa.functional-analysis ca.classical-analysis-and-odes
524 questions
2
votes
1
answer
607
views
Stein's extension operator and wave front sets
Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein ...
- 7,817
6
votes
4
answers
8k
views
Characterization of the non-negative definite functions $f(x,y)$
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called non-negative definite, if for any real numbers $x_1,\...
- 1,649
2
votes
1
answer
535
views
about decomposition of a non-negative definite operators
Hello,
Many years before, I had the following problem.
We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called separable if it ...
- 1,649
0
votes
0
answers
298
views
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
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |...
- 1,649
1
vote
2
answers
687
views
High dimensional beta integral (a typo in Stein's book "singular integrals")
Hello,
When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake:
$$
\int_{R^n} |x-y|^{-n+\alpha} |y|^{-n+\beta}=\frac{\gamma(\alpha)\gamma(\beta)}{\gamma(\alpha+\beta)},...
- 1,649
2
votes
1
answer
547
views
Equivalent references for Schwartz's book of the distribution theory
Hello,
It seems that there is no English translation of the Schwartz's book 1966. I may need to use the spaces like
$$
\dot{\mathcal{B}}(R),\quad \dot{\mathcal{B}}'(R),\quad \mathcal{B}(R),\quad \...
- 1,649
1
vote
0
answers
693
views
A question about an equivalent definition of the Schwartz distribution
Hello,
Does anyone know a reference or proof of the "if" part of the following statement?
$$
\mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\...
- 1,649
15
votes
2
answers
3k
views
What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
- 2,135
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
2
votes
1
answer
570
views
Is a polynomial positive on the sphere a sum of squares of spherical harmonic polynomials?
Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then does there exist some $m>0$ and ...
user2529
59
votes
7
answers
29k
views
Learning roadmap for harmonic analysis
In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
5
votes
3
answers
3k
views
What is the $L^p$-norm of the (uncentered) Hardy-Littlewood maximal function?
The (uncentered) Hardy-Littlewood maximal function $M(f)$ of (a locally integrable) function $f$ on $\mathbb{R}^{n}$ is defined by the rule $M(f)(x)=\sup_{\delta>0,\left|y-x\right|<\delta} \text{...
- 1,993
19
votes
4
answers
5k
views
Explicit extension of Lipschitz function (Kirszbraun theorem)
Kirszbraun theorem states that if $U$ is a subset of some Hilbert space $H_1$, and $H_2$ is another Hilbert space, and $f : U \to H_2$ is a Lipschitz-continuous map, then $f$ can be extended to a ...
- 1,839
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
4
votes
1
answer
1k
views
Sobolev-Slobodeckij spaces for p=infinity
For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then,
$
\...
- 3,217
12
votes
3
answers
3k
views
Infinitesimal generators of stochastic processes
What's the $L^1$ analogue of Stone's theorem saying that any strongly continuous 1-parameter unitary groups has a unique self-adjoint generator?
More precisely: let $X$ be a measure space ($\sigma$-...
- 22.3k
7
votes
2
answers
988
views
Missing mass conjecture
Let $n,t$ be positive integers and $p_1,p_2,\ldots,p_n$ positive numbers summing to 1. Conjecture:
$$
\sum_{i=1}^n p_i (1-p_i)^t \le \frac{n(1-1/n)^n}{t}
$$
always holds.
The motivation comes from my ...
- 6,541
24
votes
3
answers
3k
views
Can Hölder's Inequality be strengthened for smooth functions?
Is there an $\epsilon>0$ so that for every nonnegative integrable function $f$ on the reals,
$$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$
Of course, we ...
- 9,756
5
votes
1
answer
794
views
Can the Sobolev norm of order 1/2 detect "jumps"?
We are given a function $f: \mathbb R^d \to \mathbb R$. For simplicity we can assume that $f$ is smooth and compactly supported. Is the Sobolev norm of order $\frac{1}{2}$ strong enough to prove an ...
- 570
9
votes
2
answers
1k
views
Hilbert transforms of measures
Given a finite measure $\mu$ on the real line $\mathbb R$, one definition of its Hilbert transform is $(H\mu)(y) =\frac{1}{\pi}(PV)\int \frac{d\mu(x)}{x-y}$ which is known to exist almost everywhere ...
- 91
9
votes
1
answer
2k
views
Rate of convergence of smooth mollifiers
How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis ...
- 2,243
1
vote
0
answers
477
views
A norm ratio inequality
Let $y,z\in(0,1)^n$ satisfy $||y||_1 = ||z||_1=1$.
Then
$$
\frac{||z||_3}{||z||_2} \le
K_n
||z/y||_\infty
\frac{||y||_3}{||y||_2}
$$
where $z/y\in\mathbb R^n$ is the coordinate-wise quotient of $z$ ...
- 6,541
8
votes
1
answer
2k
views
Recent progress on Bochner-Riesz conjecture
Consider the family of operators $T_\delta$, $\delta \geq 0$, defined on $\mathbb{R}^n$ by
$
\widehat{T_\delta f}(\xi) = (1-|\xi|^2)_+^\delta \widehat{f}(\xi).
$
($(1-|\xi|^2)_+^\delta$ are known as ...
- 505
7
votes
1
answer
822
views
On a decomposition of L^1(G)
[EDITED by Y. Choi - I have attempted to paraphrase the original question into something a bit terser and more precise; if this is not what the original poster intended, they should make corrections ...
- 643
1
vote
1
answer
367
views
An integral which is related to Biharmonic extension
In my research, I need to evaluate an integral:
$$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}\log(|\xi^{2}|+\frac{1}{4})d\xi$$
where $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it ...
- 11
1
vote
2
answers
515
views
continuity of extension of maps along curves
Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-embedding connecting $x$ ...
- 2,935
3
votes
6
answers
1k
views
Reference for complex analysis jargon
I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts:
logarithmic capacity
transfinite diameter
Green's function of a compact ...
- 741
2
votes
2
answers
991
views
An extension of the Hardy-Littlewood-Polya inequality?
Let $x,y$ be vectors in $\mathbb{R}^n$ and let's use the notation $\hat x$ for the vector $x$ with its components sorted in increasing order.
The Hardy-Littlewood-Polya inequality states that
$$ x\...
- 6,541
18
votes
2
answers
1k
views
Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform
The Hilbert transform on the real Hilbert space $L^2(\mathbb R)$ is the singular integral operator
$$
\mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy.
$$
It satisfies $\...
- 43.2k
19
votes
1
answer
5k
views
Intuition for the Hardy space $H^1$ on $R^n$
the standard intuition for Lebesgue spaces $L^p(\mathbb R^n)$ for $p \in [1,\infty]$ are measurable functions with certain decay properties at infinity or at the singularities.
In particular, a ...
- 5,327
12
votes
1
answer
859
views
Who first found this characterization of Lebesgue integration?
Write $L^1$ for the Banach space $L^1([0, 1])$. Given $f \in L^1$, define $f_1, f_2 \in L^1$ by
$$
f_1(x) = f(x/2),
\qquad
f_2(x) = f((x + 1)/2).
$$
Let $I = \int_0^1$. Then $I$ is the unique ...
- 27.7k
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
8
votes
1
answer
678
views
Spectral theory of pseudo-differential operators
Consider a finite rank complex bundle $E$ over $S^1$ with connection $\nabla$. Let $Q_0, Q_1 \in C^\infty(S^1, E)$ be pseudo-differential operators. $Q_0$ is defined by the symbol $\sigma_0(x, \xi) =...
0
votes
1
answer
222
views
Bounding near the boundary for a Sobolev function.
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$...
- 3,217
27
votes
3
answers
5k
views
Weak and Strong Integration of vector-valued functions
This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
- 741
2
votes
1
answer
412
views
General Sobolev Inequalities
In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated:
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ ...
- 3,217
1
vote
1
answer
491
views
Bounding a smooth function near the boundary
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>0$ and that $f$ ...
- 3,217
7
votes
3
answers
1k
views
A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...
- 11.2k
1
vote
1
answer
1k
views
Laplace equation over concentric spheres
Is there a closed formula for the solution of Dirichlet problem ($\Delta u=0$) for annulus $r <|x| < R$, $x \in R^n$ (n>2), with two given boundary value functions, $f$ over $|x|=r$ and $g$ over ...
- 39
2
votes
1
answer
1k
views
Range of the Radon Transform
Let us consider the Radon transform in two dimensions:
$$\tag{1}Rf(r,\theta):=\int\limits_{-\infty}^{\infty} f(r\cos\theta-t\sin\theta,r\sin\theta+t\cos\theta) dt,$$
where $r\in\mathbb{R}$ and $0\...
- 931
8
votes
3
answers
1k
views
Fourier dimension of the sum of sets
This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets ...
- 505
4
votes
2
answers
2k
views
Inclusions of $C^{k,\alpha}$ spaces
When is $C^{k,\alpha}(\bar{\Omega})$ a
subset of
$C^{k',\alpha'}(\bar{\Omega})$?
Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
- 1,771
0
votes
2
answers
2k
views
fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
- 1
2
votes
4
answers
358
views
When do functions near F have zeros near a zero of F?
Consider a sequence of functions $F_n : \mathbb{R}^d \to \mathbb{R}^d$, a function $F: \mathbb{R}^d \to \mathbb{R}^d$, and an $\mathbf{x} \in \mathbb{R}^d$ so that $F(\mathbf{x}) = \mathbf{0}$. In ...
- 1,068
8
votes
2
answers
8k
views
Version of the Poincaré Inequality
Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality
$$\|u\|_p \le C \|\nabla u\|_p$$
($\|\cdot\|_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used here)...
- 2,270
2
votes
1
answer
1k
views
Green's function for wave equations in R² or R³
Hello,
For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we ...
- 1,649
3
votes
1
answer
2k
views
fourier transform of radon measure
hi,
assume that I have a function $q$ which is a Fourier Multiplier of order zero, i.e.
$$
\left|\left( \frac{d}{dx}\right)^nq(x)\right|\lesssim \left(\frac{1}{1+|x|}\right)^n\quad \mbox{for all ...
- 979
1
vote
1
answer
307
views
variational formulation: boundedness of the bilinear form
The simplest case of the problem I'm thinking about involves an elliptic differential operator, $Lu = -u'' + qu$, on the interval $(0,1)$, with homogeneous Dirichlet boundary conditions. I want to ...
- 343
3
votes
1
answer
362
views
Cartesian product of test function spaces
Mini introduction
Suppose $U \subset \mathbb R^n, V \subset \mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions_space#Test_function_space">test functions $f_i \in \...
1
vote
0
answers
283
views
Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
- 19