All Questions
Tagged with ap.analysis-of-pdes fourier-analysis
48 questions with no upvoted or accepted answers
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
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
$$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
- 873
7
votes
0
answers
927
views
What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?
I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi \...
- 1,051
6
votes
0
answers
211
views
Regularity of $|u|^{\alpha}$ when $u$ is Schwartz
Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). ...
5
votes
0
answers
168
views
Sobolev extension from a discrete set of points
Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define
$$...
- 501
5
votes
0
answers
211
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}...
- 1,616
5
votes
0
answers
913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
4
votes
0
answers
131
views
Systems of parabolic equations -- Petrovskii's condition
Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$.
Given a matrix field $A:Q_T\rightarrow\text{M}...
- 3,425
4
votes
0
answers
179
views
Approximation by gaussian mollification in Sobolev spaces
I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify)
$$\label{0}\tag{0}
\|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
- 230
4
votes
0
answers
135
views
Zygmund class, Schwartz class and Littlewood-Paley projection operators
I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions:
Consider the Zygmund class of functions defined as ...
- 349
4
votes
0
answers
2k
views
Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity
Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
4
votes
0
answers
131
views
Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$
Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...
- 355
4
votes
0
answers
551
views
$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?
Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra
$$A(\mathbb R):= \{f\in L^{1}(...
- 1,051
3
votes
0
answers
99
views
Rate of convergence of mollified distributions in Besov spaces with negative regularity
Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
- 293
3
votes
0
answers
156
views
The proof of Tomas's restriction theorem $\|\hat{f}(\theta)\|_2\lesssim \|f\|_p$
I am trying to understand the proof of Tomas's theorem:
The proof reads
My question:
How do we get the estimates
$$\|T_k\ast f \|_{\infty}\lesssim 2^{-(n-1)k/2}\|f\|_{1},\qquad\qquad (1)$$
$$\|T_k\...
- 143
2
votes
0
answers
65
views
Generalized Fourier transforms associated to Schroedinger operators
Let $n\geq 1$. Let $q\in C^{\infty}_0(\mathbb R^n)$ be compactly supported and consider the operator $P= -\Delta+q(x)$ on $\mathbb R^n$. We will assume that $q$ is sufficiently small so that the ...
- 4,135
2
votes
0
answers
144
views
Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$
Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$
I would like to prove (or disprove) ...
- 1,101
2
votes
0
answers
149
views
An oscillatory integral
Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates?
\begin{...
- 351
2
votes
0
answers
67
views
Asking a reference for a fact about nonlocal operators
Let assume that $u$ is smooth enough and $ -\Delta (u \phi) \in L^1(\Omega)$ for any $\phi \in C_c^{\infty}(\Omega)$. Then it easily follows that $ -\Delta u \in L^1_{\mathrm{loc}}(\Omega)$ by ...
- 371
2
votes
0
answers
162
views
Hilbert transform on weighted Sobolev spaces
Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
- 4,135
2
votes
0
answers
164
views
What are (the different aspects of) harmonic analysis good for?
Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have ...
- 2,071
2
votes
0
answers
113
views
discrete Fourier transform for composition of differential operators on a grid
This question pertains to stability analysis of finite difference methods using the discrete Fourier transform.
Suppose I have a convection diffusion equation of the form:
(1) $\hspace{.5in}u_t + \...
- 121
2
votes
0
answers
240
views
Proof of Taylor's Schwartz kernel estimate of pseudodifferential operators
I am interested in the proof of the following result which gives an estimate on the Schwartz Kernel of a $\Psi$DO. There is one aspect the proof that is not clear to me which I would like to ask the ...
- 53
2
votes
0
answers
285
views
Equation of motion of a charged string in a twisted torus.
The flat torus background:
Say we want to study the sigma model of strings (closed strings $S^1= \mathbb{R}/\mathbb{Z}$) on a flat torus (for example $T^3=\mathbb{R}^3/\mathbb{Z}^3$ with a flat ...
- 3,242
2
votes
0
answers
164
views
Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?
This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = \frac{1}{2\...
- 1,471
1
vote
0
answers
65
views
A parametrix construction for heat boundary value problem using Fourier transformation
Let $\Omega$ be a smooth bounded open subset in $\mathcal{R}^{d}$, with $d \geqslant 3
$ and $T>0$. Consider the linear parabolic initial Dirichlet boundary value problem with $f\in H^{-1}(\Omega)$...
- 61
1
vote
0
answers
108
views
Recovering phase function using Fourier decomposition
I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function
$$f = e^{i \phi(x)}. $$
I am interested in the following problem. If I know the function/distribution $...
- 151
1
vote
0
answers
166
views
Wiener Integral and its distribution
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space.
Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field.
Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...
- 193
1
vote
0
answers
180
views
A potential wrong proof of a Lemma
Consider the following lemma: Let $g \in H^s_{x,y}(S)$ where $S = \mathbb{R}^2$ or $S = \mathbb{T}^2$, and $\eta \in C^\infty(\mathbb{R})$, $\operatorname{Supp}(\eta) \subset [-2,2]$, and $\eta \equiv ...
- 159
1
vote
0
answers
76
views
Second question on a real sequence
I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
- 903
1
vote
0
answers
400
views
Calculating frequency of sound of ringing metal coin
I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
- 119
1
vote
0
answers
353
views
Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
1
vote
0
answers
105
views
Strichartz estimate for the Schrödinger equation
Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP:
$$i\partial_t ...
- 31
1
vote
0
answers
78
views
Reference (foundamental sol. and grad estimate, etc.): a particular elliptic PDE
In $\mathbb{R}^d$, consider the following equation
$$\Delta u -x\cdot \nabla u = f $$
where $f$ can be $C^\infty$ and decay like $e^{-\frac{c|x|^2}{2}}$.
I would like to know fundamental sol. to this ...
1
vote
0
answers
61
views
Convergence of PDE/PsiDE - expansion of pseudo-differential operators
I have am working with a nonlinear pseudo-differential evolution equation of the form
$$u_t + \mathcal{N}(u) + \mathcal{D}_{\epsilon} u = 0$$
where $\mathcal{N}$ is a nonlinear operator and $\mathcal{...
- 155
1
vote
0
answers
119
views
Does $u\in H^{3/2}(\Omega)$ imply continuity of $\nabla u\cdot\overrightarrow{n}$ across an interior interface?
When investigaing the regularity of certain functions, I encountered this problem:
if $u\in H^{3/2}([0,1]\times [0,1])$,
what can we say about the continuity of $\nabla u\cdot\overrightarrow{n}$
...
- 398
1
vote
0
answers
49
views
non-conical support of fundamental solution possible?
In his 1970 paper, on page 124, Hormander discusses fundamental solutions of linear PDE with constant coefficients. I notice he only discusses cases where the support $F$ of the fundamental solution ...
- 1,461
1
vote
0
answers
341
views
Reference for PDE problem book
What I need is a source of solved exercises, problems in Partial Differential Equations; to be hard enough (olympiad style) and in areas like Calderon-Zygmund theory and applications, Paley-Littlewood ...
- 85
1
vote
0
answers
92
views
Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
- 515
1
vote
0
answers
154
views
variation norm of a Fourier transform
Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in (...
- 331
1
vote
0
answers
99
views
decomposition of tempered distributions by entire analytic functions
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...
- 1,051
1
vote
0
answers
164
views
How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
- 1,051
1
vote
0
answers
440
views
A problem about Joint sine and cosine fourier transform
There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...
- 11
0
votes
0
answers
112
views
Fourier integral operators and parametrix
Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary.
Question: Is there an expression for the ...
- 593
0
votes
0
answers
32
views
On the I-method's energy increment calculation in a paper of Dodson
I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem ...
- 840
0
votes
0
answers
75
views
$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$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
0
votes
0
answers
80
views
Convergence of a infinite sum in Besov space
Suppose we have an annulus $A \subset \mathbb R^n$, which is the set $\{x|0<r \leqslant\|x\| \leqslant R\}$, $\alpha \in \mathbb R$ and $\{ u_j\}_{j\geqslant-1} $ be a seqence of smooth functions ...
- 253
0
votes
0
answers
467
views
Is the product of two Sobolev functions in L^p?
Assume that $f\in W^{\alpha-1,p}(R^n)$ with $0<\alpha<1$ and $p>2n/\alpha$.
Given another function $ g\in W^{\beta,p}(R^n)$ with $\beta>0$.
Under what conditions on $\beta$ can we get ...
- 411