All Questions
Tagged with ap.analysis-of-pdes fourier-analysis
122 questions
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
4
votes
1
answer
219
views
Poincaré inequality on compact manifolds without boundary
The question arises from a paper The heat flow for the full bosonic string, Ann. Global Anal. Geom. 50 (2016).
In line 4, Page 362, the author claimed the following inequality which looks similar to ...
- 143
2
votes
1
answer
176
views
Does $i\partial_tu = \Delta^2 u$ exhibit more or less dispersion than $i\partial_t u= \Delta u$?
Consider the initial-value problems in $d=1$
$$\begin{cases} i\partial_tu = \Delta^2 u \\
u(x,0)=u_0
\end{cases}$$
and
$$\begin{cases} i\partial_t u= \Delta u \\
u(x,0)=u_0,
\end{cases}$$
Solutions to ...
- 840
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
1
vote
1
answer
112
views
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
- 852
0
votes
1
answer
121
views
A simple bilinear estimate
Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that
$\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$.
Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$.
What is the optimal value of $t=t(\...
- 852
5
votes
1
answer
246
views
An asymmetric quadrilinear estimate
Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$
where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
- 852
0
votes
0
answers
112
views
Fourier integral operators and parametrix
Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary.
Question: Is there an expression for the ...
- 593
0
votes
0
answers
32
views
On the I-method's energy increment calculation in a paper of Dodson
I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem ...
- 840
9
votes
1
answer
639
views
Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
- 807
0
votes
1
answer
507
views
Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 101
8
votes
1
answer
496
views
A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
- 4,135
5
votes
3
answers
314
views
The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
Let $f_a(x)=e^{-|x|^a}$, $x\in \mathbb{R}^n$. Then $f \in L^p(\mathbb{R}^n)$ for every $1\leq p\leq \infty$. It is also smooth away from the origin and decays faster than any polynomial as $|x|\...
- 852
0
votes
1
answer
624
views
Does this dyadic sum converge?
Let $a\in (0,1)$ and define
$$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$
Note that rescaling $2^{j} s\mapsto s$ shows that
$$J(j)\leq 2^{-j(1+a)}\int_{0}^...
- 852
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
8
votes
2
answers
670
views
Asymptotic behavior of a certain oscillatory integral
Let $x>0$ and consider the integral
$$I(x):=\int_0^\infty \frac{e^{i r}}{r^{\frac{1}{2}}} \int_0^\infty \frac{e^{-s}}{s^{\frac{1}{2}}} \frac{r}{sx+\sqrt{sxr}+r} \, ds \, dr.$$
I am trying to ...
- 852
3
votes
1
answer
182
views
How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 325
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
0
votes
1
answer
123
views
$\|\hat{f}\|_{L^q}< \infty \implies \left\| \|\chi_{n+(-1/2, 1/2]} \widehat{f}\|_{L^p_{\xi}} \right\|_{\ell^q_n}<\infty $
Suppose that support of $f:\mathbb R \to \mathbb R$ is compact set $K\subset \mathbb R.$ Assume that $ \int_{\mathbb R} |\widehat{f}|^q d\xi <\infty.$ ($\widehat{\cdot}$ denote the Fourier ...
- 325
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
2
votes
1
answer
236
views
Approximation of Hölder functions by Fourier series
Let $Q$ be a bounded domain in $\mathbf R^N$ with smooth boundary. Let $f\in C^a(\overline{Q})$, $0<a<1$.
Denote $\psi_k(x)$ normalized eigenfunctions and $\lambda_k$ eigenvalues ($k=0,1,2\...
- 21
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
1
answer
148
views
Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?
I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator
$...
- 852
7
votes
2
answers
407
views
$L^p-L^q$ boundedness of this simple singular oscillatory integral operator
Let $0<\alpha<1$ and define
$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$
The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of
$Hf(x):=\int \frac{...
- 852
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)
$$\label{0}\tag{0}
\|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
- 230
5
votes
1
answer
212
views
Two dimensional oscillatory integral
I am having a little confusion in verifying the two dimensional oscillatory integral in Lemma 2.1 in This paper, namely
$$I_t (x,y) = \int_{\mathbb{R}^2} |\xi|^{\epsilon + i \beta} e^{i t(\xi^3 + \xi \...
- 159
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
1
answer
230
views
Why we have $f=0$
Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$.
Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
- 501
0
votes
0
answers
75
views
$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 159
1
vote
0
answers
180
views
A potential wrong proof of a Lemma
Consider the following lemma: Let $g \in H^s_{x,y}(S)$ where $S = \mathbb{R}^2$ or $S = \mathbb{T}^2$, and $\eta \in C^\infty(\mathbb{R})$, $\operatorname{Supp}(\eta) \subset [-2,2]$, and $\eta \equiv ...
- 159
3
votes
1
answer
251
views
Asymptotic behavior of a double oscillatory integral
Let $0<\theta_1,\theta_2<\pi/2$. Suppose $\psi$ is a smooth real-valued function with compact support.
Consider the oscillatory integral
$$I(t):=\int_{0}^{1}\frac{1}{(y-e^{\dot{\imath}\theta_1})
...
- 852
0
votes
1
answer
130
views
Riesz transform after linear transformation
I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation
$$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$
I ended up with ...
- 159
-3
votes
1
answer
101
views
Asking for reference about a relation related to Fourier transform [closed]
Sorry for the not-perfect question. I am asking for a reference for the following relation:
$$\int f . g. h ...= \int_{\xi_1 +\xi_2 +...=0} \hat{f}(\xi_1) \hat{g}(\xi_2)... d\xi_1 d\xi_2...$$
Could ...
- 159
2
votes
2
answers
328
views
$H^s$ norm of non-integer power of functions
Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $.
My ...
- 105
1
vote
1
answer
203
views
Explanation of a step in a work by C. E. Kenig and A.D. Ionescu
I am studying the work
Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
- 159
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
1
vote
1
answer
506
views
Fourier transform of the fractional Poisson kernel
Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows:
$$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+...
- 537
6
votes
0
answers
211
views
Regularity of $|u|^{\alpha}$ when $u$ is Schwartz
Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). ...
2
votes
1
answer
64
views
Convergence of an infinite sum, whose terms are supported in balls, in Besov space
Suppose we have a ball $B \subset \mathbb R^n$ and $\alpha >0$, and $\{ u_j\}_{j\geqslant-1} $ is a sequence of smooth functions such that the Fourier transforms $\mathcal{F}(u_j) $ are supported ...
- 253
0
votes
0
answers
80
views
Convergence of a infinite sum in Besov space
Suppose we have an annulus $A \subset \mathbb R^n$, which is the set $\{x|0<r \leqslant\|x\| \leqslant R\}$, $\alpha \in \mathbb R$ and $\{ u_j\}_{j\geqslant-1} $ be a seqence of smooth functions ...
- 253
1
vote
1
answer
1k
views
How to prove that the L-infinity norm is smaller than the Besov norm?
Suppose we have a distribution $u\in B_{\infty,\infty}^\alpha$, the Besov space with regularity coefficient $\alpha>0$. How to prove the folowing inequality?
$$
\|u\|_{L^\infty}\leqslant c\|u\|_{B_{...
- 253
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
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
5
votes
1
answer
445
views
Schwartz regularity for the density of a stochastic process
Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables
$$\begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*}$$
It ...
- 3,669
1
vote
1
answer
133
views
Integrability of fractional heat kernel
In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (...
- 109
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
$$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
- 873
1
vote
0
answers
76
views
Second question on a real sequence
I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
- 903
5
votes
1
answer
206
views
Mean value principle reversed
Suppose that $\Omega \subset \mathbb R^3$ is a domain with smooth boundary $\partial \Omega$ and suppose that $0\in \Omega$. Given any $f \in C^{\infty}(\partial \Omega)$ let $u^f$ denote the unique ...
- 4,135
2
votes
1
answer
195
views
Sufficient conditions for the convexity of the discrete Fourier transforms
Let $f : [0,2\pi] \to \mathbb{R}$ be some function. Then the discrete Fourier transform of $f$ when sampled at $2\pi i/N$ is then given by
$$
X_n := \sum_{i=0}^{N-1}\cos\left(\frac{2\pi n i}{N}\right)...
- 595
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