2
votes
0answers
88 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definition We define the Zygmund spaces $C^r_{*}$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with ...
0
votes
1answer
138 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$? [closed]

(This may be very simple question for MO; I had post it to math stack exchange few days back but I could not get any answer(or comment) to it) It is well-known that, for $f,g \in L^{1}(\mathbb R).$ ...
2
votes
1answer
129 views

To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where $\mu(\alpha)$ is a non decreasing function ...
0
votes
1answer
44 views

properties of frequency-uniform decomposition operator $\square_{k}^{\sigma}$

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
4
votes
0answers
145 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 ...
0
votes
1answer
168 views

Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
1
vote
0answers
91 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; ...
0
votes
0answers
95 views

Localized convolution operators

Take our favorite differential operator $\Delta$ on $\mathbb{R}^3$. Module a constant, we can represent its inverse (in some sense) by a convolution operator, namely $$\Delta^{-1}f=f*\frac{1}{|x|}.$$ ...
-3
votes
1answer
155 views

$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
3
votes
1answer
215 views

Uniqueness theorems related to Hardy Uncertainty Principle

Uncertainty Principles state that a function and its Fourier transform cannot be simultaneously sharply localised. A well known result due to G.H.Hardy says that if $f(x)=O(e^{-\alpha^2|x|^2})$, ...
4
votes
1answer
256 views

Is there a good way to estimate the Fourier transform of $\frac{1}{\lambda-iP(\xi)}$

Assume that P is a real valued strong elliptic polynomial, then what do we know about the following $$ K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0 $$ The ...
0
votes
1answer
139 views

On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is : ...
0
votes
1answer
283 views

Generalized Friedrichs Lemma

Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J_\epsilon$ on a compact ...
1
vote
1answer
186 views

A variational problem involving a negative fractional Soboblev norm.

I've run into the problem of trying to evaluate the following: $\max_{ \xi} \iint_{\partial B \times \partial B} \xi(x) \Phi(|x-y|) \xi(y) dS(x)dS(y)$ subject to $\int_{\partial B} \xi(y)dS(y) = 0$ ...
2
votes
0answers
220 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 ...
2
votes
0answers
146 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) = ...
3
votes
1answer
607 views

Endpoint Strichartz Estimates for the Schrödinger Equation

The non-endpoint Strichartz estimates for the (linear) Schrödinger equation: $$ \|e^{i t \Delta/2} u_0 \|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u_0\|_{L^2_x(\mathbb{R}^d)} $$ $$ 2 ...
7
votes
0answers
246 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 ...
1
vote
1answer
505 views

FFT setting the boundary conditions

I am using the FFT method to solve the two dimensional PDE on a $2\pi\times 2\pi$ square. I have boundary conditions that $u(x,0,t)=0$ and $u(x,2\pi,t)=0$. How do I go about setting these boundaries? ...
0
votes
2answers
1k 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 ...
-2
votes
1answer
557 views

General solution to ODE [closed]

Considering the following ODE : find $f(x)$ such that $$\frac{\sigma^{2}}{2}\frac{d^2}{dx^2}f(x)+a(b-x)\frac{d}{dx}f(x)-(\rho+\lambda)f(x)=-\lambda g(x) $$ Where, ...
2
votes
3answers
787 views

Posing cauchy data for the heat equation: $t=0$ a characteristic surface?

When solving the heat equation on say $\mathbb{R}$ (or $[0,2\pi]$, whichever is easier to talk about) we are posing Cauchy data on the surface $t=0$. My understanding is that $t=$constant are ...
1
vote
1answer
734 views

Elliptic regularity on bounded domains

I'm concerned with a generic uniformly elliptic operator $L$ on $\mathbb{R}^n$. If $L$ is uniformly elliptic and I am studying the equation $Lu=f$ then the way I can deduce regularity on ...
10
votes
2answers
1k views

Hypoellipticity of square root of laplacian

It is a well known result (sometimes called the Weyl lemma) that the laplacian in $\mathbb{R}^n$ is hypoelliptic, i.e. if $f$ is a distribution s.t. $\triangle(f)$ is smooth in an open set, than $f$ ...
3
votes
2answers
507 views

Ansätze for solving PDEs with wavelets

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ansätze do you know ...