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9 votes
1 answer
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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 ...
Phil Isett's user avatar
  • 2,243
9 votes
1 answer
1k 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 \...
John H's user avatar
  • 217
8 votes
0 answers
277 views

a question on the paper of Łaba and Wolff

I'm reading the paper A local smoothing estimate in higher dimensions by Izabella Łaba and Thomas Wolff. The paper can be found at J. Anal. Math. 88 (2002), 149–171, doi: 10.1007/BF02786576, arxiv: ...
Tony B's user avatar
  • 463
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 ...
Steven Heilman's user avatar
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). ...
Benjamin Pineau's user avatar
4 votes
1 answer
461 views

Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation \...
asv's user avatar
  • 21.8k
4 votes
1 answer
162 views

Uniqueness of critical points for Lipschitz perturbations of uniformly convex Hamiltonians

Consider a macroscopic free energy functional of the form $$\mathcal{F}_\beta(\mu):= \frac{1}{\beta}\int_{\mathbb{R}^d}\log(\mu)\mu dx + \int_{\mathbb{R}^d}V(x)\mu(x)dx + \iint_{(\mathbb{R}^d)^2}g(x-y)...
Matt Rosenzweig's user avatar
4 votes
2 answers
929 views

Rate of convergence of mollifiers // Sobolev norms

Following up to the question raised here, I am searching for a reference (or a simple argument) to establish (in the whole space) the following (suggested) equivalence : Given $N_1$ and $N_2$ two (...
Ayman Moussa's user avatar
  • 3,425
4 votes
0 answers
170 views

Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$

Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
user avatar
3 votes
1 answer
465 views

An inequality from Bessel potential space to Besov space

I'm not sure this question is suitable for MathOverflow. Currently, I'm reading a paper "Inhomogeneous Dirichlet Problem in Lipschitz domain" by Jerison and Kenig. I have a question on some ...
Will Kwon's user avatar
  • 323
2 votes
0 answers
206 views

Failure of Calderón–Zygmund inequality at the endpoints

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'd like to prove that the famous Calderón–Zygmund elliptic estimate $$ \norm{ \partial_{ij}u }_{L^p} \leq C \norm{\Delta u }_{L^...
Marc's user avatar
  • 457
2 votes
0 answers
320 views

About definition of NTA domain

I'm not an expert in analysis on very rough domains, such as NTA(Nontangentially Accessible Domain). Here is my question. Usually, NTA domain $\Omega$ is a domain that has inner and outer corkscrew ...
Will Kwon's user avatar
  • 323
2 votes
0 answers
126 views

On the infimium of a functional

Let $(M^n,g)$ be a closed Riemannian manifold. Define $$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$ where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
user162551's user avatar
2 votes
0 answers
88 views

Solvability of Neumann boundary problems with singular boundary data $g \in (H^{1})^{*}$

I have a question on the solvability of Neumann boundary problems with singular data. To state my question, let $\Omega$ be a bounded Lipschitz domain (open and connected) in $\mathbb{R}^n$. In the ...
Will Kwon's user avatar
  • 323
2 votes
0 answers
219 views

A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
Sean Gomes's user avatar
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\...
Analysis Now's user avatar
  • 1,471
1 vote
0 answers
104 views

Notation for right hand side of local smoothing conjecture

In Tao's "Recent progress on the restriction conjecture" On page 53, Tao introduced the local smoothing conjecture: let $u(t,x)$ be the solution to the wave equation $u_{tt}=\Delta u$, $u(0,...
Simplyorange's user avatar
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 (...
Shaoming Guo's user avatar
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}^...
Medo's user avatar
  • 852
0 votes
1 answer
140 views

Singular integral bounded by Dirichlet form?

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am ...
António Borges Santos's user avatar
0 votes
1 answer
181 views

What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data $...
Analysis Now's user avatar
  • 1,471