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Sufficient condition for interpolation

If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying: $Z_0=X$, $...
mejopa's user avatar
  • 101
1 vote
0 answers
43 views

If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?

Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be: $m(x) \cdot \text{div} ( s(x) \nabla f(x))$. What ...
Timothy Chu's user avatar
4 votes
1 answer
279 views

Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator ...
António Borges Santos's user avatar
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
1 vote
1 answer
113 views

An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$

Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e., $$ g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \quad t>0, x \in \mathbb R^d. $$ Let $f : \mathbb R^d \to \...
Akira's user avatar
  • 835
1 vote
0 answers
82 views

For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$

For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
Akira's user avatar
  • 835
1 vote
1 answer
138 views

Can functions with "big" discontinuities be in $H^1$?

How can I prove that the function: $$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
Bogdan's user avatar
  • 1,759
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 ...
TaD's user avatar
  • 101
2 votes
0 answers
180 views

Approximating $L^p$ functions by eigenfunctions of Laplacian

I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932. In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
ze min jiang's user avatar
1 vote
2 answers
181 views

Solution of $\Delta f -\frac{1}{2}hf = 0$ behaves asymptotically as $f(x) = 1 - C/|x|$

Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE: $$\Delta f -\frac{1}{2}h f = 0$$ where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ ...
JustWannaKnow's user avatar
5 votes
1 answer
311 views

Maximal operator estimates for the Schrödinger equation

Let $a>0$ and consider the operator $$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$ When $a=2$, the function $Tf$ solves the Cauchy problem ...
Medo's user avatar
  • 852
6 votes
1 answer
285 views

Distinguishing the Besov and Triebel-Lizorkin spaces

Theorem 2.3.9. in Triebel's Theory of Function Spaces states that the Besov space $B^{s_1, p_1}_{q_1} (\mathbb R^d)$ coincides with the Triebel-Lizorkin space $F^{s_2, p_2}_{q_2} (\mathbb R^d)$ if and ...
Jason Zhao's user avatar
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
7 votes
2 answers
508 views

Why is $\frac{1}{|x|^{n-2}}u(\frac{x}{|x|^2})$ harmonic if $u$ is harmonic?

I found myself trying to prove the following, but I had to compute everything explicitly. It is well known that if $u:\mathbb{R}^n\to\mathbb{R}$ is an harmonic function on $\mathbb{R}^n$, then the so-...
Gauge_name's user avatar
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 \...
Mr. Proof's user avatar
  • 159
5 votes
1 answer
281 views

de Rham theorem for tempered distributions

I am wondering if the following statement holds. If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
Will Kwon's user avatar
  • 323
1 vote
0 answers
57 views

How to show the solution map of NLS is not smooth?

Let $u(\delta, t)$ satisfy $$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$ Note that the mapping: $$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$ ...
 Analyst 's user avatar
9 votes
2 answers
1k views

Density of restrictions of harmonic functions inside a ball

Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let $$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
Ali's user avatar
  • 4,145
1 vote
0 answers
260 views

Closure of smooth functions in Besov spaces

For real numbers $\alpha > \beta$, we know there is a continuous embedding of Besov spaces $B^\alpha_{\infty,\infty}\subset B^\beta_{\infty,\infty}$. We take the closure of the intersection $C^{\...
Inuyasha's user avatar
  • 253
4 votes
0 answers
127 views

Algebra properties regarding Gevrey spaces: closed under multiplication

In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
Feng's user avatar
  • 517
3 votes
1 answer
164 views

Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?

Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
BigbearZzz's user avatar
  • 1,245
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<...
Matt Rosenzweig's user avatar
4 votes
1 answer
221 views

Is a specific product function orthogonal to all harmonic functions

Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
Ali's user avatar
  • 4,145
7 votes
0 answers
3k views

Definition of homogeneous Sobolev spaces

As we know the inhomogeneous Sobolev space (we only consider $s>0$) $${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...
Slm2004's user avatar
  • 633
2 votes
2 answers
836 views

Laplace equation on the disk with Robin boundary condition

Consider the following two dimensional Laplace equation on the unit disk $D$ with homogeneous Robin boundary condition: $$\Delta u = 0, ~~\frac{\partial u}{\partial n} = b(x) u(x)~~ \forall x \in \...
Jacob Lu's user avatar
  • 903
2 votes
1 answer
178 views

References for Neumann eigenfunctions

I am looking for references on eigenfunctions with Neumann boundary condition. In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...
sharpe's user avatar
  • 721
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
4 votes
2 answers
928 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
1 vote
0 answers
103 views

Choosing the weight in a particular definition of Besov spaces

Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" ...
leo monsaingeon's user avatar
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
7 votes
2 answers
508 views

Making the Fourier transform quantitative

I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website. I understand ...
André's user avatar
  • 225
2 votes
3 answers
303 views

Uniqueness of solution depending on constant?

I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation. I encountered the following integral equation for functions $f:[...
Andrea Tauber's user avatar
5 votes
1 answer
171 views

Invariant subspace in infinite dimensions

Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$ The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
Zorgo's user avatar
  • 177
1 vote
1 answer
737 views

$L^2$ function in Schwartz space?

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$ Such a function has the property that when multiplied with any ...
Zorgo's user avatar
  • 177
7 votes
2 answers
219 views

Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions

I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...
Saal Hardali's user avatar
  • 7,789
2 votes
1 answer
311 views

Differentiation on $[0,1]$

EDIT: Perhaps a more reasonable question after thinking about the answer I got would have been. Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
Sascha's user avatar
  • 536
5 votes
1 answer
472 views

Embedding theorem for anisotropic Sobolev spaces

Let $d=d_1+d_2$, $s_1,s_2>0$, $p>1$ and $(x_1,x_2)\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$, $(\xi_1,\xi_2)\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$. Define $$ W^{s_1,s_2}_{p}:=\left\{f: ...
Guohuan Zhao's user avatar
2 votes
1 answer
244 views

Smoothness of distributions defined by oscillation integrals

In M.A. Shubin's book Pseudodifferential Operators and Spectral Theory, we have the following statement. Let $X\subset\mathbb{R}^n$ be an open set, and fix a symbol $a\in S_{\varrho,\delta}^m(X\...
Dominic Wynter's user avatar
1 vote
1 answer
334 views

Orthonormal basis and decay

Edit: I added smoothness, hoping to simplify the problem with this additional assumption. Let me motivate this question first: In signal analysis it is often of interest to understand when a certain ...
Zinkin's user avatar
  • 501
2 votes
1 answer
1k views

Pointwise convergence implies uniform convergence?

Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like $$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$ Assume that $K\in C^{\text{bounded}...
BaoLing's user avatar
  • 329
7 votes
1 answer
489 views

When the value of a function in a point is equal to its integral average over the point's neighborhood?

It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
Grove's user avatar
  • 91
2 votes
0 answers
110 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
user avatar
1 vote
1 answer
400 views

The reproducing kernel for harmonics on compact manifolds

Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
Student's user avatar
  • 617
1 vote
0 answers
192 views

The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
Student's user avatar
  • 617
2 votes
1 answer
382 views

Is the translation/dilation of an $L^p$-multiplier again an $L^{p}$-multiplier?

Suppose that $m:\mathbb R \to \mathbb C$ satisfies: there exists $C > 0$ such that $$\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}.$$ That is, $m$ is an $L^{p}$-multiplier. Let $M(L^{p}...
Inquisitive's user avatar
  • 1,051
10 votes
3 answers
1k views

Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in technical detail) the historical development of the various subareas of analysis and partial differential equations? I'...
1 vote
0 answers
62 views

Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$. ...
JumpJump's user avatar
  • 679
6 votes
2 answers
2k views

Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some $C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$ $$ \left\Vert \left( -I+\Delta\right) ^{\...
user72012's user avatar
  • 233
3 votes
1 answer
191 views

Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$ Now, by ...
anonymous's user avatar
2 votes
1 answer
396 views

$BMO$-property via a John-Nirenberg type estimate?

Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also $$ f_B:= \frac1{|B|}\int_B f \, dx. $$ Suppose $f \in L_{\rm loc}^p(\Omega)$ for all $1<p&...
Juhana Siljander's user avatar