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Interpolation of Sobolev spaces with constraints

Let us consider a real interval $[0, L]$, with $a\in (0, L)$, and let $I_1=(0, a)$ and $I_2=(a, L)$. We denote by $H^k(I_1)$ and $H^k(I_2)$ the usual Sobolev spaces, defined for $k\in \mathbb{N}$. Now,...
rebo79's user avatar
  • 81
3 votes
1 answer
67 views

Infinite direct sum decomposition of the heat semigroup on $\mathbb R$

This question is based on a very similar question posted by me yesterday. A very nice solution was provided by Aleksei Kulikov. Here I modify my question slightly. Let $Q_t$ be the heat semigroup on $...
Ribhu's user avatar
  • 407
0 votes
0 answers
16 views

Representing a periodic strip operator as a tensor product of operators

I hope this question is not trivial, but here goes. I want to consider a bounded operator on $\mathcal{H}=\ell^2(\mathbb{Z}\times \{0,...,N-1\})$ that is a discrete Schrodinger like operator. ...
Keen-ameteur's user avatar
1 vote
1 answer
128 views

Infinite direct sum decomposition of the heat semigroup on $\mathbb{R}^n$

Consider the heat semigroup $Q_t$ on $L^2(\mathbb{R}^n)$ generated by the Laplace operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}$. Does there exist a direct sum decomposition $$\oplus_{...
Ribhu's user avatar
  • 407
0 votes
1 answer
117 views

Validity of approximation method for von Mangoldt function

I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
Brendan Thorne's user avatar
1 vote
1 answer
215 views

Compactness with respect to topology induced by total-variation distance

I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$ is ...
Andrew Luo's user avatar
2 votes
0 answers
331 views

What is the spectrum of this differential operator?

My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} \left( u(x) f(x) \right)$$ where $u(x)$ is a known but arbitrary smooth function that ...
Peter A's user avatar
  • 151
3 votes
1 answer
79 views

Closed linear span of the range of $\boldsymbol f$ and Pettis integrals of $\boldsymbol f$

Let $X$ be a noncompact locally compact topological space, let $H$ by a complex Hilbert space and let $\boldsymbol f:X\to H$ be a continuous function vanishing at infinity whose support is equal to $X$...
P. P. Tuong's user avatar
1 vote
1 answer
122 views

distance in the matrix algebra w.r.t. the nuclear norm

Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
Krzysztof's user avatar
  • 375
2 votes
1 answer
204 views

A continuous analogue of the notion of Hilbert basis

Let $X$ be a locally compact space, let $H$ be a Hilbert space and let $\beta:X\to H$ be a continuous function such that the linear subspace of $H$ spanned by $\beta(X)$ is dense in $H$. I would like ...
P. P. Tuong's user avatar
8 votes
0 answers
115 views

optimal regularity for the Neumann heat equation on Lipschitz domains

$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
leo monsaingeon's user avatar
1 vote
0 answers
72 views

How to understand "sparse graph limits"

For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph. For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
tom jerry's user avatar
  • 349
0 votes
0 answers
96 views

Sufficient condition for weak convergence in Banach spaces

The question is quite elementary but nonetheless no proof or counter example comes to mind immediately. Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
an_ordinary_mathematician's user avatar
1 vote
0 answers
55 views

Characterizing one-sided M-projections on real C*-algebras

Let $A$ be a real C*-algebra, and let $P: A \to A$ be a bounded linear projection. We say that $P$ is a left M-projection if the map $$ v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
Neal B's user avatar
  • 11
2 votes
1 answer
75 views

How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$

Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$. Assume ...
Arghya kundu's user avatar
0 votes
0 answers
100 views

Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube

A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if \begin{equation*} \dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K, \end{equation*} for any $x,y\in \...
Javier's user avatar
  • 69
2 votes
1 answer
121 views

Constructing a function $u$ such that $\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta<\infty$, but $u\notin H^{1/2}$

For $u\in \mathcal{S}'(\mathbb{R})$, define, if finite, $$\Lambda(u)^4=\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta.$$ Using the triangle inequality $|\eta-\xi|\le |\eta|...
Dispersion's user avatar
3 votes
0 answers
77 views

How linearly independent are orthonormal wavelet systems (Daubechies wavelets)?

Disclaimer: Of course, any orthonormal system of functions is linearly independent in the sense of linear algebra, but I am interested in infinite linear combinations with potentially "ugly" ...
PhoemueX's user avatar
  • 734
2 votes
0 answers
40 views

Characterization of critical point of an integral operator

I have an integral operator and I wonder how I can characterize the critical point. I'll give a simplified example so maybe people can comment on and I can maybe generalize in another question. ...
user8469759's user avatar
0 votes
0 answers
42 views

Geometric alignment of adaptive models on evolving manifolds

Let $(M_t)_{t\in[0,T]}$ be a smooth family of compact $d$-dimensional Riemannian submanifolds of $\mathbb{R}^n$. Consider a function $f_t : \mathbb{R}^n \to \mathbb{R}$ evolving over time $t \in [0,T]$...
CollisionGeometry's user avatar
4 votes
0 answers
148 views

Some questions on Hardy's spaces

In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
A beginner mathmatician's user avatar
0 votes
0 answers
54 views

Functional equations with coupled arguments and additive sructure

Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation $$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$ for all $x, y \...
Chandler Halderson's user avatar
3 votes
0 answers
108 views

A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$

Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
B.Hueber's user avatar
  • 1,171
4 votes
1 answer
132 views

Direct characterization of finite-dimensional $1$-injective Banach spaces

It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...
Martin Argerami's user avatar
7 votes
0 answers
269 views

Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by: $$ Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy $$ This operator ...
martin tassy's user avatar
3 votes
1 answer
219 views

Moment problem, ergodicity and spectral gap on the space of tempered distributions

Let $\{ S_n \}_{n=0}^\infty$ be a collection of tempered distributions where $S_0:=1$ and $S_n$ is a tempered distribution on $\mathbb{R}^n$. Just below formula [5] in p.122 of the Fröhlich paper, ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
141 views

(Sub)Optimality of random transport

Problem Setup: Consider the intervals $I_R = [a_R, b_R]$ and $I_B = [a_B,b_B]$. Let $F_R$ and $F_B$ denote the CDF of distributions with support on the intervals $I_R$ and $I_B$. I draw $k$ red and ...
yfful's user avatar
  • 25
7 votes
0 answers
249 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
Tom Solberg's user avatar
  • 4,049
0 votes
0 answers
20 views

Decomposition of measures orthogonal to the algebra $R(K_1 \times \ldots \times K_n)$ - Can it be done via projection-preserving products of bands?

See "Measures orthogonal to tensor products of function algebras" by Marek Kosiek. Here, it is described for the two-dimensional case. It uses another, more general, approach to OB Bekken's ...
S-F's user avatar
  • 63
3 votes
1 answer
287 views

Expectation comparison inequality for concave function of symmetric random variables

Suppose that $X_i$, $i\in[n]$ are independent symmetric random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
Aryeh Kontorovich's user avatar
3 votes
0 answers
117 views

Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?

Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
Stefan Schrott's user avatar
0 votes
0 answers
112 views

Vector field connecting two points

I'm now working on somehow an inverse problem of an ODE: Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t). Now there is a ...
Sqr's user avatar
  • 1
2 votes
0 answers
80 views

Surjectivity of kernel operator with kernel in $L^1(\nu \times \mu)$

Let $ \mu $ and $ \nu $ be two finite and non discrete measures. Let's begin with a well-known fact. Let $ k \in L^2(\nu \otimes \mu) $, then we can define an operator $ \tilde{T} $ as follows: $$ \...
thibault jeannin's user avatar
2 votes
1 answer
79 views

There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?

Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
Ilovemath's user avatar
  • 677
-1 votes
1 answer
86 views

how take weak derivative of norms in hilbert spaces?

Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$. Let $u∈L^2 ([0,T];V); ...
Alucard-o Ming's user avatar
2 votes
1 answer
128 views

Density of smooth functions in weighted Sobolev space

Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
Bastien's user avatar
  • 23
2 votes
1 answer
142 views

Bounded differentiation operator on compact intervals with $L^2$ norm

It is known that the differentiation operator $D$ is not bounded on $C^1([0,1])$ with $L^2$ norm (counterexample: $f(x)=x^n$). Now I am wondering whether there is an infinitely dimensional subspace ...
graham's user avatar
  • 153
3 votes
1 answer
375 views

Dimensionality reduction for total variation

Let $P_i,Q_i$, $i\in[n]$, be distributions on a finite set $\Omega$. We will use $P^\otimes_{i\in[n]}$ to denote $n$-fold products of distributions. For each $i\in[n]$, define the dimensionally-...
Aryeh Kontorovich's user avatar
4 votes
1 answer
54 views

Krein-Rutman for integral transforms: proof of convergence to leading eigenvector

Disclaimer: This is a question in functional analysis, on which I don't have much background. It arose from me trying to prove on my own a folklore result in probability theory. Consider an integral ...
Plemath's user avatar
  • 312
5 votes
1 answer
261 views

Counter example for Hadamard Differentiability

I am having a hard time while trying to fully understand Hadamard differentiability. I use the following definition taken from a German source ( Martin Brokate, "Konvexe Analysis und ...
Matthis's user avatar
  • 53
2 votes
0 answers
55 views

Distance between a Hölder function and a Sobolev ball

Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively. My question ...
Drew Brady's user avatar
7 votes
1 answer
959 views

a claim for a proof of the invariant subspace problem [closed]

Recently four mathematicians claimed to have proven the invariant subspace problem, which is the problem that states Does every bounded operator on a separable Hilbert space have a non-trivial ...
euleroid's user avatar
6 votes
1 answer
335 views

Existence of pairwise quasi-complementary but not complementary subspaces

Let $𝑋$ be an infinite-dimensional Banach space (complex or real). A subspace of $𝑋$ means a closed linear submanifold. Subspaces $M$ and $N$ of $X$ are quasi-complementary if $M\cap N=\{0\}$ and $M+...
Janko Bracic's user avatar
2 votes
0 answers
120 views

On mollifiers acting between $L^2$ and Sobolev spaces

(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.) Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by $$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
S.Z.'s user avatar
  • 505
0 votes
0 answers
78 views

What does analytic uniformly in $s$ mean?

Suppose I have a complex vector space $V$ with finite basis $\{e_{1},...,e_{s}\}$. Then, I can consider the algebra $\mathcal{U}$ of formal polynomials on the variables $e_{1},...,e_{s}$. Suppose ...
InMathweTrust's user avatar
0 votes
0 answers
80 views

Relationship between two minimization problems

Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
Cauchy's Sequence's user avatar
2 votes
1 answer
106 views

Dual space and conditions for weak convergence in Orlicz Space not having $\Delta_{2}$ property

I am interested in conditions for weak convergence on Orlicz spaces where the corresponding Young function, $\Phi:[0,\infty) \rightarrow [0,\infty)$, does not have the $\Delta_{2}$ condition, i.e. ...
user273331's user avatar
2 votes
1 answer
49 views

Is any submetrizable linear topology linearly submetrizable?

Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$. Is ...
erz's user avatar
  • 5,529
3 votes
1 answer
187 views

Is this property preserved under weak$^*$ convergence?

Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
Cauchy's Sequence's user avatar
2 votes
1 answer
89 views

Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances

Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$: $$ d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|. $$ Let $\rho$ be the Levy-Prokhorov metric on the ...
ssss nnnn's user avatar
  • 177