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2 votes
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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
3 votes
1 answer
224 views

Extension of Sobolev function defined on unit cube

Im wondering about theorems concerning extending Sobolev functions defined on the $d$-dimensional unit cube to all of $\mathbb{R}^d$. More precisely, given $f:[0,1]^d \to \mathbb{R}$ with $f\in H^k([0,...
Jjj's user avatar
  • 93
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
10 votes
1 answer
1k views

A strange Lipschitz function

Let $n \geq 3$. Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the following conditions hold? The origin is a weak Lebesgue point of $\nabla f$, in the sense that the ...
Nate River's user avatar
  • 6,215
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
0 votes
0 answers
42 views

Is this function $\mathcal{C}^1$ in the global sense?

Denote by $\mathbb{U}$ the complex unit disk. Let $\mathcal{O}$ an nonempty open subset of $\mathbb{R}^n$ $(n\geq 1)$, and $f\in\mathcal{C}^1(\mathcal{O}\times\mathbb{R},\mathbb{U})$ such that for all ...
G. Panel's user avatar
  • 449
5 votes
1 answer
174 views

Do the zeroes of some hypergeometric functions interlace?

Confluent hypergeometric functions differing from $F={}_1F_1(a,b,z)$ by $\pm1$ in either parameter $a$ or $b$ are called contiguous to $F$. For rational $a, b$, assume I know $z_0$ is a zero of $F$. ...
Sveti Ivan Rilski's user avatar
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
0 votes
1 answer
96 views

Existence of a complemented basic sequence

Let $X$ be an infinite-dimensional Banach space (complex or real). A subspace of $X$ means a closed linear submanifold. If $S$ is a non-empty subset of $X$, then $[S]$ denotes the closed linear span ...
Janko Bracic's user avatar
0 votes
0 answers
30 views

Analytic / algebraic characterization of the limiting value of the unique nonnegative root of a polynomial

I'm interested in the following problem which arises from some "random matrix theory" calculations. Let $\phi,s_1,s_2, p > 0$ with $p \in [0,1]$, and set $p_1=p$, $p_2=1-p$, and $q_k := ...
dohmatob's user avatar
  • 6,853
3 votes
1 answer
176 views

Are measurable maps with countably separated image in a Banach space always strongly measurable?

Let $(E,\|.\|)$ be a (not necessarily separable) Banach space and $\Sigma_E$ the Borel $\sigma$-algebra (w.r.t. the norm topology). Let $(\Omega,\Sigma_\Omega)$ be a measurable space (which we can ...
Packo's user avatar
  • 285
1 vote
0 answers
87 views

Convergence and sequential compactness for nonlinear operators

I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear. What kind of notions of convergence does one have for such operators? I'm specifically ...
C_Al's user avatar
  • 251
2 votes
1 answer
120 views

Difference between finite partial sums from two divergent series

Fix a sequence $(r_i)_{i\in\mathbb{N}} \subseteq (0, 1)$ such that $\lim_i r_i=0$ and $\sum_{i\in \mathbb{N}} r_i=\infty$. According to the answer in this post, for any $c>0$ there exists $N,M\in\...
Sanae Kochiya's user avatar
13 votes
2 answers
813 views

A dichotomy for everywhere differentiable eikonal functions

Let $f: \mathbb R^n \to \mathbb R$ be everywhere differentiable, with $|\nabla f| = 1$ almost everywhere. Is it true that $|\nabla f| = 0$ or $1$ everywhere?
Nate River's user avatar
  • 6,215
3 votes
0 answers
90 views

Sobolev embedding on a compact manifold without boundary

I am reading M. E. Taylor, "Partial Differential Equations III", Second Edition, Springer-Verlag, New York, (1996). In chapter 13, section 2, in Prop. 2.3 and Prop. 2.4, one finds the ...
Azam's user avatar
  • 311
3 votes
1 answer
104 views

From Wightman to HK axioms for "non-neutral (charged?)" fields

Wightman axioms deal with operator-valued distributions (Wightman fields) whose values are unbounded operators in general. On the other hand, the Haag-Kastler axioms deal with net of observables, ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
474 views

Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?

Let $$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$ be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
Balaji Mallikarjun S's user avatar
0 votes
0 answers
49 views

Kadec-Klee property of an equivalent norm on a Hilbert space

Let us consider the space $\ell_2$ with the Hilbert norm $\Vert \cdot \Vert$ and consider the following eqivalent norm: $$ \Vert (r,x) \Vert_A^2 = \Vert (r, Tx)\Vert^2 + \max \{ \Vert x \Vert, \vert r ...
PPB's user avatar
  • 85
2 votes
0 answers
103 views

A question from a proof of an inequality in Sobolev space $W^{1,1}$

I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot: Here is what I did: $$-u(x)=u(y)-u(x)=\...
Bogdan's user avatar
  • 1,759
3 votes
1 answer
327 views

Derivative norm estimates

Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$. QUESTION. Does this norm estimate hold? ...
T. Amdeberhan's user avatar
1 vote
1 answer
58 views

Proving one one condition for the Gaussian mixture model

$\textbf{Question:}$ Consider the following matrix representation for a two-component bivariate Gaussian Mixture Model (GMM): $S = \begin{bmatrix} A & X \\ X' & B \end{bmatrix}$ where $A = \...
Andyale's user avatar
  • 123
2 votes
1 answer
127 views

Strong Ditkin sets in the Fourier algebra

What is the definition of a Ditkin set (resp. a strong Ditkin set) for the Fourier algebra $A(G)$ of a locally compact (not necessarily abelian) group $G$? More specifically, let $E$ be a closed ...
Aristides's user avatar
4 votes
0 answers
140 views

Condition under a function is uniquely identifiable by the supremum values

Let $f(x),g(x)$ be two real-valued functions on $\mathbb{R}$ and $h(x,y)$ be a real-valued function on the plane. We can assume continuity (maybe piecewise differentiability also) of these functions. ...
mukhujje's user avatar
  • 271
1 vote
0 answers
67 views

regularity convolution of a $L^2$ function with $W^{1,1}$ function [closed]

Let $u\in L^2(\mathbb R)$ and $w \in W^{1,1}(\mathbb R)$, we consider the convolution $$u*v$$ Is it true that $w*u \in W^{1,2}(\mathbb R)$? What regularity can we put on $w$ for this to be true?
user3177306's user avatar
6 votes
0 answers
162 views

Dual space of local Sobolev space on a manifold

$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
Fabian Patzwaldt's user avatar
7 votes
2 answers
706 views

Poisson binomial conjecture

Let $X_i\in\{0,1\}$ be mutually independent and distributed according to $\mathrm{Bernoulli}(p_i)$ and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$, for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=...
Aryeh Kontorovich's user avatar
1 vote
1 answer
65 views

Reference dual Dirichlet space $D^1$

Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that $$ \|f\|_{A^1} ...
Scottish Questions's user avatar
0 votes
0 answers
84 views

Question on approximation of norms

Suppose that $E\in Int[L_{p},L_{q}]$ for some $1<p<q<\infty$ and $E$ is $w$-concave with $1<w<\infty$. It is well-known that for each $r\geq w$, we have $E=L_{r}\odot F_{r}$ for some ...
Sijie Luo's user avatar
3 votes
0 answers
53 views

Bounds on Besov norms for mollification of a bounded Lipschitz function

Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to ...
LittleQuestionBoy's user avatar
4 votes
3 answers
309 views

Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$

Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
Bogdan's user avatar
  • 1,759
12 votes
1 answer
402 views

Boundedness of sequences and cardinality

Let $X$ be a set of sequences of real numbers that converge to zero with the property that for any unbounded sequence of real numbers $(y_n)$, there is a sequence $(x_n)$ in $X$ for which the ...
Chris Stuart's user avatar
7 votes
1 answer
243 views

Isoperimetric inequality, but $L_p$ norm

I would like to consider the isoperimetric problem of $L_p$ norm: Given a region in $\mathbb R^2$ such that the boundary is a curve $C(x,y)$, where $\int_{C}(|\mathrm dx|^p+|\mathrm dy|^p)^{1/p}$ is a ...
JetfiRex's user avatar
  • 843
4 votes
1 answer
252 views

Show that $\Lambda_\varphi(x_n)\to \Lambda_\varphi(x)$ for an nsf weight $\varphi$ on a von Neumann algebra

Let $\varphi$ be an nsf weight on a von Neumann algebra $M$. Fix a square-integrable element $x\in \mathscr{N}_\varphi$. Put $$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
Andromeda's user avatar
  • 175
8 votes
1 answer
449 views

What do smooth signatures give you?

My background is in rough paths theory. In short, if you have an irregular function $f:[0,T]\to\mathbb R^d$ and you want to make sense of integrals $\int_s^t \cdot \ df(r)$, the right objects that are ...
user479223's user avatar
  • 1,904
15 votes
0 answers
244 views

Natural examples of Borel surjections without right inverse

As discussed in this question, in general a Borel surjection $f:\mathbb{R}\rightarrow\mathbb{R}$ may not have a Borel right inverse, namely a $g$ such that $f\circ g=id$, although there is always a ...
183orbco3's user avatar
  • 623
1 vote
1 answer
130 views

Existence of solutions to a series of integral equations

I am trying to solve the following integral equation analytically: $$ \sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T], $$ where $(f_n(t))_n$ is the unknown ...
Gustave's user avatar
  • 617
7 votes
1 answer
281 views

Norm in the minimal tensor product of C*-algebras

Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
Hannes Thiel's user avatar
  • 3,497
0 votes
0 answers
119 views

Boundedness of 2 times the unit ball

Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball $$ B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X, $$ is it necessarily ...
Chandan Biswas's user avatar
3 votes
1 answer
157 views

How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?

I have a question about the completeness of complex exponentials in function spaces. For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
Zhang Yuhan's user avatar
2 votes
0 answers
80 views

Prove uniqueness of Radon transform without using Fourier transform

The uniqueness of Radon transform can be expressed by the following claim (I assumed that the function has compact support for simplicity): If a continuous function with compact support has zero ...
Zhang Yuhan's user avatar
4 votes
0 answers
198 views

When a null uncountable set can be image of some increasing function with discontinuities on a dense countable set

Consider the following result: A: Let $f:D \to \mathbb R$ be an increasing function with discontinuities on a dense countable subset of $D$ such that the jump values sum to $\mu(D)$, where $D$ is a ...
Amir's user avatar
  • 303
1 vote
2 answers
220 views

A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$

Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$. Note that $A$ is non-empty with a Baire category argument. I ...
Ali Taghavi's user avatar
4 votes
1 answer
196 views

(Lattice approximation) Does UV stability lead to continuum limit of a subsequence?

In the context of lattice approximation, the term "UV stability" seems to be used frequently. To me, it seems like Uniform boundedness of the partition function in the limit where lattice ...
Isaac's user avatar
  • 3,477
0 votes
0 answers
36 views

Derivate involving Bessel function of second type

Let. $$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$ Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
Ryo Ken's user avatar
  • 109
3 votes
1 answer
376 views

A more general product rule for weak derivatives?

Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
Bogdan's user avatar
  • 1,759
1 vote
0 answers
51 views

Compact embeddings RKHSs into Sobolev Spaces

Let $\mathcal{H}$ be an RKHS over an open domain $\Omega \subseteq \mathbb{R}^d$. Are there conditions under which $\mathcal{H}$ can be compactly embedded in a Sobolev space $W^{s,p}(\Omega)$ for ...
Sam_the_Sampler's user avatar
1 vote
0 answers
73 views

Convexity and subdifferential monotonicity

Do you know any reference where I can find some results in this sense: Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
Bogdan's user avatar
  • 1,759

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