All Questions
12,935 questions
2
votes
2
answers
87
views
Growth of nonnegative functions satisfying $\Delta u \geq C>0$
Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality
$$ \Delta u(x) \geq C>0 $$
for all $x\in \Omega$. Furthermore, we ...
- 1,045
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+...
- 391
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{...
- 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 ...
5
votes
1
answer
219
views
Compactly supported wave packet in Schrödinger's evolution
Does the following result hold:
For any compactly supported wave packet, under free Schrödinger's evolution, it is no longer compactly supported after any finite time?
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,...
- 93
2
votes
0
answers
50
views
"Reverse monotonicity" for k-Hessian measures
My question deals with the $k$-Hessian measure $\mu_k[u]$ of a $k$-convex function $u\in \Phi_k(\Omega)$ as defined in the articles "Hessian measures I-III" by N. S. Trudinger and X. J. Wang....
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 ...
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. ...
- 21
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 ...
- 5,529
2
votes
0
answers
81
views
Asymptotic expansion of the solution of a nonlinear wave equation
I am reading the article Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime (arXiv link) by Barreto, Plamen, where they consider the modified cubic NLWE
$$-\...
- 860
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 $$ \lim_{m \to \infty}...
2
votes
0
answers
42
views
Diffusions vs elliptic operators with dkp coefficients
I am wondering if there is any literature on the relationship between diffusions and elliptic equations. In particular I am interested in literature concerning operators with Dahlberg–Kenig–Pipher ...
- 121
2
votes
1
answer
89
views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
- 177
0
votes
1
answer
96
views
Existence of a complemented basic sequence
Let $X$ be an infinite-dimensional Banach space (complex or real). A subspace of $X$ means a closed linear submanifold. If $S$ is a non-empty subset of $X$, then $[S]$ denotes the closed linear span ...
- 391
3
votes
1
answer
176
views
Are measurable maps with countably separated image in a Banach space always strongly measurable?
Let $(E,\|.\|)$ be a (not necessarily separable) Banach space and $\Sigma_E$ the Borel $\sigma$-algebra (w.r.t. the norm topology). Let $(\Omega,\Sigma_\Omega)$ be a measurable space (which we can ...
- 285
0
votes
0
answers
40
views
Is the principal eigenvalue of the clamped plate minimized by the equilateral triangle among all triangles of a given area?
Let $\Omega$ be a ``nice'' domain in $\mathbb{R}^2$
(bounded and with piecewise smooth boundary). Consider the clamped plate eigenvalue problem, given by
$$\Delta^2 u = \lambda u $$
$$ u|_{\partial \...
- 3,245
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 ...
- 251
5
votes
2
answers
384
views
On a 3D Gagliardo-Nirenberg inequality
It is well known that there exists a constant $C$ such that
$$\forall f\in C^\infty_c(\mathbb R^3), \quad
\Vert f\Vert_{L^6(\mathbb R^3)}\le C\Vert \nabla f\Vert_{L^2(\mathbb R^3)}.
\tag{$\ast$}$$
Now ...
- 16.2k
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 ...
- 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, ...
- 3,477
4
votes
0
answers
198
views
Pricing zero coupon bonds through PDE
I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book.
The idea is to model the market price of risk as a ...
- 51
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 ($...
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:
$$
\Vert (r,x) \Vert_A^2 = \Vert (r, Tx)\Vert^2 + \max \{ \Vert x \Vert, \vert r ...
- 85
2
votes
0
answers
103
views
A question from a proof of an inequality in Sobolev space $W^{1,1}$
I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot:
Here is what I did:
$$-u(x)=u(y)-u(x)=\...
- 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? ...
- 43.2k
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 ...
- 21
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. ...
- 271
6
votes
1
answer
183
views
Bounded functions satisfying $\Delta u \geq u$ on $\mathbb{R}^n$
Consider a function $u \in C^\infty(\mathbb{R}^n)$ such that
$$
\begin{cases}
u(x) > 0 & \forall x \in \mathbb{R}^n \\
\Delta u(x) \geq u(x) > 0 &\forall x \in \mathbb{R}^n
\end{...
1
vote
0
answers
67
views
regularity convolution of a $L^2$ function with $W^{1,1}$ function [closed]
Let $u\in L^2(\mathbb R)$ and $w \in W^{1,1}(\mathbb R)$, we consider the convolution
$$u*v$$
Is it true that $w*u \in W^{1,2}(\mathbb R)$?
What regularity can we put on $w$ for this to be true?
- 73
6
votes
0
answers
162
views
Dual space of local Sobolev space on a manifold
$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
2
votes
2
answers
154
views
Domains of type (A) are Lipschitz?
In this article and in the book of Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations we have the following definition of what is a domain of type (A):
There is no example of a ...
- 1,759
1
vote
1
answer
65
views
Reference dual Dirichlet space $D^1$
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that
$$
\|f\|_{A^1} ...
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 ...
- 45
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 ...
2
votes
0
answers
89
views
Malliavin calculus for the regularity of the density of the supremum of a process
I am reading Chapter 2 from Nualart's book 'The Malliavin calculus and related topics'.
Proposition 2.1.10 gives the conditions for the law of the supremum of a process to have a density. Condition (...
- 61
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 ...
- 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 ...
- 121
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 ...
- 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
$$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
- 175
1
vote
1
answer
130
views
Existence of solutions to a series of integral equations
I am trying to solve the following integral equation analytically:
$$
\sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T],
$$
where $(f_n(t))_n$ is the unknown ...
- 617
7
votes
1
answer
281
views
Norm in the minimal tensor product of C*-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
- 3,497
0
votes
0
answers
119
views
Boundedness of 2 times the unit ball
Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball
$$
B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X,
$$
is it necessarily ...
3
votes
0
answers
179
views
Green function of an elliptic operator
Let $L$ be an elliptic operator on $\Bbb C$ with
$$\DeclareMathOperator{\Img}{Im}
L^{-1}f(z)=\int_{\Bbb {C}}K_1(|z-w|^2)f(w) e^{i\cdot\Img\langle z,\overline{w}\rangle} \, dw
$$ where $K_1$ is the ...
- 109
1
vote
2
answers
220
views
A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$.
Note that $A$ is non-empty with a Baire category argument.
I ...
- 356
4
votes
1
answer
196
views
(Lattice approximation) Does UV stability lead to continuum limit of a subsequence?
In the context of lattice approximation, the term "UV stability" seems to be used frequently. To me, it seems like
Uniform boundedness of the partition function in the limit where lattice ...
- 3,477
6
votes
1
answer
206
views
Reference for $\epsilon$-regularity
I am looking for a reference for the following $\epsilon$-regularity statement: let
$(M,g)$ be a Riemannian manifold of dimension $n$,
$\Delta=dd^*+d^*d$,
$B_r$ denotes a ball of radius $r$ around a ...
- 954
0
votes
0
answers
36
views
Derivate involving Bessel function of second type
Let.
$$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(...
- 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^...
- 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 ...
