All Questions
Tagged with fa.functional-analysis real-analysis
453 questions with no upvoted or accepted answers
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
2
votes
0
answers
66
views
Existence of saddle points under a $C^0$-perturbation of a continuous function
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and has a strict maximum point $a$ and strict minimum point $b$. Define $g(x,y)=f(x)+f(y)$ and $h_\varepsilon(x,y)$ be a family of continuous ...
- 379
2
votes
0
answers
166
views
Green's function for elliptic PDE with potential
$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u) $ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic ...
2
votes
0
answers
68
views
Core for Neumann Laplacians
Let $d$ be a positive integer. We write $\mathbb{H}^d$ for the closed $d$-dimensional upper-half space: $\mathbb{H}^d=\{(x_1,\ldots,x_d) \in \mathbb{R}^d,\,x_d \ge 0\}$. We consider the Neumann ...
- 721
2
votes
0
answers
55
views
An integral average condition and its relationship with BMO, VMO, and Sobolev spaces
Let $V: \mathbb R^n \to \mathbb R^n$ be a vector field which satisfies
$$
\lim_{l \to \infty} \sup_{x \in \mathbb R^n} \left|\frac{1}{l^n} \int_{[0,l]^n}V(x+y) dy \right| = 0
$$
What is the ...
- 839
2
votes
0
answers
57
views
Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?
The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...
- 1,947
2
votes
0
answers
229
views
Weighted Sobolev norm in terms of Spherical harmonics coefficients
Let $M = [1,\infty) \times S^2$.
Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm:
$$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$
...
- 969
2
votes
0
answers
298
views
A question on convergence rates of Fourier series and strict convergence
Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the function at every point of continuity, as $n\to\infty$. The convergence rate is $O(1/n)$....
- 698
2
votes
0
answers
164
views
(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis
It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
- 1,245
2
votes
0
answers
45
views
Additivity of squared Schatten $p$-norm with respect to spatial partition
Consider a Hilbert-Schmidt operator $A$ on $L^2(\mathbb R^d)$ with integral kernel $A(x,y)$. Let $\Omega\subset \mathbb R^d$ and $1_{\Omega}(x)$ denote its characteristic function as well as the ...
- 91
2
votes
0
answers
65
views
On a question relating integral equation:
I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
- 41
2
votes
0
answers
65
views
Reference request for type of specific integral equation in two variable:
Consider the following integral equation:
$$\int_0^\infty K(t,y)\phi(t,x)dt=0$$
Here, $K(t,y)$ is a trigonometric kernel and
$\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$).
I want to find the ...
- 41
2
votes
0
answers
132
views
Green's identity with a different norm
Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...
- 820
2
votes
0
answers
44
views
Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous
Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...
- 6,853
2
votes
0
answers
71
views
Strict Riesz's rearrangement inequality when function is not nonnegative
The strict Riesz rearrangement inequality (Lieb- and Loss's book Analysis, Section 3, Theorem 3.9 ,page 93) says that if the functions $f,g,h,$ are all nonnegative and $g$ is strictly symmetric ...
- 379
2
votes
0
answers
136
views
Banach limit with added properties
Let $c=\{a:\mathbb{N}\rightarrow \mathbb{C}: \exists \alpha\in \mathbb{C} \textrm{ so that }
\lim\limits_{n\rightarrow \infty} a(n)=\alpha=:L_c(a)\}\subset \ell^\infty$, where $\ell^\infty$ is the ...
- 21
2
votes
0
answers
162
views
Bochner's formula for fractional Laplacian
Is there an analogue of the classical Bochner formula
$\frac{1}{2} \Delta |\nabla u|^2 = |\nabla^2 u|^2$ for harmonic functions that holds for $s$-harmonic functions?
- 161
2
votes
0
answers
553
views
$\ell_\infty$-norm covering number of RKHS ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$
For any $\epsilon \in (0,1)$, let $N_\infty(\epsilon, \mathcal{H}, R)$ denote the $\epsilon$-covering number of the RKHS norm ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$ with respect to the $\...
- 121
2
votes
0
answers
72
views
Product of Besov and Lorentz functions
Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound
$$
\|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
- 282
2
votes
0
answers
250
views
A sequence of compact operators, weak convergence versus strong convergence
Let $(X,\mu)$ be a finite measure space.
We consider two sequences of bounded linear operators $\{T_n\}_{n=1}^\infty$ and $\{S_n\}_{n=1}^\infty$ on $L^2(X,\mu)$. We denote by $\mathcal{L}$ the space ...
- 721
2
votes
0
answers
132
views
When does a nonnegative $C^1$ function on $[a,b]$ have finitely many zeros?
It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.
It is also known that you can have a ...
- 133
2
votes
0
answers
104
views
Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
2
votes
0
answers
76
views
wave equation with L^2 boundary data via spectral decomposition
It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation
\begin{equation}\label{pf2}
\begin{aligned}
\begin{cases}
\partial^2_{t}u- \Delta u=0\,\...
- 4,135
2
votes
0
answers
86
views
Eigenvalues of the operator $A = -v'' + B(x) v$
How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that
$$
\left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||...
- 217
2
votes
0
answers
162
views
Hilbert transform on weighted Sobolev spaces
Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
- 4,135
2
votes
0
answers
164
views
What are (the different aspects of) harmonic analysis good for?
Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have ...
- 2,071
2
votes
0
answers
115
views
Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
- 21
2
votes
0
answers
42
views
Generalized Hardy operator and Lorentz gamma spaces
I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...
- 109
2
votes
0
answers
99
views
Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...
- 192
2
votes
0
answers
89
views
Prove integral inequality for divergence-free vector fields
Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold?
$$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \...
- 839
2
votes
0
answers
126
views
Mixed partial derivatives of planar functions converging to delta distribution
Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
- 51
2
votes
0
answers
120
views
Hilbert transform on a Besov space
Consider the usual Hilbert transform of periodic functions
$$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$
We know $H$ does not map $L^\infty$ continuously to $L^\infty$. Now ...
- 903
2
votes
0
answers
158
views
Estimate involving Besov norm
When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details.
For $p\in(0,\infty)$ a $p$-variation semi-norm of a ...
- 421
2
votes
0
answers
100
views
What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?
Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. Then there exists a subsequence such that $f_{n_k} \to f$ weakly in $L^p([0,1])$. What is the weak limit of the sequence of ...
- 217
2
votes
0
answers
42
views
Analysis of coefficients for quickly vanishing analytic vector field
Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
- 749
2
votes
0
answers
160
views
Approximation in fractional Sobolev space
Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$.
How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$?
Recall that,
$$|u|^p_{W^{s,p}(\Omega)}= ...
- 1,101
2
votes
0
answers
92
views
First Dirichlet eigenvalue below second Neumann eigenvalue?
Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
- 173
2
votes
0
answers
130
views
Partition of unity in $\mathbb{R}$ with additional conditions on the derivatives
Let $K\subseteq \mathbb{R}$ be locally compact without isolated points and $X$ an infinite dimentional Banach space. Then
$$C_{0}^{(1)}(K,X)=\{ f\in C_{0}(K,X): \text{$f$ is continuously ...
- 121
2
votes
0
answers
189
views
Point wise convergence of Laplace transform and convergence of functions
Assume that functions $f_n(t), f(t)\in C_b(R_+)$. For every $\lambda >0$, we have
$$
\bigg|\int_0^\infty e^{-\lambda t}f_n(t)d t-\int_0^\infty e^{-\lambda t}f(t)d t\bigg|\leq C_\lambda n^{-1},
$$
...
- 411
2
votes
0
answers
97
views
Prove that this integral operator is onto
Let us consider the operator $T$ defined by$$\eqalign{
& T:{L^2}((a,b) \times (c,d)) \to {L^2}((c,d)) \cr
& Tf(s,x) \mapsto \int\limits_{q(x)}^{p(x)} {f(\alpha (s,x),s)ds} \cr} $$
where ...
- 617
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
2
votes
0
answers
171
views
How to use Stein-Tomas theorem to check to following inequality?
Recently, I am reading Rodnianski & Schlag
Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. In lemma 3.2, R&S said that by using Stein-Tomas theorem ...
- 429
2
votes
0
answers
216
views
Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?
Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...
- 369
2
votes
0
answers
240
views
Monotone approximation of continuous function by Lipschitz function
Let $X$ be a normed space, and $f\in C_b(X,\mathbb{R})$. How can I construct a sequence $\varphi_k,\,k\geq 1$ of bounded Lipschitz functions with $\varphi_k\downarrow f$ pointwise? Thanks
- 139
2
votes
0
answers
90
views
Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces
In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
- 5,405
2
votes
0
answers
240
views
Discrete Sobolev embedding
It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
2
votes
0
answers
77
views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
2
votes
0
answers
148
views
Approximation of functions in $L^p(R^d;L^\infty)$
Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that
$$...
- 411
2
votes
0
answers
190
views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
2
votes
0
answers
58
views
Absolute continuity of DOS measure for Schrödinger operators
Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure.
I would like to ...
- 21