Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
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For which Banach spaces is the self composition operator Lipschitz?
Let $X\subseteq \{f|f:D\rightarrow \mathbb{R}^n\}$ be a Banach space, with at least all polynomials on $D$ contained in $X$, where $D\subseteq \mathbb{R}^n$ is open and bounded.
Let $U\subseteq X\cap \...
- 183
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93
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Does Anderson localisation occur if the potential are equal in pairs?
Consider the Anderson model given by the Hamiltonian $H \in B(l^2( \mathbb{Z}^d)) $ defined by $H = - \Delta + V$ where the potential $V$ acts on a unit vector $ \vert x \rangle \in l^2( \mathbb{Z}^d)...
- 1,054
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105
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Derivative and Green function of Fractional Laplacian in a bounded domain: $(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega $?
Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does ...
- 303
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233
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Variance-based localized Rademacher complexity for RKHS unit-ball
Let $\mathscr X$ be a compact subset of $\mathbb R^d$ (e.g the unit-sphere). Let $K: \mathscr X \times \mathscr X \to \mathbb R$ be a positive kernel function and let $\mathscr H_K$ be the induced ...
- 6,853
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137
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Relative compactness... but what is the toplogy?
The following Theorem was described in a text I was reading as a compactness result. The proof is probably too advanced for me but I was just wondering with respect to what topology we have ...
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77
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Regarding an integrability property of Schwartz class function
Let $f\in\mathcal{S}(\mathbb{R^n})$, Schwartz class, satisfying
$$\int_{\mathbb{R}^n}|f(x)|e^{g(||x||)}dx<\infty, $$
where $g:[0,\infty)\to[0,\infty)$ be an increasing function satisfying $\int_0^\...
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313
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Confusion: Normal homomorphism vs. weak*-continuous vs $\sigma$-weakly continuous
$\newcommand\M{\mathcal M}
\newcommand\N{\mathcal N}
\newcommand\A{\mathcal A}
\newcommand\B{\mathcal B}$
In Takesaki [1], I find the following theorem:
Proposition 5.13. Let $\M_1,\M_2,\N_1,\N_2$ be ...
- 896
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56
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Monotonically increasing and bounded function is in $BV_{loc}$?
For any $n\in \mathbb{N}$ let $f_n:\mathbb{R}\to [0,1]$ be monotonically increasing and $\lim_{x\to -\infty} f_n(x)=0$ and $\lim_{x\to \infty} f_n(x)=1$. It follows $f_n$ is differentiable a.e..
I'm ...
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Common core for unbounded operators
Suppose that $\mathcal H$ is a Hilbert space representing some physical system, $H$ is the Hamiltonian for the system, and $A$ is some observable for the system, that is, some unbounded self-adjoint ...
- 771
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Identify input maps corresponding to control system ($A,B$), where $B = \delta_0$
The text I'm struggling with comes from "Observation and Control for Operator Semigroups" by Tucsnak and Weiss, page 119.
Take $X = L^2(0,\infty)$ and let $X_{-1}$ be the dual of the Sobolev ...
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140
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Haagerup tensor product
Let $H$ and $K$ be Hilbert spaces. Recall that a closed subspace $V\subset B(H,K)$ is called a ternary ring of operators(TRO) if $xy^*z \in V$ for all $x,y,z \in V$. Obviously a TRO is also a ...
- 1,115
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Obtaining identity from Pokhozhaev formula
From the classical Pokhozhaev formula, how can I obtain that the following identity holds for $u,v \in C^2(\bar \Omega)$?
$$
\int_\Omega (\Delta u(x,\nabla v) + \Delta v(x,\nabla v)) dx = \int_{\...
- 217
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224
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Is this mixed-Sobolev interpolation inequality valid?
By $L^p_T\dot H^s$ I mean the space $L^p([0, T] \to \dot H^s)$ where as usual
$$\|u\|_{\dot H^s}^2 = \int_{\mathbb R^d} |\hat u(\xi)|^2 |\xi|^{2s} ~d\xi.$$
Of course, we can interpolate in the time ...
- 708
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259
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Cut-off function and fractional Laplacian
Is there a smooth function $u$ such that $u = 1$ in $B_r(0)$, $u=0$ in $\mathbb R^N \setminus B_{2r}(0)$, and
$$
|\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s}...
- 99
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68
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Inequality of exponentials of Banach operators
(I have moved this question from Stackexchange).
Given the operators $\{A_j\}$ in a Banach algebra and a positive integer $p$, let
\begin{equation}
g=\exp\left(\frac{1}{n}\sum_{j=1}^p A_j\right)\quad\...
- 111
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36
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Existence and uniqueness for fractional parabolic equation with transport term
Let us consider the problem
\begin{equation}
\begin{cases}
u_t+(-\Delta)^{\sigma}u+\mathrm{div}(a(t,x)u) = 0 & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x) & \text{in } \...
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111
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Minimisation in dual Sobolev space
Say we have a function $F(\lambda) = \|f(\lambda)\|_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are ...
- 537
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191
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Dual of union of Reproducing Kernel Hilbert Spaces
I have a union of Reproducing Kernel Hilbert Spaces $\mathcal{B}$. I am interested in finding the dual of $\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the ...
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Condition on Functions on Overlapping patches of a Domain
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain. Let $\{p_i\}_{i=1}^m$ for some finite $m\in\mathbb{N}$ be the overlapping open sets of $\Omega$ such that $\Omega =\bigcup_{i=1}^mp_i$.
Now, ...
- 111
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280
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On measurability of certain group actions on spaces of bounded measurable functions
Let $\mathcal{H}$ be a separable Hilbert space equipped with a strongly continuous unitary representation of a locally compact group $G$. Denote by $\mathcal{L}^{\infty}(H)$ the space of the bounded ...
- 557
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144
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Liouville theorem for elliptic equation with advection term
How can one prove that any $L^2$ solution of
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$
is zero if $a(x)$ is a divergence-free vector field such that
$\int |\...
- 99
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61
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Discrete Orlicz space estimate
We consider the discrete LlogL space of sequences $x=(x_i)$ such that
$$\Vert x\Vert_{LL}:=\sum_i \vert x_i \log(x_i)\vert <\infty.$$
Let $x=(x_i)$ and $y=(y_i)$ two sequences in the above LlogL ...
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130
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Fractional Sobolev embedding theorem
Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds
$$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
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Normalizing constants preserve metric entropy
Suppose $\mathcal{F}=\left\{f\in L^2([a,b]): 0<\underline{c}\leq f\leq\overline{c} \right\}$. Consider the following transformation
$$\tilde{\mathcal{F}} := \left\{\frac{f}{\int f d\mu}: f\in \...
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71
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Show that the Laplacian on these domains is isospectral
Let $\Omega_i\subseteq\mathbb R^d$ be bounded and open, $A_i$ denote the weak Laplacian with domain $\mathcal D(A):=\{u\in H_0^1(\Omega_i):\Delta u\in L^2(\Omega_i)\}$ on $L^2(\Omega_i)$ and $$T_i(t)f:...
- 167
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122
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Series and solution of $-\Delta u + \lambda u = f(x)$
Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
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Is $A^{-\infty}$ a $Q$-algebra?
Let $\mathbb{D}=\{z\in \mathbb{C}: |z|<1\}$ and
$$
A^{-\infty}=\Big\{f:\mathbb{D}\rightarrow \mathbb{C}\;|\; \exists n\in \mathbb{N} \textrm{ such that }\|f\|_{-n}= \sup\limits_{z\in \mathbb{D}}\;\!...
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Gaussian measures on infinite dimensional spaces
On Zabczyk & Da Prato book about infinite dimensional SDEs they introduce the idea of Gaussian measures in infinite dimensional Banach spaces. They do so by means of Fernique theorem.
In the ...
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142
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On the uniform boundedness principle and the space of functions of bounded variation
Let $U$ be a bounded smooth domain of $\mathbb{R}^d$. We write $m$ for the Lebesgue measure on $U$. A function $f \in L^1(U,m)$ has bounded variation in $U$ if
\begin{align*}
V(f,U):=\sup \left\{\int_{...
- 721
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140
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Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?
Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form:
$$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$
where the ...
- 435
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Taut string algorithm and TV-minimization equivalence
Given real numbers $y_i's$, consider the following convex optimization problem:
$$
\min_{x_i's} \sum_{i=1}^N(y_i-x_i)^2 + \lambda\sum_{i=1}^{N-1}|x_{i+1}-x_{i}|.
$$
The paper A Direct Algorithm for 1D ...
- 1,216
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On the boundary integral of Neumann eigenfunctions
Let $v$ be an eigenfunction corresponding to the first nonzero Neumann Laplacian eigenvalue on a domain $\Omega \subset \mathbb{R}^2$. By definition, we know that $\int_{\Omega} v \, dx=0$. If $\Omega$...
- 1,350
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57
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Inequalities in special function cones
We consider the Banach space $X=C([0,1])$ endowed with the norm $\|v\|_{\infty}=\max _{t \in[0,1]}|v(t)|$ and, we define the cone
$\mathcal{C}=\{u \in X \mid u \mbox{ is concave, } u \geq 0, u(0)=u(1)=...
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Various definitions of weak solutions and continuity property
For example, consider the transport equation
$$
\partial_tu+b\cdot\partial_xu=0, u(0,x)=u_0(x),
$$
where $t\ge0,x\in\mathbb{R}^3$, $b\in W^{1,1}_{\rm{loc}}([0,\infty)\times\mathbb{R}^3)$.
It seems ...
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Derivatives of measures of bounded variation on intervals
Investigating an abstract Cauchy problem on the space of measures with bounded variation I came up with the following space:
Let $\operatorname{BV}[a,b]$ the space of all functions $f:[0, 1] \to \...
- 301
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81
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Compact imbedding for weight space
We begin with some definitions. Let $\gamma \geqslant 1,\,p \in \left[ {1,\infty } \right)$, we define
$$L_\gamma ^p\left( {0,1} \right) = \left\{ {v:\left( {0,1} \right) \to \mathbb{R}:{{\left\| v \...
- 183
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Properties of the Fourier Transform of Countably Supported Functions on $[0,1)$
Identifying $\mathbb{R}/\mathbb{Z}$ with the interval $\left[0,1\right)$, let $C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right)$ denote the set of all functions $f:\mathbb{R}/\mathbb{Z}\rightarrow\...
- 1,284
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Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
$\DeclareMathOperator\rad{rad}$
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ be compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
In $H_{\rad}^1(\mathbb{R}^3)$, by Struass estimate $|f(x)| \lesssim |x|^{-1} ...
- 429
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Is there a term for a linear operator on an $L^p$ space that "locally respects boundedness"?
Let $X$ be a Polish space, and $\mu$ a locally finite measure. Take any $p \in \{0\} \cup [1,\infty)$. We will say that a linear operator $T \colon L^p(\mu) \to L^p(\mu)$ has property $(\ast)$ if ...
- 708
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74
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Dimension dependence: boundedness result of the fractional Riesz integral
I am looking for the best known constant in the boundedness result of the fractional Riesz integral. In particular, I am interested in the dependence on the dimension $d$ and on the parameter $\alpha&...
- 778
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70
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Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix
Let $\phi:[-1,1] \to \mathbb R$ be a function such that
$\phi$ is $\mathcal C^\infty$ on $(-1,1)$.
$\phi$ is continuous at $\pm 1$.
For concreteness, and if it helps, In my specific problem I have $\...
- 6,853
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42
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Energy estimate for $\theta_t + H(\theta)_x = 0$ in $t>0, x >0$?
Consider the IBVP for $$\theta_t + H(\theta)_x = 0, \qquad t>0, \ x>0$$ with $$H(\theta) = \frac{1}{\pi} \text{pv}\int_{0}^\infty \frac{\theta(y)}{y-x} dy$$
with Dirichlet boundary conditions. ...
- 161
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64
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Embedding a normed space as a hyperplane
Let $X$ be a real normed space and suppose that $X$ is a closed hyperplane of a bigger space $\tilde X$. Given any unit vector $u$ in $\tilde X\setminus X$, consider the function $p:X\to\mathbb R$ ...
- 483
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45
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Generalizations of the Wiener Tauberian Theorem to Musielak-Orlicz spaces
Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm
$$
\|f\|_M:=\inf\left\{\lambda &...
- 5,405
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123
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Metric transforms that preserve $\ell^1$ embeddability
Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances.
Work by Schoenberg and ...
- 93
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119
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Integrable functions that may not satisfy the inversion Fourier formula
Let $f\in L^1(\mathbb{R})$. We define $\phi_f(x)=\int_{\mathbb{R}} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$ if the improper Riemann integral is finite otherwise, $\phi_f(x)=\infty$.
Does there exist ...
- 4,058
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81
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Examples of spaces which have explicit expression as colimits in $\mathrm{Top}$
$\DeclareMathOperator\Ball{Ball}$Question: What "well-known" spaces can be explicitly written down in the form $\bigcup_k \phi_k C(K_n,\mathbb{R}^m)$; where $K_n$ is a non-empty compact ...
- 5,405
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210
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The translation is continuous in $L^1(\mathbb{R}^n,d\mu)$, $d\mu=\frac{1}{1+|y|^{n+a}}dy$,$ a>0$
For any function $f\colon\mathbb{R}^n\to\mathbb{R}$, set: $\tau_hf(x):=f(x+h)$, $x,h\in\mathbb{R}^n$. Consider the following finite measure on $\mathbb{R}^n$:
$$\mu(A):=\int_A\frac{1}{1+|y|^{n+a}}\,dy$...
- 339
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74
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Fourier transform of a Sobolev function dependent on a "parameter"
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
- 339
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213
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Fractional Laplacian extension problem and uniqueness question
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. Consider the following problem:
$$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...
- 339