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Results tagged with fa.functional-analysis
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user 99469
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
4
votes
0
answers
78
views
Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t...
$
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- 825
2
votes
1
answer
211
views
Self-adjointness of generator and semigroup of an SDE
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- 6,696
0
votes
1
answer
84
views
Is the evolution family self-adjoint?
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I am reading Roland Schnaubelt's survey p …
- 6,696
2
votes
1
answer
203
views
Hölder continuity in time of heat semigroup for regular initial distribution
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Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \exp …
- 578
3
votes
1
answer
256
views
Hölder continuity in time of heat semigroup
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We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|_{\ …
- 1,190
1
vote
1
answer
167
views
Rate of convergence of mollified functions in $L^p$ norm
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Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a probabi …
0
votes
1
answer
95
views
Convergence of mollified functions in weighted $L^p$ norm
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Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a probabi …
- 5,005
2
votes
2
answers
149
views
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \...
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We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ is …
- 825
0
votes
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \...
$
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Let $d:=1, Z \sim \mathcal N (0, 1)$ and $\ell := p_1$. We write $M_1 \lesssim M_2$ if there exi …
- 825
0
votes
0
answers
44
views
Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\...
We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x- …
- 825
1
vote
2
answers
86
views
Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $...
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f ( …
- 60.6k
1
vote
1
answer
54
views
Lower bound the best $\alpha$-Hölder constant of a convolution
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f ( …
- 60.6k
2
votes
1
answer
269
views
Decompose a function into a bounded part and a Lipschitz part
Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that
$$
\sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty.
$$
Are there functions $g,h: \mathbb R^d \to \ …
- 42.8k
2
votes
1
answer
153
views
Are these two norms on localized versions of $L^p_q$ equivalent?
$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$.
Let $E$ be the space of all real-valued …
2
votes
0
answers
109
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{...
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R^ …
- 39.8k