9
votes
Gradient $L^p$ estimates for heat equation
Let me comment on what I know in an open set $\Omega$, trying to control $C$. First of all, the heat semigroup can be expressed through a kernel $p$ which is pointwise dominated by the heat kernel in $...
- 6,025
8
votes
Accepted
Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?
In other words, if $A$ is the infinitesimal generator of $T$, the mild solution of the abstract inhomogeneous Cauchy problem
$$\begin{cases}\dot v =A v +f\\v(0)=0\end{cases}$$
needs not to be $W^{1,1}...
- 60.5k
8
votes
Accepted
Literature request: Schatten class difference of semigroups
The problem is discussed in a more general setting (operator ideals in Banach spaces) for the so-called analytic semigroups (parabolic problems) in
Blunck, S.; Weis, L., Operator theoretic properties ...
- 4,729
8
votes
The contractivity of the heat semigroup in $L^p$ spaces
For the record, the result requested in the question is given in Theorem 1.3.3 in: E.B. Davies, Heat Kernels and Spectral Theory, DOI:10.1017/CBO9780511566158.
The assumptions are:
$L$ is a positive ...
- 17.2k
8
votes
What is dispersive estimate?
The intutitive picture is the following: there is a certain amount of "mass", which at time $t=0$ you might imagine concentrated in a heap near a location $x=0$. This heap evolves with time ...
- 9,007
8
votes
What is dispersive estimate?
To complete Piero's answer, let me give a definition of dispersiveness. To stay at a rather simple level, let me consider an linear evolution PDE with constant coefficients,
$$P(\partial_t,\nabla_x)u=...
- 52.3k
8
votes
Accepted
Contractivity of Neumann Laplacian
It might be helpful to point out the following conceptual reasons why ultracontractivity estimates are mainly interesting for times close to $0$.
Let us consider the following general setting: We have ...
- 12.5k
7
votes
The Bochner integral about a semigroup of bounded linear operators on a Banach space
Following ideas of John von Neumann,
J. von Neumann, Über einen Satz von Herrn M. H. Stone, Ann. Math. (2) 33, 567-573 (1932). ZBL0005.16402,
it can be shown that if a function satisfies the ...
- 4,729
7
votes
Accepted
Fractional powers of an operator
This question was considered by Functional Analysis specialists around 1960 or earlier. In the case of Banach Algebras $C(X)$ (for Hausdorff compact $X$) this gets reduced often to studying the auto-...
- 4,776
7
votes
Accepted
Vacuum region with positive measure for the Schrödinger equation
The purpose of this answer is to extend Christian Remling's answer to dimension $d = 3$. There are two steps. (N.B. below the cut I show how to replace part 1 by a different argument that works in all ...
- 39k
7
votes
What's the name of this semi-group theorem?
I don't think that there is any specific name for this result; it is typically considered a standard fact for the matrix exponential function or, more generally, for the operator exponential function.
...
- 12.5k
7
votes
Accepted
Strong positivity of Neumann Laplacian
As other users have indicated in the comments, for sufficiently smooth domains one can get it by combining, for instance, elliptic regularity with Hopf's boundary point lemma (and then go from the ...
- 12.5k
6
votes
Accepted
Angle of analyticity of semigroup
Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by $\Delta+Bx \cdot \...
- 6,025
6
votes
Accepted
Does uniform boundedness carry over from the non-negative real axis to closed sectors of $\mathbb{C}$ for analytic semigroups?
There is a one-dimensional counterexample: Consider the analytic semigroup $z \mapsto e^{iz}$. This semigroup is bounded on the non-negative real line, but it is not bounded on any sector $\Delta_\...
- 12.5k
6
votes
Accepted
Asymptotic formula for fractional Laplacian
Probabilistic approach
I have to think about the right reference, but things are very much different in the non-local case.
The solution $u = u^\epsilon$ can be written in probabilistic terms as
$$ u(...
- 17.2k
6
votes
Accepted
Koopman operators on $L^p(X)$
Yes, $C_0$-semigroups of Koopman operators and, more generally, lattice homomorphisms on $L^p$-spaces are studied in the paper "Measure-preserving semiflows and one-parameter Koopman semigroups&...
- 12.5k
5
votes
Generator of an analytic semigroup of operators
As you write it, this is just a bounded perturbation of the sectorial operator
$$\begin{pmatrix} 0 & 0 & 0 \\ 0 & \Delta & 0 \\ 0 & 0 & 0 \end{pmatrix}.$$
It is quite ...
- 4,729
5
votes
Accepted
Generation of strict contraction semigroups
Your conditions is for contraction semigroups equivalent to have uniform exponential stability, i.e., to have growth bound less than zero, see Proposition V.1.7. in
Engel, Klaus-Jochen; Nagel, Rainer, ...
- 4,729
5
votes
Accepted
Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalent norms?
First, the classical Calderón–Zygmund estimate gives
$$\left\|D^2 u\right\|_{L^p(\mathbb{R}^d)}\leq C(p,d) \left\|\Delta u\right\|_{L^p(\mathbb{R}^d)}.$$
By interpolation, $\left\|u\right\|_{W^{2,p}(...
- 166
5
votes
The contractivity of the heat semigroup in $L^p$ spaces
I will sketch the construction for a general Riemannian Manifold, thought as a triple $(M,d_g,Vol_g)$ (Smooth manifold $M$, Riemannian Metric $d_g$, volume form $Vol_g$) and at the end provide ...
- 136
5
votes
What is dispersive estimate?
The word "dispersion" means "spread". In the context of wave equations it refers to the spreading (broadening) of a wave packet. The estimate in the OP is called dispersive because ...
- 188k
5
votes
Accepted
Neumann/Robin Laplacian semigroup well-known estimate
The estimate the OP is looking for is called an ultracontractivity estimate. A characterisation of semigroups that satisfy such an estimate can be found in the Theorem on page 65, Subsection 7.3.2, of ...
- 12.5k
5
votes
Accepted
approximation of a Feller semi-group with the infinitesimal generator
You look for the concept of analytic (or entire) vector. If you have a strongly continuous group, then they are dense, otherwise it might happen that there is only the 0 vector (for nilpotent ...
- 4,729
5
votes
Accepted
Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup
Check Chapter 4, more specifically Chapter 4.2 in Ouhabaz: Analysis of heat equations on domains (2005). ZBL1082.35003. The approach there goes by the form method which I personally perceive to be ...
- 2,670
5
votes
Accepted
Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard
For the first "why": that $\hat{v}(\tau)$ is holomorphic is a version of Paley-Wiener. On the domain $\mathbb{C}_\alpha$ you have $\exp(i\tau t)$ is uniformly bounded (for any $t\in \mathbb{...
- 39k
5
votes
The contractivity of the time derivative of the heat semigroup in $L^p$ spaces
The fact that $t\partial_t e^{-tL}=-tLe^{-tL} $ is bounded in $L^p$ for $1<p<+\infty$ follows from analyticity of the heat semigroup on $L^p$; see the paper Harmonic Analysis on semigroups by ...
- 2,703
4
votes
Generator of Wiener process and its running maximum
My guess would be indeed $\tfrac{1}{2} \partial_{yy}$, with $f$ in the domain if and only if:
(a) $f$ is in $C_0(\mathbb{R}_+^2)$;
(b) $f(x, \cdot)$ is $C_0^2(\mathbb{R}_+)$ for each $x$;
(c) $\...
- 17.2k
4
votes
Accepted
Spectral representation of closed operators with finite spectral bound
I've looked it up now. The formula in question does indeed hold in the following sense:
Theorem. Let $(e^{tA})_{t \in [0,\infty)}$ be a $C_0$-semigroup on a complex Banach space $X$. Let $\omega \in \...
- 12.5k
4
votes
Accepted
Equality in spectral inclusion theorem
Here is a positive solution if the semigroup is eventually compact:
Consider, say, the open right halfplane
$$
H := \{\lambda \in \mathbb{C}: \, \operatorname{Re}\lambda > \operatorname{Re}\...
- 12.5k
4
votes
Accepted
On a property of resolvents associated with holomorphic semigroups
As pointed out in the comments, this is true for $Re\, \lambda >0$ but can fail in general. An example is the Ornstein-Uhlenbeck operator $L=D^2-xD$ in $L^2(e^{-x^2/2}\, dx)$. The $L^2$ spectrum ...
- 6,025
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