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 $...
Giorgio Metafune's user avatar
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}...
Pietro Majer's user avatar
  • 55.1k
8 votes
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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 ...
András Bátkai's user avatar
7 votes

Existence of closed operators with arbitrary dense domain of a given Banach space

This cannot happen, if, e.g., $Y$ is a countable union of finite dimensional subspaces (by the Baire category theorem). Suitable examples are given by the spaces of finite sequences in $\ell^p$. ...
shasta's user avatar
  • 186
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 ...
András Bátkai's user avatar
7 votes
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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-...
Wlod AA's user avatar
  • 4,666
7 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=...
Denis Serre's user avatar
  • 50.5k
7 votes
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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 ...
Willie Wong's user avatar
  • 35.2k
7 votes
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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 ...
Jochen Glueck's user avatar
6 votes
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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_\...
Jochen Glueck's user avatar
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 \...
Giorgio Metafune's user avatar
6 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 ...
Mateusz Kwaśnicki's user avatar
6 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 ...
Piero D'Ancona's user avatar
6 votes
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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(...
Mateusz Kwaśnicki's user avatar
6 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 ...
Jochen Glueck's user avatar
6 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. ...
Jochen Glueck's user avatar
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 ...
András Bátkai's user avatar
5 votes
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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}(...
Hengrong  Du's user avatar
5 votes

Is every $C_0$ semigroup on a Hilbert space automatically a $C_0$ group on a larger space?

An operator $A$ is a group generator if and only if both $A$ and $-A$ are semigroup generators. So under your assumptions, if we denote by $\tilde{A}$ the generator of the extension $\{\tilde{T}(t),t\...
Delio Mugnolo's user avatar
5 votes
Accepted

Existence of closed operators with arbitrary dense domain of a given Banach space

You remarked in comments that $T$ should be an unbounded operator from $X$ to $X$. If $X$ is separable then $D(T)$ has to be a Borel set in $X$. Note that $X \times X$ is a separable Banach space ...
Nate Eldredge's user avatar
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, ...
András Bátkai's user avatar
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 ...
Carlo Beenakker's user avatar
5 votes
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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 ...
Jochen Glueck's user avatar
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 ...
András Bátkai's user avatar
4 votes

A generalisation of $C_0$-semigroups

This does not answer your question only adds something to your list of examples: two parameter semigroups, as investigated by Slocinski, M. Unitary dilation of two-parameter semi-groups of ...
András Bátkai's user avatar
4 votes
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Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ generates?

Yes, there is a characterization like this. See Theorems 1-3 (p.182) in Markus Haase, MR 2183483 A functional calculus description of real interpolation spaces for sectorial operators, Studia Math. ...
András Bátkai's user avatar
4 votes

$L^p$–$L^q$ estimates for heat equation - regularizing effect

This is standard, but the argument is short enough to fit in an answer. It is not restrictive to assume that $0\in\Omega$. Denote by $Q(t,x,y)$ the heat kernel associated with the Dirichlet Laplacian ...
Piero D'Ancona's user avatar
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) $\...
Mateusz Kwaśnicki's user avatar
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 \...
Jochen Glueck's user avatar
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}\...
Jochen Glueck's user avatar

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