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Search: All articles in the CJM digital archive with keyword maximal regularity

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1. CJM 2007 (vol 59 pp. 1207)

Bu, Shangquan; Le, Christian
$H^p$-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces
We consider maximal regularity in the $H^p$ sense for the Cauchy problem $u'(t) + Au(t) = f(t)\ (t\in \R)$, where $A$ is a closed operator on a Banach space $X$ and $f$ is an $X$-valued function defined on $\R$. We prove that if $X$ is an AUMD Banach space, then $A$ satisfies $H^p$-maximal regularity if and only if $A$ is Rademacher sectorial of type $<\frac{\pi}{2}$. Moreover we find an operator $A$ with $H^p$-maximal regularity that does not have the classical $L^p$-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces $H^p(\R;X)$, in the case when $X$ is an AUMD Banach space.

Keywords:$L^p$-maximal regularity, $H^p$-maximal regularity, Rademacher boundedness
Categories:42B30, 47D06

2. CJM 2005 (vol 57 pp. 771)

Schrohe, E.; Seiler, J.
The Resolvent of Closed Extensions of Cone Differential Operators
We study closed extensions $\underline A$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted $L_p$-space. Under suitable conditions we show that the resolvent $(\lambda-\underline A)^{-1}$ exists in a sector of the complex plane and decays like $1/|\lambda|$ as $|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underline A$. As an application we treat the Laplace--Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$.

Keywords:Manifolds with conical singularities, resolvent, maximal regularity
Categories:35J70, 47A10, 58J40

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