Partial differential relations by Gromov M. PDF

By Gromov M.

ISBN-10: 0387121773

ISBN-13: 9780387121772

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Then A has a bounded H ∞ (Σϕ )-calculus for each ϕ > π/2. This result can be extended to semigroups on Lp which are dominated by a positive contraction semigroup. 1) by an interesting dilation theorem due to Fendler extending the Akcoglu dilation theorem for single operators to semigroups. We give more details. 4. r-contractive semigroups on Lp . Let S be an operator on Lp (Ω), 1 p ∞. , |Sf | T |f |, f ∈ Lp (Ω), where |f |(x) = |f (x)|, x ∈ Ω. In that case there exists a smallest positive operator |S| which dominates S.

Let ψA > ϕbip (A), ψB > ϕbip(B) be angles such that ψA + ψB < π . Assume that there exist constants 0 α < β < 1 and c 0 such that AR(λ, A) A−1 R(μ, B) − R(μ, B)A−1 c (1 + |λ|1−α )|μ|1+β for all λ ∈ Σπ−ψA and all μ ∈ Σπ−ψB . Then there exists ω ∈ R such that A + B + ω is closed and sectorial. This result can be applied to Volterra equations [MP97], to nonautonomous evolution equations [HM00a,HM00b] and also to prove maximal regularity of the Ornstein– Uhlenbeck operator [MPRS00]. 10. Interpolation spaces and BIP.

3. Characterization of BIP. Let A be a sectorial injective operator on X. 6). T HEOREM . The following assertions are equivalent: (i) D(A) ∩ R(A) is dense and Ais ∈ L(X) for all s ∈ R. (ii) The operator i log A generates a C0 -group U . In that case U (s) = Ais , s ∈ R. D EFINITION . We say that A has bounded imaginary powers and write A ∈ BIP, if A is injective, sectorial and if the two equivalent conditions of Theorem are satisfied. Note that this includes the hypothesis that D(A) ∩ R(A) be dense in X.

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Partial differential relations by Gromov M.


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