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69
On Operator–valued Fourier Multiplier Theorems
"... Abstract. We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal Lp–regularity for the generator of an analytic semigroup Tt in ter ..."
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Cited by 143 (15 self)
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Abstract. We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal Lp–regularity for the generator of an analytic semigroup Tt in terms of the R–boundedness of the resolvent of A or the semigroup Tt. 1.
the H ∞ calculus and sums of closed operators
 LeM] [Mar] [McI] [PT] [She] [Ste] [ST] [Wei] C. Le
"... Abstract. We develop a very general operatorvalued functional calculus for operators with an H ∞ −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H ∞ calculus. Using this we prove theorem of DoreVenni type on sums of commuting sect ..."
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Cited by 85 (15 self)
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Abstract. We develop a very general operatorvalued functional calculus for operators with an H ∞ −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H ∞ calculus. Using this we prove theorem of DoreVenni type on sums of commuting sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the Rboundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of L1−spaces and C(K)−spaces, to obtain some more detailed results in this setting. 1.
Maximal Regularity for Evolution Equations in L_pSpaces
, 2002
"... this paper maximal L p regularity is shown in case the semigroup e generated by \GammaA admits heat kernel estimates. The proof relies on CalderonZygmund theory, Interpolation and on the theorem of Benedek, Calder'on and Panzone. In Pruss [18] this approach is generalized to fractional evo ..."
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Cited by 39 (8 self)
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this paper maximal L p regularity is shown in case the semigroup e generated by \GammaA admits heat kernel estimates. The proof relies on CalderonZygmund theory, Interpolation and on the theorem of Benedek, Calder'on and Panzone. In Pruss [18] this approach is generalized to fractional evolution equations of parabolic type, using Poisson estimates for the kernel of the resolvent ( + A) of A rather than heat kernel estimates
The Domain of the OrnsteinUhlenbeck Operator on an L^pSpace with Invariant Measure
, 2001
"... We show that the domain of the OrnsteinUhlenbeck operator on ; dx) equals the weighted Sobolev space W ; dx), where dx is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative DoreVenni theorems. ..."
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Cited by 34 (5 self)
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We show that the domain of the OrnsteinUhlenbeck operator on ; dx) equals the weighted Sobolev space W ; dx), where dx is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative DoreVenni theorems.
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 22 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
A theorem of DoreVenni type for noncommuting operators
 Trans. Amer. Math. Soc
, 1997
"... Abstract. A theorem of the DoreVenni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations _u(t) + L(t)u(t ..."
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Cited by 18 (3 self)
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Abstract. A theorem of the DoreVenni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations _u(t) + L(t)u(t) = f(t) and evolutionary integral equations u(t) + R t 0 a(t − s)L(s)u(s)ds = g(t) which are nonautonomous. The domains of the involved operators L(t) may depend on t, but L(t)−1 is required to satisfy a certain smoothness property. The results are then applied to parabolic partial dierential and integrodierential equations. 1.
Maximal regularity for nonautonomous evolution equations
 Adv. Nonlinear Stud
"... We derive sufficient conditions, perturbation theorems in particular, for nonautonomous evolution equations to possess the property of maximal Lp regularity. 1991 Mathematics Subject Classification. 35K90, 47D06. Key words. Maximal regularity, perturbation theorems, nonautonomous parabolic evolutio ..."
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Cited by 18 (5 self)
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We derive sufficient conditions, perturbation theorems in particular, for nonautonomous evolution equations to possess the property of maximal Lp regularity. 1991 Mathematics Subject Classification. 35K90, 47D06. Key words. Maximal regularity, perturbation theorems, nonautonomous parabolic evolution equations.
IMAGINARY POWERS OF LAPLACE OPERATORS
"... Abstract. We show that if L is a secondorder uniformly elliptic operator in divergence form on R d, then C1(1 + α) d/2 ≤ ‖Liα‖L1→L1,∞ ≤ C2(1 + α) d/2. We also prove that the upper bounds remain true for any operator with the finite speed propagation property. 1. Introduction. Assume that aij ∈ ..."
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Cited by 18 (4 self)
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Abstract. We show that if L is a secondorder uniformly elliptic operator in divergence form on R d, then C1(1 + α) d/2 ≤ ‖Liα‖L1→L1,∞ ≤ C2(1 + α) d/2. We also prove that the upper bounds remain true for any operator with the finite speed propagation property. 1. Introduction. Assume that aij ∈ C ∞ (R d), aij = aji for 1 ≤ i, j ≤ d and that κI ≤ (aij) ≤ τI for some positive constants κ and τ. We define a positive selfadjoint operator L on L 2 (R d) by the formula (1) L = − ∑ ∂iaij∂j. We refer readers to [8] for the precise definition and basic properties of L. In particular, L
A solution to the problem of Lpmaximal regularity
"... Abstract. We give a negative solution to the problem of the Lpmaximal regularity on various classes of Banach spaces including Lqspaces with 1 < q 6 = 2 < +∞. 1. ..."
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Cited by 17 (2 self)
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Abstract. We give a negative solution to the problem of the Lpmaximal regularity on various classes of Banach spaces including Lqspaces with 1 < q 6 = 2 < +∞. 1.
Maximal Lpregularity for stochastic evolution equations
 SIAM J. Math. Anal
"... Abstract. We prove maximal Lpregularity for the stochastic evolution equation{ dU(t) +AU(t) dt = F (t, U(t)) dt+B(t, U(t)) dWH(t), t ∈ [0, T], U(0) = u0, under the assumption that A is a sectorial operator with a bounded H∞calculus of angle less than 1 2 pi on a space Lq(O, µ). The driving proce ..."
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Cited by 16 (8 self)
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Abstract. We prove maximal Lpregularity for the stochastic evolution equation{ dU(t) +AU(t) dt = F (t, U(t)) dt+B(t, U(t)) dWH(t), t ∈ [0, T], U(0) = u0, under the assumption that A is a sectorial operator with a bounded H∞calculus of angle less than 1 2 pi on a space Lq(O, µ). The driving process WH is a cylindrical Brownian motion in an abstract Hilbert space H. For p ∈ (2,∞) and q ∈ [2,∞) and initial conditions u0 in the real interpolation space DA(1 − 1p, p) we prove existence of unique strong solution with trajectories in