Operator-Valued Fourier Multipliers on Periodic Besov Spaces and Applications

Arendt, Wolfgang; Bu, Shangquan

doi:10.1017/s0013091502000378

Cited by 127 publications

(234 citation statements)

References 15 publications

Supporting

Mentioning

219

Contrasting

Unclassified

Order By: Relevance

“…This theorem generalizes a result of Arendt and Bu presented in [2]. As in [2], our multiplier theorem is also based on Marcinkiewicz type conditions.…”

Section: Introductionsupporting

confidence: 78%

“…In order to prove (20) and (21) we can use, due to former considerations, multiplier theorems for operators between Besov spaces. Using the techniques of [2], it is not difficult to prove the following generalization of a result presented there. Proof.…”

mentioning

confidence: 79%

“…The existence of such constants is equivalent to the uniform boundedness of the family {|λ|R(λ, −∂Φ(c))} Re λ ω ⊂ L(h 1+α (S 1 )). For details see [2]. From relation (26) we obtain…”

mentioning

confidence: 95%

See 2 more Smart Citations

A moving boundary problem for periodic Stokesian Hele–Shaw flows

Escher

Matioc²

2009

Interfaces Free Bound.

View full text Add to dashboard Cite

This paper is concerned with the motion of an incompressible, viscous fluid in a Hele-Shaw cell. The free surface is moving under the influence of gravity and the fluid is modelled using a modified Darcy law for Stokesian fluids.We combine results from the theory of quasilinear elliptic equations, analytic semigroups and Fourier multipliers to prove existence of a unique classical solution to the corresponding moving boundary problem.

show abstract

“…This theorem generalizes a result of Arendt and Bu presented in [2]. As in [2], our multiplier theorem is also based on Marcinkiewicz type conditions.…”

Section: Introductionsupporting

confidence: 78%

mentioning

confidence: 79%

See 1 more Smart Citation

A moving boundary problem for periodic Stokesian Hele–Shaw flows

Escher

Matioc²

2009

Interfaces Free Bound.

View full text Add to dashboard Cite

show abstract

“…Below, we briefly recall the definition of periodic Besov spaces in vector-valued case introduced in [3]. For the scalar case, see [10,Chapter 9] and [9].…”

Section: Introductionmentioning

confidence: 99%

“…Fourier multiplier theorems on B s pq (T; X) were recently studied in [3] motivated by the maximal regularity of periodic solutions for the Cauchy problems of first and second order.…”

Section: Introductionmentioning

confidence: 99%

Solutions of Second-Order Integro-Differential Equations on Periodic Besov Spaces

Poblete¹

2007

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

show abstract

A note on operator‐valued Fourier multipliers on Besov spaces

Kim

2005

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

Let X be a Banach space. We show that each m : R \ {0} → L(X) satisfying the Mikhlin condition sup x =0 ( m(x) + xm (x) ) < ∞ defines a Fourier multiplier on B In recent years, operator-valued Fourier multipliers on vector-valued function spaces have been extensively studied [1,3,10,16]. They are needed to establish existence and uniqueness as well as regularity for differential equations in Banach spaces. In the L p spaces case, it was Weis who found the formulation of the operator-valued Fourier multiplier theorem: let X be a Banach space and let m : R \ {0} → L(X) be a C 1 -function, assume that X is a UMD space (see [5,7]), 1 < p < ∞ and that both sets {m(x) : x = 0} and {xm (x) :where we denote by L(X) the set of all bounded linear operators on X. It is known that in Weis' result the R-boundedness may be replaced by the norm-boundedness only when X is isomorphic to a Hilbert space [4].In the vector-valued Besov spaces case, things are different. It was Amann [1] (see also Weis [17]) who discovered that the assumption of R-boundedness is not needed and no hypothesis on the geometry of the underlying Banach space is required in the Besov spaces case: let m : R → L(X) be a C 2 -function, assume thatthen for 1 ≤ p, q ≤ ∞, s ∈ R, m defines a Fourier multiplier on B s p,q (R; X) (see also [10] for further development, and [2] for the corresponding result in the periodic Besov spaces case).In this paper, we are interested in the following questions: what happens when we consider a larger class of multiplier functions m in the Besov spaces case? In view of known Fourier multiplier results in the scalar L p spaces case, one may consider the following two natural classes: the first one is the set of all

show abstract

Operator-Valued Fourier Multipliers on Periodic Besov Spaces and Applications

Cited by 127 publications

References 15 publications

A moving boundary problem for periodic Stokesian Hele–Shaw flows

A moving boundary problem for periodic Stokesian Hele–Shaw flows

Solutions of Second-Order Integro-Differential Equations on Periodic Besov Spaces

A note on operator‐valued Fourier multipliers on Besov spaces

Contact Info

Product

Resources

About