Maximal L p -regularity for Parabolic Equations, Fourier Multiplier Theorems and $H^\infty$ -functional Calculus

2004 ◽  
pp. 65-311 ◽  
Author(s):  
Peer C. Kunstmann ◽  
Lutz Weis
Keyword(s):  
Parabolic Equations ◽  
Functional Calculus ◽  
Fourier Multiplier ◽  
Multiplier Theorems

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON TRIEBEL SPACES

2005 ◽  
Vol 25 (4) ◽  
pp. 599-609 ◽  
Author(s):  
Shangquan Bu ◽  
Jin-Myong Kim
Keyword(s):  
Fourier Multiplier ◽  
Multiplier Theorems

scholarly journals Operator-valued multiplier theorems characterizing Hilbert spaces

2004 ◽  
Vol 77 (2) ◽  
pp. 175-184 ◽  
Author(s):  
Wolfgang Arendt ◽  
Shangquan Bu
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Fourier Multiplier ◽  
Multiplier Theorem ◽  
Multiplier Theorems

AbstractWe show that the operator-valued Marcinkiewicz and Mikhlin Fourier multiplier theorem are valid if and only if the underlying Banach space is isomorphic to a Hilbert space.

scholarly journals On vector-valued Fourier multiplier theorems

1989 ◽  
Vol 93 (3) ◽  
pp. 201-222 ◽  
Author(s):  
Frank Zimmermann
Keyword(s):  
Fourier Multiplier ◽  
Multiplier Theorems ◽  
Vector Valued

scholarly journals The Holomorphic Hörmander Functional Calculus

2019 ◽  
Author(s):  
Florian Pannasch
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Functional Calculus ◽  
Differential Operators ◽  
Uniform Boundedness ◽  
General Context ◽  
Multiplier Theorem ◽  
Integral Means ◽  
Imaginary Powers ◽  
Multiplier Theorems

TThe topic of this thesis is functional calculus in connection with abstract multiplier theorems. In 1960, Hörmander showed how the uniform boundedness of certain integral means of a function m in L ∞ (R^d) and its weak derivatives imply that m yields a bounded Lp -Fourier multiplier. Nowadays, this is known as the Hörmander multiplier theorem, sometimes Hörmander--Mikhlin multiplier theorem. A noteworthy detail is that a radial function m(|x|) satisfies Hörmander's condition if and only if m (|x|²) does. Hence, Hörmander's theorem is also a result on the functional calculus of the negative Laplacian -Δ. Hörmander's result has inspired a lot of research, and authors have also proven similar results for other operators such as certain Schrödinger operators, Sublaplacians on Lie groups, and later certain differential operators on spaces of homogeneous type. For us, the work of Kriegler and Weis is of particular interest. Starting with the PhD thesis of Kriegler in 2009, they showed how abstract multiplier theorems can be proven in a more general context. Namely, considering a certain class of 0-sectorial and 0-strip type operators on a general Banach space, one can construct an abstract Hörmander functional calculus based on the classical holomorphic calculus. Then, by using probalistic techniques from Banach space geometry involving so-called R-boundedness one can derive multiplier results in this generalized setting. In 2001, García-Cuerva, Mauceri, Meda, Sjögren, and Torrea proved an abstract multiplier theorem for generators of symmetric contraction semigroups, where a bounded Hörmander calculus is inferred from growth conditions on the imaginary powers of the generator. As the considered operators need not be 0-sectorial, this result is not covered by the methods of Kriegler and Weis. However, the result is based on Meda's earlier work, where he derived a bounded Hörmander if the given imaginary powers only grow polynomially fast. In this case, the operator is 0-sectorial, and Kriegler and Weis were able to recover the result while improving the order of the calculus. In this thesis, we introduce a generalized class of Hörmander functions defined on strips and sectors. Based on this and the classical holomorphic calculus, we construct a holomorphic Hörmander calculus for a class of operators which may also have strip type or angle of sectoriality greater than zero. The main result is a generalization of the multiplier theorem of García-Cuerva et al. to Banach spaces of finite cotype and Banach spaces with Pisier's property (α), where we retain and even improve the order given by Kriegler and Weis for the 0-sectorial case.

Well-posedness of fractional degenerate differential equations in Banach spaces

2019 ◽  
Vol 22 (2) ◽  
pp. 379-395
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Differential Equation ◽  
Fourier Multiplier ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Well Posedness ◽  
Multiplier Theorems ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Closed Linear Operators

Abstract We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.

scholarly journals Fourier Multiplier Theorems Involving Type and Cotype

2017 ◽  
Vol 24 (2) ◽  
pp. 583-619 ◽  
Author(s):  
Jan Rozendaal ◽  
Mark Veraar
Keyword(s):  
Fourier Multiplier ◽  
Type And Cotype ◽  
Multiplier Theorems
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