Operator Valued Fourier Multipliers

Results on Fourier multipliers are important tools in the study of partial differential equations. They represent a major step, for example, when establishing a priori estimates for solutions of parabolic evolution equations of Agmon-Douglis-Nirenberg type [1]. Among the results known, the classical theorem of Mikhlin, which guarantees the boundedness of translations-invariant operators on L p (ℝ n ), is of special importance.

[1]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[2]  M. Christ Lectures on singular integral operators , 1991 .

[3]  Herbert Amann,et al.  Operator‐Valued Fourier Multipliers, Vector ‐ Valued Besov Spaces, and Applications , 1997 .

[4]  J. Schwartz A remark on inequalities of Calderon-Zygmund type for vector-valued functions , 1961 .

[5]  A. Calderón,et al.  CONVOLUTION OPERATORS ON BANACH SPACE VALUED FUNCTIONS. , 1962, Proceedings of the National Academy of Sciences of the United States of America.

[6]  J. Bourgain Extension of a result of Benedek, Calderón and Panzone , 1984 .

[7]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[8]  H. Triebel Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[9]  Giovanni Dore,et al.  On the closedness of the sum of two closed operators , 1987 .

[10]  J. Prüss Evolutionary Integral Equations And Applications , 1993 .

[11]  P. Grisvard Elliptic Problems in Nonsmooth Domains , 1985 .

[12]  Herbert Amann,et al.  Linear and Quasilinear Parabolic Problems , 2019, Monographs in Mathematics.

[13]  L. Hörmander,et al.  Estimates for translation invariant operators inLp spaces , 1960 .

[14]  J. Torrea,et al.  Calderón-Zygmund theory for operator-valued kernels , 1986 .

[15]  T. McConnell,et al.  On Fourier multiplier transformations of Banach-valued functions , 1984 .

[16]  B. Schulze Pseudo-Differential Operators on Manifolds with Singularities , 1991 .

[17]  E. Stein Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. , 1970 .

[18]  F. Zimmermann On vector-valued Fourier multiplier theorems , 1989 .

[19]  Matthias Hieber,et al.  Integrated semigroups and differential operators onLp spaces , 1991 .