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Maximal Regularity and the Method of Fourier Multipliers

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Regularity of Difference Equations on Banach Spaces

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Abstract

Difference equations in a Banach space X of the form

$$\displaystyle{ \Delta u(n) = Au(n) + f(n) }$$
(2.0.1)

arise in several branches of mathematical physics and engineering.

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Notes

  1. 1.

    The usefulness of the fixed-point methods for applications has increased enormously by the development of efficient techniques for computing fixed points. In fact, nowadays, fixed-point arguments have become a powerful weapon in the arsenal of applied mathematicians [102].

  2. 2.

    Suppose that X and Y are Banach spaces, then X is finitely representable in Y if for all finite dimensional subspaces E of X and all λ > 1, there is a linear map T: E → Y such that, for all x ∈ E,

    $$\displaystyle{\lambda ^{-1}\vert \vert x\vert \vert _{ X} \leq \vert \vert Tx\vert \vert _{Y } \leq \lambda \vert \vert x\vert \vert _{X}.}$$

    A Banach space X is super-reflexive if it is reflexive and every Banach space that is finitely representable in X is also reflexive. This concept was introduced and studied by James in [104–106] (see [33] for additional comments).

  3. 3.

    It is clear that the R-boundedness of the countable set \(\mathcal{T}\) is independent of the order in which we enumerate its element. Thus it is interesting to note that, given any enumeration, the subset of n first members of the sequence are fully representative of all finite subsets of \(\mathcal{T}\) in view of R-boundedness.

  4. 4.

    For a bounded collection \(\mathcal{T} \subset \mathcal{B}(X,Y )\) we denote the Minkowski functional of \(abco(\mathcal{T} )\) by \(\vert \vert \cdot \vert \vert _{\mathcal{T}}: \mathcal{B}(X,Y ) \rightarrow [0,\infty ]\), \(T \rightarrow \vert \vert T\vert \vert _{\mathcal{T}} =\inf \{ t > 0: T \in t \cdot abco(\mathcal{T} )\}\) (see [174]).

  5. 5.

    The first positive result on maximal L p-regularity was obtained by Ladyzhenskaya, Solonnikov, and Ural’tseva [128], where X = L p(G), \(G \subset \mathbb{R}^{n}\) being a bounded domain with smooth boundary, A a strongly elliptic second- order differential operator with continuous coefficients, and 1 < p < ∞. The first abstract result was obtained by de Simon [62] for Hilbert spaces. Specifically, let H be a Hilbert space and A be the generator of an analytic semigroup. Then (2.3.1) has maximal L p-regularity on [0, ∞). De Simon’s proof employ Plancherel’s theorem which is known to be valid only in the Hilbert space case (see [65, 68]).

  6. 6.

    Let X be a Banach space, \((x_{k})_{k\in \mathbb{N}} \subset X\) is called a Schauder basis if, for every x ∈ X, there is a unique sequence \((a_{k})_{k\in \mathbb{N}} \subset \mathbb{C}\) such that \(x =\sum _{ k=1}^{\infty }a_{k}x_{k}\). It is called an unconditional basis if the series converges unconditionally.

  7. 7.

    A proof of J. Schwartz’s result (Theorem 2.4.2) using the Calderon–Zygmund method can be found in [18].

  8. 8.

    \(\mathcal{D}^{\prime}([0, 2\pi ]; X)\) is the set of all linear mappings \(T\) from \(\mathcal{D}([0, 2\pi ])\) into \(X\) such that \(\vert \vert T(f)\vert \vert _{X} \leq C\sum _{n\leq N}\sup _{t\in [0,2\pi ]}\vert f^{(n)}(t)\vert \) for all \(f \in \mathcal{D}([0, 2\pi ])\) and for some \(N \in \mathbb{N}\) and C > 0 independent of f. Elements in \(\mathcal{D}^{\prime}([0, 2\pi ]; X)\) are called X-valued distributions on \([0, 2\pi ]\). We use the weak topology on \(\mathcal{D}^{\prime}([0, 2\pi ]; X)\), i.e., a sequence \(T_{k}\) converges to \(T\) in \(\mathcal{D}^{\prime}([0, 2\pi ]; X)\) if and only if \(\lim _{k\rightarrow \infty }T_{k}(f) = T(f)\) for all \(f \in \mathcal{D}([0, 2\pi ])\).

  9. 9.

    We recall that a Banach space X is B-convex if it does not contain l 1 n uniformly. This is equivalent to saying that X has Fourier type 1 < p ≤ 2, i.e., the Fourier transform is a bounded linear operator from \(L^{p}(0, 2\pi; X)\) into \(l^{q}(\mathbb{Z},X)\) where 1∕p + 1∕q = 1.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, Kingsville, TX, USA

    Ravi P. Agarwal

  2. Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, Brazil

    Claudio Cuevas

  3. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Santiago, Chile

    Carlos Lizama

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Agarwal, R.P., Cuevas, C., Lizama, C. (2014). Maximal Regularity and the Method of Fourier Multipliers. In: Regularity of Difference Equations on Banach Spaces. Springer, Cham. https://doi.org/10.1007/978-3-319-06447-5_2

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