If fX is Continuous Seminorm Then F is Continuous

Seminorm

A seminorm defined on a Banach space X is a nonnegative real-valued functional |x| defined on H that satisfies the properties |αx| = |α| |x| and |x + y| ⩽ |x| + |y|.

From: Pure and Applied Mathematics , 1977

Numerical Analysis of Wavelet Methods

Albert Cohen , in Studies in Mathematics and Its Applications, 2003

Remark 3.2.1

The Besov semi-norm is often instead defined in its integral form

(3.2.14) ${\left|f\right|}_{B_{p,q}^s}:={\Vert t^{-s}{\omega }_n{\left(f,t\right)}_p\Vert }_{L^p\left(\left[0,1\right],dt/t\right)},$

i.e. ${\left({\displaystyle {\int }_0^1{\left(t^{-8}{\omega }_n{\left(f,t\right)}_p\right)}^q\frac{dt}{t}}\right)}^{1/q}$ if q <   +∞. The equivalence between the above expression and its discrete analog of (3.2.13) is easily derived from the monotonicity of the modulus of smoothness (ω_n (f, t) _p ≤ ω_n (f, t') _p if t < t'). Also note that ${\left|f\right|}_{B_{p,q}^s}~{\Vert {\left(2^{sj}{\omega }_n{\left(f,2^{-j}\right)}_p\right)}_{j\ge J}\Vert }_{{\ell }^q}$ with constants independent of J ≥ 0 such that diam(Ω) ≤ 2^−J .

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Topological Vector Spaces

Henri Bourlès , in Fundamentals of Advanced Mathematics 2, 2018

3.3.2 Semi-norms

(I) A semi-norm on E is a mapping $p:E\to {\mathrm{ℝ}}_{+}$ that satisfies the following conditions for all x, y ∈ E and all $\lambda \in \mathbb{K}$ :

(N ₁ ) p (0) = 0.

(N ₂ ) p (λx) = |  λ  | p (x).

(N ₃ ) p (x + y) ≥ p (x) + p (y) ("triangle inequality").

A semi-norm p is said to be a norm if it also satisfies the condition

(N ₄ ) p (x) = 0 ⇒ x = 0.

Any semi-norm p is a convex mapping (exercise) and determines a pseudometric d :(x, y) ↦ p (x − y). If p is a norm, then d is a metric. Let p be a semi-norm. For all α > 0, write

$B_p\left(\alpha \right)=\left\{x\in E:p\left(x\right)<\alpha \right\},B_p^c\left(\alpha \right)=\left\{x\in E:p\left(x\right)\le \alpha \right\}.$

The sets B _p (α) and B _p ^c (α) are both disked and absorbing, and the following conditions are equivalent (exercise): (i) p is continuous on E; (ii) p is continuous at 0; (iii) p is uniformly continuous; (iii) for all α > 0, B _p (α) is open in E (so is an open neighborhood of 0); (iv) for all α > 0, B _p ^c (α) is a neighborhood of 0. Given a disked and absorbing set A ⊂ E, define p _A (x) = inf {ρ : ρ > 0, x ∈ ρA} for all x ∈ E. Then p _A is a semi-norm on E (exercise), called the gauge (or Minkowski functional) of A; furthermore, B _{p A} (1)   ⊂ A  ⊂ B _{p A} ^c (1). If A is open (resp. closed) in E, then p _A is continuous (resp. lower semi-continuous, cf. Definition 2.31) and A  = B _{p A} (1) (resp. A  = B _{p A} ^c (1)) (exercise*: cf. [BKI 81], Chapter II, section 2.10, Proposition 23). Conversely, suppose that p _A is lower semi-continuous; then $p_A^{-1}\left(\left[0,1\right]\right)=p_A^{-1}\left({\complement }_{{\mathrm{ℝ}}_{+}}]1,\infty [\right)$ and $p_A^{-1}\left({\complement }_{{\mathrm{ℝ}}_{+}}]1,\infty [\right)={\complement }_E\left(p_A^{-1}\right(]1,\infty [\left)\right)$ ([P1], section 1.1.2 (VI)); but p _A ^{−   1}(]1,   ∞[) is open, so p _A ^{−   1}([0,   1]) is closed.

(II) A topological vector space is said to be semi-normed if its topology $\mathfrak{T}$ is determined by a semi-norm p, that is, if {B _p (α): α > 0} is a fundamental system of neighborhoods of 0 for $\mathfrak{T}$ (any such space is Hausdorff if and only if p is a norm).

Let E be a vector space on $\mathbb{K}$ and suppose that p ₁, p ₂ are two semi-norms on E. These semi-norms are said to be equivalent if they determine the same topology on E. The proof of claim (1) of the next theorem is an exercise. Claim (2) follows from Lemma 3.6:

Theorem 3.17

1)

The two semi-norms p ₁, p ₂ on E are equivalent if and only if there exist real numbers α > 0, β > 0 such that αp ₁ ≤ p ₂ ≤ βp ₁.

2)

On ${\mathbb{K}}^n$ , all norms are equivalent.

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Sobolev Spaces

In Pure and Applied Mathematics, 2003

6.29 Domains of Finite Width

Consider the problem of determining for what domains Ω in R ⁿ is the seminorm

$|u{|}_{m,p,\Omega }={\left({\displaystyle \sum _{|\alpha |=m}||D^{\alpha }u|{|}_{0,p,\Omega }^p}\right)}^{1/p}$

actually a norm on W ₀ ^m,p (Ω) equivalent to the standard norm

${\Vert \varphi \Vert }_{0,p,\Omega }\le K|\varphi {|}_{1,p,\Omega }.$

This problem is closely related to the problem of determining for which unbounded domains Ω the imbedding W ₀ ^m,p (Ω) → L^p (Ω) is compact because both problems depend on estimates for the L^p norm of a function in terms of L^p estimates for its derivatives.

We can easily show the equivalence of the above seminorm and norm for a domain of finite width, that is, a domain in Rⁿ that lies between two parallel planes of dimension (n − 1). In particular, this is true for any bounded domain.

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Sobolev Spaces

In Pure and Applied Mathematics, 2003

4.29(Seminorms)

For 1 ≤ p < ∞ and for integers j, 0 ≤ j ≤ m, we introduce functionals |·| _j·p on W ^m·p (Ω) as follows:

$|u{|}_{j,p}=|u{|}_{j,p,\Omega }={\left({\displaystyle \sum _{|\alpha |=j}|D^{\alpha }u\left(x\right){|}^{p}dx}\right)}^{1/p}.$

Clearly |u|_0·p = |u|_0,p = |u| _p is the norm on L^p (Ω) and

$|u{|}_{m,p}={\left({\displaystyle \sum _{j=0}^m|u{|}_{j,p}^p}\right)}^{1/p}.$

If j ≥ 1, we call |·| _{j, p} a seminorm . It has all the properties of a norm except that |u| _{j, p} = 0 need not imply u = 0 in W^m,p (Ω). For example, u may be a nonzero constant function if Ω has finite volume. Under certain circumstances which we begin to investigate in Paragraph 6.29, |·| _m·p is a norm on $W_0^{\mathrm{m,p}}$ (Ω) equivalent to the usual norm |·| _m,p . In particular, this is so if Ω is bounded.

For now we will confine our attention to these seminorms as they apply to functions in $C_0^{\infty }$ (ℝ ⁿ ).

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Topics in Multivariate Approximation and Interpolation

Angela Kunoth , in Studies in Computational Mathematics, 2006

5 Multilevel Regularization

Next we extend the least squares fitting strategy to include into the objective functional a Sobolev seminorm with positive smoothness index α as a regularizing term in the least squares functional (1) to enforce a smooth approximation. This classical approach for α ∈ {1,2} was studied, e.g., in [23] for general splines, [55] for splines with free knots, [34] for hierarchical splines, [37] for splines on triangulations, or [32] in a wavelet reformulation of the spline problem. Here we choose more generally $\mathrm{ℱ}\left(f\right):={‖f‖}_{H^{\alpha }\left(\Omega \right)}^2$ so that we consider

(14) $\mathcal{J}\left(f\right)={\displaystyle \sum _{i=1}^N{\left(z_i-f\left(x_i\right)\right)}^2+v}{‖f‖}_{H^{\alpha }\left(\Omega \right)}^2\text{,}$

where the not necessary integral α > 0 parameterizes the smoothing and v > 0 balances the fidelity of the data and the smoothness. In view of the statements made after (5), the wavelet formulation allows to represent functions in Sobolev norms in terms of sequence norms of weighted coefficients from their wavelet expansions. Accordingly, following [11], we replace ${‖\cdot ‖}_{H^{\alpha }\left(\Omega \right)}^2$ by the corresponding sequence norm on the right hand side of (14),

(15) ${‖{\displaystyle \sum _{j,\mathbf{k},\mathbf{e}}{d}_{j,\mathbf{k},\mathbf{e}}{\psi }_{j,\mathbf{k},\mathbf{e}}}‖}_{H^{\alpha }\left(\Omega \right)}^2~{\displaystyle \sum _{j,\mathbf{k},\mathbf{e}}{2}^{2\mathit{\alpha j}}{\left|d_{j,\mathbf{k},\mathbf{e}}\right|}^2\text{.}}$

The minimization of (14) using this sequence norm is now equivalent to the solution of the normal equations

(16) $\left(M+\mathit{vR}\right)\mathbf{d}=\mathbf{b}\text{.}$

Here R is a diagonal matrix with entries

(17) $R=\mathrm{diag}\left(2^{2\mathit{\alpha jo}}I,\dots ,2^{2\mathit{\alpha J}}I\right)$

if J is the highest level of approximation and the sizes of the identity matrices depend on the cardinality of Λ _j. This representation of the smoothing term in the wavelet context has some interesting properties. First, the different dyadic scales are decoupled which means that the weak decoupling between levels in M is maintained, and the effect of the regularization term boils down to a penalization of the higher frequencies. Thus, the parameter v controls the balance between fidelity and regularization while α controls the relative penalization across scales. Second, since (5) holds for the whole scale of fractional Sobolev spaces {H ^α (Ω)} _{α  ∈   [0,γ)}, one has easy access to the entire scale which reduces to a simple diagonal scaling in the wavelet framework by means of the diagonal matrix R. Third, the formulation is independent of the wavelet family. Of course, the smoothness of the wavelets imposes a limit on the maximal value of α for which (15) holds.

We have now two parameters in (14), the smoothness index α and the weight parameter v to balance the data approximation and the smoothness part which can be used to adjust as best as possible to the given data. An example of an application from photogrammetry with fixed α = 4 and v = 0.01 is provided in Figs. 4 and 5. It should be mentioned that for the linear pre-wavelets we employ the norm equivalence (15) does not hold for this high value of α. Nevertheless, the reconstruction in Fig. 5 gives the best result for this case.

Fig. 4. Left: Original data, a 3D point set consisting of 330.000 points of an industrial site (taken by Leica Cyrax 2500, Prof. Staiger, GH Essen). Middle: Vertical view of a section of the original data. Right: Sampling geometry of this section.

Fig. 5. Approximation of the data from Fig. 4 by wavelet reconstruction with regularization for v = 0.01, α = 4 and thresholding parameter ε = 10^{−   3}. Left: Reconstruction for J = 4. Middle: Wavelet coefficients of type (1, 1). Right: Reconstruction for J = 6. The final approximation contains #Λ₆ = 2623 coefficients as opposed to 16384 coefficients for the full grid.

Standard cross validation techniques start out with computing the weight parameter v for a fixed choice of α, which requires to solve an additional system of normal equations. Below we once more exploit the multilevel framework provided by wavelets and introduce a multilevel version of the cross validation procedure. This scheme turns out to be both relatively inexpensive from a computational point of view as well as adjusting well to smoothness and to localization effects.

Let us first describe the classical situation where the smoothness parameter α is fixed. Different ways to select an appropriate regularization parameter v in (14) were proposed, where one of the most popular choices are cross validation methods, see, e.g., [60]. Particularly, in the Generalized Cross Validation (GCV) method one computes v as the minimizer of the GCV potential with respect to the Euclidean norm || • ||, that is,

(18) $\mathit{GCV}\left(v\right):=\frac{{‖I-H\left(v\right)z‖}^2}{{\left(\mathrm{tr}\left(I-H\left(v\right)\right)\right)}^2}\text{,}$

where the influence matrix H(v) is defined as H(v) := A(M + vR)^{−   1} A^T. The influence matrix is never computed explicitly, since only the application of H(v) to a vector needs to be carried out. This in turn requires the solution of a linear system with matrix M + vR which is of a similar type as in (16). The trace in the denominator of (18) is usually estimated stochastically as tr(B) = u ^TB u where u is a random vector with entries 1 and −   1 and B is a quadratic symmetric matrix, see [41]. Consequently, the evaluation of GCV (v) for each v requires the solution of two linear systems of the type (16). The total computation time then depends on the number of GCV evaluations which are needed to find the minimum of GCV (v). In [63] an automatized computation of a GCV involving up to 15 evaluations of the functional is reported, together with the CPU time which is needed in the solution of a system of normal equations with penalization for a given v. We have displayed in [7] a series of bivariate examples where we have applied these GCV techniques and where the solution of each linear system costed less than one second.

The GCV technique yields an optimal v in the sense that it minimizes the variance under the assumption that the data has been corrupted by white noise, see, e.g., [59]. This theory developed for splines immediately carries over to the wavelet case[5]. Moreover, as shown in [7] the underlying heuristics can also be applied successfully to other data fitting approaches which require some regularization such as for data sets with a highly varying point distribution or for data with holes in the domain where typically overfitting artifacts occur.

The main issue in [7] is the following multilevel version of the GCV algorithm used within the context of the adaptive coarse-to-fine Algorithm. The idea is to simultaneously use the GCV with the hierarchical growth of the wavelet tree as the levels become higher. We have observed above that the wavelet representation penalizes the different dyadic levels separately with the same weight 2^2αj . In view of the structure of the diagonal matrix R given in (17), we prescribe instead a diagonal penalizing matrix with the same values v _j ≥ 0 for all entries of each level j, where the v_j are computed independently following a GCV criterion. Now at each level j the normal equations attain the form

(19) $\left(A_{{\Lambda }_j}^T{A}_{{\Lambda }_j}+R_j\right)d=A_{{\Lambda }_j}^{T}z\text{,}$

where the diagonal matrix R_j is defined component–wise as ${\left(R_j\right)}_{\mathrm{\lambda },{\mathrm{\lambda }}^{\prime }}:={\delta }_{\mathrm{\lambda },{\mathrm{\lambda }}^{\prime }}{v}_j$ for some set of scalars ${\left\{v_{j^{\prime }}\right\}}_{j_0\le j^{\prime }\le j}$ . These scalars are computed inductively at each level following a GCV criterion. At level j ₀, we define the influence matrix

$H\left(v_{j_0}\right):=A_{{\Lambda }_{j_0}}{\left(A_{{\Lambda }_{j_0}}^T{A}_{{\Lambda }_{j_0}}+R_{j_0}\right)}^{-1}{A}_{{\Lambda }_{j_0}}^T\text{,}$

and $v_{j_0}$ is obtained by minimization of the corresponding GCV potential (18). At any subsequent level j the influence matrix is defined as

(20) $H\left(v_j;v_{j_0},\dots ,v_{j-1}\right):=A_{{\Lambda }_j}{\left(A_{{\Lambda }_j}^T{A}_{{\Lambda }_j}+R_j\right)}^{-1}{A}_{{\Lambda }_j}^T$

to have v_j as only variable since the $v_{j^{\prime }}$ from previous levels j ₀  ≤ j′   < j are already computed. v_j is then determined by minimizing (18). Recall that the influence matrices are never computed explicitly as they only need to be applied to a vector. The new penalizing term can no more be interpreted as stemming from a smoothing term ${‖f‖}_{H^{\alpha }\left(\Omega \right)}^2$ since no relation between the penalizing parameters ${\left\{v_j\right\}}_{j\ge j_0}$ is prescribed. Nevertheless, this approach offers some interesting advantages. It is easily built into the coarse–to–fine growth of the tree. One can attain higher flexibility for the smoothing effect. Moreover, overfitting artifacts are typically localized in scale which enables us to disentangle the different scales. Last, the method is much cheaper as the GCV method with both α and v as free parameters from a computational point of view.

As illustration, we apply the multilevel GCV algorithm to the data set in Fig. 6 obtained from [49] which corresponds to a bathymetrical study of part of the sea floor of the Dominican Republic. As can be seen from the left image, measurements of the sea floor depth are irregularly distributed, forming lines, clusters and holes. The central plot shows a visualization of the depth of the full set of data points using piecewise linear interpolation. For the classical GCV, there is no obvious way to predict which α may give a good reconstruction. In contrast, our multilevel GCV algorithm gives a sufficiently good reconstruction in the right plot.

Fig. 6. Left: Sampling geometry with N = 10732 points. Middle: Piecewise linear interpolation. Right: Reconstruction with multilevel GCV.

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FN-Topologies and Group-Valued Measures

Hans Weber , in Handbook of Measure Theory, 2002

PROOF

(a) Since it is enough to prove that (μ _n ) is uniformly exhaustive with respect to any continuous seminorm on G. we may assume that G = (G, $\left|\right|$ ) is a seminormed group. Then the semivariation $\Vert {\mu }_n\Vert$ of μ _n is bounded by Corollary 2.7. Therefore there are numbers ∈ _n > 0 such that η:= ${\displaystyle {\sum }_{n=1}^{\infty }{\in }_n\Vert {\mu }_n\Vert }$ is a bounded function. It follows that η is an exhaustive submeasure and μ _n is η-continuous for every n. Suppose that (μ_n)_n $\mathbb{N}$ is not uniformly exhaustive. Then there is an ∈ > 0, a disjoint sequence (a_n ) _{n ∈} $\mathbb{N}$ in R and a subsequence (v_n ) _{n ∈} $\mathbb{N}$ of (μ _n ) _{n ∈} $\mathrm{\mathbb{N}***************}$ such that $\left|v_n(a_n)\right|\ge \in$ for every n ∈ $\mathbb{N}$ . By Lemma 10.2, (a_n )n ∈ $\mathbb{N}$ has a subsequence (b_n )n ∈ $\mathbb{N}$ , such that the restriction of η on S := σ({b_n : n ∈ $\mathbb{N}$ }) is σ-order continuous. Applying Theorem 10.1 to $v_{n|S}$ one obtains that ( $v_{n|S}$ )n ∈ $\mathbb{N}$ is $\eta |{}_S$ -equicontinuous. Since η(b_n ) → 0, it follows that sup _m $\left|v_m(b_n)\right|$ → 0(n → +∞), a contradiction to $\left|v_n(a_n)\right|$ ≥ ∈.

(b) follows from (a) and Corollary 7.2.

(c) Let u be the supremum of the μ _n -topologies, n ∈ $\mathbb{N}$ . Then u is σ-order continuous if μ _n ,n∈N are σ-additive (see Proposition 4.2). By (b), μ is μ-continuous, hence σ-additive.

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Functional Analysis and its Applications

Wieslaw Żelazko , in North-Holland Mathematics Studies, 2004

Theorem 2

Let A be a complex (resp. real) topological division algebra with continuous inverse and possessing a continuous p-homogeneous seminorm , 0 < p ≤ 1. Then A is topologically isomorphic to C (resp. to either of R, C, H).

The requirement that the operation x → x ⁻¹ is necessary for validity of the above result, but in the case of an F-algebra it is satisfied automatically. It follows from the following result proved by Arens in [3] and by Banach in [10] in the separable case, and in [29] in the general case.

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Real Reductive Groups I

In Pure and Applied Mathematics, 1988

Theorem

Let (π, H) be a finitely generated, admissible, Hilbert representation of G. Set V = H_K and Λ = Λ _V . There is a positive constant d such that if μ ∈(H ^∞ )′_K then there exists a continuous semi-norm, σ_μ, on H^∞ with the property that

$\left|\left(\mu \left(\pi \left(a\right)\right)\upsilon \right)\right|\le {\left(1+log\Vert a\Vert \right)}^d{a}^{\Lambda }{\sigma }_{\mu }\left(\upsilon \right)$

for v ∈ H^∞ and a ∈ Cl(A ⁺).

Let μ ∈ (H ^∞ )′_K . 4.3.3 implies that μ = σ(w) with w ∈ H_K . Lemmas 2.A.2.2 and 2.A.2.3 imply that there exists δ ∈ a* and C > 0 such that if x, y ∈ H then

$\left|〈\pi \left(a\right)x,y〉\right|\le {\text{Ca}}^{\delta }\Vert x\Vert \Vert y\Vert \text{for}a\in \text{C}1\left(A^{+}\right).$

This clearly implies that if μ ∈ (H ^∞ )′_K then there exists, σ′ _μ , a continuous semi-norm on H ^∞ such that

(1)

$\left|\left(\mu \left(\pi \left(a\right)\right)\upsilon \right)\right|\le a^{\delta }{{\sigma }^{\prime }}_{\mu }\left(\upsilon \right)\text{for}\upsilon \in H^{\infty }\text{and}a\in C1\left(A^{+}\right).$

The idea of the proof is to show that if δ(H_j ) > Λ(H_j ) then we can replace δ in (1) by δ – mα_j with m = min{1/2, μ(H_j ) – Λ(H_j )} at the cost of possibly changing the semi-norm σ′ _μ and putting in a term (1 + log ‖a‖) ^d .

Let α ∈ Δ₀. Set F = Δ₀ – {α}. If α = α_j then set H = H_j . Then a_F = R H. Set a_t = exp(tH). If a ∈ Cl(A ⁺) then a can be written uniquely in the form a = a′a_t with a = exp(Σ x_kH_k ), x_k ≥ 0, x_j = 0 and t ≥ 0.

Let q be the canonical projection of V ^˜ onto V ^˜/n _F V ^˜.

(2)

If q(μ) = 0 then there exists a continuous semi-norm t′ _μ on H ^∞ such that |μ(π(a)v)| ≤ a ^δ–α γ′ _μ (v), for a ∈ Cl(A ⁺) and v ∈ H ^∞.

Let X ₁,…, X_p be a basis of n _F consisting of root vectors for a corresponding to the roots β ₁,…, β_p respectively. Our assumption implies that μ = Σ X_kμ_k with μ_k ∈ V ^˜. Hence

$\begin{array}{cc}\left|\left(\mu \left(\pi \left(a\right)\upsilon \right)\right)\right|\hfill & =\left|\Sigma X_k{\mu }_k\left(\pi \left(a\right)\upsilon \right)\right|=\left|-\Sigma {\mu }_k\left(\pi \left(X_k\right)\pi \left(a\right)\upsilon \right)\right|\hfill \\ \hfill & \le \Sigma \left|{\mu }_k\left(\pi \left(a\right)\pi \left(\text{A}d\left(a^{-1}\right)X_k\right)\upsilon \right)\right|=\Sigma a^{-\beta }\left|{\mu }_k\left(\pi \left(a\right)\pi \left(X_k\right)\upsilon \right)\right|\hfill \\ \hfill & \le \Sigma a^{\delta -\beta }{{\sigma }^{\prime }}_{\mu }\left(\pi \left(X_k\right)\upsilon \right).\hfill \end{array}$

(2) now follows from 3.8.6(1).

Let for z ∈ C, (V ^˜/n _F )V ^˜) _z denote the generalized eigenspace for H with eigenvalue z. Let P_z be the projection of (V ^˜/n _F V ^˜) onto (V ^˜/n _F V ^˜) _z corresponding to the H-weight space decomposition. Let μ ∈ V ^˜. Then q(μ) = Σ P_zq(μ). Let μ_z ∈ V ^˜ be such that q(μ_z ) = P_zq(μ). Then μ – Σ μ_z ∈ n _F V ^˜. We now estimate μ_z (π(a)v) for each z. Set μ_z = v. Let ${\overline{v}}_1,\dots ,{\overline{v}}_p$ be a basis for U(a _F )q(v). We assume that ${\overline{v}}_1=q\left(v\right)$ . Let v_k ∈ V ^˜ be such that $q\left(v_k\right)={\overline{v}}_k$ for k ≥ 2. Now

$H{\overline{v}}_k=\sum b_{kn}{\overline{v}}_n$

and B = [b_kn ] has the property that

(3)

${\left(B-zI\right)}^p=0.$

We also note that

(4)

${\sigma }_k=H{v}_k-\sum b_{kn}{v}_n\in {\mathcal{n}}_F{V}^{\sim }.$

Let a′ ∈ Cl(A ⁺) be such that (a′) ^α = 1. We set

$F\left(t,a^{\prime };\upsilon \right)=\left[\begin{array}{c}{v}_1\left(\pi \left(a_t{a}^{\prime }\right)\upsilon \right)\\ ⋮\\ v_p\left(\pi \left(a_t{a}^{\prime }\right)\upsilon \right)\end{array}\right]$

and

$G\left(t,a^{\prime };\upsilon \right)=\left[\begin{array}{c}{\sigma }_1\left(\pi \left(a_t{a}^{\prime }\right)\upsilon \right)\\ ⋮\\ {\sigma }_p\left(\pi \left(a_t{a}^{\prime }\right)\upsilon \right)\end{array}\right]$

Then

(5)

$\left(\frac{d}{dt}\right)F\left(t,a^{\prime };\upsilon \right)=-BF\left(t,a^{\prime };\upsilon \right)-G\left(t,a^{\prime };\upsilon \right)$

This implies that

(6)

$F\left(t,a^{\prime };\upsilon \right)=exp\left(-tB\right)F\left(0,a^{\prime };\upsilon \right)-exp\left(-tB\right){\displaystyle \underset{0}{\overset{t}{\int }}}exp\left(sB\right)G\left(s,a^{\prime };\upsilon \right)ds.$

We now estimate the terms in (6). (1) implies that

(7)

‖F(0, a′; v)‖ ≤ (a′) ^δ β(v) with β a continuous semi-norm on H^∞ . (2) implies that

(8)

(9)

$\begin{array}{l}\Vert exp\left(sB\right)\Vert \le C{\left(1+\left|s\right|\right)}^p{e}^{sRez}\text{for}s\in \text{R}.\text{Here}p\le d\\ \text{(see the beginning of the proof)}\text{.}\end{array}$

This follows immediately from (3). These estimates imply that if t ≥ 0 then

$\Vert F\left(t,a^{\prime };\upsilon \right)\Vert \le C{\left(1+t\right)}^p{e}^{-tRez}{\left(a^{\prime }\right)}^{\delta }\beta \left(\upsilon \right)\cdot \left(1+{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(1+s\right)}^p{e}^{s\left(Rez+\delta \left(H\right)-1\right)}ds\right),$

for some continuous semi-norm β on H ^∞ and some positive constant C. We observe that (1 + s) ^p e ^–ɛs is bounded by a constant C for ɛ > 0 and s ≥ 0. We therefore have

(10)

$\begin{array}{c}\Vert F\left(t,a^{\prime };\upsilon \right)\Vert \le C{\left(1+t\right)}^p{e}^{-tRez}{\left(a^{\prime }\right)}^{\delta }\beta \left(\upsilon \right)+C{\left(1+t\right)}^p{e}^{t\left(\delta \left(H\right)-2/3\right)}{\left(a^{\prime }\right)}^{\delta }\beta \left(\upsilon \right)\\ \text{for}t>0.\end{array}$

Here C is a positive constant and β is a continuous semi-norm on H^∞ .

There are now two cases.

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Real Reductive Groups I

In Pure and Applied Mathematics, 1988

5.1.2 Proposition

Let (π, H) be a Hilbert representation of G. If H_K is tempered then (π, H) satisfies the weak inequality. If H_K is rapidly decreasing then (π, H) satisfies the strong inequality.

Let V = H_K . Theorem 4.3.5 implies that there exists d > 0 such that if w ∈ V then there exists a continuous semi-norm, σ _w , depending on w such that |〈π(a)v, w〉| ≤ σ _w (v)(1 + log ‖a‖) ^d a ^Λ (Λ = Λ _V ) for all a ∈ Cl(A ⁺). Let w ₁, …, w_p be a basis of the span of Kw. Let σ(v) = sup _{k ∈ K} Σ σ _w (kv). π(k)w = Σg_j (k)W_j with each g_j a continuous function on K. 〈π(k ₁ ak ₂)v, w〉 = 〈π(ak ₂)v, π(k ₁)^–1 w〉 = Σ conj(g_j ((k ₁)^–1))〈π(ak ₂)v, w〉. It follows that

(1)

If w ∈ V and v ∈ H^∞ then |〈π(k ₁ ak ₂)v, w〉| ≤ σ(v)(1 + log ‖a‖) ^d a^Λ

for a ∈ Cl(A ⁺) and k ₁, k ₂ ∈ K.

We now prove the result. Suppose that V is tempered. Then α^Λ ≤ a ^−ρ for all a ∈ Cl(A ⁺). Now, a ^–ρ ≤ Ξ(a) for all a ∈ Cl(A ⁺) (Theorem 4.5.3). Since Ξ(k ₁ gk ₂) = Ξ(g) for all k ₁, k ₂∈ K, the first assertion now follows immediately from (1).

If μ ∈ ⁺ a* then for each r > 0 there exists a positive constant C_r such that, a ^–μ ≤ C_r (1 + log ‖a‖)^–r for a ∈ Cl(A ⁺). Hence, the second assertion is also a direct consequence of (1).

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Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations

Marco Cannone , in Handbook of Mathematical Fluid Dynamics, 2005

2.5.3. Le Jan-Sznitman spaces

Recently, Le Jan and Sznitman [137,138] considered the space of tempered distributions f whose Fourier transform verifies

(116) $\underset{{\mathrm{ℝ}}^3}{\sup }|\mathit{\xi }{|}^2|\widehat{\mathit{f}}(\mathit{\xi })|<\infty .$

Now, if in the previous expression we consider ${\int }_{\mathit{\xi }\in {\mathrm{ℝ}}^3}$ instead of ${\sup }_{\mathit{\xi }\in {\mathrm{ℝ}}^3}$ , we obtain the (semi)-norm of a homogeneous Sobolev space. This is not the case: the functions whose Fourier transform is bounded define the pseudo-measure space $\mathcal{P}\mathcal{ℳ}$ of Kahane. In other words, a function f belongs to the space introduced by Le Jan and Sznitman if and only if $\mathrm{\Delta }\mathit{f}\in \mathcal{P}\mathit{ℳ}$ , Δ being the Laplacian (in three dimensions). A simple calculation (see [48]) shows that condition (116) is written, in the dyadic decomposition Δ _j of Littlewood and Paley in the form $4^{\mathit{j}}{||{\mathrm{\Delta }}_{\mathit{j}}\mathit{f}||}_{\mathcal{P}\mathit{ℳ}}=4^{\mathit{j}}{||\widehat{{\mathrm{\Delta }}_{\mathit{j}}\mathit{f}}||}_{\infty }\in {\ell }^{\infty }(\mathrm{\mathbb{Z}})$ and defines in this way "the homogeneous Besov space" ${\dot{\mathit{B}}}_{\mathcal{P}\mathit{ℳ}}^{2,\infty }$ .

Let us note that this quantity is not a norm, unless we work in S′ modulo polynomials, as we did in Section 2.2 in the case of homogeneous Besov spaces (for example, if f is a constant or, more generally a polynomial of degree 1, it is easy to see that ${\left|\mathit{\xi }\right|}^2\left|\widehat{\mathit{f}}(\mathit{\xi })\right|=0$ ). Another possibility to avoid this technical point is to ask that $\widehat{\mathit{f}}\in {\mathit{L}}_{\mathrm{loc}}^1$ . In other words, the Banach functional space relevant to our study is defined by

(117) $\mathcal{P}{\mathit{ℳ}}^2=\left\{\mathit{v}\in {\mathcal{S}}^{\prime }:\widehat{\mathit{v}}\in {\mathit{L}}_{\mathrm{loc}}^1,{||\mathit{v}||}_{\mathcal{P}{\mathit{ℳ}}^2}\equiv \underset{\mathit{\xi }\in {\mathrm{ℝ}}^3}{\sup }|\mathit{\xi }{|}^2\widehat{\mathit{v}}(\mathit{\xi })|<\infty \right\}.$

A generalization of this functional space was recently introduced in the paper by Bhattacharya, Chen, Dobson, Guenther, Orum, Ossiander, Thomann and Waymire (see [8]).

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