SPECTRALITY FOR MATRICES OF FOURIER MULTIPLIER OPERATORS ACTING IN Lp -SPACES OVER LCA GROUPS

Abstract

N. Dunford and J.T. Schwartz gave a complete characterization of those matrices of (bounded) Fourier multiplier operators acting in L ²(R^N) ⁿ which are spectral operators, [4; ch. XV]. In the present note this characterization is extended to the setting of L^P(G)ⁿ , where G is a locally compact abelian group and 1 < p < ∞; see Theorem 2.     

A trace formula for Weyl transforms

A formula for the trace of a trace class Weyl transform associated to a symbol in L¹(R²ⁿ) is given.     

A trace formula for Weyl transforms

A formula for the trace of a trace class Weyl transform associated to a symbol in L¹(R²ⁿ) is given. This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0008562.     

The multiplier for the ball and radial functions

It is shown that the multiplier for the ball is restricted weak type on radial functions in L^p($\text{R}$ⁿ) when $\text{p =}\text{2n}\text{(n + 1)}$. Interpolation then yields a theorem of Herz.     

ADMISSIBLE WAVELETS ASSOCIATED WITH THE TRANSFORM GROUP ON R

Let p be the transform group on R, then P has a natural unitary representation U onL2 (R^n). Decompose L2(R^n) into the direct sum of irreducible invariant closed subspace,s. The re-striction of U on these suhspaces is square-intagrable. In this paper the characterization of admissi-ble condition in tarrns of the Fourier transform is given. The wavelet transform is defined, and theorthogorml direct sum decomposition of function space L2 (P,du1) is obtained.     

Multipliers of L which vanish at infinity

The main results of this paper are as follows.

1.

(i) If p ≠ 2, there exists a continuous multiplier (function) of L^p(Rⁿ) which vanishes at infinity, but which is not in the closure, in the multiplier norm, of the space of Fourier transforms of integrable functions.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

The Weyl transform and bounded operators on Lp(Rn)

A multiplier theorem for the Weyl transform is proved. This theorem is used to derive sufficient conditions for the boundedness of a general operator on L^p($\text{R}$ⁿ). An application to multipliers of the Hermite expansion is given.     

Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains

In this paper, the existence of (L²(Rn),L²(Rn))-pullback attractors and (L²(Rn),H¹(Rn))-pullback attractors are proved for reaction-diffusion equation in unbounded domains.     

Null spaces and ranges for the classical and spherical Radon transforms

Let R be the classical Radon transform that integrates a function over hyperplanes in Rⁿ and let SM be the transform that integrates a function over spheres containing the origin in Rⁿ. We prove continuity results for both transforms and explicitly give the null space of R for a class of square integrable functions on the exterior of a ball in Rⁿ as well as the null space of SM for square integrable functions on a ball. We show SM: L²(Rⁿ) → L²(Rⁿ) is one-one, and we characterize the range of SM on classes of smooth functions and square integrable functions by certain moment conditions. If g(x) is a Schwartz function on Rⁿ that is zero to infinite order at x = 0, we prove moment conditions sufficient for g to be in the range of SM(C^∞(Rⁿ)). We apply our results on SM to existence and uniqueness theorems for solutions to a characteristic initial value problem for the Darboux partial differential equation.     

Weights andL Φ-boundedness of the Poisson integral operator

In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ, v(x)dx) to weak-L _Φ(R ₊ ⁿ⁺¹ ,dμ), where Φ is anN-function satisfying the Δ₂-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ,v(x)dx) toL _Φ(R ₊ ⁿ⁺¹ ,dμ) under some additional condition. Partially supported by NNSF of P.R. China     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

Translation-invariant subspaces of functions on the group of linear transformations on the real line

The characterization of right translation-invariant subspaces ofL _∞(G ^*), where , is studied. We introduce the class of multiplier functions which, in the semisimple case, play a role similar to that played by the exponentials for the real line. However, it is proved that multiplier functions ofG ^* with respect toR fail to characterize right translation-invariant subspaces ofL _∞(G ^*). That is, we construct a right translation-invariant, w^*-closed subspace ofL _∞(G ^*) which contains no multiplier function. This paper is a part of the author's Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor H. Furstenberg, to whom the author wishes to express his thanks for his helpful guidance, and valuable remarks.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

A Lower Bound for the Norm of the Minimal Residual Polynomial

Let S be a compact infinite set in the complex plane with 0∉S, and let R _n be the minimal residual polynomial on S, i.e., the minimal polynomial of degree at most n on S with respect to the supremum norm provided that R _n (0)=1. For the norm L _n (S) of the minimal residual polynomial, the limit k(S):=lim_n?￥ⁿ?{L_n(S)}\kappa(S):=\lim_{n\to\infty}\sqrt[n]{L_{n}(S)} exists. In addition to the well-known and widely referenced inequality L _n (S)≥κ(S) ⁿ , we derive the sharper inequality L _n (S)≥2κ(S) ⁿ /(1+κ(S)²ⁿ ) in the case that S is the union of a finite number of real intervals. As a consequence, we obtain a slight refinement of the Bernstein–Walsh lemma.     

Degenerate elliptic operators in one dimension

Let H be the symmetric second-order differential operator on L ₂(R) with domain ${C_c^\infty({\bf R})}Let H be the symmetric second-order differential operator on L ₂(R) with domain C_c^￥(R){C_c^\infty({\bf R})} and action Hj = -(c j^￠)^￠{H\varphi=-(c\,\varphi^{\prime})^{\prime}} where c ? W^1,2_loc(R){ c\in W^{1,2}_{\rm loc}({\bf R})} is a real function that is strictly positive on R\{0}{{\bf R}\backslash\{0\}} but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if n = n₊ún_-{\nu=\nu_+\vee\nu_-} where n_±(x)=±ò^±1_±x c^-1{\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}} then H has a unique self-adjoint extension if and only if n ? L₂(0,1){\nu\not\in L_2(0,1)} and a unique submarkovian extension if and only if n ? L_￥(0,1){\nu\not\in L_\infty(0,1)}. In both cases, the corresponding semigroup leaves L ₂(0,∞) and L ₂(−∞,0) invariant. In addition, we prove that for a general non-negative c ? W^1,￥_loc(R){ c\in W^{1,\infty}_{\rm loc}({\bf R})} the corresponding operator H has a unique submarkovian extension.     

Group invariance and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounded operators

The Hilbert and Riesz transforms can be characterized up to scalar as the translation invariant operators that satisfy additionally certain relative invariance of conformal transformation groups. In this article, we initiate a systematic study of translation invariant operators from group theoretic viewpoints, and formalize a geometric condition that characterizes specific multiplier operators uniquely up to scalar by means of relative invariance of affine subgroups. After providing some interesting examples of multiplier operators having “large symmetry”, we classify which of these examples can be extended to continuous operators on L ^p (R ⁿ ) (1 < p < ∞). T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science. A. Nilsson was partially supported by Japan Society for the Promotion of Science.     

Décomposition spectrale dans des algèbres munies d'un star-produit

Summary The purpose of this paper is to study, in intrinsic way, the Moyal's product, defined in the flat space R ²ⁿ. This product is defined here with the twisted convolution and the Fourier transform. The S(R ²ⁿ) and L²(R ²ⁿ) spaces are_*5-algebras. Because of this definition, the_*V-product of some tempered distributions is defined. Let O _M ^v be the set of multiplication operators in S(R ²ⁿ). By transposition, the S(R ²ⁿ) space is a right-module on O _M ^v . The support of f_*v g is different from the support of f·g; under large enough hypotheses, there is a Taylor's formula for the star-product function of the v variable. The ^v space of the multiplication operators in L²(R ²ⁿ) is defined here as the space of tempered distributions, the image of which is the set of bounded operators in L²(R ²ⁿ) by the Weyl map. After the study of ^v space, it is possible to show the spectral resolution of the real elements of ^v or of O _M ^v , which satisfies a, probably superfluous, hypothesis.     

Spherical uniformity and some characterizations of the Cauchy distribution

Given a fixed line L (in Rⁿ) and a uniform distribution of points (c) on the unit sphere, L(t_c), the point of intersection of L and the hyperplane P · c = 0, leads to a mapping X_n : Rⁿ → R, which is shown to have a Cauchy distribution.     

Two-Wavelet Localization Operators on Homogeneous Spaces and Their Traces

We define two-wavelet localization operators in the setting of homogeneous spaces. We prove that they are in the trace class S ₁ and give a trace formula for them. Then we show that two-wavelet operators on locally compact and Hausdorff groups endowed with unitary and square-integrable representations, general Daubechies operators and two-wavelet multipliers can be seen as two-wavelet localization operators on appropriate homogeneous spaces. Thus we give a unifying view concerning the three classes of linear operators. We also show that two-wavelet localization operators on , considered as a homogeneous space, under the action of the affine group U are two-wavelet multipliers