Algebras of Singular Integral Operators with Piecewise Continuous Coefficients on Reflexive Orlicz Spaces

We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces L_m(σ) which are generalizations of the Lebesgue spaces L_P(σ), 1 < p < ∞. We suppose that σ belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient G, we establish a Fredholm criterion and an index formula in terms of the essential range of G complemented by spiralic horns depending on the Boyd indices of L_M(σ) and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix - valued coefficients on L_Mⁿ(σ).     

Spline Approximation Methods with Uniform Meshes in Algebras of Multiplication and Convolution Operators

The purpose of this paper is to obtain necessary and sufficient conditions for maximum defect spline approximation methods with uniform meshes to be stable. The methods are applied to operators belonging to the closed subalgebra of ℒ︁ (L² (ℝ)) generated by operators of multiplication by piecewise continuous functions on ℝ and convolution operators also with piecewise continuousgenerating functions. To that purpose, a C*-algebra of sequences is introduced, which contains the special sequences of approximating operators we are interested in. There is a direct relationship between the applicability of the approximation method to a given operator and invertibility of the corresponding sequence in this C*-algebra. Exploring this relationship, applicability criteria are derived by the use of C*-algebra and Banach algebra techniques (essentialization, localization andidentification of the local algebras by means of construction of locally equivalent representations). Finally, examples are presented, including explicit conditions for the applicability of spline Galerkin methods to Wiener-Hopf operators with piecewise continuous symbols.     

Spectral Approximation and Index for Convolution Type Operators on Cones on L^{p}(\mathbb {R}^2)

We consider an algebra of operator sequences containing, among others, the approximation sequences to convolution type operators on cones acting on L^p(\mathbb R²)L^{p}(\mathbb {R}^2), with 1 < p < ∞. To each operator sequence (A_n) we associate a family of operators W_x(A_n) ? L(L^p(\mathbb R²))W_{x}(A_{n}) \in \mathcal {L}(L^{p}(\mathbb {R}^2)) parametrized by x in some index set. When all W_x(A_n) are Fredholm, the so-called approximation numbers of A_n have the α-splitting property with α being the sum of the kernel dimensions of W_x(A_n). Moreover, the sum of the indices of W_x(A_n) is zero. We also show that the index of some composed convolution-like operators is zero. Results on the convergence of the e\epsilon-pseudospectrum, norms of inverses and condition numbers are also obtained.     

The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space L _p (? ⁿ ), 1 < p < ??, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.     

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·)(\mathbbR ₊,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L ^p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR₊{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.     

The Calkin image of algebras of singular integral operators

The image in the Calkin algebra of the Banach algebra generated by all singular integral operators with piecewise continuous coefficients on some composed contour is described up to isomorphy and isometry for weighted L^P-spaces.     

Connectedness of Spectra of Toeplitz Operators on Hardy Spaces with Muckenhoupt Weights Over Carleson Curves

Harold Widom proved in 1966 that the spectrum of a Toeplitz operator T(a) acting on the Hardy space H^p(\mathbbT)H^p({\mathbb{T}}) over the unit circle \mathbbT{\mathbb{T}} is a connected subset of the complex plane for every bounded measurable symbol a and 1 < p < ∞. In 1972, Ronald Douglas established the connectedness of the essential spectrum of T(a) on H²(\mathbbT)H^2({\mathbb{T}}). We show that, as was suspected, these results remain valid in the setting of Hardy spaces H^p(Γ,w), 1 < p < ∞, with general Muckenhoupt weights w over arbitrary Carleson curves Γ.     

Strongly continuous semigroups of operators generated by systems of pseudodifferential operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

This paper considers semigroups of operators generated by pseudodifferential operators in weighted L _p -spaces of vector functions on \mathbbRⁿ {\mathbb{R}^n} (or on a compact manifold without boundary). Sufficient conditions for a semigroup to be strongly continuous and analytic are obtained, conditions for it to be completely continuous are found, and the distribution of the eigenvalues of its infinitesimal generator is examined. Also, an integral representation that singles out the principal term of the semigroup as t → 0+ is established.     

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), ${\Omega\subset{\mathbb R}^N}The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), W ì \mathbb R^N{\Omega\subset{\mathbb R}^N} , with p = N, that is, the case of Pohozhaev–Trudinger–Moser inequality. Similarly to the case p < N where the loss of compactness in W^1,p(\mathbb R^N){W^{1,p}({\mathbb R}^N)} occurs due to dilation operators u ?t^(N-p)/pu(tx){u {\mapsto}t^{(N-p)/p}u(tx)} , t > 0, and can be accounted for in decompositions of the type of Struwe’s “global compactness” and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W₀^1,N{W_0^{1,N}} over a ball in \mathbb R^N{{\mathbb R}^N} . We give a one-parameter scale of Hardy–Sobolev functionals, a “p = N”-counterpart of the H?lder interpolation scale, for p > N, between the Hardy functional ò\frac|u|^p|x|^p dx{\int \frac{|u|^p}{|x|^p}\,{\rm d}x} and the Sobolev functional ò|u|^pN/(N-mp)  dx{\int |u|^{pN/(N-mp)} \,{\rm d}x} . Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W ^1,N -norms under Hardy–Sobolev constraints.     

Multidimensional integral operators with homogeneous kernels of compact type and multiplicatively weakly oscillating coefficients

In the space L _p (? ⁿ ), 1 < p < +∞, we consider a new class of integral operators with kernels homogeneous of degree ?n, which includes the class of operators with homogeneous SO(n)-invariant kernels; we study the Banach algebra generated by such operators with multiplicatively weakly oscillating coefficients. For operators from this algebra, we define a symbol in terms of which we formulate a Fredholm property criterion and derive a formula for calculating the index. An important stage in obtaining these results is the establishment of the relationship between the operators of the class under study and the operators of one-dimensional convolution with weakly oscillating compact coefficients.     

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.     

On the Structure of the C *-Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols

We study the C ^*-algebra generated by Toeplitz operators with piece-wise continuous symbols acting on the Bergman space on the unit disk in . We describe explicitly each operator from this algebra and characterize Toeplitz operators which belong to the algebra. To the memory of G. S. Litvinchuk     

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Fredholm conditions and an index formula are obtained for Wiener-Hopf operators W(a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces where 1 < p < ∞, w is a Muckenhoupt weight on and . The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on that are continuous on and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SO_{p, w} of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calderón-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a. Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.     

On rational Drinfeld associators

We prove an estimate on denominators of rational Drinfeld associators. To obtain this result, we prove the corresponding estimate for the p-adic associators stable under the action of suitable elements of Gal([`(\mathbbQ)]/\mathbbQ){{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})} . As an application, we settle in the positive Duflo’s question on the Kashiwara–Vergne factorizations of the Jacobson element J _p (x, y) = (x + y) ^p − x ^p − y ^p in the free Lie algebra over a field of characteristic p. Another application is a new estimate on denominators of the Kontsevich knot invariant.     

The ρ-variation as an operator between maximal operators and singular integrals

The ρ-variation and the oscillation of the heat and Poisson semigroups of the Laplacian and Hermite operators (i.e. Δ and −Δ + |x|²) are proved to be bounded from into itself (from into weak- in the case p = 1) for 1 ≤ p < ∞ and w being a weight in the Muckenhoupt’s A _p class. In the case p = ∞ it is proved that these operators do not map L ^∞ into itself. Even more, they map L ^∞ into BMO but the range of the image is strictly smaller that the range of a general singular integral operator. R. Crescimbeni was partially supported by Fundación Carolina, Ministerio de Educación de la República Argentina and Universidad Nacional del Comahue. R. A. Macías and B. Viviani were partially supported by Facultad de Ingeniera Química-UNL.     

C*-algebras of singular integral operators in domains with oscillating conical singularities

 For a domain with singular points on the boundary, we consider a C ^* -algebra of operators acting in the weighted space . It is generated by the operators of multiplication by continuous functions on and the operators where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for . When combined with the local principle, this leads to describing the Fredholm operators in . Received: 21 December 2000     

An L0（F,R）-valued Function＇s Intermediate Value Theorem and Its Applications to Random Uniform Convexity

Let (Ω , F , P ) be a probability space and L0 ( F, R ) the algebra of equivalence classes of real- valued random variables on (Ω , F , P ). When L0 ( F, R ) is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0 ( F, R ) to L0 ( F, R ). As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module ( S,|| · ||) is random uniformly convex iff Lp ( S ) is uniformly convex for each fixed positive number p such that 1 p + ∞ .     

The spectral radius of matrix continuous refinement operators

A simple analytic formula for the spectral radius of matrix continuous refinement operators is established. On the space L₂^m(\mathbb R^s)L_2^m({{\mathbb R}}^s), m ≥ 1 and s ≥ 1, their spectral radius is equal to the maximal eigenvalue in magnitude of a number matrix, obtained from the dilation matrix M and the matrix function c defining the corresponding refinement operator. A similar representation is valid for the continuous refinement operators considered on spaces L _p for p ∈ [1, ∞ ), p ≠ 2. However, additional restrictions on the kernel c are imposed in this case.