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The paper is devoted to studying the Fredholmness of operators from the Banach algebra B of singular integral operators with complex conjugation and slowly oscillating coefficients on the Lebesgue spaces L p (Γ) with p ∈ (1 , ∞) over star-like curves Γ with cusps. Applying reductions to a Banach algebra of Wiener–Hopf operators with slowly oscillating coefficients on weighted Lebesgue spaces over the half-line R + and a Banach algebra of convolution type operators with piecewise slowly oscillating coefficients on weighted Lebesgue spaces over the real line R , we construct a Fredholm symbol calculus for the Banach algebra B and establish a Fredholm criterion for the operators in B . [ABSTRACT FROM AUTHOR]

We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by Mascarenhas, Santos and Seidel. [ABSTRACT FROM AUTHOR]

Let α and β be orientation-preserving diffeomorphism (shifts) of R + = ( 0 , ∞ ) onto itself with the only fixed points 0 and ∞. We establish a Fredholm criterion and calculate the index of the weighted singular integral operator with shifts ( a I − b U α ) P γ + + ( c I − d U β ) P γ − , acting on the space L p ( R + ) , where P γ ± = ( I ± S γ ) / 2 are the operators associated to the weighted Cauchy singular integral operator S γ given by ( S γ f ) ( t ) = 1 π i ∫ R + ( t τ ) γ f ( τ ) τ − t d τ with γ ∈ C satisfying 0 < 1 / p + ℜ γ < 1 , and U α , U β are the isometric shift operators given by U α f = ( α ′ ) 1 / p ( f ∘ α ) , U β f = ( β ′ ) 1 / p ( f ∘ β ) , under the assumptions that the coefficients a , b , c , d and the derivatives α ′ , β ′ of the shifts are bounded and continuous on R + and admit discontinuities of slowly oscillating type at 0 and ∞. [ABSTRACT FROM AUTHOR]

Let $${\alpha}$$ be an orientation-preserving homeomorphism of $${[0,\infty]}$$ onto itself with only two fixed points at 0 and $${\infty}$$ , whose restriction to $${\mathbb{R}_+=(0,\infty)}$$ is a diffeomorphism, and let $${U_\alpha}$$ be the corresponding isometric shift operator acting on the Lebesgue space $${L^p(\mathbb{R}_+)}$$ by the rule $${U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)}$$ . We prove criteria for the one-sided invertibility of the binomial functional operator $${aI-bU_\alpha}$$ on the spaces $${L^p(\mathbb{R}_+)}$$ , $${p\in(1,\infty)}$$ , under the assumptions that a, b and $${\alpha'}$$ are bounded and continuous on $${\mathbb{R}_+}$$ and may have slowly oscillating discontinuities at 0 and $${\infty}$$ . [ABSTRACT FROM AUTHOR]

Let $$\alpha ,\beta $$ be orientation-preserving diffeomorphism (shifts) of $$\mathbb {R}_+=(0,\infty )$$ onto itself with the only fixed points $$0$$ and $$\infty $$ and $$U_\alpha ,U_\beta $$ be the isometric shift operators on $$L^p(\mathbb {R}_+)$$ given by $$U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )$$ , $$U_\beta f=(\beta ')^{1/p}(f\circ \beta )$$ , and $$P_2^\pm =(I\pm S_2)/2$$ where is the weighted Cauchy singular integral operator. We prove that if $$\alpha ',\beta '$$ and $$c,d$$ are continuous on $$\mathbb {R}_+$$ and slowly oscillating at $$0$$ and $$\infty $$ , and then the operator $$(I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-$$ is Fredholm on $$L^p(\mathbb {R}_+)$$ and its index is equal to zero. Moreover, its regularizers are described. [ABSTRACT FROM AUTHOR]

We present a study of C ⁎ -algebras SO ( φ ) of slowly oscillating functions in the direction of filters φ on a locally compact topological group G . We show that SO ( φ ) is an m -admissible subalgebra of C ( G ) if and only if the closure of the filter φ in the LUC -compactification G LUC of G is an ideal of G LUC and that the semigroup compactification of G determined by SO ( φ ) always contains right zero elements. Using this, we characterize a new interesting C ⁎ -algebra of bounded continuous functions on G . The spectrum of this C ⁎ -algebra determines the universal semigroup compactification of G with respect to the property that the semigroup compactification contains a right zero element. We show that the topological center of this universal compactification is G . As an application of the previous results, we show that every closed ideal of G LUC contained in the ideal U ( G ) of uniform points of G LUC can be decomposed into 2 2 κ ( G ) closed left ideals of G LUC . [ABSTRACT FROM AUTHOR]

We consider the C*-algebra generated by Toeplitz operators acting on the Bergman space over the upper half-plane whose symbols depend only on the argument of the variable. This algebra is known to be commutative, and it is isometrically isomorphic to a certain algebra of bounded complex-valued functions on the real numbers. In the paper we prove that the latter algebra consists of all bounded functions ƒ that are very slowly oscillating on the real line in the sense that the composition of ƒ with sinh is uniformly continuous with respect to the usual metric. [ABSTRACT FROM AUTHOR]

Applying the theory of Calderon-Zygmund operators, we study the compactness of the commutators [aI, W0(b)] of multiplication operators aI and convolution operators W0(b) on weighted Lebesgue spaces Lp(ℝ,w) with p ∈ (l,∞) and Muckenhoupt weights w for some classes of piecewise quasicontinuous functions a ∈ PQC and b ∈ PQCp,w on the real line ℝ. Then we study two C*-algebras Z1 and Z2 generated by the operators aW0(b), where a,b are piecewise quasicontinuous functions admitting slowly oscillating discontinuities at ∞ or, respectively, quasicontinuous functions on ℝ admitting piecewise slowly oscillating discontinuities at ∞. We describe the maximal ideal spaces and the Gelfand transforms for the commutative quotient C*-algebras Ziπ = Zi/K (i = 1,2) where K is the ideal of compact operators on the space L²(ℝ), and establish the Fredholm criteria for the operators A ∈ Zi. [ABSTRACT FROM AUTHOR]

The paper investigates the convergence of the negative order Cesaro means of the Fourier series in Walsh system with monotone coefficients. It is given a description of the classes of the mentioned type series, the Cesaro means of which over all subsequences unbounded diverge on the sets of positive measure. [ABSTRACT FROM AUTHOR]

Abstract: A mapping of a semigroup T into a set X is a right zero homomorphism if for all . Let S be a discrete cancellative semigroup of cardinality , let βS be the Stone–Čech compactification of S, and let denote the ideal of βS consisting of uniform ultrafilters. We show that (a) if , then admits a continuous right zero homomorphism onto , and (b) if , then admits a continuous right zero homomorphism onto any connected compact metric space and onto a connected compact Hausdorff space of cardinality . [Copyright &y& Elsevier]

Abstract: Let a be a semi-almost periodic matrix function with the almost periodic representatives and at −∞ and +∞, respectively. Suppose is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space . We prove that if the operator with and is Fredholm on the variable Lebesgue space , then the operators and are invertible on standard Lebesgue spaces and with some exponents and lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively. [Copyright &y& Elsevier]

Suppose α is an orientation-preserving diffeomorphism (shift) of $${\mathbb {R}_+=(0,\infty)}$$ onto itself with the only fixed points 0 and ∞. In Karlovich et al. (Integr Equ Oper Theory , doi:) we found sufficient conditions for the Fredholmness of the singular integral operator with shiftacting on $${L^p(\mathbb {R}_+)}$$ with 1 < p < ∞, where $${P_\pm=(I\pm S)/2, S}$$ is the Cauchy singular integral operator, and $${W_\alpha f=f\circ\alpha}$$ is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α′ of the shift are bounded and continuous on $${\mathbb {R}_+}$$ and may admit discontinuities of slowly oscillating type at 0 and ∞. Now we prove that those conditions are also necessary. [ABSTRACT FROM AUTHOR]

Suppose α is an orientation preserving diffeomorphism (shift) of $${{\mathbb{R}}_+=(0,\infty)}$$ onto itself with the only fixed points 0 and ∞. We establish sufficient conditions for the Fredholmness of the singular integral operator with shiftacting on $${L^p({\mathbb{R}}_+)}$$ with 1 < p < ∞, where P = ( I ± S)/2, S is the Cauchy singular integral operator, and $${{{W_{\alpha}f=f\circ\alpha}}}$$ is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α′ of the shift are bounded and continuous on $${{\mathbb{R}}_+}$$ and may admit discontinuities of slowly oscillating type at 0 and ∞. [ABSTRACT FROM AUTHOR]

We prove necessary and sufficient conditions for the applicability of the finite section method to an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions and the convolution operators with symbols in the algebra generated by piecewise continuous and slowly oscillating Fourier multipliers on $${L^p(\mathbb {R})}$$, 1 < p < ∞. [ABSTRACT FROM AUTHOR]

Abstract: For every infinite discrete group G, the remainder of the Stone–Čech compactification βG of G has a natural structure of G-space. The orbit equivalence E on ( for some ) produces the following three derived equivalences on [()] , where , are E-equivalence classes containing x and y, [ ] is the smallest by inclusion equivalence on containing E such that every -equivalence class is closed, [ ] is the smallest by inclusion closed in equivalence on containing E, and the relation [ ] , , which is an equivalence if G is countable. We study the interrelations between the classes of these equivalences and the principal left ideals of the semigroup βG. [Copyright &y& Elsevier]

Abstract: We first extend slowly oscillating functions to a more general setting and investigate their properties. Then we show the existence and uniqueness of slowly oscillating solutions of parabolic equations and parabolic inverse problems. [Copyright &y& Elsevier]

Abstract: The paper proposes a unified approach to many key theorems proved in the last twenty years in different areas of abstract harmonic analysis. This approach is based on the so-called slowly oscillating functions which were introduced in coarse geometry. In addition to this method being the most natural and simple, it also leads to the generalisation of some of the results and to the achievement of some new results. Several of these results concern the topological centres of convolution algebras and semigroup compactifications. [Copyright &y& Elsevier]

Abstract: Given a function defined on the support of a ballean, we introduce the notion of slow oscillation in direction of a filter on X. We show that there exists a filter on X responsible for the rate of slow oscillation of f at infinity. We apply this result to the Stone–Čech compactifications of discrete groups. [Copyright &y& Elsevier]

Abstract: We introduce balleans as asymptotical counterparts of uniform topological spaces. Using slowly oscillating functions, for every ballean we define two compact spaces: corona and binary corona. These spaces can be considered as generalizations of the Higson''s coronas of metric spaces and the spaces of ends of groups, respectively. We consider some balleans related to an infinite group and prove some results concerning their coronas. At the end we apply these results to describe the compact right-zero semigroups which are continuous homomorphic images of , the reminder of the Stone–Čech compactification of discrete group G. [Copyright &y& Elsevier]