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.     

On the Solvability of Integral Operators with Bihomogeneous Kernels of the Compact Type and Variable Coefficients

In this paper, we consider the C ^? -algebra Ω_mult(? ⁿ ) of multiplicatively weakly oscillating functions and a new wide class of kernels of compact type, which includes the class of SO(n)-invariant kernels. For the Banach algebra generated by operators with kernels of this class and coefficients from Ω_mult(? ⁿ ), we construct a symbolic calculus, obtain necessary and sufficient conditions for the presence of the Fredholm property, and propose a method of calculating the index of families. Similar results are obtained for operators with bihomogeneous kernels of compact type and multiplicatively weakly oscillating coefficients, i.e., for operators from the tensor product \( {{\mathfrak{M}}_{p,n}}_{{_1}}\otimes {{\mathfrak{M}}_{p,n}}_{{_2}} \) .     

On the constant and step in Jackson’s inequality for best approximations by trigonometric polynomials and by Haar polynomials

We consider the C*-algebra generated by multidimensional integral operators with (?n)th-order homogeneous kernels and by the operators of multiplication by oscillating coefficients of the form |x|^iα. For this algebra, we construct an operator symbolic calculus and obtain necessary and sufficient conditions for the Fredholm property of an operator in terms of this calculus.     

Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type

The problem of getting effective Fredholm conditions for operators with bihomogeneous kernels reduces to the question of invertibility for families of operators with homogeneous kernels and to the calculation of homotopy invariants for spaces of Fredholm and invertible operators of that type. The purpose of the present paper is to study integral operators with homogeneous kernels of compact type in L _p (? ⁿ ), 1 < p < +??. The classes of homotopy equivalence for the spaces of Fredholm and invertible operators in the C*-algebra of pair operators with homogeneous kernels of compact type are calculated by means of operator K-theory.     

Volterra type integral operators with homogeneous kernels in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We consider multidimensional integral Volterra type operators with kernels homogeneous of degree (?n); the operators act in L _p -spaces with a submultiplicative weight. For these operators we obtain necessary and sufficient conditions of their invertibility. Besides, we describe the Banach algebra generated by the operators. For this algebra we construct the symbolic calculus, in terms of which we obtain an invertibility criterion of the operators.     

On the algebra of multidimensional integral operators with homogeneous SO(n)-invariant kernels and weakly radially oscillating coefficients

In this paper, we study the Banach algebra B generated by multidimensional integral operators whose kernels are homogeneous functions of degree (?n) invariant with respect to the rotation group SO(n) and by the operators of multiplication by radial weakly oscillating functions. A symbolic calculus is developed for the algebra 25. The Fredholm property and the formula for calculating the index are described in terms of this calculus.     

Weighted Oscillation and Variation Inequalities for Singular Integrals and Commutators Satisfying Hrmander Type Conditions

This paper is devoted to investigating the weighted L~p-mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish the weighted type(p, p) estimates for 1 p ∞ and the weighted weak type(1,1) estimate for the oscillation and variation operators of singular integrals with kernels satisfying certain Hormander type conditions, which contain the Riesz transforms, singular integrals with more general homogeneous kernels satisfying the Lipschitz conditions and the classical Dini's conditions as model examples. Meanwhile, we also obtain the weighted L~p-boundeness for such operators associated to the family of commutators generated by the singular integrals above with BMO(R~d)-functions.     

Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ ₁ ≤ 1 and σ ₂ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T _Ω, fractional integrals T _Ω,α and parametric Marcinkiewicz integrals μ _Ω ^ρ with variable kernels on the Hardy spaces H ^p (R ⁿ ) and weak Hardy spaces WH ^p (R ⁿ ). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces H ^p,q(R ⁿ ) for all p < q < ∞.     

On Hille-Tamarkin operators and Schatten classes

We determine the smallest Schatten class containing all integral operators with kernels inL _p(L_{p', q})^symm, where 2 <p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].     

Compact elements of the algebras PM p (G)and PF p (G) of a locally compact group

Compact and weakly compact elements of the group algebra L ¹ (G) of a locally compact group G, have been considered by a number of authors. In these investigations it has been shown that, if G is non-compact, then the only weakly compact element of L ¹ (G ) is zero. Conversely, if G is compact, then every element of L ¹ (G) is compact. For 1<p<∞, let PM _p (G)and PF _p (G) denote the closure of L ¹ (G), considered as an algebra of convolution operators on L ^p (G), with respect to the weak operator topology and the norm topology, respectively, in B(L ^p (G), b), the bounded linear operators on L ¹ (G). We study the question of characterizing compact and weakly compact elements of the algebras PM _p (G)and PF _p (G).     

Norm estimates of ω-circulant operator matrices and isomorphic operators for ω-circulant algebra

An n ×nω-circulant matrix which has a specific structure is a type of important matrix. Several norm equalities and inequalities are proved for ω-circulant operator matrices with ω = e^iθ (0 ≤ θ < 2π) in this paper. We give the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norms. Pinching type inequality is also proposed for weakly unitarily invariant norms. Meanwhile, we present that the set of ω-circulant matrices with complex entries has an idempotent basis. Based on this basis, we introduce an automorphism on the ω-circulant algebra and then show different operators on linear vector space that are isomorphic to the ω-circulant algebra. The function properties, other idempotent bases and a linear involution are discussed for ω-circulant algebra. These results are closely related to the special structure of ω-circulant matrices.     

Transformation operators for the canonical Dirac system

For canonical Dirac systems of differential equations with locally integrable coefficients, we prove the existence of transformation operators and estimate the kernels of these operators. We also give estimates for these kernels for the case in which the coefficients belong to the space L _loc ² . We establish a relationship between the kernel of the transformation operators and the potential matrix.     

Schur coefficients in several variables

In the paper we study weakly continuous Schur-class-valued maps and their associated Schur coefficient families, that we call functional Schur coefficients. A case of special interest is the family of the “slices” through the polytorus of an n-variable function in the unit ball of H^∞(Dn), which is shown to be a weakly continuous map from the polytorus into the Schur class. The continuity properties of its functional Schur coefficients are used to characterize the rational inner functions in the polydisk algebra. As a consequence we obtain extensions in several variables of the Schur-Cohn test on zeroes of polynomials. This provides in particular a necessary and sufficient condition of stability for multi-dimensional AR filters.     

The fractional Hausdorff operators on the Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper, we study the high-dimensional fractional Hausdorff operators and establish their boundedness on the real Hardy spaces H ^p (? ⁿ ) for 0 < p < 1.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poisson Semigroup Associated with Elliptic Systems

We study the infinitesimal generator of the Poisson semigroup in L ^p associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the L ^p -based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may be adapted to treat the case of higher order systems in graph Lipschitz domains.     

Sparsity and non-Euclidean embeddings

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables one to construct embeddings of ? _p ⁿ , p > 0, into various types of Banach or quasi-Banach spaces. In particular, for 0 < r < p < 2 with r ≤ 1, we construct a family of operators that embed ? _p ⁿ into $\ell _r^{(1 + \eta )n}$ , with sharp polynomial bounds in η > 0.     

Weighted inequalities for some integral operators with rough kernels

In this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-0em} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-0em} {q_i }}}$ where Ω _i : ? ⁿ → ? are homogeneous functions of degree zero, satisfying a size and a Dini condition, A _i are certain invertible matrices, and n/q ₁ +…+n/q _m = n?α, 0 ≤ α < n. We obtain the appropriate weighted L ^p -L ^q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.     

Finite Section Method for a Banach Algebra of Convolution Type Operators on {L^p(\mathbb R)} with Symbols Generated by PC and SO

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){L^p(\mathbb {R})}, 1 < p < ∞.     

Analysis of a Distinguished Laplacian on Solvable Lie Groups

We study a class of kernels associated to functions of a distinguished Laplacian on the solvable group AN occurring in the Iwasawa decomposition G = ANK of a noncompact semisimple Lie group G. We determine the maximal ideal space of a commutative subalgebra of L¹, which contains the algebra generated by the heat kernel, and we prove that the spectrum of the Laplacian is the same on all L^p spaces, 1 ≤ p < ∞. When G is complex, we derive a formula that enables us to compute the L^p norm of these kernels in terms of a weighted L^p norm of the corresponding kernels for the Euclidean Laplacian on the tangent space. We also prove that, when G is either rank one or complex, certain Hardy-Littlewood maximal operators, which are naturally associated with these kernels, are weak type (1, 1).     

The algebra of Calderón-Zygmund kernels on a homogeneous group is inverse-closed

We show that the subalgebra of convolution operators with Calderón-Zygmund kernels on a homogeneous group G is inverse-closed in the algebra of all bounded linear operators on the Hilbert space L ²(G). The main tool used is a symbolic calculus, where the convolution of distributions on the group is translated via the abelian Fourier transform into a “twisted product” of symbols on the dual to the Lie algebra g of G.