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All of the integral operators with nonintegrable kernels are given in terms of computable Hadamard's partie finie, i.e. finite part integrals [168], which can be applied to problems in applications (Guiggiani [163], Schwab et al [380]).

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on $L^p$, $1, Comment: Change of title. New abstract and new introduction

We consider the Riesz transforms ∇L−1/2, where L≡− divA(x)∇, and A is an accretive, n × n matrix with bounded measurable complex entries, defined on Rn. We establish boundedness of these operators on Lp(Rn), for the range pn < p ≤ 2, where pn = 2n/(n + 2), n ≥ 2, and we obtain a weak-type estimate at the endpoint pn. The case p = 2 was already known: it is equivalent to the solution of the square root problem of T. Kato.

We shall define a Calderon-Zygmund class \(\mathrm{CZ}^p(\Delta _{\alpha ,\beta })\) on the Jacobi hypergroup \((\mathbf{R}_+,\Delta _{\alpha ,\beta }, *)\) such that, if a function g on \(\mathbf{R}_+\) belongs to \(\mathrm{CZ}^p(\Delta _{\alpha ,\beta })\), then the convolution operator \(g*\) is bounded from \(L^q(\Delta _{\alpha ,\beta })\) to itself for \(p\le q\le 2\). Actually, we shall obtain a relation between the \(L^p\) norms of g and the Abel transform \(\mathcal{A}g\) and a transference principle between the \(L^p\) operator norms of \(g*\) and the Euclidean operator \(\phi \circledast \), where \(\phi (x)=e^{(\frac{2}{p}-1)\rho x}\mathcal{A}g(x)\). Therefore, to define the Calderon-Zygmund class \(\mathrm{CZ}^p(\Delta _{\alpha ,\beta })\), we shall obtain some conditions on g under which \(\phi \) belongs to \(\mathrm{CZ}(\mathbf{R})\). Then, \(\phi \circledast \) is bounded on \(L^q({\mathbf{R}})\) and, by the transference principle, \(g*\) is bounded on \(L^q(\Delta _{\alpha ,\beta })\).

We provide a characterization of $\mathrm{BMO}$ in terms of endpoint boundedness of commutators of singular integrals. In particular, in one dimension, we show that $\|b\|_{\mathrm{BMO}}\eqsim B$, where $B$ is the best constant in the endpoint $L\log L$ modular estimate for the commutator $[H,b]$. We provide a similar characterization of the space $\mathrm{BMO}$ in terms of endpoint boundedness of higher order commutators of the Hilbert transform. In higher dimension we give the corresponding characterization of $\mathrm{BMO}$ in terms of the first order commutators of the Riesz transforms. We also show that these characterizations can be given in terms of commutators of more general singular integral operators of convolution type.

We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces.

In this paper we show how a second order scalar uniformly elliptic equation in divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary condition. This first order system involves Hodge-Dirac operators and can be seen as a natural generalization of the Beltrami equation in the plane and we develop a theory for this equation, extending results from the plane to higher dimension. The reduction to a first order system applies both to linear as well as quasilinear second order equations and we believe this to be of independent interest. Using the first order system, we give a new representation formula of the solution of the Dirichlet problem both on simply and finitely connected domains. This representation formula involves only singular integral operators of convolution type and Neumann series there of, for which classical Calderon-Zygmund theory is applicable. Moreover, no use is made of any fundamental solution or Green's function beside fundamental solutions of constant coefficient operators. Remarkably, this representation formula applies also for solutions of the fully non-linear first order system. We hope that the representation formula could be used for numerically solving the equations. Using these tools we give a new short proof of Meyers' higher integrability theorem. Furthermore, we show that the solutions of the first order system are Holder continuous with the same Holder coefficient as the solutions of the second order equations. Finally, factorization identities and representation formulas for the higher dimensional Beurling-Ahlfors operator are proven using Clifford algebras, and certain integral estimates for the Cauchy transform is extended to higher dimensions.

In this paper, we construct two classes of rational function operators using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied.

The modular estimates for the fractional integral operators and the k-plane transforms are obtained in this paper. These estimates are obtained by using the modular estimates of Hardy operators and the modular interpolation theorem.

In this paper we study the Riesz transforms Rw related to the Weinstein operators Δw=∑i=1dxd−2α−1(∂/∂xi)(xd2α+1(∂/∂xi)). We develop for Rw a theory that runs parallel to the one for the Euclidean Riesz Transform. It is proved that the Riesz–Weinstein transform in coordinates i=1,…,d, Rwi is actually a Calderon–Zygmund singular integral operator in the sense of the associated space of homogeneous type. Moreover, our Riesz–Weinstein transform can be written as a principal value.

We select a class of operators with endpoint singularities. For these operators, we establish inequalities connecting the norms in Lebesgue spaces with weight with the norms in Holder spaces with weight. We describe specific types of the operators satisfying the conditions of the main theorem on relationship between the norms. The obtained results can be used to study the operators acting in Holder spaces with weight on the basis of the well-known results obtained for the operators acting upon the Lebesgue spaces with weight.

We study the vector-valued B-singular integral operators associated with the Laplace-Bessel differential operator $$\triangle_{B}=\sum\limits_{k=1}^{n-1}\frac{\partial^{2}}{\partial x_{k}^{2}}+(\frac{\partial^{2}}{\partial x_{n}^{2}}+\frac{2v}{x_{n}}\frac{\partial}{\partial x_{n}}) , v>0.$$ We prove the boundedness of vector-valued B-singular integral operators A from $L_{p,v}(\mathbb{R}_{+}^{n}, H_{1}) \,{\rm to}\, L_{p,v}(\mathbb{R}_{+}^{n}, H_{2}),$ 1 < p < ∞, where H1 and H2 are separable Hilbert spaces.

In this note we study the relation between oscillatory integral operators and convolution operators, and also the sharpness of L p L^p -estimates for singular oscillatory integral operators.

In this paper we introduce the weighted version of fully measurable grand Lebesgue spaces and obtain characterizations for the boundedness of maximal operator, Hilbert transform and the Hardy averaging operator on these spaces.

We give sufficient conditions on variable exponent functions p : Rn → [1,∞) for which the higher-order Riesz transforms, associated with the Ornstein–Uhlenbeck semigroup, are bounded on Lp(·) (Rn, dγ ), where γ denotes the Gaussian measure. Fil: Dalmasso, Estefanía Dafne. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina Fil: Scotto, Roberto Aníbal. Universidad Nacional del Litoral. Facultad de Ingeniería Química; Argentina

We give examples of Lebesgue spaces on which Cauchy and Cauchy–Leray–Fantappie operators are not bounded.

The aim of this paper is to obtain the description of the first and second complex interpolations of grand Lebesgue spaces. We also investigate the complex interpolation of closed subspaces satisfying the lattice property.

In approximation theory, the Lebesgue constant of an interpolation operator plays an important role. The Lebesgue constant of Berrut's interpolation operator has been extensive studied. In the present work, by introducing a new method, we obtain an optimal asymptotic Lebesgue constant of Berrut's rational interpolant at equidistant nodes.

In this paper, we give a King-type modification of the Bernstein–Kantorovich operators and study the approximation properties of these operators. We prove that the error estimation of these operators is better than the classical Bernstein–Kantorovich operators. We also give some estimations for the rate of convergence of these operators by using the modulus of continuity. Furthermore, we obtain a Voronovskaya-type asymptotic formula for these operators.

We study a new class of homogeneous operators in L2( $$ {\mathbb{R}}^2 $$ ) that, after foliation of $$ {\mathbb{R}}^2 $$ into concentric circles, are represented in fibres as singular integral operators with measurable essentially bounded coefficients. We find necessary and sufficient conditions for the invertibility of such operators and construct the operator-valued symbolic calculus for the C∗–algebra generated by such operators and operators of multiplication by multiplicatively weakly oscillating functions. We obtain a criterion for the generalized Fredholm property of operators and find effectively verifiable functional necessary conditions for the classical Fredholm property.

We study the problem of characterizing weighted inequalities on Lebesgue cones of monotone functions on the half-axis for one class of quasilinear integral operators.

The Lebesgue constant of interpolation operators plays an important role in approximation theory. Several upper bounds have been achieved on the Lebesgue constant of Berrut’s rational interpolation operator. The aim of this paper is to investigate the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes, and present a tight upper bound.

For a class of convolution integral operators whose kernels may have integrable singularities, boundedness and compactness criteria in weighted Lebesgue spaces are obtained.

We study spectral properties of operators on logarithmic perturbations of the real interpolation spaces with θ = 0 or 1. We also establish estimates for the measure of non-compactness of operators interpolated by those methods.

We introduce the notion of generalized weighted Morrey spaces and investigate the boundedness of some operators in these spaces, such as the Hardy–Littlewood maximal operator, generalized fractional maximal operators, generalized fractional integral operators, and singular integral operators. We also study their boundedness in the vector-valued setting.

The norms in weighted Holder spaces and in weighted Lebesgue spaces are different in their character and any direct connection between the norms of these spaces should not be expected. However, in this work, a special class of operators was found, for which we obtained an inequality that connects the norms of these operators acting on weighted Lebesgue spaces and acting on weighted Holder spaces. A description of such operators and a relation among parameters of these spaces are given. In particular, integral operators with local endpoint singularities belong to the considered class. These results can be used in the study of operators on weighted Holder spaces, based on known results for operators acting on weighted Lebesgue spaces.

For some questions in the theory of partial differential equations, it is useful to consider the mixed-norm Lebesgue spaces introduced by Benedek and Panzone (1962). We show that Hormander's results (1960) obtained on translation invariant operators on Lebesgue spaces extend to the mixed-norm case. We also revisit the Hormander–Mihlin multiplier theorem in this setting, which was first obtained by Lizorkin (1970), where we particularly consider the form of the constants used in the estimates. This improvement allows us to prove the boundedness of classical pseudodifferential operators on mixed-norm Lebesgue spaces, as well as generalising the H-distributions, a recently introduced (2011, by Mitrovic and the first author) extension of H-measures, to these spaces.

The authors establish the two-weight norm inequalities for the one-sided Hardy-Littlewood maximal operators in variable Lebesgue spaces. As application, they obtain the two-weight norm inequalities of variable Riemann-Liouville operator and variable Weyl operator in variable Lebesgue spaces on bounded intervals.

By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.

In this paper, we present some theorems on weighted approximation by two dimensional nonlinear singular integral operators in the following form: T λ ( f ; x , y ) = ∬ R 2 K λ ( t − x , s − y , f ( t , s ) ) d s d t , ( x , y ) ∈ R 2 , λ ∈ Λ , $${T_\lambda }(f;x,y) = \iint\limits_{{\mathbb{R}^2}}K_\lambda {(t - x,s - y,f(t,s))dsdt,\;(x,y) \in {\mathbb{R}^2},\lambda \in \Lambda ,}$$ where Λ is a set of non-negative numbers with accumulation point λ0.

We calculate weak bounds for the Hausdorff operator on Lebesgue spaces. As applications, we obtain sharp weak bounds for some integral operators, such as the fractional Hardy operators and Hilbert operators. We also conclude that the Hausdorff operator is bounded from Hardy spaces to weak Lebesgue spaces.

In the present paper, we shall give a necessary and sufficient condition for the weak/strong boundedness of generalized fractional maximal operators on generalized Orlicz–Morrey spaces. We also give necessary and sufficient conditions for the vector-valued inequalities of the Hardy–Littlewood maximal operator, generalized fractional maximal operators and singular integral operators on these function spaces.

In this work, we present a bilinear Tb theorem for singular integral operators of Calderon–Zygmund type. We prove some new accretive type Littlewood–Paley results and construct a bilinear paraproduct for a para-accretive function setting. As an application of our bilinear Tb theorem, we prove product Lebesgue space bounds for bilinear Riesz transforms defined on Lipschitz curves.

Methods of the theory of monotone operators are used to prove global theorems on the existence and uniqueness of solutions, as well as on estimates of their norms, for various classes of nonlinear integral convolution-type equations in the real Lebesgue spaces Lp(0, 1). These theorems involve nonlinear equations with potential-type kernels, including logarithmic potential-type kernels, as well as the corresponding linear integral equations within the framework of the space L2(0, 1). Corollaries illustrating the obtained results are presented.

In this paper, the behavior on the product of Lebesgue spaces is considered for the maximal operators associated with the bilinear singular integral operators whose kernels satisfy certain minimal regularity conditions.

We study the generalized fractional integral transforms associated to a measure on a quasi-metric space. We give a characterization of those measures for which these operators are bounded between L p -spaces defined on nonhomogeneous spaces. The key in the proof of one of the main theorems is the boundedness of the modified sublinear Hardy–Littlewood maximal operator in the classical Lebesgue space with general measure. We also provide necessary and sufficient conditions for some classes of integral operators to be bounded from Lorentz to Marcinkiewicz spaces.

As is well-known, because of their important applications in several branches of mathematics, such as harmonic analysis and partial differential equations, the theory of Calderon-Zygmund singular integral operators and the real-variable theory of various function spaces, which could characterize the boundedness of those operators, turn into one of the main contents of modern harmonic analysis. However, the real-variable theory of classical function spaces has been no longer suitable for characterizing the boundedness of singular integral operators associated with some more general differential operators than Laplace operators. Thus, for different operators, it has become one of the very active research fields of harmonic analysis in recent years to develop the real-variable theory of function spaces, which are suitable to those operators and could characterize the boundedness of singular integral operators associated with those operators. In this article, we study the realvariable theory of (Musielak-)Orlicz-Hardy spaces associated with some differential operators, including secondorder divergence form elliptic operators and Schrodinger operators as special cases, on n -dimensional Euclidean space R n , strongly domains of R n or metric spaces with doubling measure, and its applications to the boundedness of operators.

Fil: Ferreyra, Elida Vilma. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

In the present paper, we introduce the q-analogue of the general Gamma type operators. We establish the moments of the operators by using the q-derivatives and then prove the basic convergence theorem. Next, the Voronovskaja type theorem and some direct results for the above operators are discussed. Also, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Then, we obtain point-wise estimate using the Lipschitz type maximal function. Further, we study the A -statistical convergence of these operators. Lastly, we propose a king type modification of these operators to obtain better estimates.

A class of convolution operators on spaces of holomorphic functions related to the Hadamard multiplication theorem for power series and generalizing infinite order Euler differential operators is introduced and investigated. Emphasis is placed on questions concerning injectivity, denseness of range, and surjectivity of the operators. AMS classification: 47A05 primary; 30B40, 34A35 secondary

This paper concentrates on the Lebesgue functions of rational interpolation of non-band-limited continuous time signals. Approximation based on sampling and interpolation are cornerstones of applied mathematics. In the last years rational interpolations has been in the focus of the investigations, because they have better approximation properties then the polynomial interpolations. The Whittaker-Kotelnikov-Shannon sampling theorem is for band-limited signals and requires the a priori knowledge of the band-width. In [1], [2] new rational interpolation operators were developed for the transfer function of non-band-limited signals, which can be used also in cases when the band-width is not known a priori. The construction of these operators is based on the discrete orthogonality of the Malmquist-Takenaka systems. Combining these interpolations one can give exact interpolation on the real line for a large class of rational functions among them for the Runge test function. Our aim is to study the properties of the Lebesgue function of these rational interpolation operators.

The present paper deals with the general class of Durrmeyer type operators. Here we introduce a generalized family of the hybrid integral operators, the special cases of our operators include some well known operators as particular cases viz. Lupas–Szasz type operators, Phillips operators and the Baskakov–Szasz operators. We establish some direct results, which include the asymptotic formula and error estimation in terms of the modulus of continuity.

We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in S 1 , 0 m ( n , 2 ) and non-degenerate phase functions, from L p × L q → L r under the assumptions that m ⩽ − ( n − 1 ) ( | 1 p − 1 2 | + | 1 q − 1 2 | ) and 1 p + 1 q = 1 r . This is a bilinear version of the classical theorem of Seeger–Sogge–Stein concerning the L p boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of quasi-Banach target spaces.

We prove that a large class of operators, which arise as the projections of martingale transforms of stochastic integrals with respect to Brownian motion, as well as other closely related operators, are in fact Calderon-Zygmund operators. These operators have played an important role in studying the Lp boundedness, 1 < p < ∞, of classical Calderon-Zygmund operators such as the Beurling-Ahlfors transform and the Riesz transform. Showing that these operators are Calderon-Zygmund implies that they are not only bounded on Lp, but also satisfy weak-type inequalities. Unlike the boundedness on Lp, which can be obtained directly from the Burkholder martingale transform inequalities, the weak-type estimates do not follow from the corresponding martingale results. The reason for this is that the Lp boundedness of these operators uses conditional expectation, which unfortunately does not preserve weak-type inequalities. Instead, we represent these operators in a purely analytic fashion as integration against a kernel and obtain our result by showing that our kernel satisfies suitable estimates.

Two-weight norm estimates for sublinear integral operators involving Hardy-Littlewood maximal, Calderón-Zygmund and fractional integral operators in variable exponent Lebesgue spaces are derived. Operators and the space are defined on a quasi-metric measure space with doubling condition. The derived conditions are written in terms ofLp(·)norms and are simultaneously necessary and sufficient for appropriate inequalities for maximal and fractional integral operators mainly in the case when weights are of radial type.