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We give criteria for boundedness of the fractional integration operators of the Riemann–Liouville type with variable limits between Lebesgue spaces on the real axis.

We establish a Fredholm criterion for the operators belonging to the C ∗ -algebra generated by singular integral operators with semi-almost periodic matrix coefficients. The result is applied to Toeplitz-like operators that are perturbed by integral operators with fixed singularities at infinity, in which case it leads to an effectively verifiable Fredholm criterion together with an index formula. Our approach is based on an isomorphism theorem for C ∗ -algebras associated with C ∗ -dynamical systems and the notion of canonical generalized AP factorization for almost periodic matrix functions.

This paper is devoted to the study of four integral operators, which are generalizations and modifications of integrals of Hadamard, in the space X c p of Lebesgue measurable functions f on R + = (0, ∞) such that for c ∈ R = (−∞, ∞) [Formula: See Text] Representations for the operators are given in the form of integral transforms involving the Meijer G-function in the kernels. The mapping properties such as the boundedness, representation and range are established.

We consider expansions with respect to the multi-dimensional Hermite functions which are eigenfunctions of the harmonic oscillator L=−Δ+|x|2. For the heat-diffusion and Poisson semigroups corresponding to a self-adjoint extension L of L we investigate their boundary behaviour and mapping properties. All this is done for functions from Lp(w), 1⩽p

We study the singular integral operatordefined on all test functions f, where b is a bounded function, α ≥ 0, Ω (yʻ) is an integrable function on the unit sphere Sn-1 satisfying certain cancellation conditions. We prove that, for 1 < p < ∞, TΩ,α extends to a bounded operator from the Sobolev space to the Lebesgue space Lp with Ω being a distribution in the Hardy space Hq(Sn-1) where . The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for TΩ,α on the Hardy spaces, as well as the boundedness for the truncated maximal operator .

We prove sharp L^p-L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature assumption., 30 pages

It is shown that maximal truncations of nonconvolution L 2 -bounded sin- gular integral operators with kernels satisfying Hormander's condition are weak type (1; 1) and L p -bounded for 1 0 Rj xj 2R jk(x)jdx

By applying the way of real and functional analysis and estimating the weight functions, we build some lemmas and deduce some Hilbert-type and Hilbert-Hardy-type integral inequalities with the best possible constant factors. The equivalent forms, the reverses and the operator expressions are all considered. The composition formula of two Hilbert-Hardy-type integral operators and some examples are given.

We construct a Fredholm symbol calculus for the C ∗-algebra \({\mathfrak{B}}\) generated by the C ∗-algebra \({\mathfrak{A}}\) of two-dimensional singular integral operators with continuous coefficients on a bounded closed simply connected domain \(\overline{U}\subset\mathbb{R}^{2}\) with Liapunov boundary and by all unitary shift operators W g where g runs through a discrete solvable group G=F⋊H of diffeomorphisms of \(\overline{U}\) onto itself, where F is a commutative group of conformal mappings, H={e,γ} and γ is similar to the shift \(z\mapsto \overline{z}\). As a result, we establish a Fredholm criterion for the operators \(B\in{\mathfrak{B}}\).

In this paper we will exhibit a class of kernels generating $r$-nuclear operators. The class includes the Fox-Li and related operators. Estimates for the corresponding asymptotic behaviour of the eigenvalues are also derived.

In this note, we estimate the boundedness for singular integral operators along curves and surfaces with highly singular kernels.

The theta (t)-type oscillatory singular integral operators has been discussed. With the non-negative locally integrable weighted function, the weighted norm inequality of theta(t)-type oscillatory singular integral operators is proved, and the weighted function has replaced by action of Hardy-Littlewood maximal operators several times.

In this paper, the authors prove that some oscillatory singular integral operators of non-convolution type with non-polynomial phases are bounded from the Herz-type Hardy spaces to the Herz spaces and from the Hardy spaces associated with the Beurling algebras to the Beurling algebras in higher dimensions, even though it is well-known that these operators are not bounded from the Hardy spaces H 1(ℝ n ) into the Lebesgue space L 1 (ℝ n ).

The object of this Paper is to Present a q-analogue of the Newton's forward-difference interpolation formula. We relate the q-beta function and the q-gamma function by the q-difference operators.

We prove a Covering Theorem that allows us to prove modified norm inequalities for general maximal operators. We will also give applications to convergence of a sequence of linear operators and the differentiation of the integral.

In this paper we study the properties of positive-normal operators and show that Wey1's theorem holds for some totally positive-normal operators.

Let G be a homogeneous group. In this paper, the authors establish several general theorems for the boundedness of sublinear operators and commutators generated by linear operators and BMO(G) functions on the weighted Lebesgue space on G. The conditions of these theorems are satisfied by many important operators in analysis and these operators satisfy only some weak conditions on the size of operators and are known to be bounded in the unweighted case.

In this chapter, we shall first study two basic results in interpolation of operators in L p spaces, the Riesz–Thorin theorem and the Marcinkiewicz interpolation theorem (diagonal case). As a consequence of the former we shall prove the Hardy-Littlewood-Sobolev theorem for Riesz potentials. In this regard, we need to introduce one of the fundamental tools in harmonic analysis, the Hardy–Littlewood maximal function. In Section 2.4 we shall prove the Mihlin multiplier theorem.

In the present paper, we introduce the q-analog of the Stancu variant of Szász-Baskakov 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 a point-wise estimate using the Lipschitz type maximal function. Lastly, we study the A-statistical convergence of these operators and also, in order to obtain a better approximation, we study a King type modification of the above operators. MSC:41A25, 26A15, 40A35.

In this paper, the authors introduce the dense-image operators T as those with a wild behaviour near of the boundary of a domain G, via certain subsets. The relationship with other kinds of operators with wild behaviour is studied, proving that the new concept generalizes the earlier of omnipresent, but there is no good relationship with the strongly omnipresent operators. We obtain, among other results, that the following kinds of operators are dense-image: onto linear operators; operators with local dense range satisfying soft conditions; Volterra complex integral operators plus infinite order differential operators, multiplication operators. In addition, holomorphic selfmappings and entire functions generating dense-image right or left composition operators are completely characterized. Dirección General de Enseñanza Superior (DGES). España Junta de Andalucía

In this work we present several theorems which imply the weak type 1 with respect to the Gaussian measure for the so-called local part of certain operators associated with the Ornstein-Uhlenbeck semigroup. Particular cases of these operators are the Riesz transforms of any order and the Littlewood-Paley square function. Also, we study general results based on the “size” of the operator which ensure the strong type 1

In this paper we study the Banach algebra of singular integral operators with piecewise continuous coefficients and a Carleman orientation-reversing slowly oscillating shift on the Lebesgue space with a power weight on the unit circle. The slow oscillation of the shift derivative, in contrast to the classic assumption on its piecewise continuity, leads to the appearance of massive local spectra for the considered operators. Applying localization techniques and the theory of Mellin pseudodifferential and associated limit operators, we construct a symbol calculus for the above-mentioned operator algebra and find a Fredholm criterion and an index formula for the operators in this algebra in terms of their symbols.

Let T be a Calderon-Zygmund operator in a "non-homogeneous" space (X,d,µ), where, in particular, the measure µ may be non- doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil and A. Vol- berg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T?, which is the supremum of the trun- cated operators associated with T. Specifically, for 1 < p < 1, we obtain sucient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L p weighted inequality holds for T?. The main tool to deal with this problem is the theory of vector-valued inequalities for T? and some re- lated operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderon-Zygmund operators in non-homogeneous spaces, developed in our previous pa- per (GM). For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C? to characterize the existence of principal values for functions in weighted L p .

In this paper, we establish the boundedness of certain maximal operators along hyperspaces in Lebesgue mixed norm spaces.

The asymptotic behavior of the values of the integral of the Lebesgue function induced by interpolation at the Chebyshev roots is studied. Two leading terms in the corresponding asymptotic expansion are found explicitly.

We obtain estimates and convergence results with respect to ϕ-variation in spaces BVΦ for a class of linear integral operators whose kernels satisfy a general homogeneity condition. Rates of approximation are also obtained. As applications, we apply our general theory to the case of Mellin convolution operators, to that one of moment operators and finally to a class of operators of fractional order.

We investigate the Volterra-Stieltjes integral operators with kernels depending on two variables. Some properties of operators of such a type are established. Solvability of nonlinear integral Volterra-Stieltjes equations is also studied.

We shall study properties of box spline operators: cardinal interpolation, convolution, and the Bernstein-Schnabl operator. We prove the saturation, theorem.

In this article we give sufficient conditions on a pair of weight (w, v) for some one-sided operators to be bounded from Lp (vp) to Lp (wp). The operators we deal with include the one-sided fractional maximal operator and the one-sided singular integrals. For the first operator, necessary and sufficient conditions are known (see [8, 6]). These conditions usually amount to checking the boundedness of the operator on functions that are powers of the weights and are hard to check. Our conditions are of Ap type and are therefore easy to verify. Similar results for two-sided operators were obtained by C. Perez in [9] and [10].

If E E and F F are Banach lattices such that E E is separable and F F has the countable interpolation property, then the space of all continuous regular operators L \mathcal {L} r ( E , F ) ^r(E,F) has the Riesz decomposition property. This result is a positive answer to a conjecture posed by A. W. Wickstead.

The r-asymptotically quasi finite rank operators on Banach lattices are examples of regular Riesz operators. We characterise Riesz elements in a subalgebra of a Banach algebra in terms of Riesz elements in the Banach algebra. This enables us to characterise r-asymptotically quasi finite rank operators in terms of adjoint operators. The r-asymptotically quasi finite rank operators are also employed to study the following problem: Suppose operators S and T on a Banach lattice E satisfy 0 &#8804 S &#8804 T . If T is a Riesz operator, when is it true that S is a Riesz operator? Quaestiones Mathematicae 23(2000), 179–186

We prove an analog of the Berry-Esseen estimate for the heat kernel of second order elliptic differential operators with quasiperiodic coefficients. As an application of this result, we prove the Lp boundedness of the associated Riesz transform operators.

We introduce certain Calderon-Zygmund-type operators and discuss their boundedness on spaces such as weighted Lebesgue spaces, weighted weak Lebesgue spaces, weighted Hardy spaces and weighted weak Hardy spaces. The sharpness of some results is also investigated.

We continue the study of algebras generated by the Cauchy singular integral operator and integral operators with fixed singularities on the unit interval, started in R.Duduchava, E.Shargorodsky, 1990. Such algebras emerge when one considers singular integral operators with complex conjugation on curves with cusps. As one of possible applications of the obtained results we find an explicit formula for the local norms of the Cauchy singular integral operator on the Lebesgue spaceL 2 (Γ, ρ), where Γ is a curve with cusps of arbitrary order and ρ is a power weight. For curves with angles and cusps of order 1 the formula was already known (see R.Avedanio, N.Krupnik, 1988 and R.Duduchava, N.Krupnik, 1995).

We prove estimates for classes of singular integral operators along variable lines in the plane, for which the usual assumption of nondegenerate rotational curvature may not be satisfied. The main Lp estimates are proved by interpolating L2 bounds with suitable bounds in Hardy spaces on product domains. The L2 bounds are derived by almost-orthogonality arguments. In an appendix we derive an estimate for the Hilbert transform along the radial vector field and prove an interpolation lemma related to restricted weak type inequalities.

In this paper, we have proved some special properties of singular integral operators which are transformed from the singular integral equation defined in the interval (−1, 1), i.e., the properties of singular intergral operators at the endpoints and in the inner of (−1, 1).

Calderon-Zygmund singular integral operators have been extensively studied for almost half a century. This paper provides a context for and proof of the following result: If a Calderon-Zygmund convolution singular integral operator is bounded on the Hardy space H1 (Rn), then the homogeneous of degree zero kernel is in the Hardy space H1(Sn−1) on the sphere.

The authors establish the boundedness on the Herz spaces and the weak Herz spaces for a large class of rough singular integral operators and their corresponding fractional versions. Applications are given to Fefferman's rough singular integral operators, their fractional versions, their commutators with BMO() functions and Ricci-Stein oscillatory singular integral operators. Some new results are obtained.

We show that for integral operators of general form the norm bounds in Lorentz spaces imply certain norm bounds for the maximal function. As a consequence, the a.e. convergence for the integral operators on Lorentz spaces follows from the appropriate norm estimates.

We present bounded positivity preserving operators from Lp(ℝ) to Lq (ℝ), for 1 < p < ∞, 1/p-1/q < 1/2, which are not integral operators.

In this paper we give a further development of the results of the paper [1] and apply it to convolution operators in the spaces Lp(x) . We consider the question of extendability of the Young theorem: well known for constant p and q, to the case when they may be variable. We also potential type operators with the Kernel . In section 1 we develop some estimates for Lp(x) -norms of power functions of distance truncated to exterior of a ball of radius . Section 2 deals with convolution operators in the spaces Lp(x) and Section 3 is devoted to potential type operators.

Motivated by the theory of nonstationary linear systems a number of problems in the theory of analytic functions have analogues in the setting of upper-triangular operators, where the complex variable is replaced by a diagonal operator. In this paper we focus on the analogue of interpolation in the Hardy space H2 and study a two-sided Nudelman type interpolation problem in the framework of upper-triangular Hilbert-Schmidt operators.