The results obtained in this paper refine and supplement a Hardy-Littlewood-Sobolev type theorem on the boundedness of the Riesz potential generated by the Gegenbauer differential operator on the spaces Lp, λ proved in an earlier paper of the second author.

We study the Riesz potentials that are generated by the generalized shift operator associated to the Laplace-Bessel operator. We obtain an analog of the Sobolev-Il'in theorem for the B-Riesz potential.

We study the maximal and fractional maximal functions and Riesz potentials that are generated by the generalized shift operator associated with the Laplace-Bessel operator. We obtain some pointwise and integral estimates that give a relation between the B-maximal and B-fractional maximal functions and B-Riesz potentials and extend the available results to the objects of a more general nature. Basing on these results, we prove interpolation theorems for the B-fractional maximal functions and B-Riesz potentials.

The problem of the boundedness of a Calderon-Zygmund singular integral operator T in local Torrey-type spaces is reduced to the boundedness of the Hardy operator in weighted L-p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for the boundedness of T in local Torrey-type spaces for all admissible values of the parameters. Moreover, for a certain range of the parameters, for a genuine Calderon-Zygmund singular integral operator these sufficient conditions coincide with the necessary ones.

Taylor-Delsart formula is elaborated in the paper for functions of the generalized Gegenbauer shift. This formula is utilized to construct a version of the Gegenbauer shift modulus of smoothness of order k which for k = 1 reduces to the modulus of smoothness of the first order. By means of this modulus and Peetre's K-functional, an interpolation theorem is obtained. Equivalent normalizations are obtained for functional spaces associated with the generalized Gegenbauer shift.

В работе найдена формула Тейлора—Дельсарта для функций обобшенного сдвига ГегенбауЭра. С помошью формулы Тейлора—Дельсарта для функций обобшенного сдвига ГегенбауЭра построена некоторая величина, играюшая роль модуля гладкости k-го порядка (при к = 1 Эта величина совпадает с модулем непрерывности первого порядка), ассоциированная с обобшенным сдвигом ГегенбауЭра. С помошью Этой величины и K-функционала Петре получена интерполяционная теорема. Получены также Эквивалентые нормировки пространств функций, ассоциированных с обобшенным сдвигом ГегенбауЭра.

In this paper the generalized shift operator is considered generated by the Bessel differential operator B-n = partial derivative(2)/partial derivativex(n)(2) + gamma/x(n) partial derivative/partial derivativex(n), by means of which anisotropic Fourier-Bessel singular integral operators (B-n- anisotropic singular integral operators) are investigated. The boundedness of the anisotropic Fourier-Bessel singular integral operators is proved acting boundedly on the space L-p(gamma)(R-+(n)) equivalent to L-p(R-+(n), x(n)(gamma)dx). As well are proved limits embedding theorems on the Sobolev-Bessel space W-p,gamma(l1,...,ln) (R-+(n)).

In this paper we consider the generalized shift operator, generated by Bessel differ- ential operator B, by means of which maximal functions (B-maximal functions) and frac- tional integrals (B-fractional integrals) are investigated. The Lp(B)-boundedness result for the B-maximal function and (Lp(B),Lq(B))-boundedness result for the B-fractional integral are obtained.

In this work the Lizorkin-Triebel-Morrey type space F-p,theta,a,ae,tau(l) (G) is constructed, when the domain G C R-n satisfies the flexible lambda-horn condition and some properties of the functions from the constructed space are studied as with respect to imbedding theory.

Our goal is to obtain weighted imbedding theorems for the Sobolev spaces W-p,omega0,...,omegan(l1,...ln) (G), where G = R-+(n) or G = R-++(n) With this goal we study the behavior in weighted L-p-spaces of the anisotropic singular integral operators arising when the integral representations of Il'in and Besov are used (see(1), closed integral7.2).

In this paper we consider maximal functions, fractional maximal functions and fractional integrals which are generated by a generalized shift operator, associated with the Bessel differential operator B = A(B-1,..., B-n), B-i = partial derivative2/partial derivativex(i)(2) + gamma(i)/x(i) partial derivative/partial derivativex(i), i =1, ..., n. We present inequalities for these operators in corresponding weighted L-p-spaces. In a special case we have found necessary and sufficient conditions for pairs of weights ensuring the validity of strong type inequalities for fractional integrals.

In this work we introduce and investigate the function spaces, integral operators, generated by the Bessel differential operators, Bessel shift operators and Bessel transformations. We investigate also the boundedness of anisotropic B-maximal functions M B and the anisotropic B-Riesz potentials R on the anisotropic B-Morrey spaces L , B- BMO spaces BMO γ,α . Note that, the isotropic Hardy-Littlewood-Bessel maximal functions (B-maximal functions), Morrey-Bessel (B-Morrey) and BMO-Bessel (B-BMO) spaces were introduced and studied in [1]. We study also the anisotropic Riesz-Bessel potential (B-potential) in the anisotropic Morrey-Bessel and the anisotropic BMO-Bessel spaces. We obtain a theorem analogous to the Sobolev theorem, for the anisotropic Riesz-Bessel potential in anisotropic Morrey-Bessel spaces (see [2]).

The Hardy-Littlewood-Bessel maximal functions ( B -maximal functions), Morrey-Bessel and BMO-Bessel spaces were introduced and studied in [6]. In the present paper, we study the anisotropic Riesz-Bessel potential ( B -potential) in the Morrey-Bessel and BMO-Bessel spaces. We obtain a theorem analogous to the Sobolev theorem, for the anisotropic Riesz-Bessel potential in Morrey-Bessel spaces. We introduce a metric characteristic Ω p ,γ in the space of locally integrable functions and establish estimates connecting the characteristics of the image and preimage of the corresponding integral transform. These estimates are of independent interest. Moreover, they are used for the investigation of integral operators in different scales of Banach function spaces, in particular, in weighted L p γ -spaces. The results seem to be new even in the isotropic case.

This chapter gives the description of the boundary values in the simplest terms of harmonic analysis on the torus Tn. It applies a new technique based on the estimates connected with the Poisson kernel and on the use of the theory of vector multipliers. The chapter characterizes various classes of holomorphic functions by their radial and/or tangential properties. It describes the theorems related to the spaces of Banach-valued holomorphic functions and characterizations of the decomposition of the functions.

В терминах локальных характеристик получены оценки для интеграла типа потенциала на пространствах однородного типа (, μ). В терминах этих характеристик введены пространства которые при θ= совпадают с некоторыми весовыми пространствами . Изучено действие интегрального оператора типа потенциала на этих пространствах.An estimate is obtained for fractional integrals on spaces of homogeneous type (, μ) in terms of local characteristics. Under these terms, the spaces, coincide with some weighted -spaces for θ=. Properties of 0223 0112 V 3 integral operators of potential type in these spaces are also studied.

В терминах локальных характеристик получены оценки для интеграла типа потенциала на пространствах однородного типа (X, μ). В терминах этих характеристик введены пространства\(\Gamma _{p\theta \cdot \varphi } (X,\mu ), \Gamma _{p\theta \cdot \varphi }^ * (X,\mu ), 1< p< \infty ,1 \leqslant \theta \leqslant \infty \) которые при θ=p совпадают с некоторыми весовыми пространствамиL p, ω (X, μ). Изучено действие интегрального оператора типа потенциала на этих пространствах.