INTEGRAL OPERATORS OF HARMONIC ANALYSIS IN SPACES DEFINED IN TERMS OF LOCAL CHARACTERISTICS OF FUNCTIONS II

Study of integral operators in terms of characteristics of type Ωp goes back to the papers [1], [2], [3], where the operators of classic Fourier harmonic analysis are considered. In the present paper similar investigations are carried out for the operators of Fourier-Bessel harmonic analysis. The powerful working apparatus of Fourier-Bessel’s harmonic analysis (associated with Fourier-Bessel transform) undoubtedly are singular integral operators (SIO), maximal function, Riesz and Bessel potentials and others. Here we consider convolution structures generated


Introduction
Study of integral operators in terms of characteristics of type Ω p goes back to the papers [1], [2], [3], where the operators of classic Fourier harmonic analysis are considered.
In the present paper similar investigations are carried out for the operators of Fourier-Bessel harmonic analysis.The powerful working apparatus of Fourier-Bessel's harmonic analysis (associated with Fourier-Bessel transform) undoubtedly are singular integral operators (SIO), maximal function, Riesz and Bessel potentials and others.Here we consider convolution structures generated not by ordinary but by some special shift T y (the so called generalized shift [4]) adapted to Fourier-Bessel transform in some coordinates of a point.
Note that in some issues of mathematics, for example when studying boundary values of B-harmonic functions associated with Laplace-Bessel's differential equation x ∈ R + m+k,k , γ m+1 > 0, ..., γ m+k > 0 there arises a necessity for studying potentials and SIO generated by the generalized shift T y γ n,k in one-dimensional case by B.M. Levitan [4], [5] and called the generalized or Bessel shift operator (see the papers of I.A. Kiprinov [6] and I.A. Kiprianov, M.I.Klyuchanchev [7]).
In the paper of I.A. Kiprianov and L.A. Ivanov [8] it was proved that the integral operator as cylindrical potential (when α = 2) called the Riesz generalized potential, is the solution of the equation ∆ B m+k,k u (x) = f (x), and the problem of obtaining a priori estimations in fact was reduced to estimation of these generalized Riesz potentials and their corresponding derivatives (singular integrals).In this connection, they introduced the spaces L p,γ m+k,k .Establishment of weight L p,γ m+k,k estimations is one of the significant directions of studying integral operators of Fourier-Bessel's harmonic analysis.For SIO generated by the generalized shift T y γ n,k (in the case k = 1), this problem was first studied in the paper of I.A. Aliyev and A.Dj. Gadjiev [9] where only the case of radial weights was considered.The case of nonradial weights was first considered in the papers of S.K. Abdullayev and N.R. Karamaliyev [10], S.K. Abdullayev, A.K. Akbarov and M.K. Kerimov [11] when weight function depends only on one coordinate of space variable.The case of certain class f general weights is considered in the paper of S.K. Abdullayev, E.A. Gadjieva and F.A. Isayev [12].
In the present paper, in particular, these results are taken to the case of weights dependent on arbitrary set of s ∈ {1, ..., m + k} coordinates.
To this end, following [3], we introduce the class of operators K γ n,k (p, q) and integral characteristics of locally summable functions υ (ξ), ξ ∈ P + µ+k,k and prove the estimations associating this characteristics of the image Au with the same characteristics of the preimage u of the operator A ∈ K γ n,k (p, q).In the paper [13] in the same statement this problem was completely studied in terms of characteristics The obtained estimations becomes a starting point when studying these operators in different scales of Banach spaces determined in terms of the introduced characteristics, one of which is the scale of weight L p,γ m+k,k spaces.Note that the considered weight L p,γ m+k,k spaces with monotone increasing (decreasing) weights are expressed in terms of characteristics Ω (sx) p,µ n,k Ω * (sx) p,µ n,k .The classes as K γ n,k (p, q) subadditive operators containing in particular singular integral operators (SIO), maximal and fractional-maximal functions, the Riesz and Bessel and other operators majorized by convolution type integral operators with generalized shift T y γ n,k , were introduced for studying a wider class of harmonic analysis operators from a unified position.We also note that owing to generality of the approach, the results obtained by us contain the case of ordinary shift in all coordinates, more exactly, the case of operators of Fourier harmonic analysis.

Some Designations and Preliminary Information
Let R l be Euclidean space of dimension l, and m, k ≥ 0 be integers, be a generalized shift operator (GSO) generated by the Laplace-Bessel operator Further, we assume In designation γ n,k , n indicates the dimensionality of this vector, k the amount of its positive coordinates.
In these designations we also assume In what follows, s y ′ , γ n−s,k ′ s , ( s y ′ ) γ n−s,k ′ s and dµ n−s,k ′ s ( s y ′ ) are determined from the equalities is also determined uniquely. Assume is a space of functions summable in the p-th degree on the set G.
Further, we will repeatedly use the following easily provable properties of the generalized shift operator We need the following known inequalities.
and the existence of the left hand side follows from the existence of the right one.
Hardy-Littlerwood's theorems on maximal functions.Let 1 ≤ p ≤ q ≤ ∞, u (t) and v (t) be positive functions on the interval (0, +∞).Then for validity of the inequality , with the constant k independent of f it is necessary and sufficient to fulfill the conditions

Main Results
The operator A is said to be subadditive if for any λ, µ > 0 and any functions f and g from the domain of definition of the operator A , and C is independent of u.
In the case p = q the operators A ∈ K γ n,k (p, q) may be singular [9].Let us consider 1. Poisson's B γ n,k -integral: 4. The Riesz B m+k,2v potential It is known [13] that In what follows, C is a positive number various in different inequalities.When s ∈ {1, ..., m + k}, A * p,γ n,k ( s x) denotes the totality of all functions measurable on the set R + m+k,k and belonging to L p,γ n,k x ∈ R + m+k,k : | s x| ≤ ξ for every number ξ > 0.
Further α q,s = (s + |γ ks |) /q.For the function u ∈ A * p,γ n,k ( s x) we introduce the characteristics and the set .
The main results of the paper are given in the following three theorems.
Theorem 3.2.Let A ∈ K γ n,k (p, q), s ∈ {1, ..., m + k} and u ∈ J * p,µ n,k ( s x).Then for almost all x ∈ R + m+k,k there exists υ (x) = A (u) (x) and it holds the estimation where the constant c is independent of u and ξ.
Note that the estimation (Ω) * is new in the case of ordinary shift in all coordinates, as well.
.., m + k} and the following condition be fulfilled: Then the operator A acts from the space I * (sx) q,µ n,k (ψ) and it holds the inequality where c is independent of the function u.
Proof of lemma 4.1.

Introduce the denotations
Sequential application of the estimation where d > 0, j ≥ 1 leads to the proof of the inequality Using the Foubini theorem and then the Holder inequality in the inner integral, we get Allowing for estimation B, item a) of lemma 4.1 is proved in the case s ∈ {1, ..., m + k − 1}.The case s = m + k directly follows from the definition of I * β,s (u, ξ).Now let us prove item b) of lemma 4.1 in the case s ∈ {1, ..., m + k − 1}.First of all note that having applied the Young inequality, where r > 1, r −1 = (p ′ ) −1 + q −1 , we get Taking into account the last one, having used the Foubini theorem on reduction of multiple integral to repeated integrals, and using the passage , we get Adµ s,ks ( s y) This proves item b) of lemma 4.1 in the case s ∈ {1, ..., m + k − 1}.Now let s = m + k.For further reasonings we need to pass to spherical coordinates.
Let S + n,k ∈ x ∈ R + n,k : |x| = 1 be a unit sphere of R + n,k .Then passing to spherical coordinates centered in the origin of coordinates where dσ (θ) is the element of the area of the surface of sphere S + n,k .Then allowing for the estimation ≤ cξ (m+k+|γn,k|)/q , we get Taking into account that if s = m + k, then γ s ′ ,k ′ s + s ′ = 0, this proves item b) and completely Lemma 4.1.
Prove lemma 4.2.Let r > 1 and It is easy to show Allowing for this, passing to spherical coordinates, then and using the Dirichlet formula, we have (here we take into account that This completely proves lemma 4.2.

Proof of Main Theorems
Prove theorem 3.
Taking into account this condition A ∈ K γ n,k (p, q) and self-adjointness of the operator T y ; by item a) of lemma 4.1 we get Now let us prove estimation (Ω * ) of theorem 3.2.Take ξ > 0.
Theorem 3.2 is proved.
For further reasoningss we need two lemmas.