The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces
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The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces Vagif S. Guliyev · Fatai A. Isayev
Received: 27 August 2011 / Accepted: 7 November 2012 / Published online: 4 December 2012 © Springer Science+Business Media Dordrecht 2012
Abstract In this paper, the authors establish several general theorems for the boundedness of sublinear operators (B sublinear operators) satisfies the condition (1.2), generated by B ∂2 singular integrals on a weighted Lebesgue spaces Lp,ω,γ (Rnk,+ ), where B = ki=1 ( ∂x 2 + γi ∂ xi ∂xi
k
). The condition (1.2) are satisfied by many important operators in analysis, including B maximal operator and B singular integral operators. Sufficient conditions on weighted functions ω and ω1 are given so that B sublinear operators satisfies the condition (1.2) are bounded from Lp,ω,γ (Rnk,+ ) to Lp,ω1 ,γ (Rnk,+ ). Keywords Weighted Lebesgue space · B sublinear operator · B maximal operator · B singular integral operator · Two-weighted inequality Mathematics Subject Classification (2000) 42B20 · 42B25 · 42B35 1 Introduction Suppose that Rn is the n-dimensional Euclidean space, x = (x1 , . . . , xn ), ξ = (ξ1 , . . . , ξn ) 1 are vectors in Rn , (x, ξ ) = x1 ξ1 + · · · + xn ξn , |x| = (x, x) 2 , x = (x , x ), x = (x1 , . . . , xk ), The research of V. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/06-1, by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 110T695) and by grant of 2011-Ahi Evran University Scientific Research Projects (BAP FBA-11-13). V.S. Guliyev () · F.A. Isayev Department of Mathematics, Ahi Evran University, Kirsehir, Turkey e-mail: [email protected] F.A. Isayev e-mail: [email protected] V.S. Guliyev Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan V.S. Guliyev Baku State University, Baku 1148, Azerbaijan
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V.S. Guliyev and F.A. Isayev
x = (xk+1 , . . . , xn ), γ = (γ1 , . . . , γk ), γ1 > 0, . . . , γk > 0, (x )γ = x1 1 . . . xk k . Let Rk++ = {x ∈ Rk : x1 > 0, . . . , xk > 0}, Rnk,+ = {x ∈ Rn : x1 > 0, . . . , xk > 0}, 1 ≤ k ≤ n, Sk,+ = {x ∈ Rnk,+ : |x| = 1}. For measurable set E ⊂ Rnk,+ let |E|γ = E (x )γ dx, then |E(0, r)|γ = ω(n, k, γ ) × r n+|γ | , where ω(n, k, γ ) = |E(0, 1)|γ . n Let f ∈ Lloc 1,γ (Rk,+ ) and B be the Laplace-Bessel differential operator: γ
γ
B = B + x ,
B=
k
Bi =
Bi ,
i=1
∂2 γi ∂ + , ∂xi2 xi ∂xi
x =
n ∂2 . ∂xj2 j =k+1
The B maximal function (see [9–12]) is defined by γ 1 Mγ f (x) = sup T y |f (x)| y dy n+|γ | ω(n, k, γ )r r>0 E(0,r) and the B singular integral (see [2, 7, 15, 16, 19, 20]) defined by γ Kγ f (x) = p.v. K(y) T y f (x) y dy, Rnk,+
where K(rx) = r −n−|γ | K(x) for each r > 0, x ∈ Rnk,+ ; dσ is the element of area of the Sk,+ ; supθ∈Sk,+ |K(θ )| < ∞ and γ K(x) x dσ (x) = 0. (1.1) Sk,+
We also consider the high order Riesz-Bessel transformations (se
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