Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces

Abstract

In this paper, the necessary and sufficient conditions are found for the boundedness of the rough Bfractional integral operators from the Lorentz spaces L p,s,γ to L q,r,γ , 1 < p < q < ∞, 1 r s ∞, and from L 1,r,γ to L q,∞,γ ≡ W L q,γ , 1 < q < ∞, 1 r ∞. As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential.

On the limiting case for boundedness of the B-Riesz potential in B-Morrey spaces

“…For λ = 0, this statement was earlier proved in [2,9]. Note that in the limiting case p = n + |γ| − λ/(α) statement 1) does not hold.…”

“…. , γ k > 0 have been investigated by many researchers, see Muckenhoupt and Stein [13], Kipriyanov [10], Trimeche [18], Lyakhov [12], Stempak [17], Gadjiev and Aliev [2], Guliyev [3,4,5,6], Guliyev and Hasanov [7], Guliyev, Serbetci and Ekincioglu [9], and others. In this paper we consider the generalized shift operator generated by the Laplace-Bessel differential operator Δ B in terms of which we study the boundedness of the modified B-Riesz potential e I α,γ in the limiting case.…”

Pointwise and integral estimates for the B-Riesz potential in terms of B-maximal and B-fractional maximal functions

“…In our case X = R N n,+ , ρ(x, y) = |x − y|, β = α Q , 0 ≤ α < Q, and dμ(x) = (x ) γ dx. It is clear that this measure satisfies the doubling condition (11).…”

“…Below we will need a few lemmas for the proof of Sobolev's theorem in the limit case p = Q/α. Lemma 1 [11,14]. Let f and g be positive measurable functions on R N n,+ .…”

The Boundedness of High Order Riesz-Bessel Transformations Generated by the Generalized Shift Operator in Weighted L p,ω,γ -spaces with General Weights

“…Certain sufficient conditions on weighted functions ω and ω 1 are given so that R (k) B singular integral operator is bounded from the weighted L p,ω,γ (R n k,+ ) spaces into the weighted L p,ω 1 ,γ (R n k,+ ) spaces [7]. The methods of proof used here are closer to that in the paper of Gadjiev and Guliyev [5].…”