Riesz potential in the local Morrey–Lorentz spaces and some applications

Abstract

In this paper, the necessary and sufficient conditions are found for the boundedness of the Riesz potential {I_{\alpha}} in the local Morrey–Lorentz spaces {M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})} . This result is applied to the boundedness of particular operators such as the fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups on the local Morrey–Lorentz spaces {M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})} .

$B-$ Riezs potential in $B-$ local Morrey-Lorentz spaces

“…have been studied by many researchers [3,4,[10][11][12][13][14][15]21]. The Riesz potential connected with the Laplace-Bessel differential operator (B−Riesz potential) is generated by generalized shift operator…”

“…On local Morrey-Lorentz space, the necessary and sufficient conditions for the boundedness of the Riesz potential operator are given in [13]. On the other hand, the B− Riesz potential has been investigated in various function spaces by many mathematicians (see, for example [3,[10][11][12]).…”

Riesz potentials in the local variable Morrey-Lorentz spaces and some applications

“…These spaces are a very natural generalization of the Lorentz spaces such that M loc p,q;0 (R n ) = L p,q (R n ). Recently, in [2,13] and [14] the authors have studied the boundedness of the Hilbert transform, the Hardy-Littlewood maximal operator M and the Calderón-Zygmund operators T , and the Riesz potential I α on the local Morrey-Lorentz spaces M loc p,q;λ by using related rearrangement inequalities, respectively.…”

Maximal and Calderon-Zygmund operators on the local variable Morrey-Lorentz spaces and some applications

“…These spaces are a very natural generalization of the Lorentz spaces such that M loc p,q;0 (R n ) = L p,q (R n ). Recently, in [2,14] and [15] the authors have studied the boundedness of the Hilbert transform, the Hardy-Littlewood maximal operator M and the Calderón-Zygmund operators T , and the Riesz potential I α on the local Morrey-Lorentz spaces M loc p,q;λ by using related rearrangement inequalities, respectively. In [35], the authors give the definition of central Lorentz-Morrey space of variable exponent by the symmetric decreasing rearrangement.…”