On Generalized Fractional Integral Operators (original) (raw)
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A note on the generalized fractional integral operators
2003
We prove that the generalized fractional integral operators are bounded from a generalized Morrey space to another. Our result generalizes that of Chiarenza-Frasca [1] and links the results of Nakai [7] and of Eridani and Gunawan [3].
The Boundedness of Generalized Fractional Integral Operators on Small Morrey Spaces
Cauchy, 2024
Morrey space was first introduced by C.B. Morrey in 1938 which is the solution space of an elliptic partial differential equation. Morrey space can be said to be a generalization of Lebesgue space. Morrey spaces are generalized into generalized Morrey spaces, small Morrey spaces, weighted Morrey spaces, and Bourgain Morrrey spaces. One of the studies in Morrey space is the boundedness of operators in Morrey space. One such operator is the fractional integral operator. One of the generalizations of fractional integral operators is the generalized fractional integral operator. The generalized fractional integral operator is finite on the generalized Morrrey space. In this study, it is extended on small Morrey spaces. The small Morrey space is the set of locally Lebesgue integrable functions with norm defined supremum over radius of ball ∈ (0,1). This paper aims to prove the boundedness properties of the generality of fractional integral operators on small Morrey spaces using Hedberg-type inequality. The first, in this paper will be discuss to prove Hedberg-type inequality on small Morrey spaces using dyadic decomposition, Hölder inequality, and doubling condition. The proof method used is using Hedberg's inequality. The results imply that the generalized fractional integral operator is finite on small Morrey spaces.
FRACTIONAL INTEGRAL OPERATORS ON LEBESGUE AND MORREY SPACES
2009
In this survey paper, we will present some results on fractional integral operators, also known as the Riesz potentials, especially their boundedness property on Lebesgue spaces and (generalized) Morrey spaces, including in the non-homogeneous setting. In addition, some results on related operators -such as the Hardy-Littlewood maximal operator -and some applications will be pointed out.
Transactions of the American Mathematical Society, 2011
The action of the generalized fractional integral operators and the generalized fractional maximal operators is investigated in the framework of Morrey spaces. A typical property of the functions which belongs to Morrey spaces under pointwise multiplication by the generalized fractional integral operators and the generalized fractional maximal operators is established. The boundedness property of the fractional integral operators on the predual of Morrey spaces is shown as well. A counterexample concerning the Fefferman-Phong inequality is given by the use of the characteristic function of the Cantor set.
Characterizations for the generalized fractional integral operators on Morrey spaces
Mathematical Inequalities & Applications, 1998
We present some characterizations for the boundedness of the generalized fractional integral operators on Morrey spaces. The characterizations follow from two key estimates, one for the norm of some functions in Morrey spaces, and another for the values of the corresponding fractional integrals. We prove three theorems about necessary and sufficient conditions. We show that these theorems are independent by giving some examples. We also obtain counterparts for the weak generalized Morrey spaces.