We prove basic properties of Orlicz–Morrey spaces and give a necessary and sufficient condition for boundedness of the Hardy–Littlewood maximal operator M from one Orlicz–Morrey space to another. For example, if f ∈ L(log L)(R), then Mf is in a (generalized) Morrey space (Example 5.1). As an application of boundedness of M , we prove the boundedness of generalized fractional integral operators, improving earlier results of the author.

/pdf/orlicz-morrey-spaces-and-the-hardy-littlewood-maximal-327010u8mk.pdf

Orlicz&#8211;Morrey spaces and the Hardy&#8211;Littlewood maximal function

We consider generalized Morrey spaces ${\mathcal M}^{p(\cdot),\omega}(\Omega)$ with variable exponent $p(x)$ and a general function $\omega (x,r)$ defining the Morrey-type norm. In case of bounded sets $\Omega \subset {\mathsf R}^n$ we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type ${\mathcal M}^{p(\cdot),\omega} (\Omega)\rightarrow {\mathcal M}^{q(\cdot),\omega} (\Omega)$-theorem for the potential operators $I^{\alpha(\cdot)}$, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on $\omega(x,r)$, which do not assume any assumption on monotonicity of $\omega(x,r)$ in $r$.

/pdf/boundedness-of-the-maximal-potential-and-singular-operators-54zlra0n62.pdf

Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

The theory of generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, which Mizuhara and Nakai proposed, are equipped with a parameter and a function. The trace property is one of the main focuses of the present paper, which will clarify the role of the parameter of generalized Morrey spaces. The quarkonial decomposition is obtained as an application of the atomic decomposition. In the end, the relation between the function spaces dealt in the present paper and the foregoing researches is discussed.

Generalized Morrey spaces and trace operator

The theory of generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, which T. Mizuhara and E. Nakai proposed, are equipped with a parameter and a function. The trace property is one of the main focuses of the present paper, which will clarify the role of the parameter of generalized Morrey spaces. The quarkonial decomposition is obtained as an application of atomic decomposition. In the end, the relation between the function spaces dealt in the present paper and the foregoing researches is discussed.

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.

/pdf/characterizations-for-the-generalized-fractional-integral-27yx4cia53.pdf

Characterizations For The Generalized Fractional Integral Operators On Morrey Spaces

We prove the boundedness of the generalized fractional integral operators and their modified versions on Morrey spaces and on Campanato spaces respectively. Our approach involves the Hardy-Littlewood maximal function and Young functions.

On generalized fractional integral operators

In this paper, we shall discuss about Bessel-Riesz operators. Kurata et al. have investigated their boundedness on generalized Morrey spaces with weight. The boundedness of these operators on Lebesgue spaces and Morrey spaces will be reproved using a different approach. Moreover, we also find the norm of the operators are bounded by the norm of the kernels.

/pdf/the-boundedness-of-bessel-riesz-operators-on-morrey-spaces-4n92i13xb7.pdf

The boundedness of Bessel-Riesz operators on Morrey spaces

In this paper, we prove the boundedness of Bessel-Riesz operators on generalized Morrey spaces. The proof uses the usual dyadic decomposition, a Hedberg-type inequality for the operators, and the boundedness of Hardy-Littlewood maximal operator. Our results reveal that the norm of the operators is dominated by the norm of the kernels.

The boundedness of Bessel-Riesz operators on generalized Morrey spaces

In this paper, we consider about Bessel Riesz operator on Lebesgue and Morrey spaces in Euclidean settings, that serve as the generalization of result in arbitrary doubling measure that is a particular example of Young's inequality on weighted function spaces for convolutions. We also observe an influence of Bessel-Riesz kernel result in arbitrary doubling measure. 

The boundedness of Bessel-Riesz operators as a special case of Youngs inequality

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