Syntax Literate: Jurnal Ilmiah Indonesia pï¿½ISSN: 2541-0849 e-ISSN: 2548-1398

Vol. 8, No. 8, Agustus 2023

CHARACTERIZATIONS FOR THE FRAKSIONAL INTEGRAL OPERATOR ON CLASSIC MORREY SPACES

Sahat P. Nainggolan

Del Institute of Technology Faculty of Vocational

Abstrak

The boundedness of fractional integral operator ï¿½on ï¿½was introduced for the first time by Hardy G.H and Littlewood J.E (1928). In their evidence proof, Hardy and Littlewood used maximal operator that later known Hardy-Littlewood inequation. They proved that was limited from the Lebesgueï¿½s space ï¿½to the spaceï¿½ ï¿½with ï¿½and ï¿½In 1938, a mathematician C.B Morrey introduced one of space, namely the Morreyï¿½s space with notation . This paper will elaborate the Morreyï¿½s space and the boundedness of ï¿½toward the classic Morreyï¿½s space by benefitted the Hardy-Littlewodd maximal operator.

Keywords: Boundedness, Fractional Integral Operator, the maximal operator of Hardy Littewood, Minskowski inequation, Holder inequation, Lebesgueï¿½s space,

Introduction

In 1886, Marcell Riesz introduced one function operator known as the fractional integral operator ï¿½that is:

For example ï¿½real-valued function on ï¿½for a, ï¿½dan , fractional integral operator ï¿½defined as follows:

Furthermore, this fractional integral operator is often known as the Riesz potential. The problem studied related to the fractional integral operator above is a limitation problem. As is known, the operator ï¿½from space ï¿½to space ï¿½said to be limited, if any ï¿½such that ï¿½with ï¿½noting norms ï¿½in room . Then the operators ï¿½is said to be limited in space , If ï¿½limited from space ï¿½to space.

The limitation of an operator is a property that is expected to be met, because this property leads to conditions that are interesting to study. For example when working with differential equations or integral equations, the limitations of an operator can provide an understanding of certain physical phenomena. Meanwhile in the field of computing, computing will be much easier if you work with a limited number of operators.

For the first time the limitations of the fractional integral operator on \mathbb{R} were proved by Hardy G.H and Littlewood J.E (1928). In their proof, Hardy and Littlewood used the maximal (function) operator which became known as the Hardy-Littlewood inequality.

Furthermore, the limitations of the fractional integral operator ï¿½proved by them in one of the homogeneous spaces, namely from the Lebesgue space ï¿½to space ï¿½with ï¿½And ï¿½Connection ï¿½And ï¿½always used in proof of the limitations of the fractional integral operator. Furthermore, in 1930 the limitations in the Lebesgue space were refined by Sobolev, so that the important result he obtained was called the Hardy-Littlewood-Sobolev inequality. Several years after that, in 1937 N. Wiener reintroduced the maximal operator, but for the case of a higher dimensional Euclidean space.

In 1938, a mathematician named C.B. Morrey introduced one of the well-known spaces to date, namely the Morrey space denoted by ï¿½(Lina, 2013 hal 1). This space is often encountered when studying the Schodinger operator and potential theory where the Morrey space is an extension of the Lebesgue space. After Hardy-Littlewood-Sobolev, the limitations of the fractional integral operator ï¿½further developed by D.R. Adams in the Morrey room. This result was then proven again by Chiarenza-Frasca using the Fefferman-Stein inequality. Chiarenza-Frasca succeeded in proving the limitations of the fractional integral operator from Morrey space ï¿½to space .

Based on the description above, this paper discusses whether the fractional integral operator has the same limitations in the previous function space, namely the Lebesgue space. . Furthermore, what conditions must be met so that the fractional integral operator is confined to the Morrey space.

Research Methods

1. Morrey Room

The Morrey space is the set of all local Lebesgue integrated functions with an expansion value of a finite q-norm. To define a Morrey space, it is necessary to define a local Lebesgue space . However, before defining the local Lebesgue space, it is necessary to define the Lebesgue space first ï¿½namely the space that contains functions equipped with a q-norm whose value is up to . According to Kevin (2014:1) Lebesgue roomï¿½(named after its discoverer, Henry Lebesgue) is a scalable function space which is a natural embodiment of a finite dimensional vector space equipped with norm. Erwin Kreyszig (1978:61) defines a Lebesgue space and says that a Lebesgue space is a Banach space. The following is a definition of a Lebesgue space .

Lebesgue Room

For ï¿½Lebesgue room ï¿½contains all scalable functions f on that fulfills , with,

Example: Suppose function with . Clear that ï¿½measurable function. Because any function that is continuous almost everywhere is a measurable function. It is clear that the function is a continuous function on , consequently function ï¿½with ï¿½is a measurable function. Furthermore,

Clear ï¿½But if you pay attention, for , , Because

As for the local Lebesgue room ï¿½defined as follows:

Local Lebesgue Room

Local Lebesgue Room ï¿½with ï¿½is a space containing all scalable functions ï¿½that fulfills :

for each compact subset ï¿½If ï¿½so ï¿½is said to be locally integrated in

Based on definition 1 above, the membership requirements of ï¿½still fairly 'rough', because it only requires the finiteness of the expression . Therefore, it is necessary to add one parameter in the hope that it will refine the membership conditions ï¿½The result of refinement of the Lebesgue space ï¿½by adding one parameter, it is called a Morrey space (named after its discoverer, Charles B. Morrey, Jr). In brief, ï¿½related to the local properties of ï¿½which is defined on , whereas ï¿½related to global properties. The following is the definition of a Morrey space .