Fractional Integrals Of Functions Research Articles

RELATIONSHIP BETWEEN UPPER BOX DIMENSION OF CONTINUOUS FUNCTIONS AND ORDERS OF WEYL FRACTIONAL INTEGRAL

In this paper, we mainly investigate relationship between fractal dimension of continuous functions and orders of Weyl fractional integrals. If a continuous function defined on a closed interval is of bounded variation, its Weyl fractional integral must still be a continuous function with bounded variation. Thus, both its Weyl fractional integral and itself have Box dimension one. If a continuous function satisfies Hölder condition, we give estimation of fractal dimension of its Weyl fractional integral. If a Hölder continuous function is equal to 0 on [Formula: see text], a better estimation of fractal dimension can be obtained. When a function is continuous on [Formula: see text] and its Weyl fractional integral is well defined, a general estimation of upper Box dimension of Weyl fractional integral of the function has been given which is strictly less than two. In the end, it has been proved that upper Box dimension of Weyl fractional integrals of continuous functions is no more than upper Box dimension of original functions.

Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure

The boundedness of modified maximal operators and potentials with variable parameter in variable exponent Morrey spaces with non-doubling measure is established. Moreover, Hölder continuity properties for fractional integrals of functions in Morrey spaces with variable exponent defined on non-homogeneous spaces are investigated.

Inverse stochastic transfer principle

In (10) a direct stochastic transfer principle was introduced, which represented multiple integrals with respect to fractional Brownian mo- tion in terms of multiple integrals with respect to standard Brownian motion. The method employed in (10) involved an operator i (n) H , mapping a class of functions L 2 to L 2 . However, the operator does not map L 2 onto L 2 . Hence i (n) H is not invertible. The non-invertibility arises from the fact that i (n) H is defined in terms of n-fold Riemann-Liouville fractional integrals and it is well known that functions f 2 L p (a,b) cannot always be represented as fractional integrals of functions ' 2 L p (a,b). This paper establishes the inverse sto- chastic transfer principle. Our purpose is to explicitly define an operator, which, acting on a certain class of functions, gives the transfer principle in- verting that in (10). As a result, multiple stochastic integrals with respect to standard Brownian motion are represented in terms of multiple stochastic integrals with respect to persistent fractional Brownian motion. In order to do this, we first prove a characterization of a class of functions for which there exists ', defined on a compact set in R n , such that f is equal to the n-fold Riemann-Liouville fractional integral of '. We establish a method for computing ' and conclude the paper with an example of using the inverse transfer principle in filtering.

Integral Representations and Continuous Projectors in Some Spaces of Analytic and Pluriharmonic Functions

Abstract We consider analytic and pluriharmonic functions belonging to the classes 𝐵𝑝(Ω) and 𝑏𝑝(Ω) and defined in the ball . The theorems established in the paper make it possible to obtain some integral representations of functions of the above-mentioned classes. The existence of bounded projectors from the space 𝐿(ρ, Ω) into the space 𝐵𝑝(Ω) and from the space 𝐿(ρ, Ω) into the space 𝑏𝑝(Ω) is proved. Also, consideration is given to the existence of boundary values of fractional integrals of functions of the spaces 𝐵𝑝(Ω) and 𝑏𝑝(Ω).

On the rate of growth of the Means Mp of holomorphic and pluriharmonic functions on the ball

A. If u E bP and v is a harmonic conjugate of then b”. Theorem A was proved by using a result due to Duren et al. 12, Theorem 51 concerning fractional derivatives and fractional integrals of functions in BP. Iff(z) = CpZO a k zk is holomorphic in D and /? > 0, then the fractional derivative ft4’ and fractional integral fib, of order /? are defined, respectively, by and