A New Approach for the Fractional Integral Operator in Time Scales with Variable Exponent Lebesgue Spaces

Abstract

Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales.

Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

“…Recently, many new Hardy inequalities have been proved, including Hardy and Rellich inequalities for Bessel pairs [16], Hardy-Sobolev inequalities on hypersurfaces for Euclidean space [17], improved Hardy inequalities with exact remainder terms [18], Hardy inequalities with double singular weights [19], and Hardy inequalities for class functions in one-dimensional fractional Orlicz-Sobolev spaces [20]. A new approach for the fractional integral operator in time scales with variable exponent Lebesgue spaces is presented in [21] and n-dimensional integral-type inequalities via time scale calculus is studied in [22].…”

New Challenges Arising in Engineering Problems with Fractional and Integer Order

“…In [12], the authors proved the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator to the norm of the centered fractional maximal diamond-α integral operator Mca on the time scales in variable exponent Lebesgue spaces. This study considered problems such as the boundedness and compactness of the integral operators in relation to the time scales.…”