Local Properties and the Critical Exponent α = 1 / p
Boris Rubin · 2024
We already know that fractional integrals of order α with L p densities on a finite interval can be non-smooth if https://www.w3.org/1998/Math/MathML" display="inline"> α and satisfy Hölder&s;s condition if https://www.w3.org/1998/Math/MathML" display="inline"> α > 1 / p https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032675022/00fd4802-bb54-48bb-af17-6e1e04a3ce61/content/math4_4.tif "/> . The borderline case https://www.w3.org/1998/Math/MathML" display="inline"> α = 1 / p https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032675022/00fd4802-bb54-48bb-af17-6e1e04a3ce61/content/math4_5.tif "/> is of particular interest and motivates our study of local properties of fractional integrals in this chapter.