Pointwise Interpolation Inequalities for Riesz and Bessel Potentials *

Pointwise interpolation inequalities

ℳ I z f ( x ) ≤ c ( ℳ I ζ f ( x ) ) ℜ z / ℜ ζ ( ℳ f ( x ) ) 1 − ℜ z / ℜ ζ ,     0 < ℜ z < ℜ ζ < n , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429186875/b86f08e4-14d7-465f-ac8c-bce3ede29963/content/equ_chapter18_217_1.tif"/>

and

ℳ J z f ( x ) ≤ c ( ℳ J ζ f ( x ) ) ℜ z / ℜ ζ ( ℳ f ( x ) ) 1 − ℜ z / ℜ ζ ,         0 < ℜ z < ℜ ζ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429186875/b86f08e4-14d7-465f-ac8c-bce3ede29963/content/equ_chapter18_217_2.tif"/>

where I_z and J_z are the Riesz and Bessel potentials of order z and ℳ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429186875/b86f08e4-14d7-465f-ac8c-bce3ede29963/content/inq_chapter18_217_1.tif"/> is the Hardy-Littlewood maximal operator, are proved.