The Limiting Cases of Affine Hardy–Littlewood–Sobolev Inequalities

Key Points

• The paper aims to investigate the limiting cases of affine Hardy–Littlewood–Sobolev inequalities as α approaches n− and 0+.
• Established affine logarithmic Hardy–Littlewood–Sobolev inequalities.
• Developed affine Beckner-type logarithmic Sobolev inequalities for function pairs.
• Extended classical inequalities to the affine setting.
• Proved new affine logarithmic inequalities in the limiting regimes.
• Recovered known sharp inequalities for the cases α→n− and α→0+.