Corpus ID: 237532477

Fractional integration of summable functions: Maz'ya's $\Phi$-inequalities

  title={Fractional integration of summable functions: Maz'ya's \$\Phi\$-inequalities},
  author={Dmitriy M. Stolyarov},
We study the inequalities of the type | ∫ Rd Φ(K ∗ f)| . ‖f‖ p L1(R) , where the kernel K is homogeneous of order α − d and possibly vector-valued, the function Φ is positively p-homogeneous, and p = d/(d − α). Under mild regularity assumptions on K and Φ, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions f . 1 Hardy–Littlewood–Sobolev inequality and its modifications at the… 

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