Hardy-Littlewood-Sobolev inequality for $p=1$

Abstract

Let W be a closed dilation and translation invariant subspace of the space of R ℓ -valued Schwartz distributions in d variables. We show that if the space W does not contain distributions of the type a ⊗ δ0, δ0 being the Dirac delta, then the inequality ∥ Iα[f ]∥L d/(d−α),1 ≲ ∥f ∥L 1 holds true for functions f ∈ W ∩ L1 with a uniform constant; here Iα is the Riesz potential of order α and Lp,1 is the Lorentz space. As particular cases, this result implies the inequality ∥∇ m−1 f ∥L d/(d−1),1 ≲ ∥Af ∥L 1 , where A is a cancelling elliptic differential operator of order m, and the inequality ∥ Iα f ∥L d/(d−α),1 ≲ ∥f ∥L 1 , where f is a divergence free vector field.Bibliography: 59 titles.

On embedding of Besov spaces of zero smoothness into Lorentz spaces

“…In recent years there is a constant interest in generalization and specification of this already fine scale. For Besov-Lorentz spaces, see [13], [15], [16], and [18]. For so-called logarithmic Besov spaces, see [5], [6], [7], [8], and [9].…”

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

“…Usually, one adds translation and dilation invariant constraints for the function f that exclude the delta measures. We refer the reader to the pioneering paper [1], to more recent studies [3], [10], [14], and to the surveys [8] and [15].…”