An $L^1$-type estimate for Riesz potentials

By Armin Schikorra, Daniel Spector, Jean Van Schaftingen. 2014

58 citations recorded by the source corpus.

In this paper we establish new $L^1$-type estimates for the classical Riesz potentials of order $\alpha \in (0, N)$: \[ \|I_\alpha u\|_{L^{N/(N-\alpha)}(\mathbb{R}^N)} \leq C \|Ru\|_{L^1(\mathbb{R}^N;\mathbb{R}^N)}. \] This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space $\mathcal{H}^1(\mathbb{R}^N)$ and provides a new family of $L^1$-Sobolev inequalities for the Riesz fractional gradient.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f7aaa9462cd553d02bba315c2b7d7617b4fbce63

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Javier Cueto
2. C. Mora‐Corral
3. José Carlos Bellido
4. Giovanni E. Comi
5. Giorgio Stefani

In Mathematical Physics

1. Javier Cueto
2. C. Mora‐Corral
3. José Carlos Bellido
4. Giovanni E. Comi
5. Giorgio Stefani

Citations in the corpus

• Fractional Piola identity and polyconvexity in fractional spaces (2018, background): “Another reference also dealing with the fractional gradient in the case p = 1 is [29].”
• A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up (2018, methodology): “Thus, accordingly to the approach developed in [9,14–18], we can consider the operators ∇̃α := ∇I1−α : C c (R) → C(R;R) and d̃iv := divI1−α : C ∞ c (R ;R) → C(R).”