An L¹-type estimate for Riesz potentials

2014/11/30 by Armin Schikorra, Daniel Spector, Jean Van Schaftingen · 2 citations
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #M. Riesz extension theorem #Measure (data warehouse) #Nonlinear Partial Differential Equations #Order (exchange) #Riesz potential #Riesz representation theorem #Riesz transform #Space (punctuation) #math.AP #math.FA

paper · pdf · doi:10.4171/rmi/937

published as Revista Matemática Iberoamericana 33 (2017), no. 1, 291-304 · 13 pages, improves previous version with full spectrum of result and with elementary proof, references display correctly in this version

openalex created_date 2016/06/24 · arxiv created 2016/09/01 · openalex publication_date 2017/02/22 · arxiv updated 2017/07/04 · openalex updated_date 2026/08/05

Abstract

In this paper we establish new L¹ -type estimates for the classical Riesz potentials of order α ∈ (0, N) : ‖I_α u‖_{L^N/(N-α)(ℝ^N)} ≤ C ‖Ru‖_{L¹(ℝ^N;ℝ^N)}. This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space H¹(ℝ^N) and provides a new family of L¹ -Sobolev inequalities for the Riesz fractional gradient.

Cited by

• Riesz potential [wikipedia]
• Sobolev inequality [wikipedia]

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