A basic version of the Pólya-Szegő inequality states that if Φ is a Young function, the Φ-Dirichlet energy -- the integral of Φ(∥∇f∥) -- of a suitable function f∈V(Rn), the class of nonnegative measurable functions on Rn that vanish at infinity, does not increase under symmetric decreasing rearrangement. This fact, along with variants that apply to polarizations and to Steiner and certain other rearrangements, has numerous applications. Very general versions of the inequality are proved that hold for all smoothing rearrangements, those that do not increase the modulus of continuity of functions. The results cover all the main classes of functions previously considered: Lipschitz functions f∈V(Rn), functions f∈W1,p(Rn)∩V(Rn) (when 1≤p<∞ and Φ(t)=tp), and functions f∈W1,1loc(Rn)∩V(Rn). In addition, anisotropic versions of these results, in which the role of the unit ball is played by a convex body containing the origin in its interior, are established. Taken together, the results bring together all the basic versions of the Pólya-Szegő inequality previously available under a common and very general framework.

The Pólya-Szegő inequality for smoothing rearrangements / Gabriele Bianchi; Richard J. Gardner; Paolo Gronchi; Markus Kiderlen. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - STAMPA. - ?:(In corso di stampa), pp. 0-0. [10.1016/j.jfa.2024.110422]

The Pólya-Szegő inequality for smoothing rearrangements

Gabriele Bianchi
;
Paolo Gronchi;
In corso di stampa

Abstract

A basic version of the Pólya-Szegő inequality states that if Φ is a Young function, the Φ-Dirichlet energy -- the integral of Φ(∥∇f∥) -- of a suitable function f∈V(Rn), the class of nonnegative measurable functions on Rn that vanish at infinity, does not increase under symmetric decreasing rearrangement. This fact, along with variants that apply to polarizations and to Steiner and certain other rearrangements, has numerous applications. Very general versions of the inequality are proved that hold for all smoothing rearrangements, those that do not increase the modulus of continuity of functions. The results cover all the main classes of functions previously considered: Lipschitz functions f∈V(Rn), functions f∈W1,p(Rn)∩V(Rn) (when 1≤p<∞ and Φ(t)=tp), and functions f∈W1,1loc(Rn)∩V(Rn). In addition, anisotropic versions of these results, in which the role of the unit ball is played by a convex body containing the origin in its interior, are established. Taken together, the results bring together all the basic versions of the Pólya-Szegő inequality previously available under a common and very general framework.
In corso di stampa
?
0
0
Gabriele Bianchi; Richard J. Gardner; Paolo Gronchi; Markus Kiderlen
File in questo prodotto:
File Dimensione Formato  
arxiv_2206-09833.pdf

accesso aperto

Tipologia: Preprint (Submitted version)
Licenza: Solo lettura
Dimensione 588.01 kB
Formato Adobe PDF
588.01 kB Adobe PDF

I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1353379
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? ND
social impact