TY - JOUR
T1 - The Pólya-Szegő inequality for smoothing rearrangements
AU - Bianchi, Gabriele
AU - Gardner, Richard J.
AU - Gronchi, Paolo
AU - Kiderlen, Markus
N1 - Publisher Copyright:
© 2024 The Authors
PY - 2024/7
Y1 - 2024/7
N2 - 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∈Wloc1,1(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.
AB - 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∈Wloc1,1(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.
KW - Convex body
KW - Modulus of continuity
KW - Pólya-Szegő inequality
KW - Rearrangement
UR - http://www.scopus.com/inward/record.url?scp=85190954630&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2024.110422
DO - 10.1016/j.jfa.2024.110422
M3 - Journal article
AN - SCOPUS:85190954630
SN - 0022-1236
VL - 287
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 2
M1 - 110422
ER -