Year: 2025
Type: article
Abstract: In this note we consider a generalisation to the metric setting of the recent work (Gu and Yung in J Funct Anal 281:109075, 2021). In particular, we show that under relatively weak conditions on a metric measure space $$(X,d,\nu )$$ ( X , d , ν ) , it holds true that $$ \bigg [ \frac{u(x)-u(y)}{d(x,y)^{\frac{s}{p}}} \bigg ]_{L^p_w(X \times X, \nu \otimes \nu )} \approx \Vert u \Vert _{L^p(X,\nu )}, $$ [ u ( x ) - u ( y ) d ( x , y ) s p ] L w p ( X × X , ν ⊗ ν ) ≈ ‖ u ‖ L p ( X , ν ) , where s is a generalised dimension associated to X and $$[\cdot ]_{L^p_w}$$ [ · ] L w p is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.
Source: Revista Matemática Complutense
Authors: Stefano Buccheri, Wojciech Górny
Institutions: University of Naples Federico II, University of Warsaw
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Awards: G1443340491, G1474110325, G1874798228, Challenges for the WIDE variational principle, G3347105513, G3878859087, G4845925241, Scales and Shapes in Continuum Thermomechanics, G5339074417, G5558726797, Inhomogeneous-growth problems including a linear-growth term, G740114916, G8590477180, G8988033314
OpenAlex ID: W4416840048