TY - JOUR
T1 - Grušin operators, Riesz transforms and nilpotent Lie groups
AU - Robinson, Derek W.
AU - Sikora, Adam
N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.
PY - 2016/2/1
Y1 - 2016/2/1
N2 - We establish that the Riesz transforms of all orders corresponding to the Grušin operator (Formula presented.), and the first-order operators (∇x,xν∇y) where x∈Rn, y∈Rm, N∈N+, and ν∈{1,…,n}N, are bounded on Lp(Rn+m) for all p∈⟨1,∞⟩ and are also weak-type (1, 1). Moreover, the transforms of order less than or equal to N+1 corresponding toHN and the operators (∇x,|x|N∇y) are bounded on Lp(Rn+m) for all p∈⟨1,∞⟩. But if N is odd all transforms of order N+2 are bounded if and only if p∈⟨1,n⟩. The proofs are based on the observation that the (∇x,xν∇y) generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces Lp(Rn+m) and HN is the corresponding sublaplacian.
AB - We establish that the Riesz transforms of all orders corresponding to the Grušin operator (Formula presented.), and the first-order operators (∇x,xν∇y) where x∈Rn, y∈Rm, N∈N+, and ν∈{1,…,n}N, are bounded on Lp(Rn+m) for all p∈⟨1,∞⟩ and are also weak-type (1, 1). Moreover, the transforms of order less than or equal to N+1 corresponding toHN and the operators (∇x,|x|N∇y) are bounded on Lp(Rn+m) for all p∈⟨1,∞⟩. But if N is odd all transforms of order N+2 are bounded if and only if p∈⟨1,n⟩. The proofs are based on the observation that the (∇x,xν∇y) generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces Lp(Rn+m) and HN is the corresponding sublaplacian.
KW - 22E45
KW - 35H20
KW - 35J70
KW - 43A65
UR - http://www.scopus.com/inward/record.url?scp=84955183672&partnerID=8YFLogxK
U2 - 10.1007/s00209-015-1548-y
DO - 10.1007/s00209-015-1548-y
M3 - Article
SN - 0025-5874
VL - 282
SP - 461
EP - 472
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 1-2
ER -