TY - JOUR
AU - Murai, Takafumi
AB - PROOF OF THE BOUNDEDNESS OF COMMUTATORS BY PERTURBATION TAKAFUMI MURAI 1. Introduction Let D> (1 ^p ^ oo) denote the L space on the real line with respect to the 1-dimensional Lebesgue measure |-|. Its norm is denote d by |||| . For aeL , w e define kernels by: H(x,y) = l/(x-y), T[a](x,y) = {A{x)-A{y)}»/(x-y)* * (n > 1), C[a)(x,y) = \/{( -y) (A(x)-A(y))}, x + i where A(x) = j*a(t)dt. We simply write Kfor the singular integral operator defined by the kernel K(x,y). We say that K is bounded if it is bounded as an operator from L to itself. As is well known, (*) H is bounded, which plays an important role in harmonic analysis on the real line. The operato r C[a] plays an analogous role to that of H in harmonic analysis on the graph {(x,A(x)); xe( — oo, oo)}. It is a well-known theorem [5] that: (*)oo Q°] * bounded. Prior to this theorem, the following theorems were known [1,2,3,4]: (*)„ T [a] is bounded (n ^ 1). (*) C[a] is bounded if WaW^ is sufficiently small. David [6] deduced (*) from (*) by perturbation. The author [10] deduced (*) 00 s s 
TI - Proof of the Boundedness of Commutators by Perturbation
JF - Bulletin of the London Mathematical Society
DO - 10.1112/blms/18.4.383
DA - 1986-07-01
UR - https://www.deepdyve.com/lp/wiley/proof-of-the-boundedness-of-commutators-by-perturbation-POKSZR0RWc
SP - 383
EP - 388
VL - 18
IS - 4
DP - DeepDyve
ER -