International Congress of Mathematicians

722 C. ThieleA basic point of singular integral theory is that an estimate of the form (1.2)may prevail for 1 < p < oo with a constant C P: K instead of ||if||i on the righthand side, if K is not absolutely integrable and the integral (1.1) is only defined ina distributional (principal value) sense. The most prominent example on the realline (indeed, all operators in this article will act on functions on the real line) is theHilbert transform with K(x) = 1/x.Taking formally Fourier transforms, one can write (1.1) as multiplier operator:37(0 = K(Of(0 =: m(£)/(£). (1.3)For the purpose of this survey a sufficiently interesting class of singular integrals isdescribed in terms of the multiplier m by imposing the symbol estimatesfor a = 0,1,2. We define the dual bilinear form(d/dO a m(0 < C|£|-° (1.4)A(fi,f 2 )= (Tfi(xj)f2(x)dx= fitti) M&mfo) da (1.5)J-'Ci+C2=owhere da is the properly normalized Lebesgue measure on the hyperplane £i+£2 = 0.The natural generalization of estimate (1.2) using duality of L p spaces then takesthe form|A(/i,/2)|j),- G Z ^e a f amu y °f functions suchthat rrij := ipj is supported in the ball B(0,2 J ) of radius 2 J around 0, vanisheson B(0, 2 J_2 ), and satisfies the symbol estimates (1.4) uniformly in j. By squarefunction estimate we mean the inequalityii(£i/*^i 2 ) 1/2 iii>i,j(£i)i/>2,j(^£i) for two families ipij and ip 2 ,j as in thesquare function estimate. Then we haveA(/i,/ 2 )I YI / (Ä * '