1. INTRODUCTIO N 3 The interes t i n two-weigh t problem s fo r singula r operator s naturall y appear s from a n attemp t t o understan d whe n th e operato r i n th e Hilber t spac e ha s a n unconditional spectra l decomposition . Du e t o Werme r [71] , the followin g rigidit y claim holds : thi s unconditiona l spectra l decompositio n exist s fo r T i f an d onl y if T = S~ 1 NS^ wher e A T is a norma l operator , an d S i s a n invertibl e operato r (similarity). Thi s similarit y t o th e norma l operato r questio n receive d muc h larg e attention recentl y fo r differen t classe s o f T . W e mentio n her e [4] , [40] , [31] . I f T i s a smal l perturbatio n o f a unitar y operato r (eve n a ran k on e perturbation) , then i n genera l th e criteri a o f similarit y wit h a norma l operato r i s mor e o r les s totally open . Eve n i f T i s a contraction , th e relatio n betwee n th e spectra l dat a o f U an d N i s very subtl e i n general . Thes e kind s o f question s immediatel y becom e related t o two-weigh t problem s fo r th e Cauch y transform , a s illustrate d b y [40] . For example, [40 ] is based on a remarkable exampl e of Fedor Nazarov , which show s that th e Hunt-Muckenhoupt-Wheeden criterio n for one-weight boundednes s o f the Hilbert transfor m i s not applicabl e i n two-weigh t situations . Th e reade r wil l find more detail s i n [41] , [42 ] and i n Chapte r 15 , Chapter 1 6 of this book . Finally, le t u s explai n th e mai n difficult y o f th e theor y o f nonhomogeneou s Calderon-Zygmund operators . Roughl y speakin g th e difficult y appear s a s a resul t of a certai n degeneracy i n th e operator . W e ca n evok e th e vagu e analog y wit h subellipticity i n PDE . I n ou r case , the degenerac y appear s no t i n the kerne l o f th e operator (th e kernel is just a classical Calderon-Zygmund kernel ) bu t i n underlyin g measure. T o illustrat e th e kin d o f difficulty tha t persistentl y appears , le t u s thin k that w e need t o estimat e th e quantit y (1.1) I:=\ f [ k(x,y)f(x)g(y)dv,(x)dn{y)\. JQ JR Three possibilitie s ca n logicall y occur : 1 ) t o estimat e k i n L°° (mayb e afte r using som e sor t o f cancellation) , an d t o estimat e / i n L l (fi), g i n L 1 (//) 2 ) t o estimate k i n L 1 !/00 (thi s i s a mixe d norm , L 1 i n th e firs t variable , ° i n th e second one) , an d t o estimat e / i n L°°(/x) , g i n L x (/i) 3 ) t o estimat e k i n L 1 , an d to estimat e / i n L°°(/i) , g in L°° . In the first cas e no difficulty appears . W e need to bound / i n (1.1) by L 2 -norms of / , g. An d thi s i s not a problem, b y the Cauch y inequalit y II/IILI M Q ) 1 / 2 I I / I I L 3 , llsllt i H(R) 1/2 \\9\\L* Suppose w e wan t t o repea t somethin g lik e tha t i n th e secon d case . Firs t of all , ° nor m canno t b e estimate d b y th e L 2 one . Bu t thi s i s no t th e dif - ficulty (strangel y enough) , becaus e i n expressio n / usuall y /, g ar e ver y simple , basically constan t function s o n Q , R. I n thi s cas e w e hav e th e desire d estimates : H/HLOO M ( Q ) ~ 1 / 2 | | / I U * , \\g\\v M # ) 1 / 2 ! M I L 2 . Subsequently , w e get th e expres - sion (0)1/2 - Thi s i s a no t s o nic e expressio n becaus e measur e o f a (small ) se t Q stands in the denominator. Fo r good measures (fo r example, for Lebesgue measure ) we hav e a contro l o f thes e "smal l denominators" . Bu t fo r a n arbitrar y measure , the denominato r ca n be arbitraril y small , o r eve n vanishing. Th e onl y hop e i s tha t R C Q i n al l suc h cases . Bu t thi s i s no t s o usually . Usuall y th e mutua l positio n of Q , R i s quit e arbitrary . I n th e thir d cas e ther e ar e tw o smal l number s i n th e denominator. Thi s i s eve n worse . S o w e ar e boun d fo r disaste r i f w e reduc e th e estimate of the operator wit h kernel k to estimates o f sums of expressions of type / . But actuall y this i s exactly the most natura l wa y of estimating Calderon-Zygmun d
Previous Page Next Page