By Jean-Lin Journé (auth.)

ISBN-10: 354012313X

ISBN-13: 9783540123132

ISBN-10: 3540400338

ISBN-13: 9783540400332

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Additional info for Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón

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BMO is an isomorphism of C2 > 0 L(b) onto L(BMO) ~ that such that ,6v~|. CI <-- [[L(b)ll(Hi,m) . <= C2 for all bEBMO . In fact, it is true that llbll~,= ilL(b)ll (see page 34 [Ib[I, >= If nbadx I Q if < C2 llg. ) llbll~>__ llL(b)ll(Hl, m). I for the definition of (Hl,®)* = the calculation above, aE~ m . be any cube, and let a0EG Q be the atom constructed near the beginning of this section which satisfies inf j'Ib-~l ~C Q ~dx = [f nbaodx[ • This a0 also satisfies for any [ao{ = ~ [ f E N I'~ , llfIIHl,== >= [If[[1 IIb[[~<__ on ).

LHijP i* BMO = (HI'P) * (See [Y]. ) 37 Since which is i L . i* is an inclusionj Because L(BMO) is the canonical injection of is closed in (Hi,m) * , Banach implies that H I~= is closed in H I'p .

Suppose that uEA • let K be 59 and (ii) (x,y,u)-~Ku(x,y) K = ~_ K u d ~ A Then Proof: is measurable on determines a K CZSI T ACx A . i automatically satisfies the standard estimates~ and C(K) ~ ~ C(Ku)d ~ ~ ~ IITuIIczd~ A A For ¢>0 ~ let us write Kc(xsy) = K(x,y)X[]x_y I >¢} . e. Since A (sup I ~ nK¢ (x,Y)f(y)dyl) d~ ¢>0 = ~ (Tu),f(x)d~ ~ L20Rn, dx) , A the function of x defined by the last integral is finite for almost every x , andj when it is finite~ we may apply the dominated convergence theorem to obtain lira ~ (Ku) ¢ (x,y)f(y)dyd~ ¢~0 ~nXA = A~ [lim0 ~ n K u ( x ~ y ) f ~ ) d y ] d ~ .

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Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón by Jean-Lin Journé (auth.)


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