Abstract. This paper is a reprint of the original work by A. P. Calderón published by the Brazilian Mathematical Society (SBM) in ATAS of SBM (Rio de Janeiro), pp. [65][66][67][68][69][70][71][72][73] 1980. The original paper had no abstract, so this reprint to be truthful to the original work is published with no abstract. Mathematical subject classification: 35J25, 35Q60, 47F05, 86A20, 86A22.Key words: inverse problems, boundary value problems, identifi cation problem, elliptic equations.In this note we shall discuss the following problem. Let D be a bounded domain in R n , n ≥ 2, with Lipschitzian boundary d D, and γ be a real bounded measurable function in D with a positive lower bound. Consider the differential operatoracting on function of H 1 (D) and the quadratic form Q γ (φ) , where the functions in H 1 (R n ) , defi ned by

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We continue the investigation initiated in [8] of uniform L p bounds for the family of bilinear Hilbert transformsIn this work we show that H α,β map L p 1 (R) × L p 2 (R) into L p (R) uniformly in the real parameters α, β satisfying | α β − 1| ≥ c > 0 when 1 < p 1 , p 2 < 2 and 2 3 < p = p 1 p 2 p 1 +p 2 < ∞. As a corollary we obtain L p × L ∞ → L p uniform bounds in the range 4/3 < p < 4 for the H 1,α 's when α ∈ [0, 1).2000 Mathematics Subject Classification: Primary: 42B20, 42B25. Secondary: 46B70, 47B38. Keywords: Time-frequency analysis, bilinear Hilbert transform, uniform bounds. Grafakos, for his constructive suggestions, for his helpful advice and guidance, for his teaching, encouragement, and friendship. The author is very grateful to Christoph Thiele for his generous comments and suggestions. He is also thankful to Michael Lacey for the very helpful discussions during a visit at the Georgia Institute of Technology in 1998. The truncated trilinear formBy the decomposition of the half plane η < 2 m ξ on the ξ-η plane in [8], we only need to consider the following operator. (See [8] for the details.)where Φ 1,k,l and Φ 2,k,l are suitable functions satisfying the following properties:if l is odd, for all nonnegative integers α. Note that the function Φ 2,k,l also depends on the parameter m, but this dependence will be suppressed for notational convenience. Now we prove (5.10). Let J = [2 k J n J , 2 k J (n J + 1)] for some k J ∈ Z. Define T J := {(k, n) ∈ T : I k,n ⊂ J}. Then

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In this note, we establish certain properties of the Cauchy integral on Lipschitz curves and prove the LPboundedness of some related operators. In particular, we obtain the recent results of R. R. Coifman and Y. Meyer [(1976 Proof: We shall first consider the case in which (p(t) is infinitely differentiable, and has compact support and shall show that the operator A ,f = limeoA,,ff is bounded in L2, and has a norm which can be estimated in terms of a bound for ,o'(t) alone provided that I I Idll < a. Once this is established the results stated above will follow by applying standard results and techniques.Let then P(t) C Co-, (P'(t)I < M, and consider the opera-where Z (t) = t + iXsp(t), 0 < X < 1, andThey are well-defined, at least for f e CoO, and in this case A(X)f and B(X)f are continuous functions of t. The operator B(X) is obtained as the formal derivative of A(X) with respect to X, but since for E > 0 the integral in the definition of B(X) is the derivative of the one in the definition of A(X) and they converge uniformly asE -0, we have indeed A(X)f = A(O)f + fXB(s)f ds, f E Co.[1]On the other hand, A(X) is uniformly bounded in L2. To see this, we write the kernel of A(X) aswhere, as is readily verified, k(X,s,t) is infinitely differentiable and has a double Fourier transform h (X,u,v) which is integrable uniformly in X. Expressing k in terms of h, and using the uniform boundedness in L2 of the truncated Hilbert transform and Minkowski's integral inequality, we obtain the desired result.Our goal is to estimate the norm of B(X) in terms of A(X) and M, that is, a bound for p'(t). This in conjunction with Eq. 1 will give us an estimate for the norm of A(X) in terms of M alone.Let 01 and 02 be the open subsets of the complex plane consisting of the points lying above and below the curve F, respectively. With a function f(t) in Co-, we associate the functions Fl(w) and F2(w), analytic in 01 and°2, respectively, given by Fj)(w) = 2wdz(s), w It is not difficult to see that these functions extend as C-functions to the curve r and that Fj(z(t)) = -f(t) + A(1)f. 2Consequently we have f(t) = Fl(z(t)) + F2(z(t)) |lFj(z(t))112 < I + IIA(1)II) If12- [3] [4]We now introduce the operator