WEIGHTED L p-BOUNDEDNESS OF FOURIER SERIES WITH RESPECT TO GENERALIZED JACOBI WEIGHTS

Abstract. Let w be a generalized Jacobi weight on the interval [−1, 1] and, for each function f , let Snf denote the n-th partial sum of the Fourier series of f in the orthogonal polynomials associated to w. We prove a result about uniform boundedness of the operators Sn in some weighted L spaces. The study of the norms of the kernels Kn related to the operators Sn allows us to obtain a relation between the Fourier series with respect to different generalized Jacobi weights.

was done by Badkov ( [1]) in the case u = v by means of a direct estimation of the kernels K,, (x, y) associated with the polynomials orthogonal with respect to dp.Later, one of us ( [10]) considered the same problem, with u and v not necessarily equal; his method consists of an appropriate use of the theory of AP weights .He found conditions for (1) which generalized those obtained for u = v by Badkov .However, this result, whích we state below, follows only in the case yi > 0, i = 1, . . . .N.
At least in the case u = v (thus gi = Gi, Vi), inequality R < r requires -yi >-0 Vi.But, with this assumption, theorem 1 follows.

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Since (3) implies gp+ y > -1, the first term is bounded by (13 For the second term, let us consider separately the three cases in the statement .a) Ifg < (y+1 In this case, (12) and (13) imply: Since p[(y + 1)(1/2 -1/p) -g + 1/2] > 0, from this inequality and (11) we obtain Proo£ It is a simple consequence of proposition 4, corollary 5 and the estimate (9) .The only thing we must do is to consider each case in these results separately.
Note.Although it will not be used in what follows, corollary 6 also holds when c = +1.The proof is similar: starting from other expressions for K, (x, f l), analogous results to proposition 4 and corollary 5 can be obtained, and then corollary 6 follows.
We are now ready to prove our main result : Proof of theorem 2: a) Let us ássume first that the inequalities (2), ( 3) and (4) hold.We prove that the operators S, are uniformly bounded by induction on the number of negative exponents y¡.If y¡ > 0 di, the result is true, as for every polynomial R, since S~R = R if n is big enough.It is clear that there exists a polynomial Q such that both I QI PuP and IQIPvP are p-integrable.Let us denote u' = I QI Pu P and v' = I QI PvP.Then, for every f E LP(u'dp) fl LP(v'dp) there exists a sequence of polynomials R,, such that From this and (14) we obtain  Taking now E = {x E [-1,1] ; u(x) > Cv(x) } and f the characteristic function on E, we deduce p(E) = 0.

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f I RnQIPuPdp < CP 1' m f I RnQI Pv Pdp = CP f I f IPv'dp .