Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems

Resumen

Suppose that {φk}∞k=0 is the orthonormal system generated by the monomials {xn}∞n=0 in L2(μ), where μ is a positive measure on [−1,1]. For such a system there are the usual partial sums Snf=∑nk=0φk(∫fφ¯¯kdμ) of the Fourier series and the maximal operator S∗f=supn|Snf|, for f∈L2(μ). The first part of this paper is concerned with extending the transplantation theorem of J. E. Gilbert [Trans. Amer. Math. Soc. 145 (1969), 495–515; MR0252941 (40 #6156)] to the case of Ap-weighted Lp spaces. This is based on the observation that the estimates needed to do this are controlled by Hardy-Littlewood maximal operators, and B. Muckenhoupt had shown the boundedness of these on Ap-weighted spaces. When dealing with Jacobi polynomial series, the authors also use the boundedness of the Carleson-Hunt maximal operator on Ap-weighted spaces, as was shown by R. A. Hunt and W. S. Young [Bull. Amer. Math. Soc. 80 (1974), 274–277; MR0338655 (49 #3419)]. The Jacobi polynomial case is an extension of work of V. M. Badkov [Mat. Sb. (N.S.) 95(137) (1974), 229–262; MR0355464 (50 #7938)]. In the second part of the paper the authors find sufficient conditions for the existence of functions with almost everywhere divergent Fourier series. A result of A. Máté, P. Nevai and V. Totik [J. Approx. Theory 46 (1986), no. 3, 314–322; MR0840398 (87j:42074)] gives lower bounds for certain Lp-norms and these are combined with an argument using the uniform boundedness principle and Egorov's theorem. The remainder of the paper is taken up with obtaining similar results for Bessel systems