Quasi-greedy bases and Lebesgue-type inequalities

We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on the L-p spaces. The novelty of the paper is in obtaining better Lebesgue-type inequalities under extra assumptions on a quasi-greedy basis than known Lebesgue-type inequalities for quasi-greedy bases. We consider uniformly bounded quasi-greedy bases of L-p, 1 < p < infinity, and prove that for such bases an extra multiplier in the Lebesgue-type inequality can be taken as C(p) ln(m+1). The known magnitude of the corresponding multiplier for general (no assumption of uniform boundedness) quasi-greedy bases is of order m(vertical bar 1/2-1/p vertical bar) p not equal 2. For uniformly bounded orthonormal quasi-greedy bases we get further improvements replacing ln(m + 1) by (ln(m + 1))(1/2).