Higher summability theorems from the weighted reverse discrete inequalities

In 1972, B. Muckenhoupt [Trans. Amer. Math. Soc. {bf 165} (1972), 207--226; [msn] MR0293384 [/msn]] proved that if a weight (i.e., a non-negative, locally integrable function) w defined on a bounded interval I subset Bbb{R} satisfies frac{1}{|I|} int_{I} w(x) ,dx leq C w(x),quad text{for all} x in I, for a constant C>1, then, for every p in [1, C/(C-1)), the function w belongs to L^p(I) and frac{1}{|I|} int_{I} w^{p}(x) ,dx leq frac{C}{C-p(C-1) }left(frac{1}{|I|} int_{I} w(x),dxright)^{p}. Related properties are the reverse Hölder inequality and a result by F.~W. Gehring [Acta Math. {bf 130} (1973), 265--277; [msn] MR0402038 [/msn]] that asserts that if there exists a constant {K > 1} such that left(frac{1}{|I|} int_{I} w^{q}(x) ,dxright)^{{1}/{q}} leq K frac{1}{|I|} int_{I} w(x) ,dx, with q>1, then, for sufficiently small varepsilon > 0, left(frac{1}{|I|} int_{I} w^{q+varepsilon}(x) ,dxright)^{1/(q+varepsilon)} leq a_{q+varepsilon} left(frac{1}{|I|} int_{I} w^{q}(x) ,dxright)^{1/q} for a certain constant a_{q+varepsilon}. par The paper under review proves discrete versions of these properties where, instead of weights, non-increasing sequences are used. par For non-negative integers a and N such that N > a, let us define Bbb{I}_{a}^{N} = {a, a+1, a+2, ldots, N} subseteq Bbb{N}, and, for any sequence f: Bbb{I}_{a}^{N} to Bbb{R}^{+}, we define Cal{M} f(n) coloneq frac{1}{Lambda(n)} sum_{k=a}^{n-1} w(k) f(k), quad text{for all} n in Bbb{I}_{a}^{N}, where Lambda(n) = sum_{k=a}^{n-1} w(k). par The authors prove that if g: Bbb{I}_{a}^{N} rightarrow Bbb{R}^{+} is a non-increasing sequence such that g(n) leq beta g(n+1) for some beta>1 and for all n in Bbb{I}_{a}^{N}, and if Cal{M} g(n) leq c g(n) for some cge1 and for all n in Bbb{I}_{a}^{N}, then the inequality Cal{M} g^{p}(N) leq frac{p}{A-p(A-1)}(Cal{M} g(N))^{p} holds for p in [1, A/(A-1)), where A = cbeta^{p}; it is the discrete version of Muckenhoupt's result. The paper also includes a discrete version of Gehring's result, among other properties.