We study, for a basis of Hölderian compactly supported wavelets, the boundedness and convergence of the associated projectors on the space for some p in ]1,∞[ and some nonnegative Borel measure μ on ℝ. We show that the convergence properties are related to the criterion of Muckenhoupt.
We show that to any multi-resolution analysis of L2(R) with multiplicity d, dilation factor A (where A is an integer ≥ 2) and with compactly supported scaling functions we may associate compactly supported wavelets. Conversely, if (Ψε,j,k = Aj/2 Ψε (Ajx - k)), 1 ≤ ε ≤ E and j, k ∈ Z, is a Hilbertian basis of L2(R) with continuous compactly supported mother functions Ψε, then it is provided by a multi-resolution analysis with dilation factor A, multiplicity d = E / (A - 1) and with compactly supported...
Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average . The square function is defined as . The local version of this operator, namely the operator , is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps into itself (p > 1) and into BMO. We prove that the operator S not only maps into BMO but it also maps BMO into BMO. We also prove that the boundedness still holds...
We investigate weak type estimates for maximal functions, fractional and singular integrals in grand Lebesgue spaces. In particular, we show that for the one-weight weak type inequality it is necessary and sufficient that a weight function belongs to the appropriate Muckenhoupt class. The same problem is discussed for strong maximal functions, potentials and singular integrals with product kernels.