In this article we present a new proof of a sharp Reverse Hölder Inequality for A∞ weights. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of ...this last result, we obtain a simple proof of a sharp weighted bound for the Hardy–Littlewood maximal function involving A∞ constants:‖M‖Lp(w)⩽c(1p−1wApσA∞)1/p, where 1<p<∞, σ=w11−p and c is a dimensional constant. Our approach allows us to extend the result to the context of spaces of homogeneous type and prove a weak Reverse Hölder Inequality which is still sufficient to prove the open property for Ap classes and the Lp boundedness of the maximal function. In this latter case, the constant c appearing in the norm inequality for the maximal function depends only on the doubling constant of the measure μ and the geometric constant κ of the quasimetric.
For a normalized root system R in RN and a multiplicity function k≥0 let N=N+∑α∈Rk(α). Denote by dw(x)=∏α∈R|〈x,α〉|k(α)dx the associated measure in RN. Let F stand for the Dunkl transform. Given a ...bounded function m on RN, we prove that if there is s>N such that m satisfies the classical Hörmander condition with the smoothness s, then the multiplier operator Tmf=F−1(mFf) is of weak type (1,1), strong type (p,p) for 1<p<∞, and is bounded on a relevant Hardy space H1. To this end we study the Dunkl translations and the Dunkl convolution operators and prove that if F is sufficiently regular, for example its certain Schwartz class seminorm is finite, then the Dunkl convolution operator with the function F is bounded on Lp(dw) for 1≤p≤∞. We also consider boundedness of maximal operators associated with the Dunkl convolutions with Schwartz class functions.
Besicovitch proved that if f is an integrable function on R2 whose associated strong maximal function MSf is finite a.e., then the integral of f is strongly differentiable. On the other hand, ...Papoulis proved the existence of an integrable function on R2 (taking on both positive and negative values) whose integral is strongly differentiable but whose associated strong maximal function is infinite on a set of positive measure. In this paper, we prove that if n≥2 and if f is a measurable nonnegative function on Rn whose integral is strongly differentiable and moreover such that f(1+log+f)n−2 is integrable, then MSf is finite a.e. We also show this result is sharp by proving that, if φ is a continuous increasing function on 0,∞) such that φ(0)=0 and with φ(u)=o(u(1+log+u)n−2)(u→∞), then there exists a nonnegative measurable function f on Rn such that φ(f) is integrable on Rn and the integral of f is strongly differentiable, although MSf is infinite almost everywhere.
In this paper we establish weighted estimates for the maximal function associated with the finite type curve in the plane R2. We follow an approach used by M. Lacey to obtain sparse bounds for the ...maximal function. Further, using a different approach we obtain a characterisation of power weights for the weighted Lp boundedness of the maximal function. We also obtain analogous results for the lacunary maximal function associated with the finite type curve in the plane R2.
We prove a weighted $ L^{p} $ boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding ...operators on the weighted Lebsgue space $ L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{m}, \, \omega _{1}(x)dx, \, \omega_{2}(y)dy) $, provided that the weights $ \omega_{1} $ and $ \omega_{2} $ are certain radial weights and that the kernels are rough in the optimal space $ L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1}) $. In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted $ L^{p} $ boundedness of the related square and maximal functions. Our weighted $ L^{p} $ inequalities extend as well as generalize previously known $ L^{p} $ boundedness results.
We prove the sharp mixed Ap−A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely‖M‖Lp,q(w)≲p,q,nwAp1pσA∞1min(p,q), where σ=w11−p. Our ...method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting.
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