We study sharp growth conditions for the boundedness of Hardy-Littlewood maximal function in generalized Orlicz spaces. assume that \(\phi(x,t)\) satisfies standard continuity properties (A0), (A1) and (A2). show if is bounded from space to itself then \(\phi(x,t)/t^p\) almost increasing large \(t\) some \(p>1\). Moreover we \(L^\phi(\mathbb{R}^n)\) only \(\phi\) weakly equivalent a \(\psi\)...

Let μ be a Borel measure on Rd which may be nondoubling. The only condition that μ must satisfy is μ(Q) ≤ c0l(Q) for any cube Q ⊂ Rd with sides parallel to the coordinate axes and for some fixed n with 0 < n ≤ d. This paper is to establish the weighted norm inequality for commutators of Calderón-Zygmund operators with RBMO(μ) functions by an estimate for a variant of the sharp maximal function ...

We give a detailed proof, in the case of one space dimension, of a pointwise upper estimate for the space gradient of a temperature. The operators involved are a one-sided Hardy-Littlewood maximal in time and the Calderón sharp maximal operator in space.

We present a framework that yields a variety of weighted and vector-valued estimates for maximally modulated Calderón-Zygmund singular (and maximal singular) integrals from a single a priori weak type unweighted estimate for the maximal modulations of such operators. We discuss two approaches, one based on the good-λ method of Coifman and Fefferman [CF] and an alternative method employing the s...

For a Schwartz function f on the plane and a non-zero v ∈ R2 define the Hilbert transform of f in the direction v to be Hvf(x) = p.v. ∫ R f(x− vy) dy y Let ζ be a Schwartz function with frequency support in the annulus 1 ≤ |ξ| ≤ 2, and ζf = ζ ∗ f . We prove that the maximal operator sup|v|=1|Hvζf | maps L2 into weak L2, and Lp into Lp for p > 2. The L2 estimate is sharp. The method of proof is ...