Weighted Norm Inequalities for the Local Sharp Maximal Function

Weighted Norm Inequalities for the Local Sharp Maximal Function 469Corollary 1.Inequalities (2.1) and (2.3) are sharp in the sense that M [p]+1 ω cannot be replacedby M [p] ω.To prove Proposition 1, we follow a counterexample due to J.M. Wilson [16]. Letn = 1 and T be the Hilbert transform H . Take ω = χ (0,1) and f = (log x) −1 χ (e,e N ) .Simple computations show that |Hf |≥c log N on (0, 1),M k ω(x) ≍(log(2 +|x|))k−12 +|x|,andTherefore,while(M r f(x)≤ c1log(2 +|x|) χ {|x|≤e N } + eN/rN∫R |Hf |p ωdx ≥ c(log N) p ,)1|x| 1/r χ {|x|≥e N } .∫∫ e NR (M rf ) p M [p] dxωdx ≤ c0 (log(2 + x)) p−[p]+1 (2 + x)∫+ c eNp/r ∞(log x) [p]−1N p e N x 1+p/r dx≤ c(log N + 1/N p−[p]+1 ) ≤ c log N,which contradicts (2.7) for large N.In conclusion we observe that (2.5) allows us to obtain a sufficient condition on a pairof weights (ω, v) for which T is bounded from L p v to L p ω, that is,∫∫R |Tf|p ωdx ≤n R |f |p vdx. (2.8)nIndeed, applying the well-known Sawyer’s S p condition [11] to (2.5), we get that thecondition∫ ( ( )) ∫pM v 1−p′ χ Q (Mω/ω) p ωdx ≤ c v 1−p′ dx for all Q ⊂ R nQQis sufficient for (2.8) (see also [3, 15] for different results in this direction).3. Two LemmasHere we present two important properties of the local sharp maximal function. Recallfirst some well-known definitions.The non-increasing rearrangement of a measurable function f is defined byf ∗ (t) = sup|E|=t x∈Einf |f(x)| (0