THE MAXIMAL FUNCTION ON VARIABLE Lp SPACES

230 D. Cruz-Uribe, A. Fiorenza, and C.J. NeugebauerSince |B| ≤ 1, and sinceby Lemma 2.1,−p(x)/¯p ∗ (B Ω ) + p ∗ = ( p ∗ /p ∗ (B Ω ) )( p ∗ (B Ω ) − p(x) )≥ ( p ∗ /p ∗ (B Ω ) )( p ∗ (B Ω ) − p ∗ (B Ω ) ) ,( ∫) p∗1≤ C f(y)¯p(y) dy + Cα(x) p ∗≤ CM ( f( · )¯p( · )) (x) p ∗+ Cα(x) p ∗.|B| B ΩThis is precisely what we wanted to prove.Case 2: |x| ≤ 1 and r ≥ |x|/4. The proof is essentially the same as in theprevious case: since |x| ≤ 1, α(x) ≈ 1, so inequality (2.2) and the subsequentargument still hold.Case 3: |x| ≥ 1 and r ≥ |x|/4. Since f(x) ≥ 1, p ∗ ≥ 1 and |f| p(x),Ω ≤ 1,( ∫ ) p(x) (∫) p(x)1f(y) dy ≤ |B| −p(x) f(y) p(y) dy|B| B Ω B ΩThis completes the proof.≤ Cr −np(x) |f| p(x)p(x),Ω ≤ C|x|−np ∗≤ CM ( f( · )¯p( · )) (x) p ∗+ Cα(x) p ∗.≤ Cα(x) p ∗byDefinition 2.4. Given a function f on Ω, we define the Hardy operator HHf(x) = |B |x| (0)| −1 ∫B |x| (0)∩Ω|f(y)| dy.Lemma 2.5. Given Ω and p as in the statement of Theorem 1.5, supposethat |f| p(x),Ω ≤ 1, and |f(x)| ≤ 1, x ∈ Ω. Then for all x ∈ Ω,(2.3) Mf(x) p(x) ≤ CM ( |f( · )| p( · )/p ∗ ) (x) p ∗+ Cα(x) p ∗+ CHf(x) p(x) ,where α(x) = (e + |x|) −n .Proof. We may assume without loss of generality that f is non-negative. Weargue almost exactly as we did in the proof of Lemma 2.3. In that proof we onlyused the fact that f(x) ≥ 1 in Case 3, so it will suffice to fix x ∈ Ω, |x| ≥ 1, anda ball B containing x with radius r > |x|/4, and prove that( ∫ ) p(x) 1f(y) dy ≤ CM ( |f( · )| p( · )/p ) ∗(x) p ∗+ Cα(x) p ∗+ CHf(x) p(x) .|B| B Ω