Almost Everywhere Convergence of Riesz Means Related to Schrödinger Operator with Constant Magnetic Fields

and Applied Analysis 3 Lemma 4. For λ > 0, one has 󵄩󵄩󵄩󵄩󵄩 K δ,l,j λ f (x) 󵄩󵄩󵄩󵄩󵄩 2 2 ≤ C2 −2M(j+l) δ 2M󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩 2 2 , (19) where the constant C is independent of λ and δ. Proof. With the method similar to the proof of Lemma 4 in [9], we write h(t) = φ(t) − φ(2t) and expandm into a Taylor series around λt. Then, ?̂? δ,l,j λ (t) = ∫m δ (λ(t − 2 −(j+l) δ 2 r λ )) ĥ (r) dr = ∫m δ (λt − 2 −(j+l) δ 2 r) ĥ (r) dr = ∫RM (t, r) ĥ (r) dr, (20) where the remainder RM satisfies 󵄨󵄨󵄨󵄨RM (t, r) 󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨 D M m 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 2 −(j+l) δ 2 r 󵄨󵄨󵄨󵄨 M ≤ 2 −M(l+j) δ M |r| M . (21) But ĥ is a Schwartz function and can be integrated against |r| . Hence, 󵄨󵄨󵄨󵄨󵄨󵄨 ?̂? δ,l,j λ (t) 󵄨󵄨󵄨󵄨󵄨󵄨 ≤ CM2 −M(j+l) δ M . (22) Since ER is a resolution of the identity, we see