A Note on Non-tangential Convergence for Schrödinger Operators
AA Note on Non-tangential Convergence forSchr¨odinger Operators
Wenjuan Li, Huiju Wang, and Dunyan Yan
Abstract
The goal of this note is to establish non-tangential convergence results for Schr¨odingeroperators along restricted curves. We consider the relationship between the dimension of thiskind of approach region and the regularity for the initial data which implies convergence. Asa consequence, we obtain a upper bound for p such that the Schr¨odinger maximal functionis bounded from H s ( R n ) to L p ( R n ) for any s > n n +1) . The solution to the Schr¨odinger equation iu t − ∆ u = 0 , ( x, t ) ∈ R n × R + , (1.1)with initial datum u ( x,
0) = f, is formally written as e it ∆ f ( x ) := (cid:90) R n e i ( x · ξ + t | ξ | ) (cid:98) f ( ξ ) dξ. The problem about finding optimal s for whichlim t → + e it ∆ f ( x ) = f ( x ) a.e. (1.2)whenever f ∈ H s ( R n ) , was first considered by Carleson [4], and extensively studied by Sj¨olin[20] and Vega [21], who proved independently the convergence for s > / s < / s < n n +1) .Very recently, Du-Guth-Li [9] and Du-Zhang [11] obtained the sharp results by the polynomialpartitioning and decoupling method.The natural generalization of the pointwise convergence problem is to ask a.e. convergencealong a wider approach region instead of vertical lines. One of such problems is to consider non-tangential convergence to the initial data, it is natural to expect that more regularity on the This work is supported by the Natural Science Foundation of China (No.11871452); Natural Science Founda-tion of China (No.11701452) China Postdoctoral Science Foundation (No.2017M613193); Natural Science BasicResearch Plan in Shaanxi Province of China (No.2017JQ1009).2000
Mathematics Subject Classification : 42B20, 42B25, 35S10.
Key words and phrases : Schr¨odinger operator, Non-tangential convergence. a r X i v : . [ m a t h . C A ] S e p W. LI, H. Wanginitial data is necessary to guarantee a.e. existence of the non-tangential limit. It was shown bySj¨olin-Sj¨ogren [19] that non-tangential convergence fails for s ≤ n/
2, i.e., there exists a function f ∈ H n ( R n ) such that lim sup ( y,t ) → ( x, | x − y | <γ ( t ) ,t> | e it ∆ f ( y ) | = ∞ , for all x ∈ R n , where γ is strictly increasing and γ (0) = 0. Cho-Lee-Vargas [6] raised a ques-tion about how the size or dimension of the approach region and the regularity which impliesconvergence are related.In [6], this question is considered in the one dimensional case. More concretely, let Γ x = { x + tθ : t ∈ [ − ,
1] and θ ∈ Θ } , where Θ is a given compact set in R . In [6], they proved thatthe corresponding non-tangential convergence result holds for s > β (Θ)+14 , here β (Θ) denotes theupper Minkowski dimension of Θ. This result in [6] was established by the T T (cid:63) method and atime localizing lemma. Recently, by getting around the key localizing lemma in [6], Shiraki [18]generalized this result to a wider class of equations which includes the fractional Schr¨odingerequation.However, the above question remains open in higher dimensional case until recently. In thisarticle, we consider the non-tangential convergence problem along the approach region in R n given by Γ x = { γ ( x, t, θ ) : t ∈ [0 , , θ ∈ Θ } , where Θ is a given compact set in R n . γ is a map from R n × [0 , × Θ to R n , which satisfies γ ( x, , θ ) = x for all x ∈ R n , θ ∈ Θ, and the following (C1)-(C3) hold:(C1) For fixed t ∈ [0 , θ ∈ Θ, γ has at least C regularity in x , and there exists a constant C ≥ x, x (cid:48) ∈ R n , θ ∈ Θ, t ∈ [0 , C − | x − x (cid:48) | ≤ | γ ( x, t, θ ) − γ ( x (cid:48) , t, θ ) | ≤ C | x − x (cid:48) | ; (1.3)(C2) There exists a constant C > x ∈ R n , θ ∈ Θ, t, t (cid:48) ∈ [0 , | γ ( x, t, θ ) − γ ( x, t (cid:48) , θ ) | ≤ C | t − t (cid:48) | ; (1.4)(C3) There exists a constant C > x ∈ R n , t ∈ [0 , θ, θ (cid:48) ∈ Θ, | γ ( x, t, θ ) − γ ( x, t, θ (cid:48) ) | ≤ C | θ − θ (cid:48) | . (1.5)We consider the relationship between the dimension of Θ and the optimal s for whichlim ( y,t ) → ( x, y ∈ Γ x e it ∆ f ( y ) = f ( x ) a.e. (1.6)whenever f ∈ H s ( R n ).We first give two examples for Γ x . It is not hard to check that all the conditions mentionedabove can be satisfied if we take (E1): γ ( x, t, θ ) = x + tθ , Θ is a compact subset of the unitball in R n . When n = 1 , this is just the problem considered in [6]. Another example is (E2): γ ( x, t, θ ) = x + t θ , t θ = ( t θ , t θ , · · · , t θ n ), θ = ( θ , θ , · · · , θ n ), Θ is a compact subset in the firstquadrant away from the axis of R n . For this example, it is worth to mention that when θ is fixed,Lee-Rogers [14] have obtained that the convergence along the curve ( γ θ ( x, t ) , t ) is equivalent tothe convergence along the vertical line.on-tangential Convergence for Schr¨odinger Operators 3Figure 1. Θ = { , / , · · · , − /k, · · · , k = 1 , , · · · } .Figure 2. Γ x is consist of all black points which lie on the line y = x + t θ , θ ∈ Θ, t ∈ [0 , / t = t , there are countable black points corresponding to Θ. We try to seek theoptimal s for e it ∆ f ( y ) → f ( x ) along different green-path whose points from Γ x whenever f ∈ H s .In order to characterize the size of Θ, we introduce the so called logarithmic density or upperMinkowski dimension of Θ, β (Θ) = lim sup δ → + logN ( δ ) − logδ , where N ( δ ) is the minimum number of closed balls of diameter δ to cover Θ. It is not hard tosee that when Θ is a single point, β (Θ) = 0; when Θ is a compact subset of R n with positiveLebesgue measure, β (Θ) = n . W. LI, H. WangBy standard arguments, in order to obtain the convergence result, it is sufficient to establishthe bounded estimates for the maximal operator defined by sup ( t,θ ) ∈ (0 , × Θ | e it ∆ f ( γ ( x, t, θ )) | . Our main results are as follows. Firstly, we show the maximal operator estimate in the twodimensional case.
Theorem . When n = 2 , given B ( x , R ) ⊂ R , R (cid:46) , then for any s > β (Θ)+13 , (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ C (cid:107) f (cid:107) H s ( R ) , (1.7) whenever f ∈ H s ( R ) , where the constant C depends on s , C , C , C , and the chosen of B ( x , R ) , but does not depend on f . Then we obtain the following non-tangential convergence result.
Theorem . When n = 2 , if s > β (Θ)+13 , then lim ( y,t ) → ( x, y ∈ Γ x e it ∆ f ( y ) = f ( x ) a.e. (1.8) whenever f ∈ H s ( R ) . We notice that the convergence result obtained in Theorem 1.2 is sharp when β (Θ) = 0([9] and [3]) or β (Θ) = 2 ([19]). It is quite interesting to seek whether (1.8) is sharp when0 < β (Θ) < ∪ k Θ k with bounded overlap, for each Θ k , the size is small enough suchthat our problem can be reduced to estimate the maximal function for Schr¨odinger operatoralong certain curves, i.e. the maximal operator defined by sup t ∈ (0 , | e it ∆ f ( γ ( x, t, θ k ) | (1.9)for some θ k ∈ Θ k . The number of Θ k is determined by β (Θ). Finally, in order to get thebounded estimate for maximal function defined by (1.9), we still need the following theorem. Theorem . ([9]) For any s > / , the following bound holds: for any function f ∈ H s ( R ) , (cid:13)(cid:13)(cid:13)(cid:13) sup
3. By parabolic rescaling and time localizing lemma, inequality(1.10) is equivalent to (cid:13)(cid:13)(cid:13)(cid:13) sup 1. The range of p has been discussed inDu-Kim-Wang-Zhang [10], but the optimal range of p is still unknown. Conventions : Throughout this article, we shall use the well known notation A (cid:29) B , whichmeans if there is a sufficiently large constant G , which does not depend on the relevant param-eters arising in the context in which the quantities A and B appear, such that A ≥ GB . Wewrite A ∼ B , and mean that A and B are comparable. By A (cid:46) B we mean that A ≤ CB forsome constant C independent of the parameters related to A and B . Given R n , we write B (0 , B n (0 , 1) in R n centered at the origin for short, and the same notation isvalid for B ( x , R ). n = 2 Proof of Theorem 1.1. In order to prove Theorem 1.1, using Littlewood-Paley decomposition,we only need to show that for f with supp ˆ f ⊂ { ξ ∈ R : | ξ | ∼ λ } , λ (cid:29) (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ Cλ s + (cid:15) (cid:107) f (cid:107) L , ∀ (cid:15) > , (2.1) W. LI, H. Wangwhere s = β (Θ)+13 .We decompose Θ into subsets Θ = ∪ k Θ k with bounded overlap, where each Θ k is containedin a closed ball with diameter λ − . Then we have1 ≤ k ≤ λ β (Θ)+ (cid:15) . (2.2)We claim that for each k , (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ k | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ Cλ + (cid:15) (cid:107) f (cid:107) L . (2.3)Then inequality (2.1) follows from (2.2) and (2.3). More concretely, we have (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ (cid:18)(cid:88) k (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ k | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) (cid:19) / ≤ C (cid:18)(cid:88) k λ (cid:15) (cid:107) f (cid:107) L (cid:19) / ≤ Cλ β Θ+13 + (cid:15) (cid:107) f (cid:107) L , (2.4)which implies inequality (2.1).Now we are left to prove inequality (2.3). For this goal, we first show the following Lemma2.1. The original idea comes from Lemma 2.2 in [14]. Lemma . Assume that g is a Schwartz function whose Fourier transform is supported in { ξ ∈ R n : | ξ | ∼ λ } . If | θ − θ (cid:48) | ≤ λ − , then for each x ∈ B ( x , R ) and t ∈ (0 , , | e it ∆ g ( γ ( x, t, θ )) | ≤ (cid:88) l ∈ Z n C (1 + | l | ) n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R n e i [ γ ( x,t,θ (cid:48) )+ l λ ] · ξ + it | ξ | ˆ g ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) , (2.5) where the constant C depends on n and C in inequality (1.5). Proof. We introduce a cut-off function φ which is smooth and equal to 1 on B (0 , 2) andsupported on ( − π, π ) n . After scaling we have e it ∆ g ( γ ( x, t, θ )) = λ n (cid:90) R n e iλγ ( x,t,θ ) · η + it | λη | φ ( η )ˆ g ( λη ) dη = λ n (cid:90) R n e iλγ ( x,t,θ ) · η − iλγ ( x,t,θ (cid:48) ) · η + iλγ ( x,t,θ (cid:48) ) · η + it | λη | φ ( η )ˆ g ( λη ) dη. (2.6)Since it follows by inequality (1.5), λ | γ ( x, t, θ ) − γ ( x, t, θ (cid:48) ) | ≤ C , then by Fourier expansion, φ ( η ) e iλ [ γ ( x,t,θ ) − γ ( x,t,θ (cid:48) )] · η = (cid:88) l ∈ Z n c l ( x, t, θ, θ (cid:48) ) e i l · η , on-tangential Convergence for Schr¨odinger Operators 7where | c l ( x, t, θ, θ (cid:48) ) | ≤ C (1 + | l | ) n +1 uniformly for each l ∈ Z n , x ∈ B ( x , R ) and t ∈ (0 , | e it ∆ g ( γ ( x, t, θ )) | ≤ (cid:88) l ∈ Z n Cλ n (1 + | l | ) n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R n e i l · η + iλγ ( x,t,θ (cid:48) ) · η + it | λη | ˆ g ( λη ) dη (cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) l ∈ Z n C (1 + | l | ) n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R n e i l λ · ξ + iγ ( x,t,θ (cid:48) ) · ξ + it | ξ | ˆ g ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) , then we arrive at (2.5).By the similar argument, we can prove the following lemma. Lemma . Assume that g is a Schwartz function whose Fourier transform is supported in { ξ ∈ R n : | ξ | ∼ λ } . If | t − t (cid:48) | ≤ λ − , then for each x ∈ B ( x , R ) and θ ∈ Θ , | e it ∆ g ( γ ( x, t, θ )) | ≤ (cid:88) l ∈ Z n C (1 + | l | ) n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R n e i [ γ ( x,t (cid:48) ,θ )+ l λ ] · ξ + it | ξ | ˆ g ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) . (2.7) where the constant C depends on n and C in inequality (1.4). We now prove inequality (2.1). For fixed k , by the construction of Θ k , there is a θ k ∈ Θ k suchthat | θ − θ k | ≤ λ − holds for each θ ∈ Θ k . Then according to Lemma 2.1, for each x ∈ B ( x , R ), t ∈ (0 , 1) and θ ∈ Θ k , we have | e it ∆ f ( γ ( x, t, θ )) | ≤ (cid:88) l ∈ Z C (1 + | l | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R e iγ ( x,t,θ k ) · ξ + it | ξ | e i l λ · ξ ˆ f ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) l ∈ Z C (1 + | l | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R e iγ ( x,t,θ k ) · ξ + it | ξ | ˆ f l λ ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) l ∈ Z C (1 + | l | ) (cid:12)(cid:12)(cid:12)(cid:12) e it ∆ f l λ ( γ ( x, t, θ k )) (cid:12)(cid:12)(cid:12)(cid:12) , (2.8)where ˆ f l λ ( ξ ) = e i l λ · ξ ˆ f ( ξ ) . It follows that (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ k | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ (cid:88) l ∈ Z C (1 + | l | ) (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ k | e it ∆ f l λ ( γ ( x, t, θ k )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) W. LI, H. Wang= (cid:88) l ∈ Z C (1 + | l | ) (cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ (0 , | e it ∆ f l λ ( γ ( x, t, θ k )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ (cid:88) l ∈ Z C (1 + | l | ) λ + (cid:15) (cid:107) f l λ (cid:107) L ≤ λ + (cid:15) (cid:107) f (cid:107) L , (2.9)provided that we have proved the following lemma. Lemma . Assume that g is a Schwartz function whose Fourier transform is supported inthe annulus { ξ ∈ R : | ξ | ∼ λ } . Then for each k , (cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ (0 , | e it ∆ g ( γ ( x, t, θ k )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ Cλ + (cid:15) (cid:107) g (cid:107) L , (2.10) where the constant C is independent of k . Now let’s turn to prove Lemma 2.3. The following theorem is required. Theorem . ([14]) Let ρ : R n +1 → R n , q, r ∈ [2 , + ∞ ] , λ ≥ , supp ν ⊂ [ − , , λ ≥(cid:107) (cid:107) /nL qµ L rν , and suppose that sup x ∈ supp ( ν ) ,t ∈ supp ( ν ) | ρ ( x, t ) | ≤ M, where M > . Suppose that for a collection of boundedly overlapping intervals I of length λ − ,there exists a C > such that (cid:107) e it ∆ f ( ρ ( x, t )) (cid:107) L qµ L rν ( I ) ≤ C (cid:107) f (cid:107) L ( R n ) , whenever ˆ f is supported in { ξ ∈ R n : | ξ | ∼ λ } . Then there is a constant C n > such that (cid:107) e it ∆ f ( ρ ( x, t )) (cid:107) L qµ L rν ( ∪ I ) ≤ C n M / C (cid:107) f (cid:107) L ( R n ) whenever ˆ f is supported in { ξ ∈ R n : | ξ | ∼ λ } . Notice that in our case, for each k , we have sup ( x,t ) ∈ B ( x ,R ) × (0 , | γ ( x, t, θ k ) | ≤ sup ( x,t,θ ) ∈ B ( x ,R ) × (0 , × Θ | γ ( x, t, θ ) | . By inequality (1.4), for each ( x, t, θ ) ∈ B ( x , R ) × (0 , × Θ, | γ ( x, t, θ ) − γ ( x, , θ ) | ≤ C , then | γ ( x, t, θ ) | is uniformly bounded for ( x, t, θ ) ∈ B ( x , R ) × (0 , × Θ, and the upper boundis determined by C and the chosen of B ( x , R ), but independent of k .Therefore, according to Theorem 2.4, in order to prove Lemma 2.3, we only need to showthat for each interval I ⊂ (0 , 1) of length λ − , and any function g such that ˆ g is supported in { ξ ∈ R : | ξ | ∼ λ } , we have (cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ I | e it ∆ g ( γ ( x, t, θ k )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ Cλ + (cid:15) (cid:107) g (cid:107) L . (2.11)on-tangential Convergence for Schr¨odinger Operators 9Since I is an interval of length λ − , there exists t I ∈ I such that for each t ∈ I , | t − t I | ≤ λ − . Then by Lemma 2.2, for each x ∈ B ( x , R ), t ∈ I , we have | e it ∆ g ( γ ( x, t, θ k )) | ≤ (cid:88) l ∈ Z C (1 + | l | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R e iγ ( x,t I ,θ k ) · ξ + it | ξ | e i l λ · ξ ˆ g ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) l ∈ Z C (1 + | l | ) (cid:12)(cid:12)(cid:12)(cid:12) e it ∆ g l λ ( γ ( x, t I , θ k )) (cid:12)(cid:12)(cid:12)(cid:12) , (2.12)where ˆ g l λ ( ξ ) = e i l λ · ξ ˆ g ( ξ ) . It follows that (cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ I | e it ∆ g ( γ ( x, t, θ k )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ (cid:88) l ∈ Z C (1 + | l | ) (cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ I | e it ∆ g l λ ( γ ( x, t I , θ k )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) . (2.13)For each t I , θ k , γ t I ,θ k is at least C from R to R . By inequality (1.3), for each x ∈ R , C − ≤ |∇ x γ ( x, t I , θ k ) | ≤ C . By the same reason, for each x ∈ B ( x , R ), | γ ( x, t I , θ k ) − γ ( x , t I , θ k ) | ≤ C R, which implies γ t I ,θ k ( B ( x , R )) ⊂ B ( γ ( x , t I , θ k ) , C R ). Therefore, changes of variables and The-orem 1.3 imply that (cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ I | e it ∆ g l λ ( γ ( x, t I , θ k )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ Cλ + (cid:15) (cid:107) g l λ (cid:107) L . (2.14)Combining inequality (2.13) with inequality (2.14), we have (cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ I | e it ∆ g ( γ ( x, t, θ k )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ (cid:88) l ∈ Z C (1 + | l | ) λ + (cid:15) (cid:107) g l λ (cid:107) L ≤ Cλ + (cid:15) (cid:107) g (cid:107) L . (2.15)This completes the proof of Lemma 2.3. Proof of Theorem 1.2. The proof of Theorem 1.2 is quite standard. We write the detailsfor completeness. In fact, for any s > β (Θ)+13 , f ∈ H s ( R ), fix λ > 0, choose g ∈ C ∞ c ( R ) suchthat (cid:107) f − g (cid:107) H s ( R ) ≤ λ(cid:15) / C , C is the constant in inequality (1.7), which follows (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B ( x , R ) : sup ( t,θ ) ∈ (0 , × Θ | e it ∆ ( f − g )( γ ( x, t, θ )) | > λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ | e it ∆ ( f − g )( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L ( B ( x ,R )) ≤ C λ (cid:107) f − g (cid:107) H s ( R ) ≤ (cid:15). (2.16)Moreover, lim ( y,t ) → ( x, y ∈ Γ x e it ∆ g ( y ) = g ( x ) (2.17)uniformly for x ∈ B ( x , R ). Indeed, for each x ∈ B ( x , R ),lim sup ( y,t ) → ( x, y ∈ Γ x | e it ∆ g ( y ) − g ( x ) | ≤ lim sup ( y,t ) → ( x, y ∈ Γ x | e it ∆ g ( y ) − e it ∆ g ( x ) | + lim sup ( y,t ) → ( x, y ∈ Γ x | e it ∆ g ( x ) − g ( x ) | = lim sup ( y,t ) → ( x, y ∈ Γ x | e it ∆ g ( y ) − e it ∆ g ( x ) | + lim sup t → + | e it ∆ g ( x ) − g ( x ) | . (2.18)By mean value theorem and inequality (1.4), we have | e it ∆ ( g )( γ ( x, t, θ )) − e it ∆ g ( x ) | ≤ t (cid:90) R | ξ || ˆ g ( ξ ) | dξ, (2.19)and | e it ∆ g ( x ) − g ( x ) | ≤ t (cid:90) R | ξ | | ˆ g ( ξ ) | dξ. (2.20)Inequalities (2.18) - (2.20) imply (2.17).By (2.16) and (2.17) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B ( x , R ) : lim sup ( y,t ) → ( x, y ∈ Γ x | e it ∆ ( f )( y ) − f ( x ) | > λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B ( x , R ) : lim sup ( y,t ) → ( x, y ∈ Γ x | e it ∆ ( f − g )( y ) | > λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B ( x , R ) : | f ( x ) − g ( x ) | > λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B ( x , R ) : sup ( t,θ ) ∈ (0 , × Θ | e it ∆ ( f − g )( γ ( x, t, θ )) | > λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B ( x , R ) : | f ( x ) − g ( x ) | > λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) on-tangential Convergence for Schr¨odinger Operators 11 (cid:46) (cid:15) + 2 λ (cid:107) f − g (cid:107) H s ( R ) ≤ (cid:15) + (cid:15) C ≤ (cid:15) + (cid:15) , (2.21)since we can always assume that C ≥ 1, which implies convergence for f ∈ H s ( R ) and almostevery x ∈ B ( x , R ). By the arbitrariness of B ( x , R ), in fact we can get convergence for almostevery x ∈ R . This completes the proof of Theorem 1.2. n ≥ Proof of Theorem 1.4. We briefly explain the proof of Theorem 1.4 since most of the detailsare similar to the proof of Theorem 1.1. As in the proof of Theorem 1.1, we only need to provethat for f , supp ˆ f ⊂ { ξ ∈ R n : | ξ | ∼ λ } , λ (cid:29) (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L p ( B ( x ,R )) ≤ Cλ s + (cid:15) (cid:107) f (cid:107) L , ∀ (cid:15) > , (3.1)where s = β (Θ) p + n n +1) .We decompose Θ into subsets Θ = ∪ k Θ k with bounded overlap, where each Θ k is containedin a closed ball with diameter λ − . Then we have1 ≤ k ≤ λ β (Θ)+ (cid:15) . As in the proof of Theorem 1.1, we can show that for each k , (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ k | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L p ( B ( x ,R )) ≤ Cλ n n +1) + ( p − (cid:15)p (cid:107) f (cid:107) L . Then we have (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) L p ( B ( x ,R )) ≤ (cid:18)(cid:88) k (cid:13)(cid:13)(cid:13)(cid:13) sup ( t,θ ) ∈ (0 , × Θ k | e it ∆ f ( γ ( x, t, θ )) | (cid:13)(cid:13)(cid:13)(cid:13) pL p ( B ( x ,R )) (cid:19) /p ≤ C (cid:18)(cid:88) k λ np n +1) +( p − (cid:15) (cid:107) f (cid:107) pL (cid:19) /p ≤ Cλ β (Θ) p + n n +1) + (cid:15) (cid:107) f (cid:107) L , which implies inequality (3.1). Proof of Theorem 1.5. Taking γ ( x, t, θ ) = x + tθ , Θ is the interior of the unit ball in R n .Then we have β (Θ) = n, (3.2)and the approach regionΓ x = { γ ( x, t, θ ) : t ∈ [0 , , θ ∈ Θ } = { y : | y − x | < t, t ∈ [0 , } . (3.3)2 W. LI, H. WangAssuming that (1.10) holds true, then it follows from Theorem 1.4 and inequality (3.3) that forany s > β (Θ) p + n n +1) , lim ( y,t ) → ( x, | x − y |