A note on σ-point and nontangential convergence
aa r X i v : . [ m a t h . C A ] J a n A NOTE ON σ -POINT AND NONTANGENTIAL CONVERGENCE JAYANTA SARKAR
Abstract.
In this article, we generalize a theorem of Victor L. Shapiro concerningnontangential convergence of the Poisson integral of a L p -function. We introduce thenotion of σ -points of a locally finite measure and consider a wide class of convolutionkernels. We show that convolution integrals of a measure have nontangential limits at σ -points of the measure. We also investigate the relationship between σ -point and thenotion of the strong derivative introduced by Ramey and Ullrich. In one dimension,these two notions are the same. Introduction
In this article, by a measure µ we will always mean a complex Borel measure or asigned Borel measure such that the total variation | µ | is locally finite, that is, | µ | ( K )is finite for all compact sets K . If µ ( E ) is nonnegative for all Borel measurable sets E then µ will be called a positive measure. The notion of Lebesgue point of a measurewas defined by Saeki [3] which we recall. A point x ∈ R n is called a Lebesgue point ofa measure µ on R n if there exists L ∈ C such that(1.1) lim r → | µ − Lm | ( B ( x , r )) m ( B (0 , r )) = 0 , where B ( x , r ) denotes the open ball of radius r with center at x with respect to theEuclidean metric and m denotes the Lebesgue measure of R n . In this case, the symmetricderivative of µ at x ,(1.2) D sym µ ( x ) := lim r → µ ( B ( x , r )) m ( B (0 , r ))exists and is equal to L . The set of all of Lebesgue points of a measure µ is calledthe Lebesgue set of µ . It is not very hard to see that the Lebesgue set of a measure µ includes almost all (with respect to the Lebesgue measure) points of R n (see Proposition2.3). Given a measure µ on R n , its Poisson integral P µ on the upper half space R n +1+ = { ( x, t ) : x ∈ R n , t > } is defined by the convolution P µ ( x, t ) = Z R n P ( x − ξ, t ) dµ ( ξ ) , Mathematics Subject Classification.
Primary 31B25, 44A35; Secondary 31A20, 28A15.
Key words and phrases. σ -Point, Nontangential convergence, Convolution integral, Strong derivative. whenever the integral exists. Here, the kernel P ( x, t ) is the usual Poisson kernel of R n +1+ given by the formula P ( x, t ) = c n t ( t + k x k ) n +12 , c n = π − ( n +1) / Γ (cid:18) n + 12 (cid:19) . It is known that if the integral above exists for some ( x , t ) ∈ R n +1+ then it exists forall points in R n +1+ and defines a harmonic function in R n +1+ . In this article, we willbe concerned with the nontangential convergence of P µ or of more general convolutionintegrals. For x ∈ R n and α >
0, we define the conical region S ( x , α ) with vertex at x and aperture α by S ( x , α ) = { ( x, t ) ∈ R n +1+ : k x − x k < αt } . Definition 1.1.
A function u defined on R n +1+ or on a strip R n × (0 , t ) for some t > L ∈ C at x ∈ R n if, for every α > ( x,t ) → ( x , x,t ) ∈ S ( x ,α ) u ( x, t ) = L. It is a classical result that if f ∈ L p ( R n ), 1 ≤ p ≤ ∞ then the Poisson integral P f of f has nontangential limit f ( x ) at each Lebesgue point x of f (see [8, Theorem3.16]). In [3], Saeki generalized this result for more general class of kernels as well asfor measures instead of L p -functions (see Theorem 2.5). A natural question arises thatwhat happens to the nontangential convergence at non-Lebesgue points. To answer thisquestion, Shapiro [6] introduced the notion of σ -point of a locally integrable function. Definition 1.2.
A point x ∈ R n is called a σ -point of a locally integrable function f on R n provided the following holds: for each ǫ >
0, there exists δ > (cid:12)(cid:12)(cid:12)(cid:12)Z B ( x,r ) ( f ( ξ ) − f ( x )) dm ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ ( k x − x k + r ) n , whenever k x − x k < δ and r < δ .The set of all σ -points of f is called the σ -set of f . As observed by Shapiro, theLebesgue set of a locally integrable function is contained in the σ -set of the function [6,P.3182]. This containment is strict for some functions. In fact, Shapiro constructed afunction f ∈ L p ( R ), 1 ≤ p ≤ ∞ such that 0 is a σ -point of f but not a Lebesgue pointof f (see [6, Section 3]). Our main aim in this article is to generalize the following resultof Shapiro [6, Theorem 1] for measures. Theorem 1.3.
Let f ∈ L p ( R n ) , ≤ p ≤ ∞ . If x ∈ R n is a σ -point of f , then P f hasnontangenial limit f ( x ) at x . In the next section, we will define the notion of σ -point of a measure on R n and provea generalization of Theorem 1.3. We will consider a wide class of kernels which includesthe Poisson kernel and Gauss-Weierstrass kernel and discuss the nontangential behaviorof convolutions of these kernels with measures. -POINT AND NONTANGENTIAL CONVERGENCE 3 It is worth mentioning that Ramey-Ullrich [1] had also discussed the nontangentialbehavior of
P µ by considering the strong derivative of a positive measure µ . We willdiscuss the result of Ramey-Ullrich and the relationship between the strong derivativeand σ -point in the last section.2. nontangential convergence of convolution integrals Let us start by defining the notion of σ -point of a measure. Definition 2.1.
Let µ be a measure on R n . A point x ∈ R n is called a σ -point of µ ifthere exists L ∈ C satisfying the following: for each ǫ >
0, there exists δ > | ( µ − Lm )( B ( x, r )) | < ǫ ( k x − x k + r ) n , whenever k x − x k < δ and r < δ . In this case, we will denote D σ µ ( x ) = L . Remark 2.2.
Every Lebesgue point of the measure µ is a σ -point of µ . Moreover, D σ µ ( x ) = D sym µ ( x ), whenever x is a Lebesgue point of µ . To see this, we take aLebesgue point x ∈ R n of µ . We fix ǫ >
0. By the definition of Lebesgue point, thereexists δ > | µ − Lm | ( B ( x , r )) < ǫm ( B (0 , m ( B ( x , r )) = ǫr n , whenever 0 < r < δ , where L = D sym µ ( x ). This implies that | ( µ − Lm )( B ( x, r )) | ≤ | µ − Lm | ( B ( x, r )) ≤ | µ − Lm | ( B ( x , r + k x − x k )) < ǫ ( k x − x k + r ) n , whenever 0 < r + k x − x k < δ . This shows that x is a σ -point of µ and D σ µ ( x ) = D sym µ ( x ) . We have already mentioned in the introduction that the converse is not true.The following proposition shows that almost every points of R n is a Lebesgue pointand hence σ -point of a measure. Proposition 2.3.
The Lebesgue set of a measure µ on R n includes almost all (withrespect to the Lebesgue measure) points of R n .Proof. Let dµ = f dm + dµ s be the Radon-Nikodym decomposition of µ with respect to m , where f ∈ L loc ( R n ) and µ s ⊥ m (see [2, P.121-123]). If L f denotes the Lebesgue setof f , then we know that [8, P.12] m ( R n \ L f ) = 0 . We observe that | µ s | ⊥ m and hence by [2, Theorem 7.13], m ( R n \ A ) = 0 , where A = { x ∈ R n | D sym | µ s | ( x ) = 0 } . Consequently, R n \ ( L f ∩ A ) is of Lebesgue measure zero. Now, for x ∈ L f ∩ A and any r > | µ − f ( x ) m | ( B ( x , r )) m ( B (0 , r )) ≤ m ( B (0 , r )) Z B ( x ,r ) | f ( x ) − f ( x ) | dm ( x ) + | µ s | ( B ( x , r )) m ( B (0 , r )) . J. SARKAR
Since x ∈ L f ∩ A , each summand on the right hand side of the inequality above goesto zero as r →
0. This proves our assertion. (cid:3)
Let φ : R n → [0 , ∞ ) be radial and radially decreasing measurable function, that is, φ ( x ) = φ ( y ) , if k x k = k y k φ ( x ) ≥ φ ( y ) , if k x k < k y k , with Z R n φ ( x ) dm ( x ) = 1 . For t >
0, we consider the usual approximate identity φ t ( x ) = t − n φ (cid:16) xt (cid:17) , x ∈ R n . Given a measure µ , we define the convolution integral φ [ µ ] by(2.1) φ [ µ ]( x, t ) = µ ∗ φ t ( x ) = Z R n φ t ( x − ξ ) dµ ( ξ ) , whenever the integral exists for ( x, t ) ∈ R n +1+ . Remark 2.4.
It was proved in [3, Remark 1.4] that if µ is a measure on R n and φ is anonnegative, radially decreasing function on R n then the finiteness of | µ | ∗ φ t ( x ) impliesthe finiteness of | µ | ∗ φ t ( x ) for all ( x, t ) ∈ R n × (0 , t ). Note also that if | µ | ( R n ) is finitethen µ ∗ φ t ( x ) is well defined for all ( x, t ) ∈ R n +1+ .In addition to the above, in some of our results we will also assume that φ is strictlypositive and satisfies the following comparison condition [3, P.134].(2.2) sup (cid:26) φ t ( x ) φ ( x ) | t ∈ (0 , , k x k > (cid:27) < ∞ . It is easy to see that P ( x,
1) and the Gaussian(2.3) w ( x ) = (4 π ) − n e − k x k , x ∈ R n satisfy the comparison condition (2.2) [5, Example 2.2]. The following generalizationof the nontangential convergence of the Poisson integral P f was proved by Saeki [3,Theorem 1.5].
Theorem 2.5.
Suppose φ : R n → (0 , ∞ ) satisfies the following conditions: (1) φ is radial, radially decreasing measurable function with k φ k L ( R n ) = 1 . (2) φ satisfies the condition (2.2).Suppose µ is a measure on R n such that | µ | ∗ φ t ( x ) is finite for some t > and x ∈ R n . Then the convolution integral φ [ µ ] has nontangential limit D sym µ ( x ) at eachLebesgue point x of µ . -POINT AND NONTANGENTIAL CONVERGENCE 5 Remark 2.6.
It was shown in [3, Remark 1.6] that the theorem above fails in theabsence of condition (2.2). The assumption that x ∈ R n is a Lebesgue point is sostrong that one can prove the nontangential convergence of ψ [ µ ] (defined analogouslyto (2.1)) at x for any measurable function ψ : R n → C (not necessarily radial, radiallydecreasing) such that | ψ ( x ) | ≤ φ ( x ), for all x ∈ R n .Our main interest in this paper is to prove a generalization of Theorem 1.3 for con-volution integral of the form (2.1). Our first lemma shows that condition (2.2) can beused to reduce matters to the case of a measure µ such that | µ | ( R n ) < ∞ . Lemma 2.7.
Suppose φ is as in Theorem 2.5. If µ is a measure such that | µ | ∗ φ t (0) is finite for some t ∈ (0 , ∞ ) , then for all α > , (2.4) lim ( x,t ) → (0 , x,t ) ∈ S (0 ,α ) µ ∗ φ t ( x ) = lim ( x,t ) → (0 , x,t ) ∈ S (0 ,α ) ˜ µ ∗ φ t ( x ) , where ˜ µ is the restriction of µ on the closed ball B (0 , t ) . Moreover, if zero is a σ -pointof µ then zero is also a σ -point of ˜ µ and vice versa. In both the cases, D σ µ (0) = D σ ˜ µ (0) . Proof.
In view of Remark 2.4, without loss of generality we assume that t <
1. Wewrite for 0 < t < t , x ∈ R n ,(2.5) µ ∗ φ t ( x ) = ˜ µ ∗ φ t ( x ) + Z { ξ ∈ R n : k ξ k >t } φ t ( x − ξ ) dµ ( ξ ) . Since φ is a radial function, we will write for the sake of simplicity φ ( r ) = φ ( ξ ), whenever r = k ξ k . For any r ∈ (0 , ∞ ), we have Z r/ ≤k ξ k≤ r φ ( ξ ) dm ( ξ ) ≥ ω n − φ ( r ) Z rr/ s n − ds = A n r n φ ( r ) , where ω n − is the surface area of the unit sphere S n − and A n is a positive constantwhich depends only on the dimension. Since φ is an integrable function, the integral onthe left hand side converges to zero as r goes to zero and infinity. Hence, it follows that(2.6) lim k ξ k→ k ξ k n φ ( ξ ) = lim k ξ k→∞ k ξ k n φ ( ξ ) = 0 . We denote the integral appearing on the right-hand side of (2.5) by I ( x, t ). We fix α > < t < min { , t α } , k x − ξ k ≥ k ξ k − k x k ≥ k ξ k − k ξ k k ξ k , J. SARKAR whenever k ξ k > t and ( x, t ) ∈ S (0 , α ). Therefore, using the fact that φ is radiallydecreasing, we obtain for ( x, t ) ∈ S (0 , α ) ∩ (cid:0) R n × (0 , min { , t α } ) (cid:1) , | I ( x, tt ) | = ( tt ) − n (cid:12)(cid:12)(cid:12)(cid:12)Z { ξ ∈ R n : k ξ k >t } φ (cid:18) x − ξtt (cid:19) dµ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( tt ) − n Z { ξ ∈ R n : k ξ k >t } φ (cid:18) x − ξtt (cid:19) d | µ | ( ξ ) ≤ ( tt ) − n Z { ξ ∈ R n : k ξ k >t } φ (cid:18) ξ tt (cid:19) d | µ | ( ξ )= Z { ξ ∈ R n : k ξ k >t } (cid:16) k ξ k tt (cid:17) n φ (cid:16) ξ tt (cid:17) k ξ k n φ t ( ξ ) φ t ( ξ ) d | µ | ( ξ )(2.7)From (2.6) we get that lim t → (cid:18) k ξ k tt (cid:19) n φ (cid:18) ξ tt (cid:19) = 0 , for each fixed ξ ∈ R n . On the other hand, by the comparison condition (2.2), thereexists some positive constant C such that (cid:16) k ξ k tt (cid:17) n φ (cid:16) ξ tt (cid:17) k ξ k n φ t ( ξ ) = 2 n φ t (cid:16) ξt (cid:17) φ (cid:16) ξt (cid:17) ≤ C, for k ξ k > t and 0 < t < /
2. Since | µ | ∗ φ t (0) < ∞ , that is, φ t ∈ L ( R n , d | µ | ), by thedominated convergence theorem, it follows from (2.7) thatlim ( x,t ) → (0 , x,t ) ∈ S (0 ,α ) | I ( x, tt ) | = 0 . Consequently,lim ( x,t ) → (0 , x,t ) ∈ S (0 ,α ) Z { ξ ∈ R n : k ξ k >t } φ t ( x − ξ ) dµ ( ξ ) = lim ( x,t ) → (0 , x,t ) ∈ S (0 ,α ) I ( x, t ) = lim ( x,t ) → (0 , x,t ) ∈ S (0 ,α ) I ( x, tt − t ) = 0 , as 0 < t <
1. This proves (2.4). Suppose that D σ µ (0) = L . We take ǫ >
0. Then thereexists 0 < δ < t / | ( µ − Lm )( B ( x, r )) | < ǫ ( k x k + r ) n , whenever k x k < δ and r < δ . But for k x k < δ and r < δ , we observe that B ( x, r ) ⊂ B (0 , δ ) ⊂ B (0 , t ) . Using this observation and the definition of ˜ µ in the last inequality, we get that | (˜ µ − Lm )( B ( x, r )) | < ǫ ( k x k + r ) n , whenever k x k < δ and r < δ . This shows that D σ ˜ µ (0) = L = D σ µ (0) . -POINT AND NONTANGENTIAL CONVERGENCE 7 Proof of the converse implication is similar. (cid:3)
Before proceed to our next lemma, we recall that a real valued function f on a topo-logical space X is said to be lower semicontinuous if { x ∈ X : f ( x ) > s } is open forevery real number s [2, P.37]. Lemma 2.8.
Assume that φ : R n → [0 , ∞ ) is a radial, radially decreasing, integrablefunction. If φ is lower semicontinuous then, for every t ∈ (0 , φ (0)) B t = { x ∈ R n : φ ( x ) > t } , is an open ball centred at zero with some finite radius θ ( t ) (say).Proof. Since φ is integrable, there exists x ∈ R n such that φ ( x ) ≤ t . For any x ∈ B t , φ ( x ) > t ≥ φ ( x ) . As φ is radially decreasing, the inequality above implies that k x k < k x k and hence B t is bounded. Therefore, θ ( t ) := sup { r > B (0 , r ) ⊂ B t } < ∞ . We claim that B (0 , θ ( t )) is contained in B t . To see this, for x ∈ B (0 , θ ( t )), we take r ∈ ( k x k , θ ( t )). By the definition of θ ( t ), this implies that x ∈ B (0 , r ) ⊂ B t . On theother hand, if x ∈ B t , φ ( ξ ) ≥ φ ( x ) > t, for all ξ ∈ B (0 , k x k ) . Thus, B (0 , k x k ) ⊂ B t for all x ∈ B t . Consequently,(2.8) B (0 , θ ( t )) ⊂ B t ⊂ B (0 , θ ( t )) . We shall show that B t = B (0 , θ ( t )) or B t = B (0 , θ ( t )) . Suppose there exists ξ ∈ B t \ B (0 , θ ( t )). Then by (2.8), k ξ k = θ ( t ) and hence by radialityof φ , B t = B (0 , θ ( t )). Since φ is lower semicontinuous, B t = B (0 , θ ( t )). (cid:3) Remark 2.9.
It follows from the proof that if φ is not a lower semicontinuous functionthen B t may turn out to be a closed ball centred at origin. This can be seen from thefollowing example. Define φ : R n → (0 , ∞ ) by φ ( x ) = ( e −k x k , k x k ≤ e − k x k , k x k > . Then for any t ∈ ( e − , e − ), B t = B (0 , Theorem 2.10.
Suppose φ and µ be as in Theorem 2.5. Further assume that φ is lowersemicontinuous. If x ∈ R n is a σ -point of µ with D σ µ ( x ) = L ∈ C , then φ [ µ ] hasnontangential limit L at x . J. SARKAR
Proof.
Without loss of generality, we can assume x = 0. Indeed, we consider thetranslated measure µ = τ − x µ , where τ − x µ ( E ) = µ ( E + x ) , for all Borel subsets E ⊂ R n . Using translation invariance of the Lebesgue measure itfollows that ( µ − Lm )( B ( x, r )) = ( µ − Lm )( B ( x + x , r )) . We fix ǫ >
0. Since x is a σ -point of µ with D σ µ ( x ) = L , the equality above impliesthat there exists δ > | ( µ − Lm )( B ( x, r )) | < ǫ ( k x k + r ) n , whenever k x k < δ, r < δ. This shows that 0 is a σ -point of µ with D σ µ (0) = L . As translation commutes withconvolution, it also follows that(2.9) µ ∗ φ t ( x ) = ( τ − x µ ∗ φ t )( x ) = τ − x ( µ ∗ φ t )( x ) = µ ∗ φ t ( x + x ) , for any ( x, t ) ∈ R n × (0 , t ). We fix an arbitrary positive number α . As ( x, t ) ∈ S (0 , α )if and only if ( x + x, t ) ∈ S ( x , α ), one infers from (2.9) thatlim ( x,t ) → (0 , x,t ) ∈ S (0 ,α ) φ [ µ ]( x, t ) = lim ( ξ,t ) → ( x , ξ,t ) ∈ S ( x ,α ) φ [ µ ]( ξ, t ) . Hence, it suffices to prove the theorem under the assumption that x = 0. ApplyingLemma 2.7, we can restrict µ on B (0 , t ), if necessary, to assume that | µ | ( R n ) < ∞ .Since D σ µ (0) = L , lim r → µ ( B (0 , r )) m ( B (0 , r )) = L. Therefore, there exists a positive constant r such that | µ ( B (0 , r )) | m ( B (0 , r )) < L + 1 , for all r < r . Using finiteness of the total variation of µ , we get that | µ ( B (0 , r )) | m ( B (0 , r )) ≤ | µ | ( B (0 , r )) m ( B (0 , r )) ≤ | µ | ( R n ) m ( B (0 , r )) , for all r ≥ r . Combining above two inequalities, we obtain(2.10)
M µ (0) := sup r> | µ ( B (0 , r )) | m ( B (0 , r )) < ∞ . For each 0 < t < φ (0), we define B t = { x ∈ R n : φ ( x ) > t } . By Lemma 2.8, B t is an open ball with centre at 0 and radius θ ( t ). Clearly, θ is amonotonically decreasing function in (0 , φ (0)) and hence measurable. We also note thatfor any r ∈ (0 , ∞ ) and x ∈ R n , n ξ ∈ R n : φ (cid:18) x − ξr (cid:19) > t o -POINT AND NONTANGENTIAL CONVERGENCE 9 is an open ball with centre at x and radius rθ ( t ). Let { ( x k , t k ) } ∞ k =1 be a sequence in S (0 , α ) converging to (0 , t k ∈ (0 , t ) forall k . As R R n φ ( x ) dm ( x ) = 1, we can write µ ∗ φ t k ( x k ) − L = t − nk Z R n φ (cid:18) x k − ξt k (cid:19) dµ ( ξ ) − Lt − nk Z R n φ (cid:18) x k − ξt k (cid:19) dm ( ξ )= t − nk Z R n φ (cid:18) x k − ξt k (cid:19) d ( µ − Lm )( ξ )= t − nk Z R n Z φ (cid:16) xk − ξtk (cid:17) ds d ( µ − Lm )( ξ ) . As | µ − Lm | ∗ φ t ( x ) is finite for all ( x, t ) ∈ R n × (0 , t ), applying Fubini’s theorem onthe right hand side of the last equality, we obtain µ ∗ φ t k ( x k ) − L = t − nk Z φ (0)0 ( µ − Lm ) (cid:18)n ξ ∈ R n : φ (cid:18) x k − ξt k (cid:19) > s o(cid:19) ds = Z φ (0)0 ( µ − Lm ) ( B ( x k , t k θ ( s )))( k x k k + t k θ ( s )) n × (cid:18) k x k k + t k θ ( s ) t k (cid:19) n ds. (2.11)Since D σ µ (0) = L , lim ( x,r ) → (0 , ( µ − Lm )( B ( x, r ))( k x k + r ) n = 0 . Therefore, for each s ∈ (0 , φ (0)), integrand on the right hand side of (2.11) has limitzero as k → ∞ because k x k k /t k < α , for all k . Moreover, using (2.10), the integrand isbounded by the function s m ( B (0 , M µ (0) + L )( θ ( s ) + α ) n , s ∈ (0 , φ (0)) . In order to apply the dominated convergence theorem on the right hand side of (2.11),we need to show that this function is integrable in (0 , φ (0)). For this, it is enough toshow that the function s θ ( s ) n is integrable in (0 , φ (0)). Using a well-known formulainvolving distribution functions [2, Theorem 8.16], we observe that Z R n φ ( x ) dm ( x ) = Z φ (0)0 m ( { x ∈ R n : φ ( x ) > s } ) ds = Z φ (0)0 m ( B s ) ds = m ( B (0 , Z φ (0)0 θ ( s ) n ds. Hence, applying the dominated convergence theorem on the right hand side of (2.11) weobtain lim k →∞ µ ∗ φ t k ( x k ) = L. This completes the proof. (cid:3)
Shapiro also considered nontangential limits of Gauss-Weierstrass integral of a L p -function [6, Theorem 2]. We recall that the Gauss-Weierstrass kernel or the heat kernelof R n +1+ is given by W ( x, t ) = (4 πt ) − n e − k x k t , ( x, t ) ∈ R n +1+ . The Gauss-Weierstrass integral of a measure µ is given by the convolution W µ ( x, t ) = Z R n W ( x − y, t ) dµ ( y ) , x ∈ R n , t ∈ (0 , ∞ ) , whenever the above integral exists. Recalling (2.3), we observe that(2.12) W µ ( x, t ) = µ ∗ w √ t ( x ) , ( x, t ) ∈ R n +1+ . As an easy corollary of Theorem 2.10, we get the following generalization of the abovementioned theorem of Shapiro.
Corollary 2.11.
Suppose µ is a measure on R n such that W | µ | ( x , t ) is finite for some x ∈ R n and t > . If x ∈ R n is a σ -point of µ with D σ µ ( x ) = L ∈ C , then theGauss-Weierstrass integral W µ has nontangential limit L at x .Proof. We fix an arbitrary positive number α . We have already mentioned that w satifiesthe comparison condition (2.2). Moreover, k w k L ( R n ) = 1 (see [8, P.9]). Thus, w satisfiesall the hypothesis of Theorem 2.10. Hence, in view of (2.12), Theorem 2.10 giveslim ( x,t ) → ( x , k x − x k < √ αt W µ ( x, t ) = L. Note that S ( x , α ) ∩ { ( x, t ) ∈ R n +1+ | t < α } ⊂ { ( x, t ) ∈ R n +1+ | k x − x k < √ αt, t < α } . Using this set containment relation together with the equation above, we conclude that
W µ has nontangential limit L at x . (cid:3) We can drop the comparison condition (2.2) in Theorem 2.10 by imposing some growthcondition on µ . More precisely, we have the following. Theorem 2.12.
Let φ : R n → [0 , ∞ ) be radial, radially decreasing, lower semicontinuousfunction with k φ k L ( R n ) = 1 . Suppose µ is a measure on R n such that (2.13) | µ | ( B (0 , r )) = O ( r n ) , as r → ∞ , and that µ ∗ φ t ( x ) is finite for some x ∈ R n and t ∈ (0 , ∞ ) . If x ∈ R n is a σ -pointof µ with D σ µ ( x ) = L ∈ C , then φ [ µ ] has nontangential limit L at x .Proof. Without loss of generality, we assume that x = 0. We will use the same notationas in the proof of Theorem 2.10. From the proof of Theorem 2.10, we observe that itsuffices to prove that M µ (0) < ∞ and then the the rest of the arguments remains same.As D σ µ (0) = L , it follows that D sym µ (0) = L and hence there exists a positive constant r such that | µ ( B (0 , r )) | m ( B (0 , r )) < L + 1 , for all r ≤ r . -POINT AND NONTANGENTIAL CONVERGENCE 11 Using (2.13), we get two positive constants M and R such that | µ ( B (0 , r )) | m ( B (0 , r )) < M , for all r ≥ R . Finally, for all r ∈ ( r , R ) | µ ( B (0 , r )) | m ( B (0 , r )) ≤ | µ | ( B (0 , R )) m ( B (0 , r ))From the last three inequalities and the fact that | µ | is locally finite, we conclude that M µ (0) = sup r> | µ ( B ( x , r )) | m ( B (0 , r )) < ∞ . (cid:3) Remark 2.13.
We can drop the assumption that φ is lower semicontinuous from The-orem 2.10 and Theorem 2.12 in the following two special cases.i) x is a Lebesgue point of µ .ii) µ is absolutely continuous with respect to the Lebesgue measure m .3. σ -point and strong derivative In this section, we will discuss the relationship between σ -point of a measure and thenotion of strong derivative of a measure introduced by Ramey-Ullrich [1]. We recall thedefinition of strong derivative of a measure. Definition 3.1.
Given a measure µ on R n , we say that µ has strong derivative L ∈ C at x ∈ R n if lim r → µ ( x + rB ) m ( rB ) = L holds for every open ball B ⊂ R n . Here, rB = { rx | x ∈ B } , r >
0. The strongderivative of µ at x , if it exists, is denoted by Dµ ( x ). Note that rB ( ξ, s ) = B ( rξ, rs ). Proposition 3.2.
Let µ be a measure on R n . If x ∈ R n is a σ -point of µ with D σ µ ( x ) = L ∈ C , then the strong derivative of µ at x exists and is equal to L .Proof. We take a ball B = B ( x, s ) in R n and fix ǫ >
0. As x is a σ -point of µ , thereexists δ > | ( µ − Lm )( x + rB ) | = | ( µ − Lm ) ( B ( x + rx, rs )) | < ǫ ( k rx k + rs ) n , whenever k rx k < δ and rs < δ . This implies that (cid:12)(cid:12)(cid:12)(cid:12) µ ( x + rB ) m ( rB ) − L (cid:12)(cid:12)(cid:12)(cid:12) < ǫ ( k x k + s ) n m ( B (0 , s )) , whenever k rx k < δ and rs < δ. Taking r = min { δ k x k +1 , δs } , it follows that the last inequality holds for all r < r . Thiscompletes the proof. (cid:3) Remark 3.3.
In [1, Theorem 2.2], among other things, Ramey-Ullrich proved that if µ is a positive measure on R n with well-defined Poisson integral P µ then the strongderivative of µ at x ∈ R n is L ∈ [0 , ∞ ) if and only if P µ have nontangential limit L at x . In view of Proposition 3.2, we can deduce Theorem 2.10 for φ = P ( .,
1) and µ positive from the result of Ramey-Ullrich.The converse of Proposition 3.2 is true in one dimension. If µ is a locally finite signedmeasure on R then there is a function f : R → R of bounded variation such that thepositive and negative parts of f are right continuous and µ (( a, b ]) = f ( b ) − f ( a ) , a, b ∈ R , a < b. For more discussion on this see [7, P.281-284].
Proposition 3.4.
Suppose that µ and f as above and x ∈ R . i) The function f is differentiable at x if and only if x is a σ -point of µ . In thiscase, f ′ ( x ) = D σ µ ( x ) . ii) The function f is differentiable at x if and only if the strong derivative µ at x exists. In this case, f ′ ( x ) = Dµ ( x ) .Proof. We first prove i ). Suppose f is differentiable at x and f ′ ( x ) = L ∈ R . Fix ǫ > δ > (cid:12)(cid:12)(cid:12)(cid:12) f ( x + h ) − f ( x ) h − L (cid:12)(cid:12)(cid:12)(cid:12) < ǫ, whenever | h | < δ. For x ∈ R , r > | ( x − x ) + r | < δ and | ( x − x ) − r | < δ , we have | ( µ − Lm ) (( x − r, x + r )) | = | f ( x + r ) − f ( x − r ) − rL | = (cid:12)(cid:12)(cid:12)(cid:12) f ( x + x − x + r ) − f ( x ) x − x + r × ( x − x + r ) − ( x − x + r ) L + ( x − x − r ) L − f ( x + x − x − r ) − f ( x ) x − x − r × ( x − x − r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | x − x + r | (cid:12)(cid:12)(cid:12)(cid:12) f ( x + x − x + r ) − f ( x ) x − x + r − L (cid:12)(cid:12)(cid:12)(cid:12) + | x − x − r | (cid:12)(cid:12)(cid:12)(cid:12) f ( x + x − x − r ) − f ( x ) x − x − r − L (cid:12)(cid:12)(cid:12)(cid:12) < | x − x + r | ǫ + | x − x − r | ǫ (by (3 . . This implies that | ( µ − Lm )( B ( x, r )) | < ǫ ( | x − x | + r ) , whenever | x − x | < δ/ , r < δ/ . Thus, x is a σ -point of µ with D σ µ ( x ) = L .Conversely, we assume that x is a σ -point of µ with D σ µ ( x ) = L ∈ R and fix ǫ > δ > | ( µ − Lm )(( x − r, x + r )) | < ǫ ( | x − x | + r ) , whenever | x − x | < δ, r < δ. -POINT AND NONTANGENTIAL CONVERGENCE 13 Taking x = x + r with r > | µ (( x , x + 2 r )) − rL | = | f ( x + 2 r ) − f ( x ) − rL | = 2 r (cid:12)(cid:12)(cid:12)(cid:12) f ( x + 2 r ) − f ( x )2 r − L (cid:12)(cid:12)(cid:12)(cid:12) < ǫ, whenever r < δ . This shows that f ′ ( x +) = L . Similarly, by taking x = x − r with r > f ′ ( x − ) = L .The statement ii ) can be proved by arguing in a similar fashion. We refer the readerto [4, Remark 2.6 (2)] where it was proved under the assumption that f is monotonicallyincreasing. (cid:3) Considering real and imaginary parts of a measure, if necessary, we obtain the follow-ing corollary.
Corollary 3.5.
Suppose µ is a measure on R and x ∈ R . Then x is a σ -point of µ ifand only if µ has strong derivative at x . Moreover, D σ µ ( x ) = Dµ ( x ) . Remark 3.6. i) It is not known to us whether for a measure µ on R n , the σ -setof µ coincides with the set of points at which the strong derivative of µ exists,if n >
1. It would be surprising if it is true in higher dimensions. A heuristicreasoning behind this is the following observation. Suppose µ is a measure on R n . If 0 is a σ -point of µ then( µ − D σ (0) m )( B ( x, r )) → , as ( x, r ) → (0 , . On the other hand, existence of strong derivative at 0 only ensures( µ − Dµ (0) m )( B ( x, r )) → , as ( x, r ) → (0 , , along the rays of the form { ( rx , rt ) | r > } , where ( x , t ) ∈ R n +1+ .ii) Suppose µ and φ as in Theorem 2.10 and n = 1. If Dµ ( x ) = L then it followsfrom Proposition 3.4 and Theorem 2.10 that φ [ µ ] converges nontangentially to L . It is not known whether the same is true for dimension n >
1. However, thefollowing theorem shows that a weaker version of convergence for φ [ µ ] holds atthe points where the strong derivative Dµ exists. Theorem 3.7.
Let φ and µ be as in Theorem 2.10. Suppose µ has strong derivative L ∈ C at x ∈ R n . Then φ [ µ ]( x, t ) has limit L as ( x, t ) → ( x , along each ray through ( x , in R n +1+ . In other words, lim r → φ [ µ ]( x + rξ, rη ) = L, for each fixed ( ξ, η ) ∈ R n +1+ . Proof.
Without loss of generality, we can assume x = 0. Let ˜ µ be the restriction of µ on the ball B (0 , t ). If B ( y, τ ) is any given ball, then for all 0 < r < t ( τ + k y k ) − ,it follows that rB ( y, τ ) is contained in B (0 , t ). This in turn implies that Dµ (0) and D ˜ µ (0) are equal. Thus, in view of Lemma 2.7, without loss of generality, we can assumethat | µ | ( R n ) is finite. We will use the same notation as in the proof of Theorem 2.10.Since Dµ (0) is equal to L , it follows that D sym µ (0) is also equal to L and hence M µ (0)is finite (see the argument preceding (2.10)). We take ( ξ, η ) ∈ R n +1+ and a sequence { r k } of positive numbers converging to zero. Substituting x k = r k ξ , t k = r k η in equation(2.11), we obtain(3.3) φ [ µ ]( r k ξ, r k η ) − L = Z φ (0)0 ( µ − Lm ) ( B ( r k ξ, r k ηθ ( s )))( r k ηθ ( s )) n θ ( s ) n ds. As Dµ (0) = L , using the definition of strong derivative, we observe that for each fixed s ∈ (0 , φ (0))(3.4) lim k →∞ ( µ − Lm )( B ( r k ξ, r k ηθ ( s )))( r k ηθ ( s )) n = lim k →∞ (cid:18) µ ( r k B ( ξ, ηθ ( s ))) m ( r k B ( ξ, ηθ ( s ))) − L (cid:19) c ′ n = 0 , where c ′ n = m ( B (0 , s m ( B (0 , M µ (0) + L ) θ ( s ) n , s ∈ (0 , φ (0)) . We have seen in the proof of Theorem 2.10 that this function is integrable in (0 , φ (0)).In view of (3.4), we can now apply dominated convergence theorem on the right-handside of (3.3) to complete the proof. (cid:3)
We show by an example that the existence of limit of φ [ µ ] along every ray through( x ,
0) may not imply the existence of the strong derivative at x . Example 3.8.
Consider the measure dµ = χ [0 , dm on R . Then D sym µ (0) is 1 / Dµ does not exist at the origin (see [4, Remark 2.5]). Taking φ = P ( ., φ [ µ ]( x, t ) = 1 π Z tt + ( x − ξ ) dm ( ξ ) = 1 π (cid:18) arctan 1 − xt + arctan xt (cid:19) , ( x, t ) ∈ R n +1+ . Therefore, for each fixed ( ξ , t ) ∈ R n +1+ we havelim r → φ [ µ ]( rξ , rt ) = lim r → π (cid:18) arctan 1 − rξ rt + arctan rξ rt (cid:19) = 1 π (cid:18) π ξ t (cid:19) . This shows that φ [ µ ] has limit along every ray through the origin but the limit dependson the ray. acknowledgements The author would like to thank Swagato K. Ray for many useful discussions duringthe course of this work. The author is supported by a research fellowship from IndianStatistical Institute.
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