Operators related to the Laplacian with drift in Euclidean space - PowerPoint PPT Presentation

In D it turns out that it matters how many derivatives are taken in the v direction: Let ∂ v denote differentiation along the vector v . 5 / 23

In D it turns out that it matters how many derivatives are taken in the v direction: Let ∂ v denote differentiation along the vector v . Then write D as a sum k � ∂ i v D ′ D = k − i , i = 0 5 / 23

In D it turns out that it matters how many derivatives are taken in the v direction: Let ∂ v denote differentiation along the vector v . Then write D as a sum k � ∂ i v D ′ D = k − i , i = 0 where D ′ k − i is of order k − i and involves only differentiation in directions orthogonal to v . 5 / 23

In D it turns out that it matters how many derivatives are taken in the v direction: Let ∂ v denote differentiation along the vector v . Then write D as a sum k � ∂ i v D ′ D = k − i , i = 0 where D ′ k − i is of order k − i and involves only differentiation in directions orthogonal to v . The maximal order of differentiation along v is then q = max { i : D ′ k − i � = 0 } and 0 ≤ q ≤ k . 5 / 23

In D it turns out that it matters how many derivatives are taken in the v direction: Let ∂ v denote differentiation along the vector v . Then write D as a sum k � ∂ i v D ′ D = k − i , i = 0 where D ′ k − i is of order k − i and involves only differentiation in directions orthogonal to v . The maximal order of differentiation along v is then q = max { i : D ′ k − i � = 0 } and 0 ≤ q ≤ k . This quantity turns out to be significant. 5 / 23

Our result says that order 2 is maximal for weak type ( 1 , 1 ) , 6 / 23

Our result says that order 2 is maximal for weak type ( 1 , 1 ) , but counting only differentiations along v : 6 / 23

Our result says that order 2 is maximal for weak type ( 1 , 1 ) , but counting only differentiations along v : Theorem The Riesz transform R D = D ( − ∆ v ) − k / 2 is of weak type ( 1 , 1 ) if and only if q ≤ 2 . 6 / 23

Our result says that order 2 is maximal for weak type ( 1 , 1 ) , but counting only differentiations along v : Theorem The Riesz transform R D = D ( − ∆ v ) − k / 2 is of weak type ( 1 , 1 ) if and only if q ≤ 2 . When q ≥ 3 , R D is bounded from the Orlicz space L ( 1 + ln + L ) q 2 − 1 ( µ ) into L 1 , ∞ ( µ ) , 6 / 23

Our result says that order 2 is maximal for weak type ( 1 , 1 ) , but counting only differentiations along v : Theorem The Riesz transform R D = D ( − ∆ v ) − k / 2 is of weak type ( 1 , 1 ) if and only if q ≤ 2 . When q ≥ 3 , R D is bounded from the Orlicz space L ( 1 + ln + L ) q 2 − 1 ( µ ) into L 1 , ∞ ( µ ) , in the sense that � | f | � q 2 − 1 � 1 + ln + | f | µ { x ; | R D f ( x ) | > λ } ≤ C d µ, λ > 0 , λ λ with C = C ( v , D ) . 6 / 23

Our result says that order 2 is maximal for weak type ( 1 , 1 ) , but counting only differentiations along v : Theorem The Riesz transform R D = D ( − ∆ v ) − k / 2 is of weak type ( 1 , 1 ) if and only if q ≤ 2 . When q ≥ 3 , R D is bounded from the Orlicz space L ( 1 + ln + L ) q 2 − 1 ( µ ) into L 1 , ∞ ( µ ) , in the sense that � | f | � q 2 − 1 � 1 + ln + | f | µ { x ; | R D f ( x ) | > λ } ≤ C d µ, λ > 0 , λ λ with C = C ( v , D ) . This inequality is sharp in the sense that q cannot be replaced by any smaller number. 6 / 23

We pass to the Littlewood-Paley-Stein functions. 7 / 23

We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup ( e t ∆ v ) t > 0 7 / 23

We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup ( e t ∆ v ) t > 0 and the Poisson semigroup ( e − t √− ∆ v ) t > 0 . 7 / 23

We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup ( e t ∆ v ) t > 0 and the Poisson semigroup ( e − t √− ∆ v ) t > 0 . For the heat semigroup, the vertical Littlewood-Paley-Stein function of order k is usually defined as �� + ∞ � 1 2 dt 2 � � k 2 ∇ k e t ∆ v f ( x ) H k ( f )( x ) = � t , � � � t 0 where ∇ k means all derivatives of order k in the x variable. 7 / 23

We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup ( e t ∆ v ) t > 0 and the Poisson semigroup ( e − t √− ∆ v ) t > 0 . For the heat semigroup, the vertical Littlewood-Paley-Stein function of order k is usually defined as �� + ∞ � 1 2 dt 2 � � k 2 ∇ k e t ∆ v f ( x ) H k ( f )( x ) = � t , � � � t 0 where ∇ k means all derivatives of order k in the x variable. They are known to be bounded on L p for 1 < p < ∞ . 7 / 23

We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup ( e t ∆ v ) t > 0 and the Poisson semigroup ( e − t √− ∆ v ) t > 0 . For the heat semigroup, the vertical Littlewood-Paley-Stein function of order k is usually defined as �� + ∞ � 1 2 dt 2 � � k 2 ∇ k e t ∆ v f ( x ) H k ( f )( x ) = � t , � � � t 0 where ∇ k means all derivatives of order k in the x variable. They are known to be bounded on L p for 1 < p < ∞ . But to write a precise weak type ( 1 , 1 ) result, we use the k :th order operator D 7 / 23

We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup ( e t ∆ v ) t > 0 and the Poisson semigroup ( e − t √− ∆ v ) t > 0 . For the heat semigroup, the vertical Littlewood-Paley-Stein function of order k is usually defined as �� + ∞ � 1 2 dt 2 � � k 2 ∇ k e t ∆ v f ( x ) H k ( f )( x ) = � t , � � � t 0 where ∇ k means all derivatives of order k in the x variable. They are known to be bounded on L p for 1 < p < ∞ . But to write a precise weak type ( 1 , 1 ) result, we use the k :th order operator D and define �� + ∞ � 1 2 dt 2 � � k 2 D e t ∆ v f ( x ) H D ( f )( x ) = � t . � � t � 0 7 / 23

Again, the number q matters, to indicate the maximal order of differentiation along v : 8 / 23

Again, the number q matters, to indicate the maximal order of differentiation along v : Theorem The operator H D is of weak type ( 1 , 1 ) if and only if q ≤ 1 . 8 / 23

Again, the number q matters, to indicate the maximal order of differentiation along v : Theorem The operator H D is of weak type ( 1 , 1 ) if and only if q ≤ 1 . H D is bounded from L ( 1 + ln + L ) q 2 − 3 4 ( µ ) into When q > 1 , L 1 , ∞ ( µ ) , 8 / 23

Again, the number q matters, to indicate the maximal order of differentiation along v : Theorem The operator H D is of weak type ( 1 , 1 ) if and only if q ≤ 1 . H D is bounded from L ( 1 + ln + L ) q 2 − 3 4 ( µ ) into When q > 1 , L 1 , ∞ ( µ ) , i.e., � | f | � q 2 − 3 � 1 + ln + | f | 4 µ { x ; H D ( f )( x ) > λ } ≤ C d µ, λ > 0 , λ λ with C = C ( v , D ) . 8 / 23

Again, the number q matters, to indicate the maximal order of differentiation along v : Theorem The operator H D is of weak type ( 1 , 1 ) if and only if q ≤ 1 . H D is bounded from L ( 1 + ln + L ) q 2 − 3 4 ( µ ) into When q > 1 , L 1 , ∞ ( µ ) , i.e., � | f | � q 2 − 3 � 1 + ln + | f | 4 µ { x ; H D ( f )( x ) > λ } ≤ C d µ, λ > 0 , λ λ with C = C ( v , D ) . Here q cannot be replaced by any smaller number. 8 / 23

Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, 9 / 23

Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, by �� + ∞ � 1 2 D e − t √− ∆ v f ( x ) 2 dt 2 � � k G D ( f )( x ) = � t . � � � t 0 9 / 23

Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, by �� + ∞ � 1 2 D e − t √− ∆ v f ( x ) 2 dt 2 � � k G D ( f )( x ) = � t . � � � t 0 It behaves a bit better than H D , 9 / 23

Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, by �� + ∞ � 1 2 D e − t √− ∆ v f ( x ) 2 dt 2 � � k G D ( f )( x ) = � t . � � � t 0 It behaves a bit better than H D , indeed Theorem The operator G D is of weak type ( 1 , 1 ) if and only if q ≤ 2 . 9 / 23

Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, by �� + ∞ � 1 2 D e − t √− ∆ v f ( x ) 2 dt 2 � � k G D ( f )( x ) = � t . � � � t 0 It behaves a bit better than H D , indeed Theorem The operator G D is of weak type ( 1 , 1 ) if and only if q ≤ 2 . G D is bounded from L ( 1 + ln + L ) q 2 − 1 ( µ ) into When q > 2 , L 1 , ∞ ( µ ) , 9 / 23

Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, by �� + ∞ � 1 2 D e − t √− ∆ v f ( x ) 2 dt 2 � � k G D ( f )( x ) = � t . � � � t 0 It behaves a bit better than H D , indeed Theorem The operator G D is of weak type ( 1 , 1 ) if and only if q ≤ 2 . G D is bounded from L ( 1 + ln + L ) q 2 − 1 ( µ ) into When q > 2 , L 1 , ∞ ( µ ) , Here q cannot be replaced by any smaller number. 9 / 23

The horizontal Littlewood-Paley-Stein functions are, for the heat semigroup, 10 / 23

The horizontal Littlewood-Paley-Stein functions are, for the heat semigroup, � 1 2 dt �� + ∞ � t k ∂ k 2 � � ∂ t k e t ∆ v f ( x ) � � h k ( f )( x ) = , k ≥ 1 � � t � 0 10 / 23

The horizontal Littlewood-Paley-Stein functions are, for the heat semigroup, � 1 2 dt �� + ∞ � t k ∂ k 2 � � ∂ t k e t ∆ v f ( x ) � � h k ( f )( x ) = , k ≥ 1 � � t � 0 and also � t k ∂ k � � ∂ t k e t ∆ v f ( x ) � � H k ( f )( x ) = sup � , k ≥ 0 . � � t > 0 10 / 23

The horizontal Littlewood-Paley-Stein functions are, for the heat semigroup, � 1 2 dt �� + ∞ � t k ∂ k 2 � � ∂ t k e t ∆ v f ( x ) � � h k ( f )( x ) = , k ≥ 1 � � t � 0 and also � t k ∂ k � � ∂ t k e t ∆ v f ( x ) � � H k ( f )( x ) = sup � , k ≥ 0 . � � t > 0 They are known to be bounded on L p , 1 < p < ∞ , by general Littlewood-Paley-Stein theory. 10 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : Theorem The operator h k is of weak type ( 1 , 1 ) if and only if k ≤ 1 . 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : Theorem The operator h k is of weak type ( 1 , 1 ) if and only if k ≤ 1 . h k is bounded from L ( 1 + ln + L ) k 2 − 3 4 ( µ ) into When k > 1 , L 1 , ∞ ( µ ) , 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : Theorem The operator h k is of weak type ( 1 , 1 ) if and only if k ≤ 1 . h k is bounded from L ( 1 + ln + L ) k 2 − 3 4 ( µ ) into When k > 1 , L 1 , ∞ ( µ ) , and this is optimal. 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : Theorem The operator h k is of weak type ( 1 , 1 ) if and only if k ≤ 1 . h k is bounded from L ( 1 + ln + L ) k 2 − 3 4 ( µ ) into When k > 1 , L 1 , ∞ ( µ ) , and this is optimal. The “maximal operator” H k is a bit better: 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : Theorem The operator h k is of weak type ( 1 , 1 ) if and only if k ≤ 1 . h k is bounded from L ( 1 + ln + L ) k 2 − 3 4 ( µ ) into When k > 1 , L 1 , ∞ ( µ ) , and this is optimal. The “maximal operator” H k is a bit better: Theorem The operator H k is of weak type ( 1 , 1 ) if and only if k ≤ 2 . 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : Theorem The operator h k is of weak type ( 1 , 1 ) if and only if k ≤ 1 . h k is bounded from L ( 1 + ln + L ) k 2 − 3 4 ( µ ) into When k > 1 , L 1 , ∞ ( µ ) , and this is optimal. The “maximal operator” H k is a bit better: Theorem The operator H k is of weak type ( 1 , 1 ) if and only if k ≤ 2 . H k is bounded from L ( 1 + ln + L ) k 2 − 1 ( µ ) into When k > 2 , L 1 , ∞ ( µ ) , 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : Theorem The operator h k is of weak type ( 1 , 1 ) if and only if k ≤ 1 . h k is bounded from L ( 1 + ln + L ) k 2 − 3 4 ( µ ) into When k > 1 , L 1 , ∞ ( µ ) , and this is optimal. The “maximal operator” H k is a bit better: Theorem The operator H k is of weak type ( 1 , 1 ) if and only if k ≤ 2 . H k is bounded from L ( 1 + ln + L ) k 2 − 1 ( µ ) into When k > 2 , L 1 , ∞ ( µ ) , and this is optimal. 11 / 23

As for weak type ( 1 , 1 ) , h k behaves like H D , with k replacing q : Theorem The operator h k is of weak type ( 1 , 1 ) if and only if k ≤ 1 . h k is bounded from L ( 1 + ln + L ) k 2 − 3 4 ( µ ) into When k > 1 , L 1 , ∞ ( µ ) , and this is optimal. The “maximal operator” H k is a bit better: Theorem The operator H k is of weak type ( 1 , 1 ) if and only if k ≤ 2 . H k is bounded from L ( 1 + ln + L ) k 2 − 1 ( µ ) into When k > 2 , L 1 , ∞ ( µ ) , and this is optimal. The analogous operators defined with the Poisson semigroup are of weak type ( 1 , 1 ) for all values of k . 11 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , 12 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , Then we write x = ( x 1 , x ′ ) . 12 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , Then we write x = ( x 1 , x ′ ) . The kernels of the operators considered can be computed, via that of the heat semigroup e t ∆ e 1 , 12 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , Then we write x = ( x 1 , x ′ ) . The kernels of the operators considered can be computed, via that of the heat semigroup e t ∆ e 1 , which is 2 e − x 1 − y 1 e − t e − | x − y | 2 p t ( x , y ) = ( 4 π t ) − n . 4 t 12 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , Then we write x = ( x 1 , x ′ ) . The kernels of the operators considered can be computed, via that of the heat semigroup e t ∆ e 1 , which is 2 e − x 1 − y 1 e − t e − | x − y | 2 p t ( x , y ) = ( 4 π t ) − n . 4 t For instance, the kernel of the Riesz transform R D = D ( − ∆ e 1 ) − k / 2 12 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , Then we write x = ( x 1 , x ′ ) . The kernels of the operators considered can be computed, via that of the heat semigroup e t ∆ e 1 , which is 2 e − x 1 − y 1 e − t e − | x − y | 2 p t ( x , y ) = ( 4 π t ) − n . 4 t For instance, the kernel of the Riesz transform R D = D ( − ∆ e 1 ) − k / 2 is � + ∞ 1 2 D x p t ( x , y ) dt k R D ( x , y ) = t t . Γ( k / 2 ) 0 12 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , Then we write x = ( x 1 , x ′ ) . The kernels of the operators considered can be computed, via that of the heat semigroup e t ∆ e 1 , which is 2 e − x 1 − y 1 e − t e − | x − y | 2 p t ( x , y ) = ( 4 π t ) − n . 4 t For instance, the kernel of the Riesz transform R D = D ( − ∆ e 1 ) − k / 2 is � + ∞ 1 2 D x p t ( x , y ) dt k R D ( x , y ) = t t . Γ( k / 2 ) 0 Locally, the kernels of all our operators define singular integrals of standard Calderón-Zygmund type, 12 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , Then we write x = ( x 1 , x ′ ) . The kernels of the operators considered can be computed, via that of the heat semigroup e t ∆ e 1 , which is 2 e − x 1 − y 1 e − t e − | x − y | 2 p t ( x , y ) = ( 4 π t ) − n . 4 t For instance, the kernel of the Riesz transform R D = D ( − ∆ e 1 ) − k / 2 is � + ∞ 1 2 D x p t ( x , y ) dt k R D ( x , y ) = t t . Γ( k / 2 ) 0 Locally, the kernels of all our operators define singular integrals of standard Calderón-Zygmund type, vector-valued in the Littlewood-Paley-Stein cases. 12 / 23

Sketches of proofs It is no restriction to assume that v = e 1 = ( 1 , . . . , 0 ) , Then we write x = ( x 1 , x ′ ) . The kernels of the operators considered can be computed, via that of the heat semigroup e t ∆ e 1 , which is 2 e − x 1 − y 1 e − t e − | x − y | 2 p t ( x , y ) = ( 4 π t ) − n . 4 t For instance, the kernel of the Riesz transform R D = D ( − ∆ e 1 ) − k / 2 is � + ∞ 1 2 D x p t ( x , y ) dt k R D ( x , y ) = t t . Γ( k / 2 ) 0 Locally, the kernels of all our operators define singular integrals of standard Calderón-Zygmund type, vector-valued in the Littlewood-Paley-Stein cases. So these parts are of weak type ( 1 , 1 ) . 12 / 23

For the parts at infinity, it turns out to be enough to consider the absolute values of the kernels. 13 / 23

For the parts at infinity, it turns out to be enough to consider the absolute values of the kernels. In the Riesz case, one has the following sharp estimate. 13 / 23

For the parts at infinity, it turns out to be enough to consider the absolute values of the kernels. In the Riesz case, one has the following sharp estimate. Proposition For | x − y | > 1   � k � | x ′ − y ′ | 2 2 | R D ( x , y ) | � e − x 1 − y 1 −| x − y | | x − y | q − n − 1  .  1 + 2 | x − y | 13 / 23

For the parts at infinity, it turns out to be enough to consider the absolute values of the kernels. In the Riesz case, one has the following sharp estimate. Proposition For | x − y | > 1   � k � | x ′ − y ′ | 2 2 | R D ( x , y ) | � e − x 1 − y 1 −| x − y | | x − y | q − n − 1  .  1 + 2 | x − y | The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. 13 / 23

For the parts at infinity, it turns out to be enough to consider the absolute values of the kernels. In the Riesz case, one has the following sharp estimate. Proposition For | x − y | > 1   � k � | x ′ − y ′ | 2 2 | R D ( x , y ) | � e − x 1 − y 1 −| x − y | | x − y | q − n − 1  .  1 + 2 | x − y | The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. From this proposition, it can be proved that R D is of weak type ( 1 , 1 ) 13 / 23

For the parts at infinity, it turns out to be enough to consider the absolute values of the kernels. In the Riesz case, one has the following sharp estimate. Proposition For | x − y | > 1   � k � | x ′ − y ′ | 2 2 | R D ( x , y ) | � e − x 1 − y 1 −| x − y | | x − y | q − n − 1  .  1 + 2 | x − y | The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. From this proposition, it can be proved that R D is of weak type ( 1 , 1 ) for q = 1 and 2. 13 / 23

For the parts at infinity, it turns out to be enough to consider the absolute values of the kernels. In the Riesz case, one has the following sharp estimate. Proposition For | x − y | > 1   � k � | x ′ − y ′ | 2 2 | R D ( x , y ) | � e − x 1 − y 1 −| x − y | | x − y | q − n − 1  .  1 + 2 | x − y | The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. From this proposition, it can be proved that R D is of weak type ( 1 , 1 ) for q = 1 and 2. We will focus on the simpler case q = 1, 13 / 23

For the parts at infinity, it turns out to be enough to consider the absolute values of the kernels. In the Riesz case, one has the following sharp estimate. Proposition For | x − y | > 1   � k � | x ′ − y ′ | 2 2 | R D ( x , y ) | � e − x 1 − y 1 −| x − y | | x − y | q − n − 1  .  1 + 2 | x − y | The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. From this proposition, it can be proved that R D is of weak type ( 1 , 1 ) for q = 1 and 2. We will focus on the simpler case q = 1, which was solved in a paper by Li, Sjögren and Wu. 13 / 23

It is enough to consider points with x 1 − y 1 > 1, since the opposite case is trivial. 14 / 23

It is enough to consider points with x 1 − y 1 > 1, since the opposite case is trivial. Some manipulations of the expression in the proposition show that for q = 1 14 / 23

It is enough to consider points with x 1 − y 1 > 1, since the opposite case is trivial. Some manipulations of the expression in the proposition show that for q = 1 | R D ( x , y ) | �  | x ′ − y ′ | 2   − 1 e − 2 x 1 ( x 1 − y 1 ) − n x 1 − y 1  χ { x 1 − y 1 > 1 } 2 exp + trivial term . 1 + | x ′ − y ′ | 4 √ x 1 − y 1 14 / 23

It is enough to consider points with x 1 − y 1 > 1, since the opposite case is trivial. Some manipulations of the expression in the proposition show that for q = 1 | R D ( x , y ) | �  | x ′ − y ′ | 2   − 1 e − 2 x 1 ( x 1 − y 1 ) − n x 1 − y 1  χ { x 1 − y 1 > 1 } 2 exp + trivial term . 1 + | x ′ − y ′ | 4 √ x 1 − y 1 Here the main part is given by the parabolic region | x ′ − y ′ | < √ x 1 − y 1 . 14 / 23

this is the y plane x 15 / 23

this is the y plane 2 j 2 j-1 x 16 / 23

We split this into parts given by 2 j − 1 < x 1 − y 1 ≤ 2 j for j = 1 , . . . . 17 / 23

We split this into parts given by 2 j − 1 < x 1 − y 1 ≤ 2 j for j = 1 , . . . . Then we get a sum of kernels � 2 − j | x ′ − y ′ | 2 � − 1 2 2 − ( n − 1 ) j e − 2 x 1 2 − j exp 2 1 + 2 − j 8 2 | x ′ − y ′ | 17 / 23

We split this into parts given by 2 j − 1 < x 1 − y 1 ≤ 2 j for j = 1 , . . . . Then we get a sum of kernels � 2 − j | x ′ − y ′ | 2 � − 1 2 2 − ( n − 1 ) j e − 2 x 1 2 − j exp 2 1 + 2 − j 8 2 | x ′ − y ′ | = e − 2 x 1 2 − j 2 ψ j ( x ′ − y ′ ) , 17 / 23

We split this into parts given by 2 j − 1 < x 1 − y 1 ≤ 2 j for j = 1 , . . . . Then we get a sum of kernels � 2 − j | x ′ − y ′ | 2 � − 1 2 2 − ( n − 1 ) j e − 2 x 1 2 − j exp 2 1 + 2 − j 8 2 | x ′ − y ′ | = e − 2 x 1 2 − j 2 ψ j ( x ′ − y ′ ) , where ψ j is the normalized dilation of the function | z ′ | 2 � − 1 � z ′ ∈ R n − 1 , ψ ( z ′ ) = exp , 1 + | z ′ | 8 by a factor 2 j / 2 . 17 / 23

We split this into parts given by 2 j − 1 < x 1 − y 1 ≤ 2 j for j = 1 , . . . . Then we get a sum of kernels � 2 − j | x ′ − y ′ | 2 � − 1 2 2 − ( n − 1 ) j e − 2 x 1 2 − j exp 2 1 + 2 − j 8 2 | x ′ − y ′ | = e − 2 x 1 2 − j 2 ψ j ( x ′ − y ′ ) , where ψ j is the normalized dilation of the function | z ′ | 2 � − 1 � z ′ ∈ R n − 1 , ψ ( z ′ ) = exp , 1 + | z ′ | 8 by a factor 2 j / 2 . Clearly, ψ ∈ L 1 ( R n − 1 ) . 17 / 23

Putting things together, we see that R D is controlled by the sum over j = 1 , 2 , . . . of the operators 18 / 23

Putting things together, we see that R D is controlled by the sum over j = 1 , 2 , . . . of the operators � � T j f ( x ) = e − 2 x 1 2 − j R n − 1 ψ j ( x ′ − y ′ ) | f ( y 1 , y ′ ) | e 2 y 1 dy 1 dy ′ 2 R 18 / 23

Putting things together, we see that R D is controlled by the sum over j = 1 , 2 , . . . of the operators � � T j f ( x ) = e − 2 x 1 2 − j R n − 1 ψ j ( x ′ − y ′ ) | f ( y 1 , y ′ ) | e 2 y 1 dy 1 dy ′ 2 R = e − 2 x 1 2 − j 2 ψ j ∗ F ( x ′ ) , 18 / 23

Putting things together, we see that R D is controlled by the sum over j = 1 , 2 , . . . of the operators � � T j f ( x ) = e − 2 x 1 2 − j R n − 1 ψ j ( x ′ − y ′ ) | f ( y 1 , y ′ ) | e 2 y 1 dy 1 dy ′ 2 R = e − 2 x 1 2 − j 2 ψ j ∗ F ( x ′ ) , where the convolution is taken in R n − 1 18 / 23

Putting things together, we see that R D is controlled by the sum over j = 1 , 2 , . . . of the operators � � T j f ( x ) = e − 2 x 1 2 − j R n − 1 ψ j ( x ′ − y ′ ) | f ( y 1 , y ′ ) | e 2 y 1 dy 1 dy ′ 2 R = e − 2 x 1 2 − j 2 ψ j ∗ F ( x ′ ) , where the convolution is taken in R n − 1 and � y ′ ∈ R n − 1 . | f ( y 1 , y ′ ) | e 2 y 1 dy 1 , F ( y ′ ) = R 18 / 23