Operators related to the Laplacian with drift in Euclidean space - - PowerPoint PPT Presentation
Operators related to the Laplacian with drift in Euclidean space Peter Sjgren University of Gothenburg Joint work with Hong-Quan Li (Shanghai) 1 / 23 In R n we consider the Laplacian with drift 2 n v = + 2 v
Peter Sjögren University of Gothenburg Joint work with Hong-Quan Li (Shanghai)
1 / 23
In Rn we consider the Laplacian with drift ∆v = ∆ + 2v · ∇ =
n
∂2 ∂x2
i
+ 2vi ∂ ∂xi
where v = 0 is a fixed vector.
2 / 23
In Rn we consider the Laplacian with drift ∆v = ∆ + 2v · ∇ =
n
∂2 ∂x2
i
+ 2vi ∂ ∂xi
where v = 0 is a fixed vector. To make this operator self-adjoint in L2, we use the measure dµ(x) = e2v,xdx and L2(µ).
2 / 23
In Rn we consider the Laplacian with drift ∆v = ∆ + 2v · ∇ =
n
∂2 ∂x2
i
+ 2vi ∂ ∂xi
where v = 0 is a fixed vector. To make this operator self-adjoint in L2, we use the measure dµ(x) = e2v,xdx and L2(µ). With ∆v there are associated Riesz transforms
2 / 23
In Rn we consider the Laplacian with drift ∆v = ∆ + 2v · ∇ =
n
∂2 ∂x2
i
+ 2vi ∂ ∂xi
where v = 0 is a fixed vector. To make this operator self-adjoint in L2, we use the measure dµ(x) = e2v,xdx and L2(µ). With ∆v there are associated Riesz transforms and Littlewood-Paley-Stein functions for the corresponding heat and Poisson semigroups.
2 / 23
In Rn we consider the Laplacian with drift ∆v = ∆ + 2v · ∇ =
n
∂2 ∂x2
i
+ 2vi ∂ ∂xi
where v = 0 is a fixed vector. To make this operator self-adjoint in L2, we use the measure dµ(x) = e2v,xdx and L2(µ). With ∆v there are associated Riesz transforms and Littlewood-Paley-Stein functions for the corresponding heat and Poisson semigroups. We will study the Lp and in particular the weak type (1, 1) boundedness properties of these operators.
2 / 23
In Rn we consider the Laplacian with drift ∆v = ∆ + 2v · ∇ =
n
∂2 ∂x2
i
+ 2vi ∂ ∂xi
where v = 0 is a fixed vector. To make this operator self-adjoint in L2, we use the measure dµ(x) = e2v,xdx and L2(µ). With ∆v there are associated Riesz transforms and Littlewood-Paley-Stein functions for the corresponding heat and Poisson semigroups. We will study the Lp and in particular the weak type (1, 1) boundedness properties of these operators. Here we always refer to the measure µ.
2 / 23
In Rn we consider the Laplacian with drift ∆v = ∆ + 2v · ∇ =
n
∂2 ∂x2
i
+ 2vi ∂ ∂xi
where v = 0 is a fixed vector. To make this operator self-adjoint in L2, we use the measure dµ(x) = e2v,xdx and L2(µ). With ∆v there are associated Riesz transforms and Littlewood-Paley-Stein functions for the corresponding heat and Poisson semigroups. We will study the Lp and in particular the weak type (1, 1) boundedness properties of these operators. Here we always refer to the measure µ. Observe that µ has exponential volume growth, i.e., the measure of large balls increase exponentially with the radius.
2 / 23
In Rn we consider the Laplacian with drift ∆v = ∆ + 2v · ∇ =
n
∂2 ∂x2
i
+ 2vi ∂ ∂xi
where v = 0 is a fixed vector. To make this operator self-adjoint in L2, we use the measure dµ(x) = e2v,xdx and L2(µ). With ∆v there are associated Riesz transforms and Littlewood-Paley-Stein functions for the corresponding heat and Poisson semigroups. We will study the Lp and in particular the weak type (1, 1) boundedness properties of these operators. Here we always refer to the measure µ. Observe that µ has exponential volume growth, i.e., the measure of large balls increase exponentially with the radius. This makes standard singular integral techniques hard to apply.
2 / 23
These operators have been studied in many other settings such as Lie groups and symmetric spaces
3 / 23
These operators have been studied in many other settings such as Lie groups and symmetric spaces with measures of exponential growth.
3 / 23
These operators have been studied in many other settings such as Lie groups and symmetric spaces with measures of exponential growth. The Lp boundedness can often be verified for 1 < p < ∞, but the weak type (1, 1) is not known in many such settings.
3 / 23
These operators have been studied in many other settings such as Lie groups and symmetric spaces with measures of exponential growth. The Lp boundedness can often be verified for 1 < p < ∞, but the weak type (1, 1) is not known in many such settings. The existing results hint that the first- and second-order Riesz transforms are better than those of higher order.
3 / 23
These operators have been studied in many other settings such as Lie groups and symmetric spaces with measures of exponential growth. The Lp boundedness can often be verified for 1 < p < ∞, but the weak type (1, 1) is not known in many such settings. The existing results hint that the first- and second-order Riesz transforms are better than those of higher order. For instance, in the Ornstein-Uhlenbeck setting in Rn with the Gaussian measure,
3 / 23
These operators have been studied in many other settings such as Lie groups and symmetric spaces with measures of exponential growth. The Lp boundedness can often be verified for 1 < p < ∞, but the weak type (1, 1) is not known in many such settings. The existing results hint that the first- and second-order Riesz transforms are better than those of higher order. For instance, in the Ornstein-Uhlenbeck setting in Rn with the Gaussian measure, it is known that the weak type (1, 1) holds for the Riesz transforms of order 1 and 2 but not higher order.
3 / 23
These operators have been studied in many other settings such as Lie groups and symmetric spaces with measures of exponential growth. The Lp boundedness can often be verified for 1 < p < ∞, but the weak type (1, 1) is not known in many such settings. The existing results hint that the first- and second-order Riesz transforms are better than those of higher order. For instance, in the Ornstein-Uhlenbeck setting in Rn with the Gaussian measure, it is known that the weak type (1, 1) holds for the Riesz transforms of order 1 and 2 but not higher order. The authors proved in 2016 for the hyperbolic ball and the Laplacian with a radial drift
3 / 23
These operators have been studied in many other settings such as Lie groups and symmetric spaces with measures of exponential growth. The Lp boundedness can often be verified for 1 < p < ∞, but the weak type (1, 1) is not known in many such settings. The existing results hint that the first- and second-order Riesz transforms are better than those of higher order. For instance, in the Ornstein-Uhlenbeck setting in Rn with the Gaussian measure, it is known that the weak type (1, 1) holds for the Riesz transforms of order 1 and 2 but not higher order. The authors proved in 2016 for the hyperbolic ball and the Laplacian with a radial drift that the first- and second-order Riesz transforms are
3 / 23
Back to our setting, ∆v in Rn.
4 / 23
Back to our setting, ∆v in Rn. In 2004, Lohoué and Mustapha showed that the Riesz transforms ∇k(−∆v)−k/2 of any order k ≥ 1 are bounded on Lp(µ), 1 < p < ∞.
4 / 23
Back to our setting, ∆v in Rn. In 2004, Lohoué and Mustapha showed that the Riesz transforms ∇k(−∆v)−k/2 of any order k ≥ 1 are bounded on Lp(µ), 1 < p < ∞. The authors, together with Y.-R. Wu, proved in 2016 that the first-order transform ∇(−∆v)−1/2 is of weak type (1, 1).
4 / 23
Back to our setting, ∆v in Rn. In 2004, Lohoué and Mustapha showed that the Riesz transforms ∇k(−∆v)−k/2 of any order k ≥ 1 are bounded on Lp(µ), 1 < p < ∞. The authors, together with Y.-R. Wu, proved in 2016 that the first-order transform ∇(−∆v)−1/2 is of weak type (1, 1). We will consider a slightly more general Riesz transform of order k ≥ 1.
4 / 23
Back to our setting, ∆v in Rn. In 2004, Lohoué and Mustapha showed that the Riesz transforms ∇k(−∆v)−k/2 of any order k ≥ 1 are bounded on Lp(µ), 1 < p < ∞. The authors, together with Y.-R. Wu, proved in 2016 that the first-order transform ∇(−∆v)−1/2 is of weak type (1, 1). We will consider a slightly more general Riesz transform of order k ≥ 1. Let D =
aα∂α be a homogeneous differential operator with constant coefficients, not all 0.
4 / 23
Back to our setting, ∆v in Rn. In 2004, Lohoué and Mustapha showed that the Riesz transforms ∇k(−∆v)−k/2 of any order k ≥ 1 are bounded on Lp(µ), 1 < p < ∞. The authors, together with Y.-R. Wu, proved in 2016 that the first-order transform ∇(−∆v)−1/2 is of weak type (1, 1). We will consider a slightly more general Riesz transform of order k ≥ 1. Let D =
aα∂α be a homogeneous differential operator with constant coefficients, not all 0. Then define the Riesz transform RD = D(−∆v)− k
2 . 4 / 23
In D it turns out that it matters how many derivatives are taken in the v direction:
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.
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 D =
k
∂i
v D′ k−i,
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 D =
k
∂i
v D′ k−i,
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 D =
k
∂i
v D′ k−i,
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 D =
k
∂i
v D′ k−i,
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 RD = 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 RD = D(−∆v)−k/2 is of weak type (1, 1) if and only if q ≤ 2. When q ≥ 3, RD is bounded from the Orlicz space L(1 + ln+ L)
q 2 −1(µ) into L1,∞(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 RD = D(−∆v)−k/2 is of weak type (1, 1) if and only if q ≤ 2. When q ≥ 3, RD is bounded from the Orlicz space L(1 + ln+ L)
q 2 −1(µ) into L1,∞(µ), in the sense that
µ {x; |RDf(x)| > λ} ≤ C |f| λ
λ q
2 −1
dµ, λ > 0, 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 RD = D(−∆v)−k/2 is of weak type (1, 1) if and only if q ≤ 2. When q ≥ 3, RD is bounded from the Orlicz space L(1 + ln+ L)
q 2 −1(µ) into L1,∞(µ), in the sense that
µ {x; |RDf(x)| > λ} ≤ C |f| λ
λ q
2 −1
dµ, λ > 0, 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 (et∆v)t>0
7 / 23
We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup (et∆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 (et∆v)t>0 and the Poisson semigroup (e−t√−∆v)t>0. For the heat semigroup, the vertical Littlewood-Paley-Stein function of
Hk(f)(x) = +∞
k 2 ∇ket∆vf(x)
t 1
2
, 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 (et∆v)t>0 and the Poisson semigroup (e−t√−∆v)t>0. For the heat semigroup, the vertical Littlewood-Paley-Stein function of
Hk(f)(x) = +∞
k 2 ∇ket∆vf(x)
t 1
2
, where ∇k means all derivatives of order k in the x variable. They are known to be bounded on Lp for 1 < p < ∞.
7 / 23
We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup (et∆v)t>0 and the Poisson semigroup (e−t√−∆v)t>0. For the heat semigroup, the vertical Littlewood-Paley-Stein function of
Hk(f)(x) = +∞
k 2 ∇ket∆vf(x)
t 1
2
, where ∇k means all derivatives of order k in the x variable. They are known to be bounded on Lp for 1 < p < ∞. But to write a precise weak type (1, 1) result, we use the k:th order
7 / 23
We pass to the Littlewood-Paley-Stein functions. They are based on the heat semigroup (et∆v)t>0 and the Poisson semigroup (e−t√−∆v)t>0. For the heat semigroup, the vertical Littlewood-Paley-Stein function of
Hk(f)(x) = +∞
k 2 ∇ket∆vf(x)
t 1
2
, where ∇k means all derivatives of order k in the x variable. They are known to be bounded on Lp for 1 < p < ∞. But to write a precise weak type (1, 1) result, we use the k:th order
HD(f)(x) = +∞
k 2 D et∆vf(x)
t 1
2
.
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 HD 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 HD is of weak type (1, 1) if and only if q ≤ 1. When q > 1, HD is bounded from L(1 + ln+ L)
q 2 − 3 4 (µ) into
L1,∞(µ),
8 / 23
Again, the number q matters, to indicate the maximal order of differentiation along v:
Theorem
The operator HD is of weak type (1, 1) if and only if q ≤ 1. When q > 1, HD is bounded from L(1 + ln+ L)
q 2 − 3 4 (µ) into
L1,∞(µ), i.e., µ {x; HD(f)(x) > λ} ≤ C |f| λ
λ q
2 − 3 4
dµ, λ > 0, with C = C(v, D).
8 / 23
Again, the number q matters, to indicate the maximal order of differentiation along v:
Theorem
The operator HD is of weak type (1, 1) if and only if q ≤ 1. When q > 1, HD is bounded from L(1 + ln+ L)
q 2 − 3 4 (µ) into
L1,∞(µ), i.e., µ {x; HD(f)(x) > λ} ≤ C |f| λ
λ q
2 − 3 4
dµ, λ > 0, 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 GD(f)(x) = +∞
k 2 D e−t√−∆v f(x)
t 1
2
.
9 / 23
Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, by GD(f)(x) = +∞
k 2 D e−t√−∆v f(x)
t 1
2
. It behaves a bit better than HD,
9 / 23
Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, by GD(f)(x) = +∞
k 2 D e−t√−∆v f(x)
t 1
2
. It behaves a bit better than HD, indeed
Theorem
The operator GD 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 GD(f)(x) = +∞
k 2 D e−t√−∆v f(x)
t 1
2
. It behaves a bit better than HD, indeed
Theorem
The operator GD is of weak type (1, 1) if and only if q ≤ 2. When q > 2, GD is bounded from L(1 + ln+ L)
q 2 −1(µ) into
L1,∞(µ),
9 / 23
Using instead the Poisson semigroup, we define the Littlewood-Paley-Stein function similarly, by GD(f)(x) = +∞
k 2 D e−t√−∆v f(x)
t 1
2
. It behaves a bit better than HD, indeed
Theorem
The operator GD is of weak type (1, 1) if and only if q ≤ 2. When q > 2, GD is bounded from L(1 + ln+ L)
q 2 −1(µ) into
L1,∞(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, hk(f)(x) = +∞
∂tk et∆vf(x)
t 1
2
, k ≥ 1
10 / 23
The horizontal Littlewood-Paley-Stein functions are, for the heat semigroup, hk(f)(x) = +∞
∂tk et∆vf(x)
t 1
2
, k ≥ 1 and also Hk(f)(x) = sup
t>0
∂tk et∆vf(x)
k ≥ 0.
10 / 23
The horizontal Littlewood-Paley-Stein functions are, for the heat semigroup, hk(f)(x) = +∞
∂tk et∆vf(x)
t 1
2
, k ≥ 1 and also Hk(f)(x) = sup
t>0
∂tk et∆vf(x)
k ≥ 0. They are known to be bounded on Lp, 1 < p < ∞, by general Littlewood-Paley-Stein theory.
10 / 23
As for weak type (1, 1), hk behaves like HD,
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
Theorem
The operator hk is of weak type (1, 1) if and only if k ≤ 1.
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
Theorem
The operator hk is of weak type (1, 1) if and only if k ≤ 1. When k > 1, hk is bounded from L(1 + ln+ L)
k 2 − 3 4 (µ) into
L1,∞(µ),
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
Theorem
The operator hk is of weak type (1, 1) if and only if k ≤ 1. When k > 1, hk is bounded from L(1 + ln+ L)
k 2 − 3 4 (µ) into
L1,∞(µ), and this is optimal.
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
Theorem
The operator hk is of weak type (1, 1) if and only if k ≤ 1. When k > 1, hk is bounded from L(1 + ln+ L)
k 2 − 3 4 (µ) into
L1,∞(µ), and this is optimal. The “maximal operator” Hk is a bit better:
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
Theorem
The operator hk is of weak type (1, 1) if and only if k ≤ 1. When k > 1, hk is bounded from L(1 + ln+ L)
k 2 − 3 4 (µ) into
L1,∞(µ), and this is optimal. The “maximal operator” Hk is a bit better:
Theorem
The operator Hk is of weak type (1, 1) if and only if k ≤ 2.
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
Theorem
The operator hk is of weak type (1, 1) if and only if k ≤ 1. When k > 1, hk is bounded from L(1 + ln+ L)
k 2 − 3 4 (µ) into
L1,∞(µ), and this is optimal. The “maximal operator” Hk is a bit better:
Theorem
The operator Hk is of weak type (1, 1) if and only if k ≤ 2. When k > 2, Hk is bounded from L(1 + ln+ L)
k 2 −1(µ) into
L1,∞(µ),
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
Theorem
The operator hk is of weak type (1, 1) if and only if k ≤ 1. When k > 1, hk is bounded from L(1 + ln+ L)
k 2 − 3 4 (µ) into
L1,∞(µ), and this is optimal. The “maximal operator” Hk is a bit better:
Theorem
The operator Hk is of weak type (1, 1) if and only if k ≤ 2. When k > 2, Hk is bounded from L(1 + ln+ L)
k 2 −1(µ) into
L1,∞(µ), and this is optimal.
11 / 23
As for weak type (1, 1), hk behaves like HD, with k replacing q:
Theorem
The operator hk is of weak type (1, 1) if and only if k ≤ 1. When k > 1, hk is bounded from L(1 + ln+ L)
k 2 − 3 4 (µ) into
L1,∞(µ), and this is optimal. The “maximal operator” Hk is a bit better:
Theorem
The operator Hk is of weak type (1, 1) if and only if k ≤ 2. When k > 2, Hk is bounded from L(1 + ln+ L)
k 2 −1(µ) into
L1,∞(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 = e1 = (1, . . . , 0),
12 / 23
Sketches of proofs It is no restriction to assume that v = e1 = (1, . . . , 0), Then we write x = (x1, x′).
12 / 23
Sketches of proofs It is no restriction to assume that v = e1 = (1, . . . , 0), Then we write x = (x1, x′). The kernels of the operators considered can be computed, via that of the heat semigroup et∆e1,
12 / 23
Sketches of proofs It is no restriction to assume that v = e1 = (1, . . . , 0), Then we write x = (x1, x′). The kernels of the operators considered can be computed, via that of the heat semigroup et∆e1, which is pt(x, y) = (4πt)− n
2 e−x1−y1 e−t e− |x−y|2 4t
.
12 / 23
Sketches of proofs It is no restriction to assume that v = e1 = (1, . . . , 0), Then we write x = (x1, x′). The kernels of the operators considered can be computed, via that of the heat semigroup et∆e1, which is pt(x, y) = (4πt)− n
2 e−x1−y1 e−t e− |x−y|2 4t
. For instance, the kernel of the Riesz transform RD = D(−∆e1)−k/2
12 / 23
Sketches of proofs It is no restriction to assume that v = e1 = (1, . . . , 0), Then we write x = (x1, x′). The kernels of the operators considered can be computed, via that of the heat semigroup et∆e1, which is pt(x, y) = (4πt)− n
2 e−x1−y1 e−t e− |x−y|2 4t
. For instance, the kernel of the Riesz transform RD = D(−∆e1)−k/2 is RD(x, y) = 1 Γ(k/2) +∞ t
k 2 Dx pt(x, y) dt
t .
12 / 23
Sketches of proofs It is no restriction to assume that v = e1 = (1, . . . , 0), Then we write x = (x1, x′). The kernels of the operators considered can be computed, via that of the heat semigroup et∆e1, which is pt(x, y) = (4πt)− n
2 e−x1−y1 e−t e− |x−y|2 4t
. For instance, the kernel of the Riesz transform RD = D(−∆e1)−k/2 is RD(x, y) = 1 Γ(k/2) +∞ t
k 2 Dx pt(x, y) dt
t . 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 = e1 = (1, . . . , 0), Then we write x = (x1, x′). The kernels of the operators considered can be computed, via that of the heat semigroup et∆e1, which is pt(x, y) = (4πt)− n
2 e−x1−y1 e−t e− |x−y|2 4t
. For instance, the kernel of the Riesz transform RD = D(−∆e1)−k/2 is RD(x, y) = 1 Γ(k/2) +∞ t
k 2 Dx pt(x, y) dt
t . 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 = e1 = (1, . . . , 0), Then we write x = (x1, x′). The kernels of the operators considered can be computed, via that of the heat semigroup et∆e1, which is pt(x, y) = (4πt)− n
2 e−x1−y1 e−t e− |x−y|2 4t
. For instance, the kernel of the Riesz transform RD = D(−∆e1)−k/2 is RD(x, y) = 1 Γ(k/2) +∞ t
k 2 Dx pt(x, y) dt
t . 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 |RD(x, y)| e−x1−y1−|x−y| |x − y|
q−n−1 2
1 + |x′ − y′|2 |x − y| k
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 |RD(x, y)| e−x1−y1−|x−y| |x − y|
q−n−1 2
1 + |x′ − y′|2 |x − y| k
2
. 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 |RD(x, y)| e−x1−y1−|x−y| |x − y|
q−n−1 2
1 + |x′ − y′|2 |x − y| k
2
. The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. From this proposition, it can be proved that RD 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 |RD(x, y)| e−x1−y1−|x−y| |x − y|
q−n−1 2
1 + |x′ − y′|2 |x − y| k
2
. The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. From this proposition, it can be proved that RD 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 |RD(x, y)| e−x1−y1−|x−y| |x − y|
q−n−1 2
1 + |x′ − y′|2 |x − y| k
2
. The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. From this proposition, it can be proved that RD 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 |RD(x, y)| e−x1−y1−|x−y| |x − y|
q−n−1 2
1 + |x′ − y′|2 |x − y| k
2
. The kernels of the Littlewood-Paley-Stein functions satisfy similar estimates. From this proposition, it can be proved that RD 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 x1 − y1 > 1, since the opposite case is trivial.
14 / 23
It is enough to consider points with x1 − y1 > 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 x1 − y1 > 1, since the opposite case is trivial. Some manipulations of the expression in the proposition show that for q = 1 |RD(x, y)| e−2x1(x1 − y1)− n
2 exp
−1 4
|x′−y′|2 x1−y1
1 + |x′−y′|
√x1−y1
χ{x1−y1>1} + trivial term.
14 / 23
It is enough to consider points with x1 − y1 > 1, since the opposite case is trivial. Some manipulations of the expression in the proposition show that for q = 1 |RD(x, y)| e−2x1(x1 − y1)− n
2 exp
−1 4
|x′−y′|2 x1−y1
1 + |x′−y′|
√x1−y1
χ{x1−y1>1} + trivial term. Here the main part is given by the parabolic region |x′ − y′| < √x1 − y1.
14 / 23
this is the y plane
x
15 / 23
this is the y plane
x 2 j 2 j-1
16 / 23
We split this into parts given by 2j−1 < x1 − y1 ≤ 2j for j = 1, . . . .
17 / 23
We split this into parts given by 2j−1 < x1 − y1 ≤ 2j for j = 1, . . . . Then we get a sum of kernels e−2x1 2− j
2 2− (n−1)j 2
exp
8 2−j |x′ − y′|2 1 + 2− j
2 |x′ − y′|
We split this into parts given by 2j−1 < x1 − y1 ≤ 2j for j = 1, . . . . Then we get a sum of kernels e−2x1 2− j
2 2− (n−1)j 2
exp
8 2−j |x′ − y′|2 1 + 2− j
2 |x′ − y′|
2 ψj(x′ − y′), 17 / 23
We split this into parts given by 2j−1 < x1 − y1 ≤ 2j for j = 1, . . . . Then we get a sum of kernels e−2x1 2− j
2 2− (n−1)j 2
exp
8 2−j |x′ − y′|2 1 + 2− j
2 |x′ − y′|
2 ψj(x′ − y′),
where ψj is the normalized dilation of the function ψ(z′) = exp
8 |z′|2 1 + |z′|
z′ ∈ Rn−1, by a factor 2j/2.
17 / 23
We split this into parts given by 2j−1 < x1 − y1 ≤ 2j for j = 1, . . . . Then we get a sum of kernels e−2x1 2− j
2 2− (n−1)j 2
exp
8 2−j |x′ − y′|2 1 + 2− j
2 |x′ − y′|
2 ψj(x′ − y′),
where ψj is the normalized dilation of the function ψ(z′) = exp
8 |z′|2 1 + |z′|
z′ ∈ Rn−1, by a factor 2j/2. Clearly, ψ ∈ L1(Rn−1).
17 / 23
Putting things together, we see that RD is controlled by the sum over j = 1, 2, . . .
18 / 23
Putting things together, we see that RD is controlled by the sum over j = 1, 2, . . .
Tjf(x) = e−2x1 2− j
2
|f(y1, y′)| e2y1 dy1 dy′
18 / 23
Putting things together, we see that RD is controlled by the sum over j = 1, 2, . . .
Tjf(x) = e−2x1 2− j
2
|f(y1, y′)| e2y1 dy1 dy′ = e−2x1 2− j
2 ψj ∗ F(x′), 18 / 23
Putting things together, we see that RD is controlled by the sum over j = 1, 2, . . .
Tjf(x) = e−2x1 2− j
2
|f(y1, y′)| e2y1 dy1 dy′ = e−2x1 2− j
2 ψj ∗ F(x′),
where the convolution is taken in Rn−1
18 / 23
Putting things together, we see that RD is controlled by the sum over j = 1, 2, . . .
Tjf(x) = e−2x1 2− j
2
|f(y1, y′)| e2y1 dy1 dy′ = e−2x1 2− j
2 ψj ∗ F(x′),
where the convolution is taken in Rn−1 and F(y′) =
|f(y1, y′)| e2y1 dy1, y′ ∈ Rn−1.
18 / 23
Putting things together, we see that RD is controlled by the sum over j = 1, 2, . . .
Tjf(x) = e−2x1 2− j
2
|f(y1, y′)| e2y1 dy1 dy′ = e−2x1 2− j
2 ψj ∗ F(x′),
where the convolution is taken in Rn−1 and F(y′) =
|f(y1, y′)| e2y1 dy1, y′ ∈ Rn−1. Notice that FL1(Rn−1) = fL1(Rn,dµ).
18 / 23
This allows us to estimate the level set of Tjf:
19 / 23
This allows us to estimate the level set of Tjf: µ
2 e−2x1 ψj ∗ F(x′) > λ
This allows us to estimate the level set of Tjf: µ
2 e−2x1 ψj ∗ F(x′) > λ
∞
−∞
e2x1 mn−1
j 2 λ e2x1
19 / 23
This allows us to estimate the level set of Tjf: µ
2 e−2x1 ψj ∗ F(x′) > λ
∞
−∞
e2x1 mn−1
j 2 λ e2x1
The change of variable s = 2
j 2 λ e2x1 shows that this equals 19 / 23
This allows us to estimate the level set of Tjf: µ
2 e−2x1 ψj ∗ F(x′) > λ
∞
−∞
e2x1 mn−1
j 2 λ e2x1
The change of variable s = 2
j 2 λ e2x1 shows that this equals
2−j/2 λ ∞ mn−1{x′ : ψj ∗ F(x′) > s} ds
19 / 23
This allows us to estimate the level set of Tjf: µ
2 e−2x1 ψj ∗ F(x′) > λ
∞
−∞
e2x1 mn−1
j 2 λ e2x1
The change of variable s = 2
j 2 λ e2x1 shows that this equals
2−j/2 λ ∞ mn−1{x′ : ψj ∗ F(x′) > s} ds = 2−j/2 λ ψj ∗ FL1(Rn−1)
19 / 23
This allows us to estimate the level set of Tjf: µ
2 e−2x1 ψj ∗ F(x′) > λ
∞
−∞
e2x1 mn−1
j 2 λ e2x1
The change of variable s = 2
j 2 λ e2x1 shows that this equals
2−j/2 λ ∞ mn−1{x′ : ψj ∗ F(x′) > s} ds = 2−j/2 λ ψj ∗ FL1(Rn−1) ≤ 2−j/2 λ ψjL1(Rn−1) FL1(Rn−1) = 2−j/2 λ ψL1(Rn−1) fL1(Rn,dµ).
19 / 23
This allows us to estimate the level set of Tjf: µ
2 e−2x1 ψj ∗ F(x′) > λ
∞
−∞
e2x1 mn−1
j 2 λ e2x1
The change of variable s = 2
j 2 λ e2x1 shows that this equals
2−j/2 λ ∞ mn−1{x′ : ψj ∗ F(x′) > s} ds = 2−j/2 λ ψj ∗ FL1(Rn−1) ≤ 2−j/2 λ ψjL1(Rn−1) FL1(Rn−1) = 2−j/2 λ ψL1(Rn−1) fL1(Rn,dµ). By summing in j, we get the weak type (1,1) of RD, for q = 1.
Some hints for the case q = 2
20 / 23
Some hints for the case q = 2 We now have the kernel e−2x1(x1 − y1)
1−n 2 exp
−1 4
|x′−y′|2 x1−y1
1 + |x′−y′|
√x1−y1
χ{x1−y1>1},
20 / 23
Some hints for the case q = 2 We now have the kernel e−2x1(x1 − y1)
1−n 2 exp
−1 4
|x′−y′|2 x1−y1
1 + |x′−y′|
√x1−y1
χ{x1−y1>1}, where the exponent of x1 − y1 is worse than before.
20 / 23
Some hints for the case q = 2 We now have the kernel e−2x1(x1 − y1)
1−n 2 exp
−1 4
|x′−y′|2 x1−y1
1 + |x′−y′|
√x1−y1
χ{x1−y1>1}, where the exponent of x1 − y1 is worse than before. With a dyadic splitting, one shows that it is enough to consider the parabolic region |x′ − y′| < √x1 − y1.
20 / 23
Some hints for the case q = 2 We now have the kernel e−2x1(x1 − y1)
1−n 2 exp
−1 4
|x′−y′|2 x1−y1
1 + |x′−y′|
√x1−y1
χ{x1−y1>1}, where the exponent of x1 − y1 is worse than before. With a dyadic splitting, one shows that it is enough to consider the parabolic region |x′ − y′| < √x1 − y1. Writing g(y) = e2y1 f(y) ∈ L1(dy),
20 / 23
Some hints for the case q = 2 We now have the kernel e−2x1(x1 − y1)
1−n 2 exp
−1 4
|x′−y′|2 x1−y1
1 + |x′−y′|
√x1−y1
χ{x1−y1>1}, where the exponent of x1 − y1 is worse than before. With a dyadic splitting, one shows that it is enough to consider the parabolic region |x′ − y′| < √x1 − y1. Writing g(y) = e2y1 f(y) ∈ L1(dy), we get the operator Sg(x) = e−2x1
(x1 − y1)
1−n 2
g(y1, y′) dy′ dy1.
20 / 23
Some hints for the case q = 2 We now have the kernel e−2x1(x1 − y1)
1−n 2 exp
−1 4
|x′−y′|2 x1−y1
1 + |x′−y′|
√x1−y1
χ{x1−y1>1}, where the exponent of x1 − y1 is worse than before. With a dyadic splitting, one shows that it is enough to consider the parabolic region |x′ − y′| < √x1 − y1. Writing g(y) = e2y1 f(y) ∈ L1(dy), we get the operator Sg(x) = e−2x1
(x1 − y1)
1−n 2
g(y1, y′) dy′ dy1. It must be proved that S is bounded from L1(dy) into L1(µ).
20 / 23
x
21 / 23
x
22 / 23
23 / 23