hardy spaces associated with semi-groups of linear...
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New Perspectives in the Theory of Function Spacesand their Applications
(Będlewo, September 17 – 23, 2017)
Hardy spacesassociated with semi-groups of linear operators
Jacek DziubańskiInstytut Matematyczny, Uniwersytet Wrocławski
J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy space Hp(RN)
Hp(RN ) comes from study holomorphic functions in R2+
For a function u(x+ iy) defined in upper half-plane x ∈ R, y > 0,we define the non-tangential maximal function
u∗(x) = sup|x−x′|<y
|u(x′ + iy)|
J. Dziubański (Wrocław) Function Spaces and their Applications
Burkholder Gundy Silverstein (TAMS 1971)
A real-valued harmonic function u(x, y) = u(x+ iy), y > 0, is areal part of a holomorphic function F (z) = u(z) + iv(z) whichsatisfies (for fixed 0 < p <∞)
supy>0
∫R|F (x+ iy)|p dx <∞ (Lp)
if and only ifu∗(x) = sup
|x−x′|<y|u(x′ + iy)|
belongs to Lp(R).
J. Dziubański (Wrocław) Function Spaces and their Applications
N -dimensional theory: Stein and Weiss, On the theory of harmonicfunctions of several variables. I. The theory of Hp-spaces, Acta Math. 1960 andFefferman and Stein, Hp spaces of several variables, Acta Math. 1972.
holomorphy system of conjugate harmonic functions:u = (u0, u1, ..., uN ) ∈ (CR), uj = uj(x0, x1, ..., xN ), x0 > 0, if
∂xjui = ∂xiuj ,
N∑j=0
∂xjuj = 0, (CR)
supx0>0
∫RN
|u(x0, x1, ..., xN )|p dx1dx2...dxN <∞. (Lp)
1− 1N < p <∞, u0(x0, x1, ..., xN )-harmonic,
then ∃u = (u0, u1, ..., uN ) satisfying (CR) and (Lp) iff
u∗0(x) = sup|x−x′|<x0
|u0(x0,x′)| ∈ Lp(RN ).
J. Dziubański (Wrocław) Function Spaces and their Applications
N -dimensional theory: Stein and Weiss, On the theory of harmonicfunctions of several variables. I. The theory of Hp-spaces, Acta Math. 1960 andFefferman and Stein, Hp spaces of several variables, Acta Math. 1972.
holomorphy system of conjugate harmonic functions:u = (u0, u1, ..., uN ) ∈ (CR), uj = uj(x0, x1, ..., xN ), x0 > 0, if
∂xjui = ∂xiuj ,
N∑j=0
∂xjuj = 0, (CR)
supx0>0
∫RN
|u(x0, x1, ..., xN )|p dx1dx2...dxN <∞. (Lp)
1− 1N < p <∞, u0(x0, x1, ..., xN )-harmonic,
then ∃u = (u0, u1, ..., uN ) satisfying (CR) and (Lp) iff
u∗0(x) = sup|x−x′|<x0
|u0(x0,x′)| ∈ Lp(RN ).
J. Dziubański (Wrocław) Function Spaces and their Applications
N -dimensional theory: Stein and Weiss, On the theory of harmonicfunctions of several variables. I. The theory of Hp-spaces, Acta Math. 1960 andFefferman and Stein, Hp spaces of several variables, Acta Math. 1972.
holomorphy system of conjugate harmonic functions:u = (u0, u1, ..., uN ) ∈ (CR), uj = uj(x0, x1, ..., xN ), x0 > 0, if
∂xjui = ∂xiuj ,
N∑j=0
∂xjuj = 0, (CR)
supx0>0
∫RN
|u(x0, x1, ..., xN )|p dx1dx2...dxN <∞. (Lp)
1− 1N < p <∞, u0(x0, x1, ..., xN )-harmonic,
then ∃u = (u0, u1, ..., uN ) satisfying (CR) and (Lp) iff
u∗0(x) = sup|x−x′|<x0
|u0(x0,x′)| ∈ Lp(RN ).
J. Dziubański (Wrocław) Function Spaces and their Applications
Then limx0→0 u0(x0,x) exists in the sense of distributions and:
Hp(RN ) = {f : f = limx0→0
u0(x0,x), (u0, u1, ..., uN ) ∈ (CR)∩(Lp)}
Several equivalent definitions of Hp(RN )
by means of
the Poisson maximal function
other maximal functions including the heat maximal function
Riesz transforms
square functions
atoms (Coifman (Studia Math. 1974), Latter (Studia Math.1978))
J. Dziubański (Wrocław) Function Spaces and their Applications
Then limx0→0 u0(x0,x) exists in the sense of distributions and:
Hp(RN ) = {f : f = limx0→0
u0(x0,x), (u0, u1, ..., uN ) ∈ (CR)∩(Lp)}
Several equivalent definitions of Hp(RN )
by means of
the Poisson maximal function
other maximal functions including the heat maximal function
Riesz transforms
square functions
atoms (Coifman (Studia Math. 1974), Latter (Studia Math.1978))
J. Dziubański (Wrocław) Function Spaces and their Applications
maximal functions characterizations
f ∈ H1(Rn) if and only if
‖f‖H1,heat = ‖ supt>0|et∆f(x)|‖L1(Rn,dx) <∞
equivalently,
‖f‖H1,Poisson = ‖ supt>0|e−t
√−∆f(x)|‖L1(Rn,dx) <∞
other maximal functions
J. Dziubański (Wrocław) Function Spaces and their Applications
Riesz transform characterization
Rjf(x) = −∂j(−∆)−1/2f(x) = − ic
∫RN
ξj|ξ|eix·ξFf(ξ)dξ
f ∈ H1(Rn) if and only if
‖f‖H1,Riesz = ‖f‖L1(Rn) +
n∑j=1
‖Rj‖L1(Rn) <∞
square function characterization
square function: Sf(x) =( ∫∫
|x−y|<t |t2∆et
2∆f(y)|2 dydt|B(x,t)|t
)1/2
f ∈ H1(Rn) if and only if
‖f‖H1,square = ‖Sf‖L1(Rn) <∞
J. Dziubański (Wrocław) Function Spaces and their Applications
atomic decomposition
Fix 1 < q ≤ ∞. A function a is an (1, q)-atom for the Hardy spaceH1(Rd) if there is a ball B such thatsupp a ⊂ B (support condition);
‖a‖Lq ≤ |B|1q−1 (size condition);∫
a = 0 (cancellation condition).Then ‖a‖H1 ≤ Cq,n for every atom a.The atomic norm ‖f‖H1
atom(Rd) is defined as
‖f‖H1atom(Rd) = inf
∑j
|λj |,
where the infimum is taken over all decompositions f =∑
j λjaj .
Theorem (Coifman, Latter)
The spaces H1(Rd) and H1atom(Rd) coincide and the norms
‖f‖H1(Rd) and ‖f‖H1atom(Rd) are equivalent.
J. Dziubański (Wrocław) Function Spaces and their Applications
atomic decomposition
Fix 1 < q ≤ ∞. A function a is an (1, q)-atom for the Hardy spaceH1(Rd) if there is a ball B such thatsupp a ⊂ B (support condition);
‖a‖Lq ≤ |B|1q−1 (size condition);∫
a = 0 (cancellation condition).Then ‖a‖H1 ≤ Cq,n for every atom a.The atomic norm ‖f‖H1
atom(Rd) is defined as
‖f‖H1atom(Rd) = inf
∑j
|λj |,
where the infimum is taken over all decompositions f =∑
j λjaj .
Theorem (Coifman, Latter)
The spaces H1(Rd) and H1atom(Rd) coincide and the norms
‖f‖H1(Rd) and ‖f‖H1atom(Rd) are equivalent.
J. Dziubański (Wrocław) Function Spaces and their Applications
atomic decomposition
Fix 1 < q ≤ ∞. A function a is an (1, q)-atom for the Hardy spaceH1(Rd) if there is a ball B such thatsupp a ⊂ B (support condition);
‖a‖Lq ≤ |B|1q−1 (size condition);∫
a = 0 (cancellation condition).Then ‖a‖H1 ≤ Cq,n for every atom a.The atomic norm ‖f‖H1
atom(Rd) is defined as
‖f‖H1atom(Rd) = inf
∑j
|λj |,
where the infimum is taken over all decompositions f =∑
j λjaj .
Theorem (Coifman, Latter)
The spaces H1(Rd) and H1atom(Rd) coincide and the norms
‖f‖H1(Rd) and ‖f‖H1atom(Rd) are equivalent.
J. Dziubański (Wrocław) Function Spaces and their Applications
atomic decomposition
Fix 1 < q ≤ ∞. A function a is an (1, q)-atom for the Hardy spaceH1(Rd) if there is a ball B such thatsupp a ⊂ B (support condition);
‖a‖Lq ≤ |B|1q−1 (size condition);∫
a = 0 (cancellation condition).Then ‖a‖H1 ≤ Cq,n for every atom a.The atomic norm ‖f‖H1
atom(Rd) is defined as
‖f‖H1atom(Rd) = inf
∑j
|λj |,
where the infimum is taken over all decompositions f =∑
j λjaj .
Theorem (Coifman, Latter)
The spaces H1(Rd) and H1atom(Rd) coincide and the norms
‖f‖H1(Rd) and ‖f‖H1atom(Rd) are equivalent.
J. Dziubański (Wrocław) Function Spaces and their Applications
Remark : generalizations to
spaces of homogeneous type: Coifman and Weiss (Bull. AMS1977), Macias and Segovia (Advances in Math. 1979),Uchiyama (TAMS 1980);
homogeneous nilpotent Lie groups: Folland and Stein (1982),Christ and Geller (Duke Math. J. 1984) Riesz transformcharacterization of the space by applying ideas of Uchiyama ofconstructive Fefferman Stein decomposition of BMO functions(if time permits: application of Christ and Geller result to characterize by
relevant Riesz transforms the H1 space associated with the Grushin operator on
Rn+1
L = −∆x − |x|2∂2
∂y2, x ∈ Rn, y ∈ R);
weighted Hardy spaces: Garcia-Cuerva (Dissertationes Math.1979), Stromberg and Torchinsky (1989),
Hardy spaces for non-doubling measures: Mauceri, Meda,Sjogren, Vallarino, Fu, Lin, Yang, Yang,...
J. Dziubański (Wrocław) Function Spaces and their Applications
product Hardy spaces: Chang and R. Fefferman, Merryfiled
Hardy spaces associated with semigroups of linear operatorsTake a semigroup etA of linear operators acting on functionspaces on a metric measure space (X, d, dµ). Consider
H1max = {f : ‖f‖H1
max= ‖ sup
t>0|etAf |‖L1(dµ) <∞}
H1square = {f : ‖f‖H1
max= ‖ sup
t>0|Sf |‖L1(dµ) <∞}
Sf(x) =( ∫∫
d(x,y)<t |t2Aet
2Af(y)|2 dydtµ(B(x,t))|t
)1/2
P. Auscher, F. Bernicot, J. Betancor, A. Bui, W. Czaja, S. Dekel, X.Duong, G. Garrigos, E. Harboure, S. Hofmann, , G. Kerkyacharian,G. Kyriazis, Z.J. Lou, G. Lu, T. Martinez, G. Mauceri, S.Mayboroda, A. McIntosh, S. Meda, D. Mitrea, M. Mitrea, E.Ouhabaz, P. Petrushev, M. Picardello, M. Preisner, F. Ricci, E.Russ, L. Song, A. Sikora, P. Sjogren, J.L. Torrea, D. Yang, L. Yan,J. M. Vallarino, Y. Xu, J. Zhao, J. Zienkiewicz,....
J. Dziubański (Wrocław) Function Spaces and their Applications
Application of local Hardy spaces
D. Goldberg (Duke Math. J. 1979),origin: study harmonic functions in {z = x+ iy : 0 < y < 1}notion of local (real) Hardy space on RN with dx: fix ` > 0,maximal functions:
M`f(x) = sup0<t<`
|e−t√−∆f(x)|, ‖f‖h1
`,√−∆
= ‖M`f‖L1 ,
M`f(x) = sup0<t<`
|et2∆f(x)|, ‖f‖h1`
= ‖M`f‖L1
M(∆−`−2)f(x) = supt>0|et(∆−`−2)f(x)|, ‖f‖H1
(∆−`−2)= ‖M(∆−`−2)f‖L1
‖ supt>` |et2(∆−`−2)f |‖L1 ≤ ‖f‖L1
‖ sup0<t<` |et2(∆−`−2)f − et2∆f |‖L1 ≤ C‖f‖L1 .
local Riesz transforms:Rj,`f = f ∗Rj,`, Rj,` =xj
|x|N+1χB(0,`)(x),‖f‖Riesz, ` = ‖f‖L1 +
∑‖Rj,`f‖L1 .
J. Dziubański (Wrocław) Function Spaces and their Applications
Atomic decomposition: local atoms:a(x) is a local `-atom for h1
` if eithera = |B(y0, `)|−1χB(y0,`)
or there is a ball B = B(y0, r), r ≤ `, such that• supp a ⊂ B• ‖a‖L∞ ≤ |B|−1
•∫a = 0.
All norms are equivalent and with comparing constantsindependent of `.
Important property of h1` : If sup |ϕ| ≤ 1 and sup |∇ϕ| ≤ `−1,
then h1` 3 f 7→ ϕf is bounded on h1
` with bound independent of `.
J. Dziubański (Wrocław) Function Spaces and their Applications
Atomic decomposition: local atoms:a(x) is a local `-atom for h1
` if eithera = |B(y0, `)|−1χB(y0,`)
or there is a ball B = B(y0, r), r ≤ `, such that• supp a ⊂ B• ‖a‖L∞ ≤ |B|−1
•∫a = 0.
All norms are equivalent and with comparing constantsindependent of `.
Important property of h1` : If sup |ϕ| ≤ 1 and sup |∇ϕ| ≤ `−1,
then h1` 3 f 7→ ϕf is bounded on h1
` with bound independent of `.
J. Dziubański (Wrocław) Function Spaces and their Applications
Application to Schrodinger semigroups
Tt = et(∆−V ), V ≥ 0 on Rd.Joint works with Jacek Zienkiewiczcontinued with G.Garrigos, M.Preisner, J.L. Torrea, T.Martinez,...
J. Dziubański (Wrocław) Function Spaces and their Applications
H1∆−V = {f ∈ L1 : ‖f‖H1
∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.
The Feynman-Kac formula:
Ttf(x) = Ex(e−
∫ t0 V (bs) dsf(bt)
), bt(ω) is Brownian motion for et∆.
Ttf(x) = et(∆−V )f(x) =
∫Rd
Kt(x, y)f(y) dy,
0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.
Does the presence of the potential V reflect that the factore−
∫ t0 V (bs) ds gives additional decay for large t such that the
maximal function supt>T |Ttf(x)| is bounded on L1 for fsupported in a compact set K?If so, can one expect local atoms in our atomic decompositions?If d ≥ 3, then bt is transient. Does the factor give an essentialdecay for colorredsmall potentials V ?If d = 1, 2, then bt is recurrent and perhaps even for locallysupported potentials the factor gives sufficient decay.
J. Dziubański (Wrocław) Function Spaces and their Applications
H1∆−V = {f ∈ L1 : ‖f‖H1
∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.
The Feynman-Kac formula:
Ttf(x) = Ex(e−
∫ t0 V (bs) dsf(bt)
), bt(ω) is Brownian motion for et∆.
Ttf(x) = et(∆−V )f(x) =
∫Rd
Kt(x, y)f(y) dy,
0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.
Does the presence of the potential V reflect that the factore−
∫ t0 V (bs) ds gives additional decay for large t such that the
maximal function supt>T |Ttf(x)| is bounded on L1 for fsupported in a compact set K?If so, can one expect local atoms in our atomic decompositions?If d ≥ 3, then bt is transient. Does the factor give an essentialdecay for colorredsmall potentials V ?If d = 1, 2, then bt is recurrent and perhaps even for locallysupported potentials the factor gives sufficient decay.
J. Dziubański (Wrocław) Function Spaces and their Applications
H1∆−V = {f ∈ L1 : ‖f‖H1
∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.
The Feynman-Kac formula:
Ttf(x) = Ex(e−
∫ t0 V (bs) dsf(bt)
), bt(ω) is Brownian motion for et∆.
Ttf(x) = et(∆−V )f(x) =
∫Rd
Kt(x, y)f(y) dy,
0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.
Does the presence of the potential V reflect that the factore−
∫ t0 V (bs) ds gives additional decay for large t such that the
maximal function supt>T |Ttf(x)| is bounded on L1 for fsupported in a compact set K?If so, can one expect local atoms in our atomic decompositions?If d ≥ 3, then bt is transient. Does the factor give an essentialdecay for colorredsmall potentials V ?If d = 1, 2, then bt is recurrent and perhaps even for locallysupported potentials the factor gives sufficient decay.
J. Dziubański (Wrocław) Function Spaces and their Applications
H1∆−V = {f ∈ L1 : ‖f‖H1
∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.
The Feynman-Kac formula:
Ttf(x) = Ex(e−
∫ t0 V (bs) dsf(bt)
), bt(ω) is Brownian motion for et∆.
Ttf(x) = et(∆−V )f(x) =
∫Rd
Kt(x, y)f(y) dy,
0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.
Does the presence of the potential V reflect that the factore−
∫ t0 V (bs) ds gives additional decay for large t such that the
maximal function supt>T |Ttf(x)| is bounded on L1 for fsupported in a compact set K?If so, can one expect local atoms in our atomic decompositions?If d ≥ 3, then bt is transient. Does the factor give an essentialdecay for colorredsmall potentials V ?If d = 1, 2, then bt is recurrent and perhaps even for locallysupported potentials the factor gives sufficient decay.
J. Dziubański (Wrocław) Function Spaces and their Applications
H1∆−V = {f ∈ L1 : ‖f‖H1
∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.
The Feynman-Kac formula:
Ttf(x) = Ex(e−
∫ t0 V (bs) dsf(bt)
), bt(ω) is Brownian motion for et∆.
Ttf(x) = et(∆−V )f(x) =
∫Rd
Kt(x, y)f(y) dy,
0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.
Does the presence of the potential V reflect that the factore−
∫ t0 V (bs) ds gives additional decay for large t such that the
maximal function supt>T |Ttf(x)| is bounded on L1 for fsupported in a compact set K?If so, can one expect local atoms in our atomic decompositions?If d ≥ 3, then bt is transient. Does the factor give an essentialdecay for colorredsmall potentials V ?If d = 1, 2, then bt is recurrent and perhaps even for locallysupported potentials the factor gives sufficient decay.
J. Dziubański (Wrocław) Function Spaces and their Applications
H1∆−V = {f ∈ L1 : ‖f‖H1
∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.
The Feynman-Kac formula:
Ttf(x) = Ex(e−
∫ t0 V (bs) dsf(bt)
), bt(ω) is Brownian motion for et∆.
Ttf(x) = et(∆−V )f(x) =
∫Rd
Kt(x, y)f(y) dy,
0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.
Does the presence of the potential V reflect that the factore−
∫ t0 V (bs) ds gives additional decay for large t such that the
maximal function supt>T |Ttf(x)| is bounded on L1 for fsupported in a compact set K?If so, can one expect local atoms in our atomic decompositions?If d ≥ 3, then bt is transient. Does the factor give an essentialdecay for colorredsmall potentials V ?If d = 1, 2, then bt is recurrent and perhaps even for locallysupported potentials the factor gives sufficient decay.
J. Dziubański (Wrocław) Function Spaces and their Applications
Plan of the talk
Hardy spaces associated with Schodinger operators havinglocal atoms in atomic decompositions
Hardy spaces associated with Schrodinger operators withsmall potentials (all atoms have cancellation property)
Hardy spaces for semigroups with upper and lower Gaussianbounds (remark)
Hardy spaces associated with Grushin operator (Riesztransform characterization)
Hardy spaces for Dunkl operators (kernels have no Gaussianbounds)
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples of Hardy spaces for Schrodinger operatorswith local atoms.
Let Q = {Qj}∞j=1 be a family of closed cubes in Rd with disjointinteriors such that:Rd \
⋃∞j=1Qj is of Lebesgue measure zero;
If Qi, Qj ∈ Q are neighbors, then diameters d(Qi) and d(Qj) arecomparable.Then we may take β > 0 small such that the family {Q∗∗∗∗j }∞j=1
has finite covering property, where Q∗ = (1 + β)Q.
A function a is an H1Q-atom if there exists Q ∈ Q such that
either a = |Q|−11Q
or a is the classical atom with support contained in Q∗ (thatis, ∃Q′ ⊂ Q∗ such that supp a ⊂ Q′,
∫a = 0, |a| ≤ |Q′|−1).
Question. Can one state conditions on V and Q that guaranteeH1
∆−V = H1Q?
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples of Hardy spaces for Schrodinger operatorswith local atoms.
Let Q = {Qj}∞j=1 be a family of closed cubes in Rd with disjointinteriors such that:Rd \
⋃∞j=1Qj is of Lebesgue measure zero;
If Qi, Qj ∈ Q are neighbors, then diameters d(Qi) and d(Qj) arecomparable.Then we may take β > 0 small such that the family {Q∗∗∗∗j }∞j=1
has finite covering property, where Q∗ = (1 + β)Q.
A function a is an H1Q-atom if there exists Q ∈ Q such that
either a = |Q|−11Q
or a is the classical atom with support contained in Q∗ (thatis, ∃Q′ ⊂ Q∗ such that supp a ⊂ Q′,
∫a = 0, |a| ≤ |Q′|−1).
Question. Can one state conditions on V and Q that guaranteeH1
∆−V = H1Q?
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples of Hardy spaces for Schrodinger operatorswith local atoms.
Let Q = {Qj}∞j=1 be a family of closed cubes in Rd with disjointinteriors such that:Rd \
⋃∞j=1Qj is of Lebesgue measure zero;
If Qi, Qj ∈ Q are neighbors, then diameters d(Qi) and d(Qj) arecomparable.Then we may take β > 0 small such that the family {Q∗∗∗∗j }∞j=1
has finite covering property, where Q∗ = (1 + β)Q.
A function a is an H1Q-atom if there exists Q ∈ Q such that
either a = |Q|−11Q
or a is the classical atom with support contained in Q∗ (thatis, ∃Q′ ⊂ Q∗ such that supp a ⊂ Q′,
∫a = 0, |a| ≤ |Q′|−1).
Question. Can one state conditions on V and Q that guaranteeH1
∆−V = H1Q?
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples of Hardy spaces for Schrodinger operatorswith local atoms.
Let Q = {Qj}∞j=1 be a family of closed cubes in Rd with disjointinteriors such that:Rd \
⋃∞j=1Qj is of Lebesgue measure zero;
If Qi, Qj ∈ Q are neighbors, then diameters d(Qi) and d(Qj) arecomparable.Then we may take β > 0 small such that the family {Q∗∗∗∗j }∞j=1
has finite covering property, where Q∗ = (1 + β)Q.
A function a is an H1Q-atom if there exists Q ∈ Q such that
either a = |Q|−11Q
or a is the classical atom with support contained in Q∗ (thatis, ∃Q′ ⊂ Q∗ such that supp a ⊂ Q′,
∫a = 0, |a| ≤ |Q′|−1).
Question. Can one state conditions on V and Q that guaranteeH1
∆−V = H1Q?
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples of Hardy spaces for Schrodinger operatorswith local atoms.
Let Q = {Qj}∞j=1 be a family of closed cubes in Rd with disjointinteriors such that:Rd \
⋃∞j=1Qj is of Lebesgue measure zero;
If Qi, Qj ∈ Q are neighbors, then diameters d(Qi) and d(Qj) arecomparable.Then we may take β > 0 small such that the family {Q∗∗∗∗j }∞j=1
has finite covering property, where Q∗ = (1 + β)Q.
A function a is an H1Q-atom if there exists Q ∈ Q such that
either a = |Q|−11Q
or a is the classical atom with support contained in Q∗ (thatis, ∃Q′ ⊂ Q∗ such that supp a ⊂ Q′,
∫a = 0, |a| ≤ |Q′|−1).
Question. Can one state conditions on V and Q that guaranteeH1
∆−V = H1Q?
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples of Hardy spaces for Schrodinger operatorswith local atoms.
Let Q = {Qj}∞j=1 be a family of closed cubes in Rd with disjointinteriors such that:Rd \
⋃∞j=1Qj is of Lebesgue measure zero;
If Qi, Qj ∈ Q are neighbors, then diameters d(Qi) and d(Qj) arecomparable.Then we may take β > 0 small such that the family {Q∗∗∗∗j }∞j=1
has finite covering property, where Q∗ = (1 + β)Q.
A function a is an H1Q-atom if there exists Q ∈ Q such that
either a = |Q|−11Q
or a is the classical atom with support contained in Q∗ (thatis, ∃Q′ ⊂ Q∗ such that supp a ⊂ Q′,
∫a = 0, |a| ≤ |Q′|−1).
Question. Can one state conditions on V and Q that guaranteeH1
∆−V = H1Q?
J. Dziubański (Wrocław) Function Spaces and their Applications
Let Kt(x, y) be the integral kernel of et(∆−V ).We impose two assumptions on V and Q mainly: (∃C, ε, δ > 0)
supy∈Q∗
∫K2nd(Q)2(x, y)dx ≤ Cn−1−ε for Q ∈ Q, n ∈ N; (D)
∫ 2t
0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C
(t
d(Q)2
)δfor x ∈ Rd, Q ∈ Q, t ≤ d(Q)2,
(K)
hs(x) = (4πs)−d/2e−|x|2/(4s).
Theorem. (J.D. and J. Zienkiewicz).
Assume (D) and (K), then
f ∈ H1∆−V ⇐⇒ f ∈ H1
Q and ‖f‖H1Q. ‖f‖H1
∆−V. ‖f‖H1
Q.
J. Dziubański (Wrocław) Function Spaces and their Applications
Let Kt(x, y) be the integral kernel of et(∆−V ).We impose two assumptions on V and Q mainly: (∃C, ε, δ > 0)
supy∈Q∗
∫K2nd(Q)2(x, y)dx ≤ Cn−1−ε for Q ∈ Q, n ∈ N; (D)
∫ 2t
0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C
(t
d(Q)2
)δfor x ∈ Rd, Q ∈ Q, t ≤ d(Q)2,
(K)
hs(x) = (4πs)−d/2e−|x|2/(4s).
Theorem. (J.D. and J. Zienkiewicz).
Assume (D) and (K), then
f ∈ H1∆−V ⇐⇒ f ∈ H1
Q and ‖f‖H1Q. ‖f‖H1
∆−V. ‖f‖H1
Q.
J. Dziubański (Wrocław) Function Spaces and their Applications
Let Kt(x, y) be the integral kernel of et(∆−V ).We impose two assumptions on V and Q mainly: (∃C, ε, δ > 0)
supy∈Q∗
∫K2nd(Q)2(x, y)dx ≤ Cn−1−ε for Q ∈ Q, n ∈ N; (D)
∫ 2t
0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C
(t
d(Q)2
)δfor x ∈ Rd, Q ∈ Q, t ≤ d(Q)2,
(K)
hs(x) = (4πs)−d/2e−|x|2/(4s).
Theorem. (J.D. and J. Zienkiewicz).
Assume (D) and (K), then
f ∈ H1∆−V ⇐⇒ f ∈ H1
Q and ‖f‖H1Q. ‖f‖H1
∆−V. ‖f‖H1
Q.
J. Dziubański (Wrocław) Function Spaces and their Applications
Let Kt(x, y) be the integral kernel of et(∆−V ).We impose two assumptions on V and Q mainly: (∃C, ε, δ > 0)
supy∈Q∗
∫K2nd(Q)2(x, y)dx ≤ Cn−1−ε for Q ∈ Q, n ∈ N; (D)
∫ 2t
0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C
(t
d(Q)2
)δfor x ∈ Rd, Q ∈ Q, t ≤ d(Q)2,
(K)
hs(x) = (4πs)−d/2e−|x|2/(4s).
Theorem. (J.D. and J. Zienkiewicz).
Assume (D) and (K), then
f ∈ H1∆−V ⇐⇒ f ∈ H1
Q and ‖f‖H1Q. ‖f‖H1
∆−V. ‖f‖H1
Q.
J. Dziubański (Wrocław) Function Spaces and their Applications
Meaning of the conditions (D) and (K).The condition (D)
(∃C, ε > 0) supy∈Q∗
∫K2nd(Q)2(x, y)dx ≤ Cn−1−ε, Q ∈ Q, n ∈ N;
says that the semigroup is ”small”for t > d(Q)2 and y ∈ Q∗.It implies for example that for f such that supp f ⊂ Q∗
‖ supt>d(Q)2
|et(∆−V )f(x)| ‖L1(dx) ≤ C‖f‖L1 ,
J. Dziubański (Wrocław) Function Spaces and their Applications
Meaning of the conditions (D) and (K).The condition (D)
(∃C, ε > 0) supy∈Q∗
∫K2nd(Q)2(x, y)dx ≤ Cn−1−ε, Q ∈ Q, n ∈ N;
says that the semigroup is ”small”for t > d(Q)2 and y ∈ Q∗.It implies for example that for f such that supp f ⊂ Q∗
‖ supt>d(Q)2
|et(∆−V )f(x)| ‖L1(dx) ≤ C‖f‖L1 ,
J. Dziubański (Wrocław) Function Spaces and their Applications
The condition (K)
(∃C, δ > 0)
∫ 2t
0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C
(t
d(Q)2
)δfor x ∈ Rd, Q ∈ Q,t ≤ d(Q)2,
implies that the semigroup {et(∆−V )}t>0 is close to the heatsemigroup et∆ for 0 < t < d(Q)2 when we act by these semigroupson functions f supported by Q∗:
‖ sup0<t<d(Q)2
|(et(∆−V ) − et∆)f(x)| ‖L1 ≤ C‖f‖L1
J. Dziubański (Wrocław) Function Spaces and their Applications
The condition (K)
(∃C, δ > 0)
∫ 2t
0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C
(t
d(Q)2
)δfor x ∈ Rd, Q ∈ Q,t ≤ d(Q)2,
implies that the semigroup {et(∆−V )}t>0 is close to the heatsemigroup et∆ for 0 < t < d(Q)2 when we act by these semigroupson functions f supported by Q∗:
‖ sup0<t<d(Q)2
|(et(∆−V ) − et∆)f(x)| ‖L1 ≤ C‖f‖L1
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples.
• d ≥ 3, V satisfies the reverse Holder inequality with exponentq > d/2, that is,(
1
|B|
∫BV (y)q dy
)1/q
≤ C 1
|B|
∫BV (y) dy for every ball B.
Define Q by: Q ∈ Q if and only if Q is the maximal dyadic cubefor which
d(Q)2|Q|−1
∫QV (y) dy ≤ 1.
Then the conditions (D) and (K) are true.Every nonnegative polynomial satisfies the reverse Holder inequalityThe integral appeared in the article of C. Fefferman (joint workwith D.H. Phong) about the uncertainty principle (Bull. AMS1983). Then it was used by Z. Shen in his work aboutboundedness of the Riesz transforms ∂xjL
−1/2 on Lp, p > 1.
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples.
• d ≥ 3, V satisfies the reverse Holder inequality with exponentq > d/2, that is,(
1
|B|
∫BV (y)q dy
)1/q
≤ C 1
|B|
∫BV (y) dy for every ball B.
Define Q by: Q ∈ Q if and only if Q is the maximal dyadic cubefor which
d(Q)2|Q|−1
∫QV (y) dy ≤ 1.
Then the conditions (D) and (K) are true.Every nonnegative polynomial satisfies the reverse Holder inequalityThe integral appeared in the article of C. Fefferman (joint workwith D.H. Phong) about the uncertainty principle (Bull. AMS1983). Then it was used by Z. Shen in his work aboutboundedness of the Riesz transforms ∂xjL
−1/2 on Lp, p > 1.
J. Dziubański (Wrocław) Function Spaces and their Applications
Examples.
• d ≥ 3, V satisfies the reverse Holder inequality with exponentq > d/2, that is,(
1
|B|
∫BV (y)q dy
)1/q
≤ C 1
|B|
∫BV (y) dy for every ball B.
Define Q by: Q ∈ Q if and only if Q is the maximal dyadic cubefor which
d(Q)2|Q|−1
∫QV (y) dy ≤ 1.
Then the conditions (D) and (K) are true.Every nonnegative polynomial satisfies the reverse Holder inequalityThe integral appeared in the article of C. Fefferman (joint workwith D.H. Phong) about the uncertainty principle (Bull. AMS1983). Then it was used by Z. Shen in his work aboutboundedness of the Riesz transforms ∂xjL
−1/2 on Lp, p > 1.
J. Dziubański (Wrocław) Function Spaces and their Applications
• The Hardy space H1L associated with one-dimensional
Schrodinger operator −L was studied in Czaja-Zienkiewicz.For any nonnegative V ∈ L1
loc(R) the collection Q of maximaldyadic intervals Q of R that are defined by the stopping timecondition
|Q|∫
16QV (y) dy ≤ 1, (1)
fulfils (D) and (K) for certain small β > 0.
J. Dziubański (Wrocław) Function Spaces and their Applications
• The Hardy space H1L associated with one-dimensional
Schrodinger operator −L was studied in Czaja-Zienkiewicz.For any nonnegative V ∈ L1
loc(R) the collection Q of maximaldyadic intervals Q of R that are defined by the stopping timecondition
|Q|∫
16QV (y) dy ≤ 1, (1)
fulfils (D) and (K) for certain small β > 0.
J. Dziubański (Wrocław) Function Spaces and their Applications
• V (x) = |x|−2, d ≥ 3. Then for Q being the Whitneydecomposition of Rd \ {0} that consists of dyadic cubes theconditions (D) and (K) hold.
J. Dziubański (Wrocław) Function Spaces and their Applications
More examples.Observation 1. For ` > 0 let Q`(Rn) be a covering of Rn bycubes of length `. Assume that V1 na Rd and Q conditions (D) i(K) hold.Consider potential V (x1, x2) = V1(x1), x1 ∈ Rd, x2 ∈ Rn, andfamily of cubes Q = {Q1 ×Q2 : Q1 ∈ Q, Q2 ∈ Qd(Q1)(Rn)} onRd+n.Then the pair (V, Q) satisfies (D) i (K).
J. Dziubański (Wrocław) Function Spaces and their Applications
More examples.Observation 1. For ` > 0 let Q`(Rn) be a covering of Rn bycubes of length `. Assume that V1 na Rd and Q conditions (D) i(K) hold.Consider potential V (x1, x2) = V1(x1), x1 ∈ Rd, x2 ∈ Rn, andfamily of cubes Q = {Q1 ×Q2 : Q1 ∈ Q, Q2 ∈ Qd(Q1)(Rn)} onRd+n.Then the pair (V, Q) satisfies (D) i (K).
J. Dziubański (Wrocław) Function Spaces and their Applications
More examples.Observation 1. For ` > 0 let Q`(Rn) be a covering of Rn bycubes of length `. Assume that V1 na Rd and Q conditions (D) i(K) hold.Consider potential V (x1, x2) = V1(x1), x1 ∈ Rd, x2 ∈ Rn, andfamily of cubes Q = {Q1 ×Q2 : Q1 ∈ Q, Q2 ∈ Qd(Q1)(Rn)} onRd+n.Then the pair (V, Q) satisfies (D) i (K).
J. Dziubański (Wrocław) Function Spaces and their Applications
More examples.Observation 1. For ` > 0 let Q`(Rn) be a covering of Rn bycubes of length `. Assume that V1 na Rd and Q conditions (D) i(K) hold.Consider potential V (x1, x2) = V1(x1), x1 ∈ Rd, x2 ∈ Rn, andfamily of cubes Q = {Q1 ×Q2 : Q1 ∈ Q, Q2 ∈ Qd(Q1)(Rn)} onRd+n.Then the pair (V, Q) satisfies (D) i (K).
J. Dziubański (Wrocław) Function Spaces and their Applications
More examples.Observation 1. For ` > 0 let Q`(Rn) be a covering of Rn bycubes of length `. Assume that V1 na Rd and Q conditions (D) i(K) hold.Consider potential V (x1, x2) = V1(x1), x1 ∈ Rd, x2 ∈ Rn, andfamily of cubes Q = {Q1 ×Q2 : Q1 ∈ Q, Q2 ∈ Qd(Q1)(Rn)} onRd+n.Then the pair (V, Q) satisfies (D) i (K).
J. Dziubański (Wrocław) Function Spaces and their Applications
Proof of Observation 1 follows from the fact that the integralkernel for the semigroup generated byL = ∆− V1(x1) = ∆Rd + ∆Rn − V1(x1) is the product of integralkernels kernels of semigroups generated by L1 = ∆Rd − V1(x1) onRd and ∆Rn on Rn;
Tt((x1, x2); (y1, y2)) = T 1t (x1; y1)ht(x2 − y2),
which satisfy (D) i (K).
J. Dziubański (Wrocław) Function Spaces and their Applications
Proof of Observation 1 follows from the fact that the integralkernel for the semigroup generated byL = ∆− V1(x1) = ∆Rd + ∆Rn − V1(x1) is the product of integralkernels kernels of semigroups generated by L1 = ∆Rd − V1(x1) onRd and ∆Rn on Rn;
Tt((x1, x2); (y1, y2)) = T 1t (x1; y1)ht(x2 − y2),
which satisfy (D) i (K).
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Observation 2. Let V1, V2 be nonnegative potentials on Rd,which together with families of cubes Q1 i Q2 satisfy (D) i (K).Assume additionally that Q1 i Q2 consist of dyadic cubes.For Q1 ∈ Q1, Q2 ∈ Q2 the cubes are either disjoint or onecontains another. Let Q1 ∧Q2 denotes the smaller one.Then the family Q = {Q1 ∧Q2, Q1 ∈ Q1, Q2 ∈ Q2} covers Rdand satisfy (D) i (K) for V = V1 + V2.Proof. Let Tt(x, y), T 1
t (x, y), T 2t (x, y) be the integral kernels for
Schrodinger semigroups with potentials V, V1, V2 respectively.Then Tt(x, y) ≤ min(T 1
t (x, y), T 2t (x, y)).
Assume that Q1 = Q1 ∧Q2. Then (K) is trivial.We check (D):
supy∈Q∗1
∫T2nd(Q1)2(x, y) dy ≤ sup
y∈Q∗1
∫T 1
2nd(Q1)2(x, y) dy ≤ Cn−1−ε.
J. Dziubański (Wrocław) Function Spaces and their Applications
Thoerem (J.D. and M.Preisner)
In the case of (D) and (K) the space H1∆−V is characterized by the
Riesz transforms Rj∂xj (−∆ + V )−1/2, that is,
‖f‖H1∆−V∼ ‖f‖L1 +
d∑j=1
‖Rjf‖L1 .
J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces for small potentials
The smallest nontrivial non-negative potential is a compactlysupported bounded function V .Let L = −∆ + V in Rd, d ≥ 3, V ∈ Cc(Rd).
With J. Zienkiewicz we showed that there is a function w such that0 < δ ≤ w ≤ 1, et(∆−V )w = w (w is L-harmonic )and the mapping
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms.In other words: every atom satisfies the cancellation condition∫a(x)w(x) dx = 0.
E. Stein asked question: For which potentials V ≥ 0 on Rd, d ≥ 3there is a function w such that 0 < δ ≤ w ≤ 1 such that
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms?J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces for small potentials
The smallest nontrivial non-negative potential is a compactlysupported bounded function V .Let L = −∆ + V in Rd, d ≥ 3, V ∈ Cc(Rd).
With J. Zienkiewicz we showed that there is a function w such that0 < δ ≤ w ≤ 1, et(∆−V )w = w (w is L-harmonic )and the mapping
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms.In other words: every atom satisfies the cancellation condition∫a(x)w(x) dx = 0.
E. Stein asked question: For which potentials V ≥ 0 on Rd, d ≥ 3there is a function w such that 0 < δ ≤ w ≤ 1 such that
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms?J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces for small potentials
The smallest nontrivial non-negative potential is a compactlysupported bounded function V .Let L = −∆ + V in Rd, d ≥ 3, V ∈ Cc(Rd).
With J. Zienkiewicz we showed that there is a function w such that0 < δ ≤ w ≤ 1, et(∆−V )w = w (w is L-harmonic )and the mapping
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms.In other words: every atom satisfies the cancellation condition∫a(x)w(x) dx = 0.
E. Stein asked question: For which potentials V ≥ 0 on Rd, d ≥ 3there is a function w such that 0 < δ ≤ w ≤ 1 such that
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms?J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces for small potentials
The smallest nontrivial non-negative potential is a compactlysupported bounded function V .Let L = −∆ + V in Rd, d ≥ 3, V ∈ Cc(Rd).
With J. Zienkiewicz we showed that there is a function w such that0 < δ ≤ w ≤ 1, et(∆−V )w = w (w is L-harmonic )and the mapping
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms.In other words: every atom satisfies the cancellation condition∫a(x)w(x) dx = 0.
E. Stein asked question: For which potentials V ≥ 0 on Rd, d ≥ 3there is a function w such that 0 < δ ≤ w ≤ 1 such that
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms?J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces for small potentials
The smallest nontrivial non-negative potential is a compactlysupported bounded function V .Let L = −∆ + V in Rd, d ≥ 3, V ∈ Cc(Rd).
With J. Zienkiewicz we showed that there is a function w such that0 < δ ≤ w ≤ 1, et(∆−V )w = w (w is L-harmonic )and the mapping
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms.In other words: every atom satisfies the cancellation condition∫a(x)w(x) dx = 0.
E. Stein asked question: For which potentials V ≥ 0 on Rd, d ≥ 3there is a function w such that 0 < δ ≤ w ≤ 1 such that
H1L 3 f 7→ wf ∈ H1(Rd)
is an isomorphisms?J. Dziubański (Wrocław) Function Spaces and their Applications
Assume that d ≥ 3, V ≥ 0, V ∈ L1loc(Rd), Kt(x, y) kernels of
et(∆−V ).
Theorem. J.D. J. Zienkiewicz
The following are equivalent:(a) There is an ∆− V -harmonic function w: 0 < c ≤ w(x) ≤ C;(b)
∫Kt(x, y) dy ≥ δ > 0
(c) The global Kato norm
‖V ‖K = supx∈Rd
∫V (y)
|x− y|d−2dy = ‖∆−1V ‖L∞ is bounded;
(d) ct−d/2e−C|x−y|2/t ≤ Kt(x, y) (Gaussian lower bounds).
(e) there is a function 0 < δ ≤ w(x) ≤ 1 such thatH1
∆−V 3 f 7→ w(x)f(x) ∈ H1(Rd) is an isomorphism.
J. Dziubański (Wrocław) Function Spaces and their Applications
Theorem (J.D. & J.Z.)
Assume that ‖V ‖K <∞, d ≥ 3.Then the operators
L1/2(−∆)−1/2 and (−∆)1/2L−1/2
are bounded on L1.Moreover,
H1L 3 f 7→ (−∆)1/2L−1/2f ∈ H1(Rd)
is isomorphism of the Hardy spaces with the inverse
H1(Rd) 3 f 7→ L1/2(−∆)−1/2f ∈ H1L.
J. Dziubański (Wrocław) Function Spaces and their Applications
Corollary (J.D. & J.Z.)
The Hardy space H1L admits characterization by the Riesz
transforms Rj = ∂xjL−1/2, that is, an L1 function f belongs to
the Hardy space H1L if and only if Rjf ∈ L1.
Proof. Based on:
(∂xj (−∆)−1/2)︸ ︷︷ ︸(classical Riesz transform)
(−∆)1/2L−1/2︸ ︷︷ ︸(isomorphismH1
L→ H1(Rd))
= ∂xjL−1/2︸ ︷︷ ︸
(Riesz transform for L)
(∂xjL−1/2)︸ ︷︷ ︸
(Riesz transform for L)
(L1/2(−∆)−1/2)︸ ︷︷ ︸(isomorphismH1(Rd)→ H1
L)
= ∂xj (−∆)−1/2︸ ︷︷ ︸(classical Riesz transform)
J. Dziubański (Wrocław) Function Spaces and their Applications
Proof: ∂xjL−1/2f ∈ L1 implies f ∈ H1
L.
(∂xj (−∆)−1/2)︸ ︷︷ ︸(classical Riesz transform)
(−∆)1/2L−1/2︸ ︷︷ ︸(isomorphismH1
L→ H1(Rd))
= ∂xjL−1/2︸ ︷︷ ︸
(Riesz transform for L)
Assume that∂xjL
−1/2f ∈ L1 j = 1, ..., d,
then(−∆)1/2L−1/2f ∈ H1(Rd),
so f ∈ H1L, because (∂xj (−∆)−1/2) is an isomorphisms.
J. Dziubański (Wrocław) Function Spaces and their Applications
Proof: ∂xjL−1/2f ∈ L1 implies f ∈ H1
L.
(∂xj (−∆)−1/2)︸ ︷︷ ︸(classical Riesz transform)
(−∆)1/2L−1/2︸ ︷︷ ︸(isomorphismH1
L→ H1(Rd))
= ∂xjL−1/2︸ ︷︷ ︸
(Riesz transform for L)
Assume that∂xjL
−1/2f ∈ L1 j = 1, ..., d,
then(−∆)1/2L−1/2f ∈ H1(Rd),
so f ∈ H1L, because (∂xj (−∆)−1/2) is an isomorphisms.
J. Dziubański (Wrocław) Function Spaces and their Applications
Proof: ∂xjL−1/2f ∈ L1 implies f ∈ H1
L.
(∂xj (−∆)−1/2)︸ ︷︷ ︸(classical Riesz transform)
(−∆)1/2L−1/2︸ ︷︷ ︸(isomorphismH1
L→ H1(Rd))
= ∂xjL−1/2︸ ︷︷ ︸
(Riesz transform for L)
Assume that∂xjL
−1/2f ∈ L1 j = 1, ..., d,
then(−∆)1/2L−1/2f ∈ H1(Rd),
so f ∈ H1L, because (∂xj (−∆)−1/2) is an isomorphisms.
J. Dziubański (Wrocław) Function Spaces and their Applications
The case of compactly supported potentials in dimension 2.
Assume that V is a nonnegative nonzero compactly supportedC2-function in R2.
Theorem (J.D. & J.Z.)
There exists a regular L-harmonic weight w such that
C−1 ln(2 + |x|) ≤ w(x) ≤ C ln(2 + |x|),
|∇w(x)| ≤ C(1 + |x|)−1,
and the space H1L,max admits atomic decomposition with atoms
satisfying: supp a ⊂ B, ‖a‖∞ ≤ |B|−1,∫a(x)w(x) dx = 0.
J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces for semigroups with Gaussian bounds
Considered by many authors: B. Ahn, P. Auscher, F. Bernicot, S.Dekel, X.T. Duong, G. Garrigos, S. Hofmann, J. Huang, G.Kerkyacharian, G. Kyriazis, G. Liu, Y. Liu, T. Martinez, A.McIntosh, S. Mayboroda, D. Mitrea, M. Mitrea, P. Petrushev, M.Preisner, J. Torrea, L. Yan, Y. Xu, D. Yang, J. Zhao, ...
J. Dziubański (Wrocław) Function Spaces and their Applications
(X, d(x, y)µ) is a metric space with doubling Borel measure:µ(B(x, 2r)) ≤ Cµ(B(x, t)) such that for very x ∈ X(0,∞) : t 7→ µ(B(x, t)) ∈ (0,∞) is 1-1 and onto
e−tLf(x) =∫X Kt(x, y)f(y) dµ(y) semigroup on L2(µ), L∗ = L;
c
µ(B(x,√t))e−Cd(x,y)2/t ≤ Kt(x, y) ≤ C
µ(B(x,√t)e−cd(x,y)2/t
Theorem, J.D. and M. Preisner:
There is a function 0 < δ ≤ w(x) ≤ 1 such that the Hardy spaceH1L,max admits atomic decomposition with atoms:• supp a ⊂ B(y0, r);• ‖a‖∞ ≤ µ(B(y0, r))
−1
•∫X a(x)w(x) dµ(x) = 0.
We do not assume any regularity condition on Kt(x, y)!
J. Dziubański (Wrocław) Function Spaces and their Applications
(X, d(x, y)µ) is a metric space with doubling Borel measure:µ(B(x, 2r)) ≤ Cµ(B(x, t)) such that for very x ∈ X(0,∞) : t 7→ µ(B(x, t)) ∈ (0,∞) is 1-1 and onto
e−tLf(x) =∫X Kt(x, y)f(y) dµ(y) semigroup on L2(µ), L∗ = L;
c
µ(B(x,√t))e−Cd(x,y)2/t ≤ Kt(x, y) ≤ C
µ(B(x,√t)e−cd(x,y)2/t
Theorem, J.D. and M. Preisner:
There is a function 0 < δ ≤ w(x) ≤ 1 such that the Hardy spaceH1L,max admits atomic decomposition with atoms:• supp a ⊂ B(y0, r);• ‖a‖∞ ≤ µ(B(y0, r))
−1
•∫X a(x)w(x) dµ(x) = 0.
We do not assume any regularity condition on Kt(x, y)!
J. Dziubański (Wrocław) Function Spaces and their Applications
(X, d(x, y)µ) is a metric space with doubling Borel measure:µ(B(x, 2r)) ≤ Cµ(B(x, t)) such that for very x ∈ X(0,∞) : t 7→ µ(B(x, t)) ∈ (0,∞) is 1-1 and onto
e−tLf(x) =∫X Kt(x, y)f(y) dµ(y) semigroup on L2(µ), L∗ = L;
c
µ(B(x,√t))e−Cd(x,y)2/t ≤ Kt(x, y) ≤ C
µ(B(x,√t)e−cd(x,y)2/t
Theorem, J.D. and M. Preisner:
There is a function 0 < δ ≤ w(x) ≤ 1 such that the Hardy spaceH1L,max admits atomic decomposition with atoms:• supp a ⊂ B(y0, r);• ‖a‖∞ ≤ µ(B(y0, r))
−1
•∫X a(x)w(x) dµ(x) = 0.
We do not assume any regularity condition on Kt(x, y)!
J. Dziubański (Wrocław) Function Spaces and their Applications
The Gaussian bounds ⇒ ∃ 0 < δ ≤ w(x) ≤ 1 and α > 0:
∣∣∣ Kt(x, y)
w(x)w(y)− Kt(x, z)
w(x)w(z)
∣∣∣ . µ(B(x,√t))−1
(d(y, z)√t
)αexp−
cd(x,y)2
t
whenever d(y, z) ≤√t and
∫X Kt(x, y) dµ(y) = w(x).
The proof uses spectral functional calculi for L.So, we may apply Uchiyama’s theorem about maximal functioncharacterization of atomic Hardy spaces on spaces of homogeneoustype by introducing the new kernel and space:
Tt(x, y) =Kt(x, y)
w(x)w(y)on the space (X, d(x, y), w(x)2dµ(x)),
∫XTt(x, y)w(y)2 dµ(y) = 1.
J. Dziubański (Wrocław) Function Spaces and their Applications
The Gaussian bounds ⇒ ∃ 0 < δ ≤ w(x) ≤ 1 and α > 0:
∣∣∣ Kt(x, y)
w(x)w(y)− Kt(x, z)
w(x)w(z)
∣∣∣ . µ(B(x,√t))−1
(d(y, z)√t
)αexp−
cd(x,y)2
t
whenever d(y, z) ≤√t and
∫X Kt(x, y) dµ(y) = w(x).
The proof uses spectral functional calculi for L.So, we may apply Uchiyama’s theorem about maximal functioncharacterization of atomic Hardy spaces on spaces of homogeneoustype by introducing the new kernel and space:
Tt(x, y) =Kt(x, y)
w(x)w(y)on the space (X, d(x, y), w(x)2dµ(x)),
∫XTt(x, y)w(y)2 dµ(y) = 1.
J. Dziubański (Wrocław) Function Spaces and their Applications
So,f ∈ H1
L,max ⇔ supt>0
∣∣∣ ∫X Kt(x,y)w(x)w(y)
f(y)w(y)w(y)2 dµ(y)
∣∣∣ ∈ L1(dµ)︸ ︷︷ ︸L1(w(x)2dµ)
f(x)
w(x)∈ H1
atom(X, d,w(x)2dµ(x))
f(x)
w(x)=∑
λj aj(x)
with aj(x) atoms satisfying∫aj(x)w(x)2 dµ(x) = 0
f(x) =∑j
λj aj(x)w(x) =∑j
λjaj(x), aj(x) = aj(x)w(x),
∫aj(x)w(x) dµ(x) =
∫aj(x)w(x)w(x) dµ(x) = 0.
J. Dziubański (Wrocław) Function Spaces and their Applications
So,f ∈ H1
L,max ⇔ supt>0
∣∣∣ ∫X Kt(x,y)w(x)w(y)
f(y)w(y)w(y)2 dµ(y)
∣∣∣ ∈ L1(dµ)︸ ︷︷ ︸L1(w(x)2dµ)
f(x)
w(x)∈ H1
atom(X, d,w(x)2dµ(x))
f(x)
w(x)=∑
λj aj(x)
with aj(x) atoms satisfying∫aj(x)w(x)2 dµ(x) = 0
f(x) =∑j
λj aj(x)w(x) =∑j
λjaj(x), aj(x) = aj(x)w(x),
∫aj(x)w(x) dµ(x) =
∫aj(x)w(x)w(x) dµ(x) = 0.
J. Dziubański (Wrocław) Function Spaces and their Applications
So,f ∈ H1
L,max ⇔ supt>0
∣∣∣ ∫X Kt(x,y)w(x)w(y)
f(y)w(y)w(y)2 dµ(y)
∣∣∣ ∈ L1(dµ)︸ ︷︷ ︸L1(w(x)2dµ)
f(x)
w(x)∈ H1
atom(X, d,w(x)2dµ(x))
f(x)
w(x)=∑
λj aj(x)
with aj(x) atoms satisfying∫aj(x)w(x)2 dµ(x) = 0
f(x) =∑j
λj aj(x)w(x) =∑j
λjaj(x), aj(x) = aj(x)w(x),
∫aj(x)w(x) dµ(x) =
∫aj(x)w(x)w(x) dµ(x) = 0.
J. Dziubański (Wrocław) Function Spaces and their Applications
So,f ∈ H1
L,max ⇔ supt>0
∣∣∣ ∫X Kt(x,y)w(x)w(y)
f(y)w(y)w(y)2 dµ(y)
∣∣∣ ∈ L1(dµ)︸ ︷︷ ︸L1(w(x)2dµ)
f(x)
w(x)∈ H1
atom(X, d,w(x)2dµ(x))
f(x)
w(x)=∑
λj aj(x)
with aj(x) atoms satisfying∫aj(x)w(x)2 dµ(x) = 0
f(x) =∑j
λj aj(x)w(x) =∑j
λjaj(x), aj(x) = aj(x)w(x),
∫aj(x)w(x) dµ(x) =
∫aj(x)w(x)w(x) dµ(x) = 0.
J. Dziubański (Wrocław) Function Spaces and their Applications
Grushin operator
L = −∆x−|x|2∂2
∂x′2, x = (x, x′) = (x1, ..., xn, x
′) ∈ Rn×R = Rn+1
ρ(x,y) control distance for ∂x1 , ..., ∂xn , x1∂x′ , ..., xn∂x′
(Rn+1, ρ(x,y), dx) space of homogeneous typeMLf(x) = supt>0 |e−tLf(x)| max function for the heat semigroupH1L = {f ∈ L1 : ‖f‖H1
L= ‖MLf‖L1 <∞}
Riesz transforms: Rj = ∂xjL−1/2, Rj+n = xj∂x′L
−1/2,j = 1, ..., n.
Theorem (J.D. and K. Jotsaroop).
The space H1L admits atomic decomposition with atoms for the
space of homogeneous type.The Riesz transforms characterizes the space H1
L:‖f‖H1
L∼ ‖f‖L1 +
∑2nk=1 ‖Rkf‖L1 .
J. Dziubański (Wrocław) Function Spaces and their Applications
Grushin operator
L = −∆x−|x|2∂2
∂x′2, x = (x, x′) = (x1, ..., xn, x
′) ∈ Rn×R = Rn+1
ρ(x,y) control distance for ∂x1 , ..., ∂xn , x1∂x′ , ..., xn∂x′
(Rn+1, ρ(x,y), dx) space of homogeneous typeMLf(x) = supt>0 |e−tLf(x)| max function for the heat semigroupH1L = {f ∈ L1 : ‖f‖H1
L= ‖MLf‖L1 <∞}
Riesz transforms: Rj = ∂xjL−1/2, Rj+n = xj∂x′L
−1/2,j = 1, ..., n.
Theorem (J.D. and K. Jotsaroop).
The space H1L admits atomic decomposition with atoms for the
space of homogeneous type.The Riesz transforms characterizes the space H1
L:‖f‖H1
L∼ ‖f‖L1 +
∑2nk=1 ‖Rkf‖L1 .
J. Dziubański (Wrocław) Function Spaces and their Applications
Grushin operator
L = −∆x−|x|2∂2
∂x′2, x = (x, x′) = (x1, ..., xn, x
′) ∈ Rn×R = Rn+1
ρ(x,y) control distance for ∂x1 , ..., ∂xn , x1∂x′ , ..., xn∂x′
(Rn+1, ρ(x,y), dx) space of homogeneous typeMLf(x) = supt>0 |e−tLf(x)| max function for the heat semigroupH1L = {f ∈ L1 : ‖f‖H1
L= ‖MLf‖L1 <∞}
Riesz transforms: Rj = ∂xjL−1/2, Rj+n = xj∂x′L
−1/2,j = 1, ..., n.
Theorem (J.D. and K. Jotsaroop).
The space H1L admits atomic decomposition with atoms for the
space of homogeneous type.The Riesz transforms characterizes the space H1
L:‖f‖H1
L∼ ‖f‖L1 +
∑2nk=1 ‖Rkf‖L1 .
J. Dziubański (Wrocław) Function Spaces and their Applications
Grushin operator
L = −∆x−|x|2∂2
∂x′2, x = (x, x′) = (x1, ..., xn, x
′) ∈ Rn×R = Rn+1
ρ(x,y) control distance for ∂x1 , ..., ∂xn , x1∂x′ , ..., xn∂x′
(Rn+1, ρ(x,y), dx) space of homogeneous typeMLf(x) = supt>0 |e−tLf(x)| max function for the heat semigroupH1L = {f ∈ L1 : ‖f‖H1
L= ‖MLf‖L1 <∞}
Riesz transforms: Rj = ∂xjL−1/2, Rj+n = xj∂x′L
−1/2,j = 1, ..., n.
Theorem (J.D. and K. Jotsaroop).
The space H1L admits atomic decomposition with atoms for the
space of homogeneous type.The Riesz transforms characterizes the space H1
L:‖f‖H1
L∼ ‖f‖L1 +
∑2nk=1 ‖Rkf‖L1 .
J. Dziubański (Wrocław) Function Spaces and their Applications
Two proofs of the Riesz transform characterization of H1
• subharmonicity + maximum principle for subharmonic functions• Uchiyama’s constructive Fefferman-Stein decomposition ofBMO(Rd)Riesz transform characterization implies the Fefferman-Steindecomposition of BMO functions:every g ∈ VMO can be written g = g0 +
∑dj=1Rjgj
with∑d
j=0 ‖gj‖∞ ≤ C‖g‖BM0
Uchiyama did opposite: proved the Fefferman-Stein decompositionof VMO functions and then concluded the Riesz transformcharacterization of H1(Rd). Actually he characterized systems(R0, R1, ..., Rm) of singular integrals which characterize H1(Rd).
Ideas of Uchiyama were extended on nilpotent Lie groups by Christand Geller (Duke Math.J. 1984).
J. Dziubański (Wrocław) Function Spaces and their Applications
Two proofs of the Riesz transform characterization of H1
• subharmonicity + maximum principle for subharmonic functions• Uchiyama’s constructive Fefferman-Stein decomposition ofBMO(Rd)Riesz transform characterization implies the Fefferman-Steindecomposition of BMO functions:every g ∈ VMO can be written g = g0 +
∑dj=1Rjgj
with∑d
j=0 ‖gj‖∞ ≤ C‖g‖BM0
Uchiyama did opposite: proved the Fefferman-Stein decompositionof VMO functions and then concluded the Riesz transformcharacterization of H1(Rd). Actually he characterized systems(R0, R1, ..., Rm) of singular integrals which characterize H1(Rd).
Ideas of Uchiyama were extended on nilpotent Lie groups by Christand Geller (Duke Math.J. 1984).
J. Dziubański (Wrocław) Function Spaces and their Applications
Two proofs of the Riesz transform characterization of H1
• subharmonicity + maximum principle for subharmonic functions• Uchiyama’s constructive Fefferman-Stein decomposition ofBMO(Rd)Riesz transform characterization implies the Fefferman-Steindecomposition of BMO functions:every g ∈ VMO can be written g = g0 +
∑dj=1Rjgj
with∑d
j=0 ‖gj‖∞ ≤ C‖g‖BM0
Uchiyama did opposite: proved the Fefferman-Stein decompositionof VMO functions and then concluded the Riesz transformcharacterization of H1(Rd). Actually he characterized systems(R0, R1, ..., Rm) of singular integrals which characterize H1(Rd).
Ideas of Uchiyama were extended on nilpotent Lie groups by Christand Geller (Duke Math.J. 1984).
J. Dziubański (Wrocław) Function Spaces and their Applications
Two proofs of the Riesz transform characterization of H1
• subharmonicity + maximum principle for subharmonic functions• Uchiyama’s constructive Fefferman-Stein decomposition ofBMO(Rd)Riesz transform characterization implies the Fefferman-Steindecomposition of BMO functions:every g ∈ VMO can be written g = g0 +
∑dj=1Rjgj
with∑d
j=0 ‖gj‖∞ ≤ C‖g‖BM0
Uchiyama did opposite: proved the Fefferman-Stein decompositionof VMO functions and then concluded the Riesz transformcharacterization of H1(Rd). Actually he characterized systems(R0, R1, ..., Rm) of singular integrals which characterize H1(Rd).
Ideas of Uchiyama were extended on nilpotent Lie groups by Christand Geller (Duke Math.J. 1984).
J. Dziubański (Wrocław) Function Spaces and their Applications
First we prove Fefferman-Stein decomposition of compactlysupported BMO(Rn+1, ρ, dx) functions:
g = g0 +
2n∑k=1
Rkgk, with2n∑k=0
‖gk‖∞ < C‖g‖BMO, gj ∈ L2
To this end we relate L with ∆ = −∑n
j=1(X2j + Y 2
j ) on theHeisenberg group Hn via the unitary representation
π(a,b,t)f(x) = π(a,b,t)f(x, x′) = f(x+ b, x′ + t+1
2a · b+ a · x),
π∆ = L, πXj = xj∂x′ , πYj = ∂xj , πXj(∆)−1/2 = xj∂x′L−1/2, ..
and use approach of Christ and Geller (Singular integralcharacterizations of Hardy spaces on homogeneous groups, DukeMath. J. 1984) based on Uchiyama’s ideas of constructiveFefferman-Stein decomposition BMO functions.Rothschild and Stein, Acta 1976.
J. Dziubański (Wrocław) Function Spaces and their Applications
First we prove Fefferman-Stein decomposition of compactlysupported BMO(Rn+1, ρ, dx) functions:
g = g0 +
2n∑k=1
Rkgk, with2n∑k=0
‖gk‖∞ < C‖g‖BMO, gj ∈ L2
To this end we relate L with ∆ = −∑n
j=1(X2j + Y 2
j ) on theHeisenberg group Hn via the unitary representation
π(a,b,t)f(x) = π(a,b,t)f(x, x′) = f(x+ b, x′ + t+1
2a · b+ a · x),
π∆ = L, πXj = xj∂x′ , πYj = ∂xj , πXj(∆)−1/2 = xj∂x′L−1/2, ..
and use approach of Christ and Geller (Singular integralcharacterizations of Hardy spaces on homogeneous groups, DukeMath. J. 1984) based on Uchiyama’s ideas of constructiveFefferman-Stein decomposition BMO functions.Rothschild and Stein, Acta 1976.
J. Dziubański (Wrocław) Function Spaces and their Applications
First we prove Fefferman-Stein decomposition of compactlysupported BMO(Rn+1, ρ, dx) functions:
g = g0 +
2n∑k=1
Rkgk, with2n∑k=0
‖gk‖∞ < C‖g‖BMO, gj ∈ L2
To this end we relate L with ∆ = −∑n
j=1(X2j + Y 2
j ) on theHeisenberg group Hn via the unitary representation
π(a,b,t)f(x) = π(a,b,t)f(x, x′) = f(x+ b, x′ + t+1
2a · b+ a · x),
π∆ = L, πXj = xj∂x′ , πYj = ∂xj , πXj(∆)−1/2 = xj∂x′L−1/2, ..
and use approach of Christ and Geller (Singular integralcharacterizations of Hardy spaces on homogeneous groups, DukeMath. J. 1984) based on Uchiyama’s ideas of constructiveFefferman-Stein decomposition BMO functions.Rothschild and Stein, Acta 1976.
J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces in the Dunkl setting
Dunkl = deformation of Fourier
Parameter : k≥0 k=0RN 3 x Euclidean space, for 0 6= α ∈ RN ,
σα(x)= x− 2〈α, x〉|α|2
α reflection in the hyperplane α⊥
R root system: finite set of vectors α 6= 0: σα(R) = R, ∀ α∈RG (Coxeter) group generated by reflections σα, α ∈ R.
J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces in the Dunkl setting
Dunkl = deformation of Fourier
Parameter : k≥0 k=0RN 3 x Euclidean space, for 0 6= α ∈ RN ,
σα(x)= x− 2〈α, x〉|α|2
α reflection in the hyperplane α⊥
R root system: finite set of vectors α 6= 0: σα(R) = R, ∀ α∈RG (Coxeter) group generated by reflections σα, α ∈ R.
J. Dziubański (Wrocław) Function Spaces and their Applications
Hardy spaces in the Dunkl setting
Dunkl = deformation of Fourier
Parameter : k≥0 k=0RN 3 x Euclidean space, for 0 6= α ∈ RN ,
σα(x)= x− 2〈α, x〉|α|2
α reflection in the hyperplane α⊥
R root system: finite set of vectors α 6= 0: σα(R) = R, ∀ α∈RG (Coxeter) group generated by reflections σα, α ∈ R.
J. Dziubański (Wrocław) Function Spaces and their Applications
A1A1 A2
B2 G2 I25
J. Dziubański (Wrocław) Function Spaces and their Applications
multiplicity function
k :R→ [0,+∞) G–invariant: k(σα) = k(α),∀σ ∈ G,α ∈ R
Assumptions: R is reduced: R ∩ Rα = {−α, α}..and R is normalized: |α|2 = 2
Dunkl operator (Dunkl derivative)
Dξf(x) = ∂ξf(x) +∑
α∈Rk(α)
2〈α,ξ〉〈α,x〉
{f(x)− f(σαx)
}Dunkl operator in dimension one and in the product case
Df(x) = f ′(x) + kx
{f(x)−f(−x)
}Dejf(x) = ∂ejf(x) +
kjxj
{f(x)−f(σjx)
}σj(x1, ..., xj , ..., xN ) = (x1, ...,−xj , ..., xN ).
If e1, ..., eN fixed orthonormal basis, then Dj = Dej
J. Dziubański (Wrocław) Function Spaces and their Applications
multiplicity function
k :R→ [0,+∞) G–invariant: k(σα) = k(α),∀σ ∈ G,α ∈ R
Assumptions: R is reduced: R ∩ Rα = {−α, α}..and R is normalized: |α|2 = 2
Dunkl operator (Dunkl derivative)
Dξf(x) = ∂ξf(x) +∑
α∈Rk(α)
2〈α,ξ〉〈α,x〉
{f(x)− f(σαx)
}Dunkl operator in dimension one and in the product case
Df(x) = f ′(x) + kx
{f(x)−f(−x)
}Dejf(x) = ∂ejf(x) +
kjxj
{f(x)−f(σjx)
}σj(x1, ..., xj , ..., xN ) = (x1, ...,−xj , ..., xN ).
If e1, ..., eN fixed orthonormal basis, then Dj = Dej
J. Dziubański (Wrocław) Function Spaces and their Applications
multiplicity function
k :R→ [0,+∞) G–invariant: k(σα) = k(α),∀σ ∈ G,α ∈ R
Assumptions: R is reduced: R ∩ Rα = {−α, α}..and R is normalized: |α|2 = 2
Dunkl operator (Dunkl derivative)
Dξf(x) = ∂ξf(x) +∑
α∈Rk(α)
2〈α,ξ〉〈α,x〉
{f(x)− f(σαx)
}Dunkl operator in dimension one and in the product case
Df(x) = f ′(x) + kx
{f(x)−f(−x)
}Dejf(x) = ∂ejf(x) +
kjxj
{f(x)−f(σjx)
}σj(x1, ..., xj , ..., xN ) = (x1, ...,−xj , ..., xN ).
If e1, ..., eN fixed orthonormal basis, then Dj = Dej
J. Dziubański (Wrocław) Function Spaces and their Applications
Basic properties. Preliminaries.
Commuting: DξDη = DηDξ
Skewsymmetry:∫RN (Dξf(x))g(x)wk(x) dx = −
∫RN f(x)(Dξg(x))wk(x)dx;
measure: dwk(x) = wk(x) dx, wk(x) =∏α∈R |〈α, x〉|k(α);
wk is G-invariant: wk(x) = wk(σx) for σ ∈ G;doubling: wk(B(x, 2r)) ≤ Cwk(B(x, r));
Dunkl Laplacian ∆k : fix an orthonormal basis e1, ..., eN of RN ;
∆kf(x) =
N∑j=1
D2ejf(x)
= ∆f(x) + 2∑α∈R+
k(α)
〈α, x〉∂αf(x)
−∑α∈R+
k(α)|α|2 f(x)− f(σαx)
〈α, x〉.
J. Dziubański (Wrocław) Function Spaces and their Applications
Basic properties. Preliminaries.
Commuting: DξDη = DηDξ
Skewsymmetry:∫RN (Dξf(x))g(x)wk(x) dx = −
∫RN f(x)(Dξg(x))wk(x)dx;
measure: dwk(x) = wk(x) dx, wk(x) =∏α∈R |〈α, x〉|k(α);
wk is G-invariant: wk(x) = wk(σx) for σ ∈ G;doubling: wk(B(x, 2r)) ≤ Cwk(B(x, r));
Dunkl Laplacian ∆k : fix an orthonormal basis e1, ..., eN of RN ;
∆kf(x) =
N∑j=1
D2ejf(x)
= ∆f(x) + 2∑α∈R+
k(α)
〈α, x〉∂αf(x)
−∑α∈R+
k(α)|α|2 f(x)− f(σαx)
〈α, x〉.
J. Dziubański (Wrocław) Function Spaces and their Applications
Heat and Poisson semigroups
et∆kf(x) =
∫RN
ht(x, y)f(y) dwk(y),
ht(x, y) is not Gaussian!
e−t√−∆kf(x) =
∫RN
Pt(x, y)f(y) dwk(y),
Ptf solves the initial (boundary value problem):(∂2t + ∆k)u(t, x) = 0, u(0, x) = f(x).
Knowledge about the kernels ht(x, y) and Pt(x, y) was rather poorfor our purposes, so our first task was to derive estimates for them.
Purpose: study L =: (∂2t + ∆k)-harmonic functions in the
half-space (0,∞)× RN and conjugate harmonic functions.
This part is a project with J.-Ph. Anker and A. Hejna
Motivation: Muckenhoupt and Stein (TAMS 1965).J. Dziubański (Wrocław) Function Spaces and their Applications
Heat and Poisson semigroups
et∆kf(x) =
∫RN
ht(x, y)f(y) dwk(y),
ht(x, y) is not Gaussian!
e−t√−∆kf(x) =
∫RN
Pt(x, y)f(y) dwk(y),
Ptf solves the initial (boundary value problem):(∂2t + ∆k)u(t, x) = 0, u(0, x) = f(x).
Knowledge about the kernels ht(x, y) and Pt(x, y) was rather poorfor our purposes, so our first task was to derive estimates for them.
Purpose: study L =: (∂2t + ∆k)-harmonic functions in the
half-space (0,∞)× RN and conjugate harmonic functions.
This part is a project with J.-Ph. Anker and A. Hejna
Motivation: Muckenhoupt and Stein (TAMS 1965).J. Dziubański (Wrocław) Function Spaces and their Applications
Heat and Poisson semigroups
et∆kf(x) =
∫RN
ht(x, y)f(y) dwk(y),
ht(x, y) is not Gaussian!
e−t√−∆kf(x) =
∫RN
Pt(x, y)f(y) dwk(y),
Ptf solves the initial (boundary value problem):(∂2t + ∆k)u(t, x) = 0, u(0, x) = f(x).
Knowledge about the kernels ht(x, y) and Pt(x, y) was rather poorfor our purposes, so our first task was to derive estimates for them.
Purpose: study L =: (∂2t + ∆k)-harmonic functions in the
half-space (0,∞)× RN and conjugate harmonic functions.
This part is a project with J.-Ph. Anker and A. Hejna
Motivation: Muckenhoupt and Stein (TAMS 1965).J. Dziubański (Wrocław) Function Spaces and their Applications
Heat and Poisson semigroups
et∆kf(x) =
∫RN
ht(x, y)f(y) dwk(y),
ht(x, y) is not Gaussian!
e−t√−∆kf(x) =
∫RN
Pt(x, y)f(y) dwk(y),
Ptf solves the initial (boundary value problem):(∂2t + ∆k)u(t, x) = 0, u(0, x) = f(x).
Knowledge about the kernels ht(x, y) and Pt(x, y) was rather poorfor our purposes, so our first task was to derive estimates for them.
Purpose: study L =: (∂2t + ∆k)-harmonic functions in the
half-space (0,∞)× RN and conjugate harmonic functions.
This part is a project with J.-Ph. Anker and A. Hejna
Motivation: Muckenhoupt and Stein (TAMS 1965).J. Dziubański (Wrocław) Function Spaces and their Applications
Unify notation: x0 = t, D0 = ∂e0 , L = ∂2e0 + ∆k =
∑Nj=0D
2j .
u = (u0, u1, ..., uN ), satisfies Cauchy-Riemann equations if
Dju` = D`uj ,
N∑j=0
Djuj = 0, (CR)
Then ∆kuj = 0.
‖u‖H1k
= supx0>0
∫RN
|u(x0, x1, ..., xN )| dwk(x1, ..., xN ) <∞. (L1)
J. Dziubański (Wrocław) Function Spaces and their Applications
Unify notation: x0 = t, D0 = ∂e0 , L = ∂2e0 + ∆k =
∑Nj=0D
2j .
u = (u0, u1, ..., uN ), satisfies Cauchy-Riemann equations if
Dju` = D`uj ,
N∑j=0
Djuj = 0, (CR)
Then ∆kuj = 0.
‖u‖H1k
= supx0>0
∫RN
|u(x0, x1, ..., xN )| dwk(x1, ..., xN ) <∞. (L1)
J. Dziubański (Wrocław) Function Spaces and their Applications
Relation with Riesz transforms:
Rjf(x) = −Dj(−∆k)−1/2f(x).
(Px0f(x), R1Px0f(x), ..., RNPx0f(x)) satisfies (CR)
Our results are:
Theorem 1.
Let u0 be L-harmonic. Then there is a vector u = (u0, u1, ..., uN )satisfying (CR) and (L1) if and only if
u∗0(x) = sup|x′−x|<x0
|u0(x0,x)|
belongs to L1(dwk), x = (x1, ..., xN ) ∈ RN and
‖u‖H1k∼ ‖u∗‖L1(dwk).
J. Dziubański (Wrocław) Function Spaces and their Applications
Relation with Riesz transforms:
Rjf(x) = −Dj(−∆k)−1/2f(x).
(Px0f(x), R1Px0f(x), ..., RNPx0f(x)) satisfies (CR)
Our results are:
Theorem 1.
Let u0 be L-harmonic. Then there is a vector u = (u0, u1, ..., uN )satisfying (CR) and (L1) if and only if
u∗0(x) = sup|x′−x|<x0
|u0(x0,x)|
belongs to L1(dwk), x = (x1, ..., xN ) ∈ RN and
‖u‖H1k∼ ‖u∗‖L1(dwk).
J. Dziubański (Wrocław) Function Spaces and their Applications
Real Hardy space H1∆k
If (u0, u1, ..., uN ) ∈ H1k, then limx0→0 u0(x0,x) exists in L1(dwk).
H1∆k
(RN ) = {f(x) = limx0→0 u0(x0,x) : (u0, u1, ..., uN ) ∈ H1k}
Theorem 2. Characterizations of H1∆k
by
Riesz transforms: f ∈ H1∆k
iff Rjf ∈ L1(dwk);
square functions:
Sf(x) =( ∫∫
|x−y|<t |t2∆ke
t2∆kf(y)|2 dwk(y)dtwk(B(x,t))t
)1/2
other maximal functions (heat semigroup)
atomic decompositions
J. Dziubański (Wrocław) Function Spaces and their Applications
Thank you
J. Dziubański (Wrocław) Function Spaces and their Applications