Hardy spaces associated with semi-groups of linear...

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)|

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).

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

|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 ).

∂xjui = ∂xiuj ,

N∑j=0

∂xjuj = 0, (CR)

supx0>0

u∗0(x) = sup|x−x′|<x0

|u0(x0,x′)| ∈ Lp(RN ).

∂xjui = ∂xiuj ,

N∑j=0

∂xjuj = 0, (CR)

supx0>0

u∗0(x) = sup|x−x′|<x0

|u0(x0,x′)| ∈ Lp(RN ).

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))

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))

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

Riesz transform characterization

Rjf(x) = −∂j(−∆)−1/2f(x) = − ic

ξj|ξ|eix·ξFf(ξ)dξ

‖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

‖f‖H1,square = ‖Sf‖L1(Rn) <∞

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

|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.

|λj |,

j λjaj .

|λj |,

j λjaj .

|λj |,

j λjaj .

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

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,...

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

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,....

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 .

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 `.

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 `.

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,...

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) =

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.

H1∆−V = {f ∈ L1 : ‖f‖H1

∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.

Ttf(x) = Ex(e−

Ttf(x) = et(∆−V )f(x) =

Kt(x, y)f(y) dy,

0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.

H1∆−V = {f ∈ L1 : ‖f‖H1

∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.

Ttf(x) = Ex(e−

Ttf(x) = et(∆−V )f(x) =

Kt(x, y)f(y) dy,

0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.

H1∆−V = {f ∈ L1 : ‖f‖H1

∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.

Ttf(x) = Ex(e−

Ttf(x) = et(∆−V )f(x) =

Kt(x, y)f(y) dy,

0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.

H1∆−V = {f ∈ L1 : ‖f‖H1

∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.

Ttf(x) = Ex(e−

Ttf(x) = et(∆−V )f(x) =

Kt(x, y)f(y) dy,

0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.

H1∆−V = {f ∈ L1 : ‖f‖H1

∆−V= ‖ supt>0 |et(∆−V )f |‖L1 <∞}.

Ttf(x) = Ex(e−

Ttf(x) = et(∆−V )f(x) =

Kt(x, y)f(y) dy,

0 ≤ Kt(x, y) ≤ (4πt)−d/2e−|x−y|2/4t.

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)

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?

∫a = 0, |a| ≤ |Q′|−1).

∆−V = H1Q?

∫a = 0, |a| ≤ |Q′|−1).

∆−V = H1Q?

∫a = 0, |a| ≤ |Q′|−1).

∆−V = H1Q?

∫a = 0, |a| ≤ |Q′|−1).

∆−V = H1Q?

∫a = 0, |a| ≤ |Q′|−1).

∆−V = H1Q?

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

)δfor x ∈ Rd, Q ∈ Q, t ≤ d(Q)2,

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

supy∈Q∗

∫ 2t

0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C

)δfor x ∈ Rd, Q ∈ Q, t ≤ d(Q)2,

hs(x) = (4πs)−d/2e−|x|2/(4s).

f ∈ H1∆−V ⇐⇒ f ∈ H1

∆−V. ‖f‖H1

supy∈Q∗

∫ 2t

0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C

)δfor x ∈ Rd, Q ∈ Q, t ≤ d(Q)2,

hs(x) = (4πs)−d/2e−|x|2/(4s).

f ∈ H1∆−V ⇐⇒ f ∈ H1

∆−V. ‖f‖H1

supy∈Q∗

∫ 2t

0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C

)δfor x ∈ Rd, Q ∈ Q, t ≤ d(Q)2,

hs(x) = (4πs)−d/2e−|x|2/(4s).

f ∈ H1∆−V ⇐⇒ f ∈ H1

∆−V. ‖f‖H1

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 ,

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 ,

The condition (K)

(∃C, δ > 0)

∫ 2t

0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C

)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

The condition (K)

(∃C, δ > 0)

∫ 2t

0(1Q∗∗∗V ) ∗ hs(x)ds ≤ C

)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

Examples.

• d ≥ 3, V satisfies the reverse Holder inequality with exponentq > d/2, that is,(

∫BV (y)q dy

≤ C 1

∫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.

Examples.

∫BV (y)q dy

≤ C 1

d(Q)2|Q|−1

∫QV (y) dy ≤ 1.

−1/2 on Lp, p > 1.

Examples.

∫BV (y)q dy

≤ C 1

d(Q)2|Q|−1

∫QV (y) dy ≤ 1.

−1/2 on Lp, p > 1.

• 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.

• 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.

• 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.

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).

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).

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).

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.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

t (x, y), T 2t (x, y)).

supy∈Q∗1

y∈Q∗1

∫T 1

2nd(Q1)2(x, y) dy ≤ Cn−1−ε.

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 .

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

H1L 3 f 7→ wf ∈ H1(Rd)

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.

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.

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)︸︷︷︸

(L1/2(−∆)−1/2)︸︷︷︸(isomorphismH1(Rd)→ H1

= ∂xj (−∆)−1/2︸︷︷︸(classical Riesz transform)

Proof: ∂xjL−1/2f ∈ L1 implies f ∈ H1

L→ H1(Rd))

= ∂xjL−1/2︸︷︷︸

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.

L→ H1(Rd))

= ∂xjL−1/2︸︷︷︸

Assume that∂xjL

−1/2f ∈ L1 j = 1, ..., d,

then(−∆)1/2L−1/2f ∈ H1(Rd),

L→ H1(Rd))

= ∂xjL−1/2︸︷︷︸

Assume that∂xjL

−1/2f ∈ L1 j = 1, ..., d,

then(−∆)1/2L−1/2f ∈ H1(Rd),

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.

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, ...

(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;

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))

•∫X a(x)w(x) dµ(x) = 0.

We do not assume any regularity condition on Kt(x, y)!

•∫X a(x)w(x) dµ(x) = 0.

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

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.

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

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.

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)

w(x)∈ H1

atom(X, d,w(x)2dµ(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.

So,f ∈ H1

L,max ⇔ supt>0

∣∣∣ ∈ L1(dµ)︸︷︷︸L1(w(x)2dµ)

w(x)∈ H1

w(x)=∑

λj aj(x)

f(x) =∑j

λj aj(x)w(x) =∑j

So,f ∈ H1

L,max ⇔ supt>0

∣∣∣ ∈ L1(dµ)︸︷︷︸L1(w(x)2dµ)

w(x)∈ H1

w(x)=∑

λj aj(x)

f(x) =∑j

λj aj(x)w(x) =∑j

So,f ∈ H1

L,max ⇔ supt>0

∣∣∣ ∈ L1(dµ)︸︷︷︸L1(w(x)2dµ)

w(x)∈ H1

w(x)=∑

λj aj(x)

f(x) =∑j

λj aj(x)w(x) =∑j

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 .

Grushin operator

L = −∆x−|x|2∂2

∂x′2, x = (x, x′) = (x1, ..., xn, x

′) ∈ Rn×R = Rn+1

L= ‖MLf‖L1 <∞}

−1/2,j = 1, ..., n.

L:‖f‖H1

L∼ ‖f‖L1 +

∑2nk=1 ‖Rkf‖L1 .

Grushin operator

L = −∆x−|x|2∂2

∂x′2, x = (x, x′) = (x1, ..., xn, x

′) ∈ Rn×R = Rn+1

L= ‖MLf‖L1 <∞}

−1/2,j = 1, ..., n.

L:‖f‖H1

L∼ ‖f‖L1 +

∑2nk=1 ‖Rkf‖L1 .

Grushin operator

L = −∆x−|x|2∂2

∂x′2, x = (x, x′) = (x1, ..., xn, x

′) ∈ Rn×R = Rn+1

L= ‖MLf‖L1 <∞}

−1/2,j = 1, ..., n.

L:‖f‖H1

L∼ ‖f‖L1 +

∑2nk=1 ‖Rkf‖L1 .

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).

∑dj=1Rjgj

with∑d

j=0 ‖gj‖∞ ≤ C‖g‖BM0

∑dj=1Rjgj

with∑d

j=0 ‖gj‖∞ ≤ C‖g‖BM0

∑dj=1Rjgj

with∑d

j=0 ‖gj‖∞ ≤ C‖g‖BM0

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.

g = g0 +

2n∑k=1

Rkgk, with2n∑k=0

j=1(X2j + Y 2

2a · b+ a · x),

g = g0 +

2n∑k=1

Rkgk, with2n∑k=0

j=1(X2j + Y 2

2a · b+ a · x),

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.

σα(x)= x− 2〈α, x〉|α|2

A1A1 A2

B2 G2 I25

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) +

{f(x)−f(σjx)

}σj(x1, ..., xj , ..., xN ) = (x1, ...,−xj , ..., xN ).

If e1, ..., eN fixed orthonormal basis, then Dj = Dej

Dξf(x) = ∂ξf(x) +∑

α∈Rk(α)

2〈α,ξ〉〈α,x〉

{f(x)− f(σαx)

Df(x) = f ′(x) + kx

{f(x)−f(−x)

{f(x)−f(σjx)

}σj(x1, ..., xj , ..., xN ) = (x1, ...,−xj , ..., xN ).

Dξf(x) = ∂ξf(x) +∑

α∈Rk(α)

2〈α,ξ〉〈α,x〉

{f(x)− f(σαx)

Df(x) = f ′(x) + kx

{f(x)−f(−x)

{f(x)−f(σjx)

}σj(x1, ..., xj , ..., xN ) = (x1, ...,−xj , ..., xN ).

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+

〈α, x〉∂αf(x)

−∑α∈R+

k(α)|α|2 f(x)− f(σαx)

〈α, x〉.

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+

〈α, x〉∂αf(x)

−∑α∈R+

k(α)|α|2 f(x)− f(σαx)

〈α, x〉.

Heat and Poisson semigroups

et∆kf(x) =

ht(x, y)f(y) dwk(y),

ht(x, y) is not Gaussian!

e−t√−∆kf(x) =

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

et∆kf(x) =

e−t√−∆kf(x) =

et∆kf(x) =

e−t√−∆kf(x) =

et∆kf(x) =

e−t√−∆kf(x) =

Unify notation: x0 = t, D0 = ∂e0 , L = ∂2e0 + ∆k =

∑Nj=0D

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

|u(x0, x1, ..., xN )| dwk(x1, ..., xN ) <∞. (L1)

Unify notation: x0 = t, D0 = ∂e0 , L = ∂2e0 + ∆k =

∑Nj=0D

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

|u(x0, x1, ..., xN )| dwk(x1, ..., xN ) <∞. (L1)

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).

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).

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

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

other maximal functions (heat semigroup)

atomic decompositions

Thank you

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