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Ball Average Characterizations of Function Spaces

Dachun Yang

(Joint work)

[email protected]

School of Mathematical Sciences

Beijing Normal University

September 17 to 23, 2017 / NPFSA-2017, B edlewo, Poland

Ball Average Characterizations of Function Spaces – p. 1/50

Outline

§I Pointwise characterizations of Besov and Triebel-Lizorkinspaces with smoothness not greater than 1

§II Ball average characterizations of second order Sobolevspaces

§III Ball average characterizations of second order Besov andTriebel-Lizorkin spaces

§IV Further remarks


Main Motivation

◮ Since there exists no differential structure on a generalmetric measure space, it is still an open question how tointroduce function spaces with smoothness on such a setting.

◮ Find new characterizations of well-known function spacesso that these new characterizations can be used as thedefinitions of the corresponding function spaces on metricmeasure spaces .


§I. Pointwise characterizations ofBesov and Triebel-Lizorkin spaceswith smoothness not greater than1


Homogeneous Sobolev SpacesWm,p(Rn) / §I

◮ Let m ∈ N := {1, 2, . . .}, Z+ := N ∪ {0} & p ∈ (1,∞).

• f ∈ Wm,p(Rn) (homogeneous Sobolev space ) ⇐⇒f ∈ S ′(Rn) (Schwartz distribution ) and ∂γf ∈ Lp(Rn) for all|γ| = m; moreover,

‖f‖Wm,p(Rn) :=∑

|γ|=m

‖∂γf‖Lp(Rn).

• Homogeneous : for any λ ∈ (0,∞) and f ∈ Wm,p(Rn),

‖f(λ·)‖Wm,p(Rn) = λm−n/p‖f‖Wm,p(Rn).


Inhomogeneous Sobolev SpacesWm,p(Rn) / §I

◮ Let m ∈ N := {1, 2, . . .}, Z+ := N ∪ {0} & p ∈ (1,∞).

• f ∈ Wm,p(Rn) ⇐⇒ f ∈ Lp(Rn) and ∂γf ∈ Lp(Rn) for all|γ| ≤ m; moreover,

‖f‖Wm,p(Rn) :=∑

|γ|≤m

‖∂γf‖Lp(Rn).

[‖f‖Wm,p(Rn) ∼ ‖f‖Lp(Rn) + ‖f‖Wm,p(Rn)

]


Besov SpacesBαp, q(R

n) / §I

Let ϕ0 ∈ S(Rn) (Schwartz functions ) satisfy

(1.1) ϕ0(x) = 1 if |x| ≤ 1 and ϕ0(y) = 0 if |y| ≥ 3/2,

and let

(1.2) ϕ(j)(x) := ϕ0(2−jx)− ϕ0(2

−j+1x), ∀x ∈ Rn, ∀ j ∈ Z.

Then ∑

j∈Z

ϕ(j)(x) = 1, ∀x ∈ Rn \ {~0n}.

Let (Triebel, 83 book )

S∞(Rn) :=

{f ∈ S(Rn) :

∫

Rn

f(x)xα dx = 0, ∀α ∈ Zn+

}.


Besov SpacesBαp, q(R

n) / §I

◮ If α ∈ (0, ∞) & p, q ∈ (0, ∞], then f ∈ Bαp, q(R

n)

(homogeneous Besov space ) ⇐⇒ f ∈ S ′∞(Rn) (dual of S∞(Rn))

such that

‖f‖Bαp, q(Rn) :=

∞∑

j=−∞

2jαq∥∥∥∥(ϕ(j)f

)∨∥∥∥∥q

Lp(Rn)

1/q

=:

∥∥∥∥∥∥

{2jα

∥∥∥∥(ϕ(j)f

)∨∥∥∥∥Lp(Rn)

}

j∈Z

∥∥∥∥∥∥ℓq

< ∞.

◮ In what follows, for any function or distribution g, g (or g∨)denotes its Fourier (or inverse Fourier ) transform.


Triebel -Lizorkin SpacesF αp, q(R

n) / §I◮ If α ∈ (0, ∞) & p, q ∈ (0, ∞], then f ∈ Fα

p, q(Rn)

(homogeneous Triebel-Lizorkin space ) ⇐⇒ f ∈ S ′∞(Rn) such

that ‖f‖Fαp, q(Rn) < ∞, where

‖f‖Fαp, q(Rn) :=

∥∥∥∥∥∥{2jα(ϕ(j)f)∨}j∈Z

∥∥∥ℓq

∥∥∥Lp(Rn)

, p < ∞

and

‖f‖Fα∞, q(Rn) := sup

x∈Rn

m∈Z

2mn

∫

B(x, 2−m)

∞∑

j=m

|2jα(ϕ(j)f)∨(y)|q dy

1

q

.

◮ F 0∞,2(R

n) = BMO(Rn), F 0p,2(R

n) = Hp(Rn), p ∈ (0,∞) &

Fmp,2(R

n) = Wm,p(Rn), m ∈ Z+, p ∈ (1,∞).Ball Average Characterizations of Function Spaces – p. 9/50

Bαp, q(R

n) & F αp, q(R

n) / §I

Let ϕ0 ∈ S(Rn) be as in (1.1) and let ϕ(0) := ϕ0 andϕ(j) := ϕ(j) for any j ∈ N, where ϕ(j) is as in (1.2). Then

∞∑

j=0

ϕ(j)(x) = 1, ∀x ∈ Rn.

◮ The inhomogeneous Besov space Bαp, q(R

n) &Triebel-Lizorkin space Fα

p, q(Rn) are defined via replacing

S ′∞(Rn) and {ϕ(j)}j∈Z in Bα

p, q(Rn) & Fα

p, q(Rn), respectively,

by S ′(Rn) (Schwartz distributions ) and {ϕ(j)}j∈Z+. Moreover,

‖f‖Fα∞, q(Rn)

:= supx∈Rn

m∈Z

2mn

∫

B(x, 2−m)

∞∑

j=max{m,0}

|2jα(ϕ(j)f)∨(y)|q dy

1

q

.


Hajłasz-Sobolev Spaces (1) / §I

• (X , d, µ): X — nonempty set;d — quasi metric, namely, d(x, y) ≤ A[d(x, z) + d(z, y)];µ — regular Borel measure

• p ∈ (1,∞), s ∈ (0, 1]

• The homogeneous fractional Hajłasz-Sobolev space M s,p(X ) isdefined to be the set of all measurable functionsf ∈ Lp

loc (X ) for which there exist a 0 ≤ g ∈ Lp(X ) and a setE ⊂ X of measure zero such that, for any x, y ∈ X \ E,

(1.3) |f(x)− f(y)| ≤ [d(x, y)]s[g(x) + g(y)].

• Denote by D(f) the class of all non-negative Borel


Hajłasz-Sobolev Spaces (2) / §I

measurable functions g satisfying (1.3). Moreover, define

‖f‖Ms,p(X ) := infg∈D(f)

{‖g‖Lp(X )}.

Let M s,p(X ) := Lp(X ) ∩ M s,p(X ) and, for any f ∈ M s,p(X ),let

‖f‖Ms,p(X ) := ‖f‖Lp(X ) + ‖f‖Ms,p(X ).

Remarks:

◮ M1,p(X ) & M1,p(X ) were introduced by Hajłasz [H96].

◮ M s,p(X ) & M s,p(X ) when s ∈ (0, 1) were introduced byHu [Hu03] for subsets (fractals) of Rn and Yang [Y03] formetric measure spaces.


Hajłasz-Sobolev Spaces (3) / §I◮ It was proved in [H96] that

M1,p(Rn) = W 1,p(Rn) = F 1p,2(R

n)

and in [Y03] that, when s ∈ (0, 1),

M s,p(Rn) = F sp,∞(Rn) % F s

p,2(Rn).

(There exists a gap for Triebel-Lizorkin spaces .)

• [H96] P. Hajłasz , Sobolev spaces on an arbitrary metricspace, Potential Anal. 5 (1996), 403-415 .

• [Hu03] J. Hu , A note on Hajłasz-Sobolev spaces onfractals, J. Math. Anal. Appl. 280 (2003), 91-101 .

• [Y03] D. Yang , New characterizations of Hajłasz-Sobolevspaces on metric spaces, Sci. China Ser. A 46 (2003), 675-689 .


Fractional s−Hajłasz Gradient / §I

• [KYZ11] P. Koskela, D. Yang & Y. Zhou , Pointwisecharacterizations of Besov and Triebel-Lizorkin spaces andquasiconformal mappings, Adv. Math. 226 (2011), 3579-3621 .

◮ Definition . Let s ∈ (0, ∞) and u be a measurable functionon X . A sequence of nonnegative measurable functions,~g := {gk}k∈Z, is called a fractional s-Hajłasz gradient of u ifthere exists E ⊂ X with µ(E) = 0 such that, for any k ∈ Zand x, y ∈ X \ E satisfying 2−k−1 ≤ d(x, y) < 2−k,

|u(x)− u(y)| ≤ [d(x, y)]s[gk(x) + gk(y)].

Denote by Ds(u) the collection of all fractional s-Hajłaszgradients of u.


M sp, q(X ) & N s

p, q(X ) / §I

• The homogeneous Hajłasz-Triebel-Lizorkin space M sp, q(X ) is

defined to be the space of all measurable functions u suchthat

‖u‖Msp, q(X ) := inf

~g∈Ds(u)

∥∥∥∥∥∥{gj}j∈Z

∥∥∥ℓq

∥∥∥Lp(X )

< ∞.

• The homogeneous Hajłasz-Besov space Nsp, q(X ) is defined to

be the space of all measurable functions u such that

‖u‖Nsp, q(X ) := inf

~g∈Ds(u)

∥∥∥∥{‖gj‖Lp(X )

}j∈Z

∥∥∥∥ℓq< ∞.


RD-Spaces (1) / §I

• A triple (X , d, µ): X is a non-empty set, d a quasi-metric(usually, for simplicity, metric), and µ a regular Borelmeasure.

◮ A space of homogenous type of Coifman-Weiss: ifµ-doubling (µ(B(x, 2r)) . µ(B(x, r))).

◮ An RD-space if µ is both doubling and reverse-doubling(µ(B(x, 2r)) ≥ C0µ(B(x, r)) and C0 > 1).

◮ There exist many examples of RD-spaces. Especially,all connected spaces of homogeneous type are RD-spaces.


RD-Spaces (2) / §I

• [HMY06] Y. Han, D. Muller & D. Yang , Littlewood-Paleycharacterizations for Hardy spaces on spaces ofhomogeneous type, Math. Nachr. 279 (2006), 1505-1537 .

• [HMY08] Y. Han, D. Muller & D. Yang , A theory of Besov andTriebel-Lizorkin spaces on metric measure spaces modeledon Carnot-Carathéodory spaces, Abstr. Appl. Anal. 2008 Art.ID 893409, 250 pp .

• [MY09] D. Muller & D. Yang , A difference characterization ofBesov and Triebel-Lizorkin spaces on RD-spaces, ForumMath. 21 (2009), 259-298 .

• D. Yang & Y. Zhou , New properties of Besov andTriebel-Lizorkin spaces on RD-spaces, Manuscripta Math. 134(2011), 59-90.


Theorem 1.5 / §I

◮ [KYZ11] Theorem . Let X be an RD-space with the upperdimension n.(i) If s ∈ (0, 1), p ∈ (n/(n+ s), ∞) and q ∈ (n/(n+ s), ∞],then M s

p, q(X ) = F sp, q(X ).

(ii) If s ∈ (0, 1), p ∈ (n/(n+ s), ∞) and q ∈ (0, ∞], thenNs

p, q(X ) = Bsp, q(X ).

• F sp, q(X ) & Bs

p, q(X ) were studied in [HMY08] & [MY09].

• Applications to the invariance under quasiconformalmappings of the certain function spaces.

◮ More recent related papers: H. Koch, P. Koskela, E. Saksman& T. Soto [JFA, 2014], M. Bonk, E. Saksman & T. Soto [arXiv:1411.5906 or Indiana Univ. Math. J. (to appear) or D. Yang, W.Yuan & Y. Zhou [JGA, 2017].


More Papers onM sp, q(X ) & N s

p, q(X ) / §I

◮ A. Gogatishvili, P. Koskela & Y. Zhou , Characterizations ofBesov and Triebel-Lizorkin spaces on metric measurespaces, Forum Math. 25 (2013), 787-819 .

◮ T. Heikkinen & H. Tuominen , Approximation by Hölderfunctions in Besov and Triebel-Lizorkin spaces, Constr.Approx. 44 (2016), 455-482 .

◮ T. Heikkinen, L. Ihnatsyeva & H. Tuominen , Measure densityand extension of Besov and Triebel-Lizorkin functions,J. Fourier Anal Appl. 22 (2016), 334-382 .

◮ T. Heikkinen, P. Koskela & H. Tuominen , Approximation andquasicontinuity of Besov and Triebel-Lizorkin functions,Trans. Amer. Math. Soc. 369 (2017), 3547-3573 .


§II. Ball average characterizations ofsecond order Sobolev spaces


Theorem of [AMV12] / §II

◮ For any t ∈ (0,∞), g ∈ L1loc (R

n) and x ∈ Rn, let

Bt(g)(x) :=1

|B(x, t)|

∫

B(x,t)

g(y) dy.

◮ ([AMV12]) Let p ∈ (1,∞). Then f ∈ W 2,p(Rn) if and only iff ∈ Lp(Rn) and there exists g ∈ Lp(Rn) such that

G(f, g)(·) :=

{∫ ∞

0

∣∣∣∣Bt(f)(·)− f(·)

t2− Bt(g)(·)

∣∣∣∣2dt

t

} 1

2

∈Lp(Rn).

◮ theory of Vector-valued C-Z operators , fine estimates

◮ [AMV12] R. Alabern, J. Mateu & J. Verdera , A newcharacterization of Sobolev spaces on Rn, Math. Ann. 354(2012), 589-626.


Lusin-Area Funct. Charact. (1) / §II

◮ ([HYY15]) (i) If p ∈ [2,∞), then f ∈ W 2,p(Rn) if and only iff ∈ Lp(Rn) and there exists g ∈ Lp(Rn) such that

S(f, g)(·):=

{∫ ∞

0

∫

B(·,t)

∣∣∣∣Bt(f)(y)− f(y)

t2−Bt(g)(y)

∣∣∣∣2

dydt

tn+1

} 1

2

∈ Lp(Rn).

(ii) If p ∈ (1, 2) and n ∈ {1, 2, 3}, then f ∈ W 2,p(Rn) if andonly if f ∈ Lp(Rn) and there exists g ∈ Lp(Rn) such thatS(f, g) ∈ Lp(Rn).

• [HYY15] Z. He, D. Yang & W. Yuan , Littlewood-Paleycharacterizations of second-order Sobolev spaces viaaverages on balls, Canadian Math. Bull. 59 (2016), 104-108 .


Lusin-Area Funct. Charact. (2) / §II

◮ ([DLYY]) (i) Let n ∈ [4,∞) ∩ N and p ∈ ( 2n4+n , 2). Then

f ∈ W 2,p(Rn) if and only if f ∈ Lp(Rn) and there existsg ∈ Lp(Rn) such that S(f, g) ∈ Lp(Rn).

(ii) Let n ∈ [5,∞) ∩ N and p ∈ (1, 2n4+n). Then the conclusion

of (i) does not hold true.

◮ In (i), if n = 4, then p ∈ (1, 2). The conclusion of (i) is nearsharp .

◮ Instead of a vector-valued C-Z operator, use a series ofvector-valued C-Z operators

• [DLYY] F. Dai, J. Liu, D. Yang & W. Yuan , Littlewood-Paleycharacterizations of fractional Sobolev spaces via averageson balls, Proc. Roy. Soc. Edinburgh Sect. A. (to appear).


G∗λ Characterization (1) / §II

◮ ([HYY15]) (i) If p ∈ [2,∞) and λ ∈ (1,∞), thenf ∈ W 2,p(Rn) ⇐⇒ f ∈ Lp(Rn) and ∃ g ∈ Lp(Rn) such that

G∗λ(f, g)(·) :=

{∫ ∞

0

∫

Rn

∣∣∣∣Bt(f)(y)− f(y)

t2−Bt(g)(y)

∣∣∣∣2

×

(t

t+ | · −y|

)λn

dydt

tn+1

} 1

2

∈ Lp(Rn).

(ii) If p ∈ (1, 2), λ ∈ (2/p,∞) and n ∈ {1, 2, 3}, thenf ∈ W 2,p(Rn) ⇐⇒ f ∈ Lp(Rn) and ∃ g ∈ Lp(Rn) such thatG∗λ(f, g) ∈ Lp(Rn).


G∗λ Characterization (2) / §II

◮ ([DLYY]) (i) Let n ∈ [4,∞) ∩ N, p ∈ ( 2n4+n , 2) and

λ ∈ (2/p,∞). Then f ∈ W 2,p(Rn) if and only if f ∈ Lp(Rn)and there exists g ∈ Lp(Rn) such that G∗

λ(f, g) ∈ Lp(Rn).

(ii) Let n ∈ [5,∞) ∩ N, p ∈ (1, 2n4+n) and λ ∈ (2/p,∞). Then

the conclusion of (i) does not hold true.

◮ In (i), if n = 4, then p ∈ (1, 2). The conclusion of (i) is nearsharp .

◮ It is still unclear on the endpoint case p = 2n4+n .


Pointwise Characterization (1) / §II

◮ ([DGYY15]) Let p ∈ (1,∞). Then f ∈ W 2,p(Rn)

⇐⇒ f ∈ Lp(Rn) and ∃ 0 ≤ g ∈ Lp(Rn) and C0 > 0 such that,for any t ∈ (0,∞) and almost every x ∈ Rn,

|f(x)−Bt(f)(x)| ≤ C0t2g(x)

⇐⇒ f ∈ Lp(Rn) and

supt∈(0,∞)

‖f − Bt(f)‖Lp(Rn)

t2=: C1 < ∞.

• Not known for spaces of homogenous type.

• [DGYY15] F. Dai, A. Gogatishvili, D. Yang & W. Yuan ,Characterizations of Sobolev spaces via averages on balls,Nonlinear Anal. 128 (2015), 86-99 .



◮ ([DGYY15]) Let p ∈ (1,∞). Then f ∈ W 2,p(Rn)

⇐⇒ f ∈ Lp(Rn) and ∃ 0 ≤ g ∈ Lp(Rn) and C, C > 0 suchthat, for any t ∈ (0,∞) and almost every x ∈ Rn,

∣∣Bt

(f −B

Ct(f)

)(x)

∣∣ ≤ Ct2g(x)

⇐⇒ f ∈ Lp(Rn) and ∃ 0 ≤ g ∈ Lp(Rn) and c, C, C > 0 suchthat, for any t ∈ (0,∞) and almost every x ∈ Rn,

Bt

(|f −B

Ct(f)|

)(x) ≤ Ct2Bct(g)(x).

◮ The second equivalence also holds true on spaces ofhomogeneous type .



◮ ([DGYY15]) Let p ∈ (1,∞), q ∈ [1, p), c ∈ (0,∞) andK ∈ (0,∞]. Then f ∈ W 2,p(Rn)

⇐⇒ f ∈ Lp(Rn) and

f ♯,Kc,q (·) := supt∈(0,K)

t−2 {Bt (|f − Bct(f)|q) (·)}1/q ∈ Lp(Rn).


A Key Lemma / §II

Let ϕ ∈ S(Rn) and C ∈ (0,∞) be a constant. Then

limt→0+

ϕ−Bt(ϕ)

t2= −

1

2(n+ 2)∆ϕ

and

limt→0+

Bt

(ϕ− B

Ct(ϕ)

t2

)(·) = −

C2

2(n+ 2)∆ϕ(·)

with convergence in S(Rn).


Further Results / §II

◮ For any ℓ ∈ N, t ∈ (0,∞), f ∈ L1loc (R

n) and x ∈ Rn, let

Bℓ,t(f)(x) := −2(2ℓℓ

)ℓ∑

j=1

(−1)j(

2ℓ

ℓ− j

)Bjt(f)(x).

(Binomial coefficients; Observe that B1,t(f) = Bt(f).)

◮ All aforementioned characterizations of W 2,p(Rn) viapointwise inequalities remain true for W 2ℓ,p(Rn), with ℓ ∈ Nand p ∈ (1,∞), if we replace Bt(f) by Bℓ,t(f) therein.

◮ ([CYYZ]) Also true for Morrey-Sobolev spaces.

• [CYYZ] D.-C. Chang, D. Yang, W. Yuan & J. Zhang , Somerecent developments of high order Sobolev-type spaces, J.Nonlinear Convex Anal. 17 (2016), 1831-1865 .


Open Questions / §II

◮ On spaces of homogeneous type (or even smooth domainsof Rn), whether or not these Sobolev spaces coincide ? (Wenow have several different definitions.)

◮ On spaces of homogeneous type, whether or notfractional Sobolev spaces contain the knownHajłasz-Sobolev spaces or the known Newton-Sobolevspaces?

◮ For analysis on metric measure spaces, any applications ?

............


§III. Ball average characterizations ofsecond order Besov andTriebel-Lizorkin spaces


Littlewood-Paley Characterization (1) / §III

◮ ([YYZ13, DGYY15]) Let α ∈ (0, 2) and q ∈ (1, ∞].

(i) If p ∈ (1,∞), then f ∈ Fαp, q(R

n) if and only if f ∈ Lp(Rn)

and∥∥∥∥∥∥{2kα |f − B2−k(f)|

}k∈Z+

∥∥∥ℓq

∥∥∥Lp(Rn)

< ∞.

(ii) If p = ∞, then f ∈ Fα∞, q(R

n) if and only if f ∈ C(Rn) and

supℓ∈Z

x∈Rn

B2−ℓ

∑

k≥max{ℓ,0}

2kαq∣∣∣∣f −B2−k(f)

∣∣∣∣q (x)

1

q

<∞.

◮ C(Rn): the space of all uniformly continuous boundedfunctions



◮ ([YYZ13, DGYY15]) Let α ∈ (0, 2), p ∈ (1,∞] andq ∈ (0, ∞]. Then f ∈ Bα

p, q(Rn) if and only if f ∈ Lp(Rn) when

p < ∞, or f ∈ C(Rn) when p = ∞, and

∞∑

j=0

2jαq ‖f − B2−j(f)‖qLp(Rn)

1

q

< ∞.

• [YYZ13] D. Yang, W. Yuan & Y. Zhou , A new characterizationof Triebel-Lizorkin spaces on Rn, Publ. Mat. 57 (2013), 57-82 .

• [DGYY15] F. Dai, A. Gogatishvili, D. Yang & W. Yuan ,Characterizations of Besov and Triebel-Lizorkin spaces viaaverages on balls, J. Math. Anal. Appl. 433 (2016), 1350-1368 .



◮ Lusin-area type function: For any f ∈ L1loc (R

n) andx ∈ Rn,

Ar(f)(x) :=

{∞∑

k=1

2kαq [B2−k (|f − B2−k(f)|r) (x)]q

r

} 1

q

.

◮ ([CLYY15]) Let α ∈ (0, 2), p ∈ (1,∞), q ∈ (1,∞] andr ∈ [1, q). Then f ∈ Fα

p,q(Rn) if and only if f ∈ Lp(Rn) and

Ar(f) ∈ Lp(Rn).

◮ [CLYY15] D.-C. Chang, J. Liu, D. Yang & W. Yuan ,Littlewood-Paley characterizations of Hajłasz-Sobolev andTriebel-Lizorkin spaces via averages on balls, Potential Anal.46 (2017), 22-259.



◮ ([CLYY15]) Let α ∈ (0, 2) and p ∈ (1,∞), q ∈ (1,∞].

(i) If f ∈ Lp(Rn) and Aq(f) ∈ Lp(Rn), then f ∈ Fαp,q(R

n).

(ii) If p ∈ [q,∞) and α ∈ (0, 2), or p ∈ (1, q) andα ∈ (n(1/p− 1/q), 2), then f ∈ Fα

p,q(Rn) implies that

f ∈ Lp(Rn) and Aq(f) ∈ Lp(Rn).

◮ In case when q = 2 and α < 1, i. e., Fαp,2(R

n) = Wα,p(Rn),then (ii) is not true when α < n(1/p− 1/2). But, whathappens when α = n(1/p− 1/q)?



◮ Part of aforementioned results are also true forBα,τp,q (Rn) & Fα,τ

p,q (Rn) (see [ZSYY]) andTriebel-Lizorkin-Morrey spaces Eα

u,q,p(Rn) (see [ZZYH]).

◮ [ZSYY] C. Zhuo, W. Sickel, D. Yang & W. Yuan ,Characterizations of Besov-type andTriebel-Lizorkin-type spaces via averages on balls,Canad. Math. Bull. 60 (2017), 655-672 .

◮ [ZZYH] J. Zhang, C. Zhuo, D. Yang and Z. He ,Littlewood-Paley characterizations ofTriebel-Lizorkin-Morrey spaces via ball averages,Nonlinear Anal. 150 (2017), 76-103 .

◮ A new idea is to introduce the local Hardy-Littlewoodmaximal function.


Pointwise Characterization (1) / §III

◮ ([YY15]) Let α ∈ (0,∞) and f ∈ L1loc (R

n). A sequence~g := {gj}j≥0 of non-negative measurable functions is calledan α-order Hajłasz type gradient sequence of f if, for eachj, there exists a set Ej ⊂ Rn with measure zero such that

|f(x)−B2−jf(x)| ≤ 2−jαgj(x), ∀x ∈ Rn \ Ej .

Each gj satisfying the above is called an α-order Hajłasztype gradient of f at level j.

◮ [YY15] D. Yang & W. Yuan , Pointwise characterizations ofBesov and Triebel-Lizorkin spaces in terms of averages onballs, Trans. Amer. Math. Soc. 369 (2017), 7631-7655 .



◮ ([YY15]) Let α ∈ (0, 2) and p, q ∈ (1,∞]. Thenf ∈ Fα

p,q(Rn) if and only if f ∈ Lp(Rn) when p ∈ (0,∞) or

f ∈ C(Rn) when p = ∞, and there exists an α-order Hajłasztype gradient sequence ~g := {gk}

∞k=0 of f such that

∥∥∥∥∥∥

{∞∑

k=0

2kαq|gk|q

}1/q∥∥∥∥∥∥Lp(Rn)

< ∞, p < ∞

and

supℓ∈Z

x∈Rn

B2−ℓ

∑

k≥max{ℓ,0}

2kαq|gk|q

(x)

1/q

< ∞, p = ∞.



◮ ([YY15]) Let α ∈ (0, 2), p ∈ (1,∞] and q ∈ (0,∞]. Thenf ∈ Bα

p,q(Rn) if and only if f ∈ Lp(Rn) when p ∈ (0,∞), or

f ∈ C(Rn) when p = ∞, and there exists an α-order Hajłasztype gradient sequence ~g := {gk}

∞k=0 of f such that

{∞∑

k=0

2kαq ‖gk‖Lp(Rn)

}1/q

< ∞.

◮ Let ℓ ∈ N. All aforementioned pointwise characterizationsof Bα

p,q(Rn) and Fα

p,q(Rn) remain true when α ∈ (0, 2ℓ) if we

replace {f − B2−j(f)}j by {f −Bℓ,2−j(f)}j.

◮ Applications?


§IV. Further remarks


Newton Spaces (1) / §IV

◮ N. Shanmugalingam , Newtonian spaces: An extension ofSobolev spaces to metric measure spaces, Rev. Mat.Iberoamericana 16 (2000), 243-279 .

◮ Defined via upper gradients (J. Heinonen & P. Koskela[Acta Math., 1998]; P. Koskela & P. MacManus [StudiaMath., 1998]). Instead of straight lines by curves γ:|f(γ(a))− f(γ(b))| ≤

∫γ g ds.

◮ Advantage: Strong locality (If a function is constant on ameasurable set , then we can take upper gradient to be zeroalmost everywhere on that set ; however, we cannot take theHajłasz gradient to be zero almost everywhere on that set .)

◮ N. Shanmugalingam, D. Yang & W. Yuan , Newton-Besovspaces and Newton-Triebel-Lizorkin spaces, Positivity 19(2015), 177-220.


Newton Spaces (2) / §IV

◮ Newtonian-type Orlicz-Sobolev spaces on metric measurespaces

• H. Tuominen , Orlicz-Sobolev spaces on metricmeasure spaces, Dissertation, University of Jyv askyl a,Jyv askyl a, 2004. Ann. Acad. Sci. Fenn. Math. Diss. No. 135(2004), 86 pp .

◮ Hajłasz-type Orlicz-Sobolev spaces and Newtonian-typeOrlicz-Sobolev spaces

• T. Ohno & T. Shimomura , Musielak-Orlicz-Sobolevspaces on metric measure spaces, CzechoslovakMath. J. 65 (140) (2015), 435-474 .


Sphere Average Charact. / §IV

◮ P. Hajłasz & Z. Liu , A Marcinkiewicz integral typecharacterization of the Sobolev space, Publ. Mat. 61(2017), 83-104.

• Let p ∈ (1,∞). Then f ∈ W 1,p(Rn) if and only iff ∈ Lp(Rn) and

∫ ∞

0

∣∣∣∣∣f(·)−1

|S(·, t)|

∫

S(·,t)

f(y) dσ(y)

∣∣∣∣∣

2dt

t3

1/2

∈ Lp(Rn),

where S(x, t) denotes the sphere centered at x withthe radius t. (Not so useful for metric measure spaces )


Weighted Sobolev Spaces / §IV

• A new and simplified proof of the characterization ofW 1,p(Rn) with p ∈ (1,∞) ([Theorem 1, AMV12])

◮ S. Sato , Littlewood-Paley operators and Sobolev spaces,Illinois J. Math. 58 (2014), 1025-1039 .

• Generalize [Theorem 1, AMV12] to the weightedcase: Wα,p

w (Rn), α ∈ (0, 2), p ∈ (1,∞) and w ∈ Ap(Rn)

◮ S. Sato , Littlewood-Paley equivalence andhomogeneous Fourier multipliers, Integral EquationsOperator Theory 87 (2017), 15-44 .

◮ S. Sato , Spherical square functions of Marcinkiewicztype with Riesz potentials, Arch. Math. (Basel) 108 (2017),415-426.


Generalized Means / §IV◮ Let Φ be a bounded radial function on Rn with compact

support satisfying∫Rn Φ(x) dx = 1.

◮ Generalized means:

Gt(g)(x) :=

∫

Rn

1

tnΦ(x− y

t

)g(y) dy.

If let Φ := 1|B(~0n,1)|

χB(~0n,1), then Gt(g) = Bt(g).

• S. Sato, F. Wang, D. Yang & W. Yuan , GeneralizedLittlewood-Paley characterizations of fractionalSobolev spaces, Commun. Contemp. Math. (to appear).[only α ∈ (0, 2]]

• Y. Zhang, D.-C. Chang & D. Yang , GeneralizedLittlewood-Paley characterizations of Triebel-Lizorkinspaces, J. Nonlinear Convex Anal. 18 (2017),1171-1190. [only α ∈ (0, 2)]


Morrey-Sobolev Spaces / §IV

• Morrey -Sobolev Spaces on Metric Measure Spaces

◮ Let 0 < p ≤ q ≤ ∞. Recall that the Morrey space Mqp(X )

is defined to be the space of all measurable functions fon X such that

‖f‖Mqp(X ) := sup

B⊂X[µ(B)]1/q−1/p

[∫

B

|f(x)|p dµ(x)

]1/p< ∞,

where the supremum is taken over all balls B in X .

◮ Replace Lp(X ) by Mqp(X )

◮ Y. Lu, D. Yang & W. Yuan , Morrey-Sobolev spaces onmetric measure spaces, Potential Anal. 41 (2014), 215-243 .


Haroske and Triebel / §IV

• Triebel [T10] introduced the higher version ofHajłasz-Sobolev spaces on Rn via higher differences , andsome very interesting applications are given in [T11] and[HT11]:

◮ [HT11] D. D. Haroske & H. Triebel , Embeddings of functionspaces: a criterion in terms of differences, Complex Var.Elliptic Equ. 56 (2011), 931-944.

◮ [T10] H. Triebel , Sobolev-Besov spaces of measurablefunctions, Studia Math. 201 (2010), 69-86.

◮ [T11] H. Triebel , Limits of Besov norms, Arch. Math. 96(2011), 169-175.


Sobolev Spaces Associated withOperators / §IV

◮ L. Yan & D. Yang , New Sobolev spaces via generalizedPoincaré inequalities on metric measure spaces, Math.Z. 255 (2007), 133-159.

◮ S. Hofmann, S. Mayboroda & A. McIntosh , Second orderelliptic operators with complex bounded measurablecoefficients in Lp, Sobolev and Hardy spaces, Ann. Sci.Ecole Norm. Sup. (4) 44 (2011), 723-800 .

◮ F. Bernicot, T. Coulhon & F. Dorothee , Sobolev algebrasthrough heat kernel estimates, J. Ec. polytech. Math. 3(2016), 99-161.

◮ J. Zhang, D.-C. Chang & D. Yang , Characterizations ofSobolev spaces associated to operators satisfyingoff-diagonal estimates on balls, Math. Methods Appl. Sci.40 (2017), 2907-2929.


Lecture Notes in Math. 2182, 2017 / §IV

Thank you for your attention.Ball Average Characterizations of Function Spaces – p. 50/50

ball average characterizations of function spacesnpfsa2017.uni-jena.de/l_notes/yang.pdf · ball...

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