Endpoint regularity of the dyadic and the fractional maximal function

Endpoint regularity of the dyadic and the fractional maximal function

Julian Weigt

1/16

Background

Forf :R^d →Rthe uncentered Hardy-Littlewood maximal function is defined by

Mf(x) = sup

B3x

f_B with f_B = 1

L(B) ˆ

B

|f|.

The Hardy-Littlewood maximal function theorem:

kMfk_Lp(R^d)≤C_d,pkfk_Lp(R^d) if and only ifp >1 Juha Kinnunen (1997):

k∇Mfk_Lp(R^d)≤C_d,pk∇fk_Lp(R^d) ifp >1

Question (Haj lasz and Onninen 2004) Is it true that

k∇Mfk_L1(R^d)≤C_dk∇fk_L1(R^d)?

Background

Forf :R^d →Rthe uncentered Hardy-Littlewood maximal function is defined by

Mf(x) = sup

B3x

f_B with f_B = 1

L(B) ˆ

B

|f|.

The Hardy-Littlewood maximal function theorem:

kMfk_Lp(R^d)≤C_d,pkfk_Lp(R^d) if and only ifp >1 Juha Kinnunen (1997):

k∇Mfk_Lp(R^d)≤C_d,pk∇fk_Lp(R^d) ifp >1

Question (Haj lasz and Onninen 2004) Is it true that

k∇Mfk_L1(R^d)≤C_dk∇fk_L1(R^d)?

1/16

Background

Forf :R^d →Rthe uncentered Hardy-Littlewood maximal function is defined by

Mf(x) = sup

B3x

f_B with f_B = 1

L(B) ˆ

B

|f|.

The Hardy-Littlewood maximal function theorem:

kMfk_Lp(R^d)≤C_d,pkfk_Lp(R^d) if and only ifp >1 Juha Kinnunen (1997):

k∇Mfk_Lp(R^d)≤C_d,pk∇fk_Lp(R^d) ifp >1

Question (Haj lasz and Onninen 2004) Is it true that

k∇Mfk_L1(R^d)≤C_dk∇fk_L1(R^d)?

Progress on classical Hardy-Littlewood

d = 1 [Tanaka 2002, Aldaz+P´erez L´azaro 2007]

block decreasing f [Aldaz+P´erez L´azaro 2009]

centered M,d = 1 [Kurka 2015]

radial f [Luiro 2018]

characteristic f [W 2020]

3/16

The dyadic and the fractional maximal function

Forα∈(0,d) define the uncentered fractional Hardy-Littlewood maximal function by

M_αf(x) = sup

B3x

r(B)^αf_B, wherer(B) is the radius ofB.

The corresponding bound is k∇M_αfk_Ld/(d−α)(R^d)≤C_d,αk∇fk₁

The version withp >1 follows the same way as for α= 0 in Kinnunen (1997). The dyadic maximal function is

M^df(x) = sup

Qdyadic cube,x∈Q

f_Q The corresponding bound is

varM^df ≤C_dvarf.

The dyadic and the fractional maximal function

Forα∈(0,d) define the uncentered fractional Hardy-Littlewood maximal function by

M_αf(x) = sup

B3x

r(B)^αf_B,

wherer(B) is the radius ofB. The corresponding bound is k∇M_αfk_Ld/(d−α)(R^d)≤C_d,αk∇fk₁

The version withp >1 follows the same way as for α= 0 in Kinnunen (1997).

The dyadic maximal function is M^df(x) = sup

Qdyadic cube,x∈Q

f_Q The corresponding bound is

varM^df ≤C_dvarf.

3/16

The dyadic and the fractional maximal function

Forα∈(0,d) define the uncentered fractional Hardy-Littlewood maximal function by

M_αf(x) = sup

B3x

r(B)^αf_B,

wherer(B) is the radius ofB. The corresponding bound is k∇M_αfk_Ld/(d−α)(R^d)≤C_d,αk∇fk₁

The version withp >1 follows the same way as for α= 0 in Kinnunen (1997). The dyadic maximal function is

M^df(x) = sup

Qdyadic cube,x∈Q

f_Q

The corresponding bound is

varM^df ≤C_dvarf.

The dyadic and the fractional maximal function

Forα∈(0,d) define the uncentered fractional Hardy-Littlewood maximal function by

M_αf(x) = sup

B3x

r(B)^αf_B,

wherer(B) is the radius ofB. The corresponding bound is k∇M_αfk_Ld/(d−α)(R^d)≤C_d,αk∇fk₁

The version withp >1 follows the same way as for α= 0 in Kinnunen (1997). The dyadic maximal function is

M^df(x) = sup

Qdyadic cube,x∈Q

f_Q The corresponding bound is

4/16

Progress on related operators

convolution, d = 1 [Carneiro+Svaiter 2013]

uncentered fractional, α >1 [Kinnunen+Saksman 2003, Carneiro+Madrid 2017]

fractional smooth convolution [Beltran+Ramos+Saari 2018]

fractional lacunary [Beltran+Ramos+Saari 2018]

convolution, radial f [Carneiro+Svaiter 2019]

fractional, radialf [Beltran+Madrid 2019]

dyadic [W 2020]

fractional [W 2020]

related: Continuity off 7→ ∇Mf as W^1,1(R^d)→L¹(R^d) (does not follow from boundedness), boundedness on other spaces, local maximal operator.

Progress on related operators

convolution, d = 1 [Carneiro+Svaiter 2013]

uncentered fractional, α >1 [Kinnunen+Saksman 2003, Carneiro+Madrid 2017]

fractional smooth convolution [Beltran+Ramos+Saari 2018]

fractional lacunary [Beltran+Ramos+Saari 2018]

convolution, radial f [Carneiro+Svaiter 2019]

fractional, radialf [Beltran+Madrid 2019]

dyadic [W 2020]

fractional [W 2020]

related: Continuity off 7→ ∇Mf as W^1,1(R^d)→L¹(R^d) (does not follow from boundedness), boundedness on other spaces, local maximal operator.

4/16

Progress on related operators

convolution, d = 1 [Carneiro+Svaiter 2013]

uncentered fractional, α >1 [Kinnunen+Saksman 2003, Carneiro+Madrid 2017]

fractional smooth convolution [Beltran+Ramos+Saari 2018]

fractional lacunary [Beltran+Ramos+Saari 2018]

convolution, radial f [Carneiro+Svaiter 2019]

fractional, radialf [Beltran+Madrid 2019]

dyadic [W 2020]

fractional [W 2020]

related: Continuity off 7→ ∇Mf as W^1,1(R^d)→L¹(R^d) (does not follow from boundedness), boundedness on other spaces, local maximal operator.

1. The Dyadic Maximal Operator

By the coarea formula

and{f > λ} ⊂ {M^df > λ} up to measure zero

varM^df = ˆ

R

H^d−1(∂{M^df > λ})dλ

≤ ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ}) +H^d−1(∂{f > λ})dλ

Since ˆ

R

H^d−1(∂{f > λ})dλ= varf it suffices to bound the first summand.

5/16

1. The Dyadic Maximal Operator

By the coarea formula and{f > λ} ⊂ {M^df > λ} up to measure zero

varM^df = ˆ

R

H^d−1(∂{M^df > λ})dλ

≤ ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ}) +H^d−1(∂{f > λ})dλ

Since ˆ

R

H^d−1(∂{f > λ})dλ= varf it suffices to bound the first summand.

1. The Dyadic Maximal Operator

By the coarea formula and{f > λ} ⊂ {M^df > λ} up to measure zero

varM^df = ˆ

R

H^d−1(∂{M^df > λ})dλ

≤ ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ}) +H^d−1(∂{f > λ})dλ

Since ˆ

R

H^d−1(∂{f > λ})dλ= varf it suffices to bound the first summand.

6/16

Definition

Q is maximal forλ <f_Q if for allP )Q we havef_P ≤λ.

GivenQ, let λ_Q be the smallest suchλ. ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ})dλ

= ˆ

R

H^d−1(∂[

{maximal Q :f_Q > λ} \ {f > λ})dλ

≤ ˆ

R

X

maximalQ:f_Q>λ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<λ˜Q

+ X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

where ”L(Q∩ {f >˜λ_Q}) = 2^−d−2L(Q)”.

Definition

Q is maximal forλ <f_Q if for allP )Q we havef_P ≤λ.

GivenQ, let λ_Q be the smallest suchλ.

ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ})dλ

= ˆ

R

H^d−1(∂[

{maximal Q :f_Q > λ} \ {f > λ})dλ

≤ ˆ

R

X

maximalQ:f_Q>λ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<λ˜Q

+ X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

where ”L(Q∩ {f >˜λ_Q}) = 2^−d−2L(Q)”.

6/16

Definition

Q is maximal forλ <f_Q if for allP )Q we havef_P ≤λ.

GivenQ, let λ_Q be the smallest suchλ.

ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ})dλ

= ˆ

R

H^d−1(∂[

{maximal Q :f_Q > λ} \ {f > λ})dλ

≤ ˆ

R

X

maximalQ:f_Q>λ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<λ˜Q

+ X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

where ”L(Q∩ {f >˜λ_Q}) = 2^−d−2L(Q)”.

Definition

Q is maximal forλ <f_Q if for allP )Q we havef_P ≤λ.

GivenQ, let λ_Q be the smallest suchλ.

ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ})dλ

= ˆ

R

H^d−1(∂[

{maximal Q :f_Q > λ} \ {f > λ})dλ

≤ ˆ

R

X

maximalQ:f_Q>λ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<λ˜Q

+ X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

where ”L(Q∩ {f >˜λ_Q}) = 2^−d−2L(Q)”.

6/16

Definition

Q is maximal forλ <f_Q if for allP )Q we havef_P ≤λ.

GivenQ, let λ_Q be the smallest suchλ.

ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ})dλ

= ˆ

R

H^d−1(∂[

{maximal Q :f_Q > λ} \ {f > λ})dλ

≤ ˆ

R

X

maximalQ:f_Q>λ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<λ˜Q

+ X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

where ”L(Q∩ {f >˜λ_Q}) = 2^−d−2L(Q)”.

Definition

Q is maximal forλ <f_Q if for allP )Q we havef_P ≤λ.

GivenQ, let λ_Q be the smallest suchλ.

ˆ

R

H^d−1(∂{M^df > λ} \ {f > λ})dλ

= ˆ

R

H^d−1(∂[

{maximal Q :f_Q > λ} \ {f > λ})dλ

≤ ˆ

R

X

maximalQ:f_Q>λ

H^d−1(∂Q\ {f > λ})dλ

= ˆ

R

X

Q:λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

=

ˆ X

+ X

H^d−1(∂Q\ {f > λ})dλ

7/16

High density case

Proposition (High density) ForL(Q∩E)≥εL(Q) we have

H^d−1(∂Q\E).εH^d−1(Q∩∂E).

ˆ

R

X

Q:λQ<λ<˜λQ

H^d−1(∂Q\ {f > λ})dλ

. ˆ

R

X

Q:λQ<λ<˜λQ

H^d−1(Q∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ

= varf

High density case

Proposition (High density) ForL(Q∩E)≥εL(Q) we have

H^d−1(∂Q\E).εH^d−1(Q∩∂E).

ˆ

R

X

Q:λQ<λ<˜λQ

H^d−1(∂Q\ {f > λ})dλ

. ˆ

R

X

Q:λQ<λ<˜λQ

H^d−1(Q∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ

= varf

7/16

High density case

Proposition (High density) ForL(Q∩E)≥εL(Q) we have

H^d−1(∂Q\E).εH^d−1(Q∩∂E).

ˆ

R

X

Q:λQ<λ<˜λQ

H^d−1(∂Q\ {f > λ})dλ

. ˆ

R

X

Q:λQ<λ<˜λQ

H^d−1(Q∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ

= varf

High density case

Proposition (High density) ForL(Q∩E)≥εL(Q) we have

H^d−1(∂Q\E).εH^d−1(Q∩∂E).

ˆ

R

X

Q:λQ<λ<˜λQ

H^d−1(∂Q\ {f > λ})dλ

. ˆ

R

X

Q:λQ<λ<˜λQ

H^d−1(Q∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ

E

Q

L(Q∩E)≥εL(Q)

=⇒ H^d−1(∂Q\E).εH^d−1(Q∩∂E)

Low density case

Estimate ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

≤ ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

We have to estimate this by varf.

9/16

Low density case

Estimate ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

≤ ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

We have to estimate this by varf.

Low density case

Estimate ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

≤ ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

We have to estimate this by varf.

9/16

Low density case

Estimate ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q\ {f > λ})dλ

≤ ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

We have to estimate this by varf.

Relative isoperimetric inequality

ForL(Q∩E)≤ L(Q)/2 the relative isoperimetric inequality states L(Q∩E)^d−1d .H^d−1(Q∩∂E)

Proposition

(f_Q −λ˜_Q)L(Q). ˆ

R

X

P(Q:¯λP<λ<fP

L(P ∩ {f > λ})dλ

whereP is maximal above ¯λ_P and

”L(P ∩ {f >¯λP}) = 2⁻¹L(P)”

”L(Q∩ {f >˜λ_Q}) = 2^−d−2L(Q)”

10/16

Relative isoperimetric inequality

ForL(Q∩E)≤ L(Q)/2 the relative isoperimetric inequality states L(Q∩E)^d−1d .H^d−1(Q∩∂E)

Proposition

(f_Q −λ˜_Q)L(Q). ˆ

R

X

P(Q:¯λP<λ<fP

L(P ∩ {f > λ})dλ

whereP is maximal above ¯λ_P and

”L(P ∩ {f >¯λP}) = 2⁻¹L(P)”

”L(Q∩ {f >λ˜_Q}) = 2^−d−2L(Q)”

fQ

f_P1

fP2

P3

fP3

12/16

X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

. ˆ

R

X

Q

l(Q)⁻¹ X

P(Q:¯λP<λ<fP

L(P∩ {f > λ})dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P∩ {f > λ})X

Q)P

l(Q)⁻¹dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P∩ {f > λ}) l(P)⁻¹dλ

≤ ˆ

R

X

P:¯λP<λ<fP

L(P∩ {f > λ})^d−1d dλ

. ˆ

R

X

P:¯λP<λ<fP

H^d−1(P∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ= varf

X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

. ˆ

R

X

Q

l(Q)⁻¹ X

P(Q:¯λP<λ<fP

L(P∩ {f > λ})dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})X

Q)P

l(Q)⁻¹dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P∩ {f > λ}) l(P)⁻¹dλ

≤ ˆ

R

X

P:¯λP<λ<fP

L(P∩ {f > λ})^d−1d dλ

. ˆ

R

X

P:¯λP<λ<fP

H^d−1(P∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ= varf

12/16

X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

. ˆ

R

X

Q

l(Q)⁻¹ X

P(Q:¯λP<λ<fP

L(P∩ {f > λ})dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})X

Q)P

l(Q)⁻¹dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ}) l(P)⁻¹dλ

≤ ˆ

R

X

P:¯λP<λ<fP

L(P∩ {f > λ})^d−1d dλ

. ˆ

R

X

P:¯λP<λ<fP

H^d−1(P∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ= varf

X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

. ˆ

R

X

Q

l(Q)⁻¹ X

P(Q:¯λP<λ<fP

L(P∩ {f > λ})dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})X

Q)P

l(Q)⁻¹dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ}) l(P)⁻¹dλ

≤ ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})^d−1d dλ

. ˆ

R

X

P:¯λP<λ<fP

H^d−1(P∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ= varf

12/16

X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

. ˆ

R

X

Q

l(Q)⁻¹ X

P(Q:¯λP<λ<fP

L(P∩ {f > λ})dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})X

Q)P

l(Q)⁻¹dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ}) l(P)⁻¹dλ

≤ ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})^d−1d dλ

. ˆ

R

X

P:¯λP<λ<fP

H^d−1(P∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ= varf

X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

. ˆ

R

X

Q

l(Q)⁻¹ X

P(Q:¯λP<λ<fP

L(P∩ {f > λ})dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})X

Q)P

l(Q)⁻¹dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ}) l(P)⁻¹dλ

≤ ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})^d−1d dλ

. ˆ

R

X

P:¯λP<λ<fP

H^d−1(P∩∂{f > λ})dλ ˆ

12/16

X

Q

(f_Q−˜λ_Q)H^d−1(∂Q)

. ˆ

R

X

Q

l(Q)⁻¹ X

P(Q:¯λP<λ<fP

L(P∩ {f > λ})dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})X

Q)P

l(Q)⁻¹dλ

= ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ}) l(P)⁻¹dλ

≤ ˆ

R

X

P:¯λP<λ<fP

L(P ∩ {f > λ})^d−1d dλ

. ˆ

R

X

P:¯λP<λ<fP

H^d−1(P∩∂{f > λ})dλ

≤ ˆ

R

H^d−1(∂{f > λ})dλ= varf

2. The fractional maximal operator

Recall 0< α <d and

M_αf(x) = sup

B3x

r(B)^αf_B.

Then for almost everyx ∈R^d the supremum is attained in some ballB with x∈B. Denote byB_α the set of all optimal balls forf. Want to show

k∇M_αfk_d/(d−α)≤C_d,αk∇fk₁.

Kinnunen and Saksman (2003) For an optimal ballB for x we have

|∇M_αf(x)| ≤(d −α)r(B)^α−1f_B. Conclude

|∇M_αf(x)|. sup

B∈Bα,x∈B

r(B)^α−1f_B =:M_α,−1f(x).

13/16

2. The fractional maximal operator

Recall 0< α <d and

M_αf(x) = sup

B3x

r(B)^αf_B.

Then for almost everyx ∈R^d the supremum is attained in some ballB with x∈B. Denote byB_α the set of all optimal balls forf.

Want to show

k∇M_αfk_d/(d−α)≤C_d,αk∇fk₁.

Kinnunen and Saksman (2003) For an optimal ballB for x we have

|∇M_αf(x)| ≤(d −α)r(B)^α−1f_B. Conclude

|∇M_αf(x)|. sup

B∈Bα,x∈B

r(B)^α−1f_B =:M_α,−1f(x).

2. The fractional maximal operator

Recall 0< α <d and

M_αf(x) = sup

B3x

r(B)^αf_B.

Then for almost everyx ∈R^d the supremum is attained in some ballB with x∈B. Denote byB_α the set of all optimal balls forf. Want to show

k∇M_αfk_d/(d−α)≤C_d,αk∇fk₁.

Kinnunen and Saksman (2003) For an optimal ballB for x we have

|∇M_αf(x)| ≤(d −α)r(B)^α−1f_B. Conclude

|∇M_αf(x)|. sup

B∈Bα,x∈B

r(B)^α−1f_B =:M_α,−1f(x).

13/16

2. The fractional maximal operator

Recall 0< α <d and

M_αf(x) = sup

B3x

r(B)^αf_B.

Then for almost everyx ∈R^d the supremum is attained in some ballB with x∈B. Denote byB_α the set of all optimal balls forf. Want to show

k∇M_αfk_d/(d−α)≤C_d,αk∇fk₁.

Kinnunen and Saksman (2003) For an optimal ballB for x we have

|∇M_αf(x)| ≤(d−α)r(B)^α−1f_B.

Conclude

|∇M_αf(x)|. sup

B∈Bα,x∈B

r(B)^α−1f_B =:M_α,−1f(x).

2. The fractional maximal operator

Recall 0< α <d and

M_αf(x) = sup

B3x

r(B)^αf_B.

Then for almost everyx ∈R^d the supremum is attained in some ballB with x∈B. Denote byB_α the set of all optimal balls forf. Want to show

k∇M_αfk_d/(d−α)≤C_d,αk∇fk₁.

Kinnunen and Saksman (2003) For an optimal ballB for x we have

|∇M_αf(x)| ≤(d−α)r(B)^α−1f_B. Conclude

14/16

1. Make disjoint ˆ

(M_α,−1f)^d−αd = ˆ

sup

B∈Bα

(r(B)^α−1f_B)^d−αd 1_B

.α,c1,c2

ˆ X

B∈Be_α

(r(B)^α−1fB)^d−αd 1B

∼_α X

B∈Beα

(f_BH^d−1(∂B))^d−αd

whereBe_α⊂ B_α such that for two ballsB,C ∈Bewe have c1B∩c1C =∅, or r(C)<r(B) andfC >c2fB.

Ifα−1≥0: Vitali covering argument suffices. Ifα−1<0: Use that ifB,C ∈ B_α with C ⊂B and

r(C)<r(B)/N we haver(C)^αf_C >r(B)^αf_B and thus f_C >N^αf_B.

1. Make disjoint ˆ

(M_α,−1f)^d−αd = ˆ

sup

B∈Bα

(r(B)^α−1f_B)^d−αd 1_B .α,c1,c2

ˆ X

B∈Be_α

(r(B)^α−1fB)^d−αd 1B

∼_α X

B∈Beα

(f_BH^d−1(∂B))^d−αd

whereBe_α⊂ B_α such that for two ballsB,C ∈Bewe have c1B∩c1C =∅, or r(C)<r(B) andfC >c2fB.

Ifα−1≥0: Vitali covering argument suffices. Ifα−1<0: Use that ifB,C ∈ B_α with C ⊂B and

r(C)<r(B)/N we haver(C)^αf_C >r(B)^αf_B and thus f_C >N^αf_B.

14/16

1. Make disjoint ˆ

(M_α,−1f)^d−αd = ˆ

sup

B∈Bα

(r(B)^α−1f_B)^d−αd 1_B .α,c1,c2

ˆ X

B∈Be_α

(r(B)^α−1fB)^d−αd 1B

∼_α X

B∈Beα

(f_BH^d−1(∂B))^d−αd

whereBe_α⊂ B_α such that for two ballsB,C ∈Bewe have c1B∩c1C =∅, or r(C)<r(B) andfC >c2fB.

Ifα−1≥0: Vitali covering argument suffices.

Ifα−1<0: Use that ifB,C ∈ B_α with C ⊂B and

r(C)<r(B)/N we haver(C)^αf_C >r(B)^αf_B and thus f_C >N^αf_B.

1. Make disjoint ˆ

(M_α,−1f)^d−αd = ˆ

sup

B∈Bα

(r(B)^α−1f_B)^d−αd 1_B .α,c1,c2

ˆ X

B∈Be_α

(r(B)^α−1fB)^d−αd 1B

∼_α X

B∈Beα

(f_BH^d−1(∂B))^d−αd

whereBe_α⊂ B_α such that for two ballsB,C ∈Bewe have c1B∩c1C =∅, or r(C)<r(B) andfC >c2fB.

Ifα−1≥0: Vitali covering argument suffices.

Ifα−1<0: Use that ifB,C ∈ B_α with C ⊂B and

14/16

1. Make disjoint ˆ

(M_α,−1f)^d−αd = ˆ

sup

B∈Bα

(r(B)^α−1f_B)^d−αd 1_B .α,c1,c2

ˆ X

B∈Be_α

(r(B)^α−1fB)^d−αd 1B

∼_α X

B∈Beα

(f_BH^d−1(∂B))^d−αd

whereBe_α⊂ B_α such that for two ballsB,C ∈Bewe have c1B∩c1C =∅, or r(C)<r(B) andfC >c2fB.

Ifα−1≥0: Vitali covering argument suffices.

Ifα−1<0: Use that ifB,C ∈ B_α with C ⊂B and

r(C)<r(B)/N we haver(C)^αf_C >r(B)^αf_B and thus f_C >N^αf_B.

2. Reduce to dyadic

X

B∈Be_α

(fBH^d−1(∂B))^d−αd

≤

X

B∈B˜α

fBH^d−1(∂B) _d−α^d

.α

X

Q∈Q˜_α

f_QH^d−1(∂Q) _d−α^d

≤C_d,α(varf)^d−αd

where l(Q)∼_αr(B) and fQ ∼_α fB so that alsoc_αQ∩c_αP =∅, or l(P)<l(Q) andf_P >2f_Q.

15/16

2. Reduce to dyadic

X

B∈Be_α

(fBH^d−1(∂B))^d−αd ≤

X

B∈B˜α

fBH^d−1(∂B) _d−α^d

.α

X

Q∈Q˜_α

f_QH^d−1(∂Q) _d−α^d

≤C_d,α(varf)^d−αd

where l(Q)∼_αr(B) and fQ ∼_α fB so that alsoc_αQ∩c_αP =∅, or l(P)<l(Q) andf_P >2f_Q.

2. Reduce to dyadic

X

B∈Be_α

(fBH^d−1(∂B))^d−αd ≤

X

B∈B˜α

fBH^d−1(∂B) _d−α^d

.α

X

Q∈Q˜_α

f_QH^d−1(∂Q) _d−α^d

≤C_d,α(varf)^d−αd

where l(Q)∼_αr(B) and fQ ∼_α fB so that alsoc_αQ∩c_αP =∅, or l(P)<l(Q) andf_P >2f_Q.

15/16

2. Reduce to dyadic

X

B∈Be_α

(fBH^d−1(∂B))^d−αd ≤

X

B∈B˜α

fBH^d−1(∂B) _d−α^d

.α

X

Q∈Q˜_α

f_QH^d−1(∂Q) _d−α^d

≤C_d,α(varf)^d−αd

where l(Q)∼_αr(B) and fQ ∼_α fB so that alsoc_αQ∩c_αP =∅, or l(P)<l(Q) andf_P >2f_Q.

2. Reduce to dyadic

X

B∈Be_α

(fBH^d−1(∂B))^d−αd ≤

X

B∈B˜α

fBH^d−1(∂B) _d−α^d

.α

X

Q∈Q˜_α

f_QH^d−1(∂Q) _d−α^d

≤C_d,α(varf)^d−αd

where l(Q)∼_αr(B) and fQ ∼_α fB so that alsoc_αQ∩c_αP =∅, or l(P)<l(Q) andf_P >2f_Q.

16/16

Thank you