Weighted boundedness of multilinear maximal function using Dirac - - PowerPoint PPT Presentation

Weighted boundedness of multilinear maximal function using Dirac deltas

Abhishek Ghosh

(Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) Indian Institute of Technology Kanpur, India

May 20-24, 2019

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 1 / 20

Motivation

For a locally integrable function f on Rn, Hardy-Littlewood Maximal function is defined as

M f(x) = sup

Q∋x

1 |Q|

• Q

|f(y)|dy.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 2 / 20

Motivation

For a locally integrable function f on Rn, Hardy-Littlewood Maximal function is defined as

M f(x) = sup

Q∋x

1 |Q|

• Q

|f(y)|dy.

It maps L1(Rn) to L1,∞(Rn). Classical proof depends on Covering lemmas(Vittali). By interpolation, it maps Lp(Rn) to itself for

1 < p < ∞.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 2 / 20

Introduction

{Kj}j be family of locally integrable functions and define the following

maximal operator

K∗ f(x) = sup

j

• Kj ∗ f(x)
• .

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 3 / 20

Introduction

{Kj}j be family of locally integrable functions and define the following

maximal operator

K∗ f(x) = sup

j

• Kj ∗ f(x)
• .

Miguel de Guzm´ an(1981): If K′

js are integrable, then K∗ is weak type

(1, 1) if and only if K∗ is weak type (1, 1) over Dirac deltas.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 3 / 20

Introduction

{Kj}j be family of locally integrable functions and define the following

maximal operator

K∗ f(x) = sup

j

• Kj ∗ f(x)
• .

Miguel de Guzm´ an(1981): If K′

js are integrable, then K∗ is weak type

(1, 1) if and only if K∗ is weak type (1, 1) over Dirac deltas. K∗ is weak type (1, 1) if and only if ∃ C > 0 such that, for any set of distinct

points a1, . . . , aN and for each λ > 0

• x : sup

j

• N
• i=1

Kj(x − ai)

• > λ
• ≤ C N

λ .

(Guzm´ an)

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 3 / 20

Contd.

• H. Carlsson(1984): M is weak type (1, 1) over Dirac deltas using the

principle of induction.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 4 / 20

Contd.

• H. Carlsson(1984): M is weak type (1, 1) over Dirac deltas using the

principle of induction. Let φ =

N

• i=1

biδai and for any λ > 0, define Eλ := {x : Mφ(x) > λ}.

Then

|Eλ| ≤ 2n λ

N

• i=1

|bi|.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 4 / 20

Contd.

• H. Carlsson(1984): M is weak type (1, 1) over Dirac deltas using the

principle of induction. Let φ =

N

• i=1

biδai and for any λ > 0, define Eλ := {x : Mφ(x) > λ}.

Then

|Eλ| ≤ 2n λ

N

• i=1

|bi|.

This provides another proof for weak (1, 1) boundedness.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 4 / 20

Contd.

• M. Trinidad Men´

arguez and F. Soria(1992): If Kj ≥ 0, then the constant in the weak (1, 1) inequality is same as the constant in the following inequality

• x : sup

j

• N
• i=1

Kj(x − ai)

• > λ
• ≤ C N

λ .

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 5 / 20

Contd.

• M. Trinidad Men´

arguez and F. Soria(1992): If Kj ≥ 0, then the constant in the weak (1, 1) inequality is same as the constant in the following inequality

• x : sup

j

• N
• i=1

Kj(x − ai)

• > λ
• ≤ C N

λ .

This method played a crucial role in obtaining the best constant in the weak type (1, 1) inequality for the centred Hardy-Littlewood maximal

• perator on R in the works of Men´

arguez and Soria(1992), Manfredi and Soria, J.M. Aldaz(1998) and finally it is settled by

• A. D. Melas(2003).

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 5 / 20

Weighted Ineq.

Theorem (B. Muckenhoupt,1972)

M is of weighted weak-type (1, 1) if and only if w ∈ A1 and when 1 < p < ∞, M is weighted strong type (p, p) if and only if w ∈ Ap.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 6 / 20

Weighted Ineq.

Theorem (B. Muckenhoupt,1972)

M is of weighted weak-type (1, 1) if and only if w ∈ A1 and when 1 < p < ∞, M is weighted strong type (p, p) if and only if w ∈ Ap. w ∈ A1 iff ∃ C > 0 such that for all cubes Q

1 |Q|

• Q w ≤ C ess inf

Q w.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 6 / 20

Weighted Ineq.

Theorem (B. Muckenhoupt,1972)

M is of weighted weak-type (1, 1) if and only if w ∈ A1 and when 1 < p < ∞, M is weighted strong type (p, p) if and only if w ∈ Ap. w ∈ A1 iff ∃ C > 0 such that for all cubes Q

1 |Q|

• Q w ≤ C ess inf

Q w.

w ∈ Ap iff ∃ C > 0 sup

Q

1

|Q|

• Q w

1 |Q|

• Q w−

1 p−1

p−1 ≤ C.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 6 / 20

Weighted Ineq.

Theorem (B. Muckenhoupt,1972)

M is of weighted weak-type (1, 1) if and only if w ∈ A1 and when 1 < p < ∞, M is weighted strong type (p, p) if and only if w ∈ Ap. w ∈ A1 iff ∃ C > 0 such that for all cubes Q

1 |Q|

• Q w ≤ C ess inf

Q w.

w ∈ Ap iff ∃ C > 0 sup

Q

1

|Q|

• Q w

1 |Q|

• Q w−

1 p−1

p−1 ≤ C.

Classical proofs depend on Calder´

• n-Zygmund decomposition.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 6 / 20

Contd.

• D. Termini and C. Vitanza: Extended Guzm´

an’s method for A1 weights.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 7 / 20

Contd.

• D. Termini and C. Vitanza: Extended Guzm´

an’s method for A1 weights. Given w ∈ C(Rn), M is weak type (1, 1) with respect to w if and

• nly if

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 7 / 20

Contd.

• D. Termini and C. Vitanza: Extended Guzm´

an’s method for A1 weights. Given w ∈ C(Rn), M is weak type (1, 1) with respect to w if and

• nly if for any φ =

N

• i=1

biδai and λ > 0, we have w({x : Mφ(x) > λ}) Cw,n λ

N

• i=1

|bi|w(ai).

(*)

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 7 / 20

Contd.

• D. Termini and C. Vitanza: Extended Guzm´

an’s method for A1 weights. Given w ∈ C(Rn), M is weak type (1, 1) with respect to w if and

• nly if for any φ =

N

• i=1

biδai and λ > 0, we have w({x : Mφ(x) > λ}) Cw,n λ

N

• i=1

|bi|w(ai).

(*) Given w ∈ C(Rn), M satisfies (*) on linear combination of Dirac deltas if and only if w ∈ A1.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 7 / 20

Contd.

Theorem (Termini and Vitanza, 1989) Hardy-Littlewood maximal function is weak type (1, 1) with respect to w if and only if w ∈ A1.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 8 / 20

Contd.

Theorem (Termini and Vitanza, 1989) Hardy-Littlewood maximal function is weak type (1, 1) with respect to w if and only if w ∈ A1.

• M. Trinidad Men´

arguez extended this for more general class of maxi- mal convolution operators.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 8 / 20

Main Result

Let {K1

i } and {K2 i } be families of locally integrable kernels defined on

Rn, Ti(f1, f2)(x) =

• K1

i ∗ f1

• (x)
• K2

i ∗ f2

• (x).

Consider the bilinear maximal operator defined by

T ∗( f1, f2)(x) = sup

i∈N

|Ti( f1, f2)(x)|.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 9 / 20

Contd.

Let w1, w2, and v be continuous weight functions and q > 0. Assume that

1

K1

i , K2 i ∈ L∞(Rn) and K1 i (. − y), K2 i (. − y) ∈ L1(v) for all y ∈ Rn.

2

Given ǫ > 0, a ball B ⊆ Rn and i ∈ N, there are

γ1 = γ(ǫ, i, B) and γ2 = γ(ǫ, i, B) such that

• B

|K j

i (x − y1) − K j i (x − y2)|v(x)dx < ǫ

whenever |y1 − y2| < γj for j = 1, 2.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 10 / 20

Main Result

Theorem (-, Shrivastava, Shuin) Under the hypothesis the following are equivalent

T ∗ is bounded from L1(w1) × L1(w2) to Lq,∞(v).

For any set of distinct points {al}N

l=1 and {bk}L k=1 and for any λ > 0,

∃ C > 0 such that v

• x ∈ Rn : T ∗(

N

• l=1

δal,

L

• k=1

δbk)(x) > λ

• ≤ C

w,n

λq

N

• l=1

w1(al) q L

• k=1

w2(bk) q .

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 11 / 20

Applications

Multilinear fractional maximal function(Kabe Moen, 2009): For 0 ≤ α < 2n, the multilinear fractional maximal function is de- fined as follows

Mα( f1, f2)(x) = sup

r>0 2

• i=1

1 |B(x, r)|1− α

2n

• B(x,r)

|fi(y)|dy.

For α = 0, the corresponding operator is the multilinear Hardy-Littlewood maximal operator defined by Lerner et al(2009).

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 12 / 20

Contd.

Mα( f1, f2)(x) = sup

ri∈Q+ |Kri ∗ f1(x)||Kri ∗ f2(x)|

= sup

i≥1

|Ki ∗ f1(x)||Ki ∗ f2(x)|

where Ki = Kri(x) =

χB(0,ri)(x) |B(0,ri)|1− α

2n . Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 13 / 20

Contd.

Mα( f1, f2)(x) = sup

ri∈Q+ |Kri ∗ f1(x)||Kri ∗ f2(x)|

= sup

i≥1

|Ki ∗ f1(x)||Ki ∗ f2(x)|

where Ki = Kri(x) =

χB(0,ri)(x) |B(0,ri)|1− α

2n .

|Ki(x − y2) − Ki(x − y1)| = 1 |B(0, ri)|1− α

2n χB∗ i

where B∗

i = B(y1, ri) ∪ B(y2, ri) \ B(y1, ri) ∩ B(y2, ri).

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 13 / 20

Contd.

Lemma (-, Shrivastava, Shuin) Let (w1, w2, v) be continuous weight functions. Suppose that for any set of distinct points {al}N

l=1 and {bk}L k=1 and for any λ > 0, ∃ C > 0 such that

v

• x ∈ Rn : Mα(

N

• l=1

δal,

L

• k=1

δbk)(x) > λ

• ≤ C

λq

N

• l=1

w1(al) q L

• k=1

w2(bk) q

holds, then the bilinear fractional maximal function Mα is bounded from

L1(w1) × L1(w2) to Lq,∞(v).

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 14 / 20

Contd.

Lemma (-, Shrivastava, Shuin) Let (w1, w2, v) be continuous weight functions. Then Mα is bounded from

L1(w1) × L1(w2) to Lq,∞(v) on finite sum of Dirac deltas if and only if (w1, w2, v) ∈ A

1,α.

We say (w1, w2, v) ∈ A

1,α if and only if

1 |B|

• B

v(x)dx ≤ C(ess inf

B w1(x))q(ess inf B w2(x))q

for all balls B ⊆ Rn and q =

n 2n−α.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 15 / 20

Contd.

For continuous weighs, Mα is bounded from L1(w1) × L1(w2) to Lq,∞(v) if and only if (w1, w2, v) ∈ A

1,α.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 16 / 20

Contd.

For continuous weighs, Mα is bounded from L1(w1) × L1(w2) to Lq,∞(v) if and only if (w1, w2, v) ∈ A

1,α.

From continuous to arbitrary weights:

(w1, w2) ∈ A

1,α if and only if v w,α, wq 1, wq 2 ∈ A1, where v w,α = wq 1wq 2 and

q =

n 2n−α.

Take

w = (w1, w2) ∈ A

1,α. For ǫ > 0, define

wǫ = (w1,ǫ, w2,ǫ) and v

w,ǫ by

wq

i,ǫ(x) :=

1 |B(x, ǫ)|

• B(x,ǫ)

wq

i (y)dy

for i = 1, 2 and v

w,ǫ(x) = wq 1,ǫ(x)wq 2,ǫ(x).

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 16 / 20

Contd.

Lemma (-, Shrivastava, Shuin) Let

w = (w1, w2) ∈ A

1,α and ǫ > 0. Then

wǫ = (w1,ǫ, w2,ǫ) ∈ A

1,α with a

constant independent of ǫ.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 17 / 20

Contd.

Lemma (-, Shrivastava, Shuin) Let

w = (w1, w2) ∈ A

1,α and ǫ > 0. Then

wǫ = (w1,ǫ, w2,ǫ) ∈ A

1,α with a

constant independent of ǫ. Theorem (-, Shrivastava, Shuin) Let (w1, w2, v) be weights. The bilinear maximal function Mα is bounded from L1(w1) × L1(w2) to Lq,∞(v) if and only if

1 |B|

• B

v(x)dx ≤ C(ess inf

B w1(x))q(ess inf B w2(x))q

for all balls B ⊆ Rn.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 17 / 20

References I

• H. Carlsson, A new proof of the Hardy-Littlewod maximal theorem. Bull.

London Math. Soc. 16(1994) 595-596. Miguel de. Guzm´ an, Real variable methods in Fourier analysis. Vol 46, North-Holland Mathematics Studies, Amsterdam, 1981.

• A. K. Lerner; S. Ombrosi; C. Perez; R. H. Torres; R. Trujillo-Gonzalez, New

maximal functions and multiple weights for the multilinear Calderon-Zygmund theory. Adv. Math. 220(2009) 1222-1264.

• A. D. Melas, The best constant for the centered Hardy-Littlewood maximal
• inequality. Ann. of Math. (2) 157 (2003), no. 2, 647–688.
• M. Trinidad Menarguez; F. Soria, Weak type (1, 1) inequalities of maximal

covolution operators. Rend. Circ. Mat. Palermo 41 (1992), 342–352.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 18 / 20

References II

• K. Moen, Weighted inequalities for multilinear fractional integral operators.
• Collect. Math. 60 (2009), no. 2, 213-238. 42B25 (26D10).
• B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function.
• Trans. Amer. Math. Soc. 165 (1972), 207-226. 46E30 (26A86 42A40).
• D. Termini; C. Vitanza, Weighted estimates for the Hardy-Littlewood maximal
• perator and Dirac deltas. Bull. London Math. Soc 22(1990) 367-374.
• A. Ghosh; S. Shrivastava; K. Shuin, Weighted boundedness of multilinear

maximal function using Dirac deltas. Rendiconti del Circolo Matematico di Palermo Series 2 . DOI:https://doi.org/10.1007/s12215-019-00401-8.

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 19 / 20

Thank You!!!

Abhishek Ghosh (Joint work with Prof. Saurabh Shrivastava and Kalachand Shuin) (Indian Institute of Technology Kanpur, India) Probability and Analysis-2019 May 20-24, 2019 20 / 20