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 R n , Hardy-Littlewood Maximal function is defined as � 1 | f ( y ) | dy . M f ( x ) = sup | Q | Q ∋ x 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 2 / 20
Motivation For a locally integrable function f on R n , Hardy-Littlewood Maximal function is defined as � 1 | f ( y ) | dy . M f ( x ) = sup | Q | Q ∋ x Q It maps L 1 ( R n ) to L 1 , ∞ ( R n ) . Classical proof depends on Covering lemmas(Vittali). By interpolation, it maps L p ( R n ) 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 { K j } j be family of locally integrable functions and define the following maximal operator K ∗ f ( x ) = sup � � � K j ∗ f ( x ) � . � � j 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 { K j } j be family of locally integrable functions and define the following maximal operator K ∗ f ( x ) = sup � � � K j ∗ f ( x ) � . � � j j s are integrable, then K ∗ is weak type an(1981): If K ′ Miguel de Guzm´ (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 { K j } j be family of locally integrable functions and define the following maximal operator K ∗ f ( x ) = sup � � � K j ∗ f ( x ) � . � � j j s are integrable, then K ∗ is weak type an(1981): If K ′ Miguel de Guzm´ (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 a 1 , . . . , a N and for each λ > 0 N � ≤ C N � � � � � > λ �� �� x : sup λ . K j ( x − a i ) (Guzm´ an) � � � � � j i = 1 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. N Let φ = � b i δ a i and for any λ > 0 , define E λ : = { x : M φ ( x ) > λ } . i = 1 Then N | E λ | ≤ 2 n � | b i | . λ i = 1 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. N Let φ = � b i δ a i and for any λ > 0 , define E λ : = { x : M φ ( x ) > λ } . i = 1 Then N | E λ | ≤ 2 n � | b i | . λ i = 1 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 K j ≥ 0 , then the constant in the weak (1 , 1) inequality is same as the constant in the following inequality N � ≤ C N � � � � � > λ �� �� x : sup K j ( x − a i ) λ . � � � � � j i = 1 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 K j ≥ 0 , then the constant in the weak (1 , 1) inequality is same as the constant in the following inequality N � ≤ C N � � � � > λ �� � �� x : sup K j ( x − a i ) λ . � � � � � j i = 1 This method played a crucial role in obtaining the best constant in the weak type (1 , 1) inequality for the centred Hardy-Littlewood maximal operator 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 ∈ A 1 and when 1 < p < ∞ , M is weighted strong type ( p , p ) if and only if w ∈ A 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 6 / 20
Weighted Ineq. Theorem (B. Muckenhoupt,1972) M is of weighted weak-type (1 , 1) if and only if w ∈ A 1 and when 1 < p < ∞ , M is weighted strong type ( p , p ) if and only if w ∈ A p . w ∈ A 1 iff ∃ C > 0 such that for all cubes Q � 1 Q w . Q w ≤ C ess inf | 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 6 / 20
Weighted Ineq. Theorem (B. Muckenhoupt,1972) M is of weighted weak-type (1 , 1) if and only if w ∈ A 1 and when 1 < p < ∞ , M is weighted strong type ( p , p ) if and only if w ∈ A p . w ∈ A 1 iff ∃ C > 0 such that for all cubes Q � 1 Q w . Q w ≤ C ess inf | Q | w ∈ A p iff ∃ C > 0 � p − 1 � 1 � � 1 Q w − � 1 � ≤ C . sup Q w p − 1 | Q | | Q | 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 6 / 20
Weighted Ineq. Theorem (B. Muckenhoupt,1972) M is of weighted weak-type (1 , 1) if and only if w ∈ A 1 and when 1 < p < ∞ , M is weighted strong type ( p , p ) if and only if w ∈ A p . w ∈ A 1 iff ∃ C > 0 such that for all cubes Q � 1 Q w . Q w ≤ C ess inf | Q | w ∈ A p iff ∃ C > 0 � p − 1 � 1 � � 1 Q w − � 1 � ≤ C . sup Q w p − 1 | Q | | Q | Q Classical proofs depend on Calder´ on-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 A 1 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 A 1 weights. Given w ∈ C ( R n ) , M is weak type (1 , 1) with respect to w if and only 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 A 1 weights. Given w ∈ C ( R n ) , M is weak type (1 , 1) with respect to w if and N only if for any φ = � b i δ a i and λ > 0 , we have i = 1 N w ( { x : M φ ( x ) > λ } ) � C w , n � | b i | w ( a i ) . (*) λ i = 1 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 A 1 weights. Given w ∈ C ( R n ) , M is weak type (1 , 1) with respect to w if and N only if for any φ = � b i δ a i and λ > 0 , we have i = 1 N w ( { x : M φ ( x ) > λ } ) � C w , n � | b i | w ( a i ) . (*) λ i = 1 Given w ∈ C ( R n ) , M satisfies (*) on linear combination of Dirac deltas if and only if w ∈ A 1 . 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 ∈ A 1 . 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
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