Toán tử tích phân loại hardy và các giao hoán tử của chúng trên một số không gian hàm tt tiếng anh

MINISTY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ||||||| * ||||||| NGUYEN THI HONG THE HARDY TYPE OPERATORS AND THEIR COMMUTATORS ON FUNCTIONAL SPACES Speciality: Integral and Differential Equations Code: 9.46.01.03 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Ha Noi - 2019 This thesis has been completed at the Hanoi National University of Education Scientific Advisor: Prof PhD Sci Nguyen Minh Chuong PhD Ha Duy Hung Referee 1: Prof.PhD.Sci Vu Ngoc Phat, Institute of Mathematics, VAST Referee 2: Assoc.Prof.PhD Khuat Van Ninh, Hanoi Padagogical University Referee 3: Assoc.Prof.PhD Tran Dinh Ke, Hanoi National University of Education The thesis shall be defended before the University level Thesis Assessment Council at on The thesis can be found in the National Library and the Library of Hanoi National University of Education dp d p INTRODUCTION MOTIVATION AND OUTLINE One of the core problems in harmonic analysis is to study the boundedness of an operator T on some functional or distributional spaces jjT fjjY (1) CjjfjjX ; where C is a constant, X; Y are functional or distributional spaces with corresponding norms k kX ; k kY This question arises from natural problems in investigations about analysis, functional theory, partial di erential equations For instance, we consider Riesz operator J de ned by Z y) J (f)(x) = Rd f( x j j y d dy (2) where p< d p then J and q = d q d bounded from L (R ) to L (R ) One important application of this results is the theorem embedding SobolevGagliardo-Nirenberg: W 1;p d q d 1 (R ) ,! L (R ), with p q p ; p = p d One of the main problems in these thesis is study (2) for one particular class of integrals and their commutators This operator class includes or closely relates with a lot of classical important operators such as: Hardy operator, maximal Calderon operators, Riemann-Lioville operators on line, in cases one dimension The estimations in form of (1) is called Hardy's inequality Hardy's integral inequality and it's discrete version appeared about 1920, related with the p continuity of the Hardy operators on L spaces One of the main motiva-tions due to these results is began from Hilbert's inequality The mathemati-cian Hilbert, while researching for the solutions of some integral equations, due PP ambn to research the convergence of the double series in form of n=1 m=1 m + n In nP P aman 1915, Hardy started that the is convergent i m+n and =1 m=1 n=1 X A a nn n are convergent, where An = a1 + n=1 X A n n ; + an Hence, we can rewrite in the form p + p + below: if f L (R ), for < p < then Hf L (R ), where Z0 x Hf(x) = x f(t)dt: (3) In 1920 G Hardy demonstrated the integral inequality below 0x p p fp(x)dx: x p Z Z f(t)dt p dx (4) Z where < p < 1, f is a nonnegative measurable function on (0; 1), and the p best contant is p Hardy operator is one case of the class Hausdor operators, appeared in the problem for numeral series and exponent series with fundamental in the research of Siskakis, and Li yand-Moricz in the real eld Their Hausdor operators in form of Z H ;A(f)(x) = (u)f(xA(u))du; (5) d R d where be measurable function on R and A = A(u) = (aij(u)) be matrix, with order is d d and aij(u) be measurable function of u: In particular, when (u) = [0;1](u), A(u) = u then H ;A turn in to classical Hardy operators above A natural question arises, which spaces replace X; Y spaces and which condition for , matrix A then (1) is true with T = H ;A Moreover, which is the best constant C in (1)? The rst question has attracted attention of a lot mathematician over the world and list some results of K Andersen, E Li yand, F M•oricz, D.S Fan However, the necessary condition about the boundedness to be given are not su cient conditions and the question about the best constant in each cases is not easy to answer The second question about the best constant in estimation in form of (1) for the class average operator has two directions: The rst is for average operator class on the globular in form of Z d H(f)(x) = (6) f(y)dy; x R n f0g: jj d x d jyj a.e t Zp q b CBMO! Q d p d+ B_ ! q ; (Qpd) ! B kbkCBMO!q2 (Qpd): real numbers such that < q < q1 < 1, ? Let s : Zp ! Qp be a measurable function ? or js(t)jp < a.e t Zp We assume that p;b ;s is determined as a _ q; d B ! Q p if and only if B is nite Then the commutator U _q; n bounded operator from B ! Q p to Remark 2.3 As we know, commutators of Hardy operators are "more singu-lar" than corresponding Hardy operators This problem is not di erent on in cases the central Morrey spaces In fact, when js(t)j p < almost everywhere t Zp then B is nite implies A < In other word, the example below given that A is nite does imply B < 1 Indeed, choose s(t) = pt, (t) = , then A < 1; B = 1: j j pt p1+(d+ ) (logp pt j j q p) 1 q < 1, = q + q2 ; < < 0; q1 < ? < 0; < < d and = + 2: Let s : Z p ! Qp be a measur-able function such that s(t) 6= almost everywhere If C is nite, then for q ; d p;b any b CBMO !2 (Q p), the corresponding commutator U ;s is bounded Theorem 2.4 Let < q < q1 _q; _ q; d d from B ! 1 (Q p) to B ! (Q p) and we have U p;b (2 + pd+ c ) q; log k s(t2) p ;skB_! (t)dt: (Q )!B_!q; (Qpd) pd Here c is a constant de ned as in Lemma 2.1 , j j j j C b q; C k kCBMO! = R Zp ? f j (Q ) pd max 1; s(t) : gj j (d+ ) jp s(t) (d+ ) 12 Chapter P BOUNDS OF -ADIC MULTILINEAR HARDY -CESARO OPERATORS AND THEIR COMMUTATORS IN P -ADIC FUNCTIONAL SPACES In this chapter, we study the norm of the p-adic weighted multilinear Hardy- Cesaro operators on product of Lebesgue spaces and the spaces of Morrey types First, we introduce the motivation due to the problem In sequel, using the schema proof the results are developed from the schema in previous chapter, combination with the methods has used in mulltilinear analysis on the real eld or on the local compact group The commutator problem of p-adic Hardy- Cesaro operators has studies in this chapter The researching method is the real variable method of Coifman(1976) Besides, we establish the estimation of di erent of two functions in CBM O p space, hence we obtained the estimation on L for the average integral operators The di erence is for the singular integral operators, we usually used John-Nirenberg, but in here, we used immediate estimate by inequalities such as Minkowski's inequality, H•older inequality The contents of this chapter is written on the paper in the author's works related to the thesis that has been published 3.1 Motivation Due to the reasons in the introduce part, we investigate the p-adic weighted multilinear Hardy- Cesaro operators on some functional spaces in p-adic eld 3.2 Bounds of the p-adic weighted multilinear Hardy- Cesaro oper-ators on the product of Lebesgue spaces and spaces of Morrey types To proof the main results we need some de nitions and lemmas below 3.2.1 Some definitions and lemmas We introduce and investigate the p-adic weighted multilinear Hardy- Cesaro operators de ned as follow: 13 m; n positive integer numbers and : De nition 3.1 Let be ? n m p ! Zp ! Qp be measurable The adic [0; +1), =(1 m) : multilinear Hardy-Cesaro ;s ! s s ? n Z operator Up;m;n , which de nes on f we ig hte d = (f ; : : : ; f m ! d m ! Qp ! C vector of measurable functions, by ; U s p;m;n s ? Z (Zp ! (f ; : : : ; f m )(x) = ! ): p ; : : :; s m Y k=1 ) f k (s (t)x) (t)dt; k n (3.1) s ; : : : ; s m) ! where = ( Remark 3.1 When m = n = 1; U p;m;n ! is reduced to U p ;s ;s by Hung(2014) has been investigated In this chapter, if not explicitly stated otherwise, q; ; q i; j are real numbers, q < 1, qj < 1, j > d for each j = 1; : : : ; m so that = q and p k q1 q1 = The weights !k W q + + qm ; qm + + q (3.2) : (3.3) m ; k = 1; : : : ; m, set m q kY !(x) = ! =1 q k k (x): (3.4) p W! It is obvious that ! W p De nition 3.2 We say that (!1; : : : ; !m) satis es the m Y !(S0) q !k(S0) qk condition if : (3.5) k=1 Example 3.1 For p example, (!1; : : : ; !m) where !k(x) = jxjp k for k = 1; : : : ; m satis es the W! condition Through out this paper, s1; : : : ; sm are measurable functions from Z ! Qp and we denote by s the vector (s1; : : : ; sm) p Lemma 3.1 Let ! W ; > fr; (x) = r then the function jj if x p < if x p 1: p !(S0) p r= j j1=r > 0: ? n p into 14 3.2.2 The main results p Theorem 3.1 Assume that (!1; : : : ; !m) satis es W! condition and there exists constant > such that jsk(t1; : : : ; tn)jp minfjt1jp ; : : : ; jtnjp g holds ? n p for every k = 1; : : : ; m and for almost everywhere (t 1; : : : ; tn) Z Then there exists a constant C such that the inequality q jj U p;m;n(f ; : : : ; f )jj ;! s m m d Y C L!(Qp) k=1 jjfkjjLq!kk (Qdp) (3.6) holds for any measurable f1; : : : ; fm if and only if m A Z := (Zp) k=1 j ? d+ k j Y n qk (t)dt < : sk(t) p (3.7) Moreover, A is the best constant C in (3.6) Remark 3.2 When m = n = 1, we obtained the theorem 3.1 of Hung(2014) Note that the inequality (13) for two sequences nonnegative real numbers, is immediate consequence of Theorem 3.1 of Hung(2014) Theorem 3.2 Let q; qk < 1; ; k; be as in (3.2), (3.3) such that k q k < k < for k = 1; : : : ; m Assume condition We set = We assume that d+ 1+ + d +d + Z that (!1; ; !m) satis es W! d+ m m: m Y k (d+ ) ? n B= (Z p ) =1 jsk(t)jp k (3.8) (t)dt < 1; k m and (!(B0)) d Here B0 is the ball fx Qp that the inequality : jxjp ; ! m 1+ kqk k =1 jj U p;m;n(f ; : : : ; f )jj s Y 1+ q q (!k(B0)) : qk (3.9) 1g Then, there exists a constant C such q; p Q L! ( m d ) C Y k=1 jjf jj q ; k L !kk k (Qdp) (3.10) holds for any measurable f1; : : : ; fm Moreover, the best constant C in (3.10) equals B 15 Theorem 3.3 Let q; qk; ; k; k be as in Theorem 3.2 with q; qk > and p conditions (3.2), (3.3) are hold Assume that (! ; ; ! ) satis es conW! dition Then U _ q ; m m B!m p;m;n s _ q; d (Qp) to B! (Qp) Moreover, jj q; Up;m;n s (Qp) jj ;! B !1 _1 q; qm; m B !m d _ q ; 1 is determined as a bounded operator from B !1 ;! d m (Qp) B ! _ d ! =B : (Qp) _ d (Qp) d (3.11) Remark 3.3 When m = n = 1, from Theorem 3.2 and 3.3, we obtained Theorem 2.1 in chapter of this thesis 3.3 The commutator of weighted bilinear Hardy- Cesaro operators In p-adic eld, the commutator of operation of Hardy types have researched by Fu, Lu, Wu, Chuong, Hung, We have the commutators of the weighted q d bilinear Hardy- Cesaro operators with the symbol in CBM O! (Q p) 3.3.1 Commutator and lemma We de ned the commutator of weighted bilinear Hardy- Cesaro operators as follow: ? n p De nition 3.3 Let n N; : Z ? n p ! [0; 1); s1; s2 : Z ! Qp; b1; b2, be d d p and f1; f2 : Q p ! C be measurable p;n The commutator of weighted bilinear Hardy- Cesaro operator U ! is de ned locally integrable functions on Q functions ;s as: ! Up;n; b ;! s Z ( (f ; f )(x) = 2 Zp? n ) ! k=1 Y fk(sk(t)x) C Z (Zp) We set (3.12) k=1 Y = ! k=1 jj (t)dt: (bk(x) bk(sk(t)x)) ! D Z (Z p ) = ? j n Y j ?n j k=1 Y sk(t) p (d+ k) k Y Remark 3.4 D2 < does not imply C2 < Remark 3.5 C2 < does not imply D2 < logp (3.13) (t)dt: j sk(t) p ! k=1 = (d+ k) k sk(t) p ! (t)dt: (3.14) 16 3.3.2 The main results Theorem 3.4 Let < q < qk < 1; < pk < 1; p + q q1 and q2 + + p1 p2 q + q1 + Assume that !(x) = !1 q q p1 q2 !2 1+ q !(B0) < k < 0; k = 1; such that 1 1 = and = k 1+ kqk !k(B0) q + qk kY q p2 q1 ; = q1 + p1 + q q2 q + q2 p + pk =1 p CBM O!2 (Qp) then U (ii) If (b1; b2) q ; d 1 _ q1; is bounded from B!1 ;! s for any b = _ are nite then for any b = (b1; b2) CBM O!1 (Qp) p;2;n; b! d _ q ; CBM O! p1 _ q; and = j ! s _ B 1 a.e p t q; 1 CBM k ! (Qp) < d _ B q; t 2 ( Q p) ? + q2 , for each d n ! (Qp) to B! _ q; d (Qp): b p d O! (Qp ), ! ;! U p;2;n; is s is nite < k < 0; k = 1; such that + p1 + p2 = Then is + Furthermore, suppose that jsk(t)jp > a.e t !2 d 1 1 q q1 p B (Qp) to B! (Qp) then D2 = k _ q2; d d Corollary 3.1 Let < q < qk < 1; < pk < 1; p Z (Qp) d (Q p ) d 2 bounded from B!1 (Qp) B!2 () d p1 (i) If both C2 and D2 Q p _ q; ( to B k d ; ) if and only if D2 p;2;n; U b Zp ? n or ! ;! s is nite bounded from 17 Chapter MULTILINEAR HARDY CESARO OPERATOR AND COMMUTATOR ON THE PRODUCT OF THE SPACES OF HERZ TYPES In this chapter, we study the boundedness of the weighted multilinear Hardy-Cesaro operator on the product of Herz and Morrey-Herz spaces First, we introduce the motivation due to the problem In sequel, method of Xiao(2001) has used in the work of Fu, Wu, Hung, Chuong, Ky, , the technology from multilinear analysis, we obtained the estimate on the boundedness of this op-erators on the product of Herz and Morrey-Herz spaces Finally, due to the real variable method of Coifman(1976), the method estimates on multilinear analysis and the key lemmas and schema of research of Fu, Gong, Lu, Yawn, , and the special case of Hung, Ky, we estimate the boundedness of their com-mutators from the product of central Morrey spaces to the central Morrey spaces with symbol in the Lipschitz space The contents of this chapter is written on the paper in the author's works related to the thesis that has been published 4.1 Motivation Our problem is investigate the weighted multilinear Hardy- Cesaro oper-ators on on the product of Herz and Morrey-Herz spaces The multilinear version of the weighted multilinear Hardy-Cesaro operator was considered by Hung, Ky(2015) de ned as n De nition 4.1 Let m; n N; n : [0; 1] ! [0; 1), s1; : : : ; sm : [0; 1] ! R be measurable functions The weighted multilinear Hardy-Cesaro operator U is de ned by Z ;! Um;n f ! s where = (f ; : : : ; f [0;1]n ! f m ( ! x) = m f s k ), s = (s ; : : : ; s !, ;s ! k=1 Y m;n m ): ( k( ) tx) t( )dt; (4.1) 18 4.2 Boundedness of the weighted multilinear Hardy-Cesaro opera-tor on the product of Herz and Morrey-Herz spaces 4.2.1 Some definitions and lemmas We would like to recall the de nition of homogeneous weights introduced by Chuong, Hung(2014) De nition 4.2 Let be a real number Let W be the set of all functions d d ! on R , which are measurable, !(x) > for almost everywhere x R , < !(y)d (y) < 1, and are absolutely homogeneous of degree , that is R S d d !(tx) = jtj !(x), for all t R n f0g; x R : We remark that W = jxj S W contains strictly the set of power weights !(x) = For our convenience, we give some common notation throughout this part Let > 0; ; ; 1; : : : ; m be real numbers, 1; : : : ; m > d, < p < 1, q < 1, pi; qi < with i = 1; : : : ; m and ; 1 + q1 + q2 Sd = ( Y m = ; p1 m + p2 + + :::; 1 = p ; q1 satisfy m 1 pm +q2 + = fx R + qm = q ; d + + = q ; + + + m = : Sd : jxj = 1g qm d for all i = 1; : : : ; m, and we set d ) : Functions !i belong to W i + + 1; !(x) = Lemma 4.1 Let p and (fk)k n on [0; 1] Then X Lemma 4.2 If f q ! qi (x): (4.2) i i=1 Obviously, ! W k=1 m fk(t)dt Z [0;1] be nonnegative and measurable functions p Zn n [0;1] B _; @ MK (!) then f k k p;q k q;! 4.2.2 The main results p fk (t) !1=p p dt k=1 X C 2k( ) f A ; k kMK_p;q (!) n Theorem 4.1 (i) Let s1(t); : : : ; sm(t) 6= almost everywhere in [0; 1] and A1 = Zn [0;1] m i=1 Y jsi(t)j i d qi i +i + ! (t)dt < 1: (4.3) 19 Suppose that p < or < p < and at least one of Then m ! kU ;s m;n ! !( ! A C; f )kMK_ ; (!) p;q Y k fi 1; k : : : ; m is positive i pi;qi MK ; _ (! ) i i : (4.4) i=1 Here 8m 2j k Q > > if p < +1 kj < ; = > k=1 m > p :(2 1=p 1) 2j k kj + if < p < and > 0: k=1 QC!! Conversely, let < p < 1, < i < for i = 1; : : : ; m Suppose m Q m;n i is de ned as a bounded operator from ;! s =1 (ii) that U Then (4.3) holds and m;n kU !k m ; i; i ;s Q i=1 MK _ (!i)!MK pi;qi _ p;q m ! = i=1(2 ! ; Q 1) p (2 1) 1=pi 1=p m i Q q( 1=q ) D _ i (!i) to MKp;q (!) ; ; ;! (4.5) ! m ipi i pi;qi A (!) where D MK ; _ (q ( ))1=qi (!(Sd)) i ii i=1 Q 1=q qi( i i) 1=qi (q()) 1=q m (1 2) Q i=1 = 1=qi (!i(Sd)) Theorem 4.2 (i) If p < 1, s1(t); : : : ; sm(t) 6= almost everywhere in n ! [0; 1] and (t)dt < 1; (4.6) d i A2 = Z n qi i m i=1 Y [0;1] then jsi(t)j m k k ;! U m;n ( A _ Kq !) s k=1 m is bounded from U Q i =1 _ i K qi j k n _ ;p i2 m Q K _ i _ (!i )!K q n E = (mp) p i i s (4.8) qi ;p 2q 1=q q ;! A (!) Where 1=p (!i) ; : : : ; t g for i = 1; : : : ; m and U m;n = 1=p (4.7) i i (!) Then (4.6) holds and E! : (!i) to Kq !k m fi +1Y i _ ;p m;n ;s t f i k ! kk i=1 Kq ;p j (!) (ii) Suppose that j s (t ; : : : ; t )j i m 2Y ;p f i + m i=1 Y 2qi i qi 1=qi m i i 1=q (!(Sd)) (! (S )) 1=qi d : : i = Q Q i=1 20 Remark 4.1 When = = m = 0, we obtain the boundedness and bounds for multilinear Hardy-Cesaro operator on the product of the Lebesgue spaces However, the results are worse than those obtained by Hung, Ky(2015) m;n In Theorem 3.1 of Hung, Ky(2015) given that the norm of U to L is exactly L!m ! pm [0;1]n R p i=1 from L! ;! jsi(t)j (t)dt: i m d+ i Q p1 s i q Remark 4.2 The result of Theorem 4.2 implies that in cases exists positive constant such that jsi(t1; : : : ; tn)j minft1 ; : : : ; tng with i = 1; : : : ; m (with the H m operators then this condition obviously true), then U m ; iQ an bounded operator from MKpi;qi _ i m;n ! is de ned as _ i ;s ; (!i) to MKp;q (!) then the necessary =1 and su cient condition is A2 nite The consequence consist of Theorem and Theorem of Gong, Fu and Ma(2014), Moreover, to get necessary condition, the authors need the condition = = m, p1 = = pm and q1 = = qm However, our result does not need this condition Similarly, for the results of Morrey-Herz, our results in Theorem 4.1 are better than the results of Gong, Fu and Ma Remark 4.3 In Theorem 4.1 we consider the cases when < p < 1, this idea derived from the paper of J Kuang(2008), in his paper, the author estimates the norm of V on the Herz space Thus, our results are better than the cor-responding results of Gong, Fu, Ma(2014) and is extented in cases multilinear of their work of Kuang, Liu, Fu, Chuong, Duong, 4.3 Commutator of the weighted multilinear Hardy-Cesaro opera-tor 4.3.1 Some definitions Commutators of U m;n n ! ;s due to Coifmann-Rochberg-Weiss, de ned as Let n m; n N, : [0; 1] ! [0; 1), s1; : : : ; sm : [0; 1] ! R, b1; : : : ; bm be locally d d integrable functions on R and f1; : : : ; fm : R ! C be measurable functions The commutator of weighted multilinear Hardy-Cesaro operator U ned as [0;1]n ;! Um;n; s ! b f ! x Z ( ) := ! m k=1 f s t x ( k( ) ) ! is de- ;s ! m k=1 k m;n b x ( k( ) b s tx k ( k( ) )) t dt: () (4.9) According to the idea of Tang, Xue, Zhou(2011), we consider the symbol belongs to Lipschitz functions, de ned as 21 n De nition 4.3 Suppose that < < The Lipschitz space Lip ( R ) is de ned n as the set of all functions f : R ! C such that n f := sup jf(x) f(y)j < : (4.10) n jx yj x;y2R ;x6=y jj jj (R ) Lip 4.3.2 The main results The main results of this part includesTheorem 4.3 and Corollary 4.1 below Theorem 4.3 Let < < 1, = q 1 m + + i > d, q qi < 1, ri, pi < 1, < < 1, i 0i; m, for i = 1; : : : ; m with = + + m > d, 1 1 p p q + + , and q = + + p = + + m, + r Suppose that bi Lip i and !i as in (4.2) for i = 1; : : : ; m = +r1 + m m n Functions s1(t); : : : ; sm(t) 6= almost everywhere t [0; 1] such that Z ! [0;1]n i=1 m + Y MK pm;qm m;n; s ; _ ;! _ 0; (!) p;q Here When m = n = 1, !1 = j ; when < p < and > or when m m X = i=1 Xd+ i i=1 j, s1(t) t, let U i (4.12) : ri b 1;1 ;b =U ! ! , we obtain the ;s B0 following result a measurable function, < Corollary 4.1 Let : [0; 1] ! [0; 1) be b < 1, d Lip (R ), q2 q1 < If A = Z0 then U b is bounded from MK t q1 _ p;q d (1 t) (t)dt < 1; to MK _ p;q ; 2 , where (4.13) = + +d ; 2011, Tang, Xue, Zhou to obtain the boundedness _ MK 2; , the authors required a su cient condition on p;q2 11 C= Z t q1(t)dt < 1: of U b q2 q1 1 from MK _ ; to p;q1 that d Since t 1, then A C In fact by choosing (t) = d q1 is determined as a bounded operator from MKp1;q1 m m p < and ! (! ) to MK m (4.11) b Then the commutator U _ (t)dt < 1: jsi(t)j d qi i + i i j1 si(t)j i = then it is easy to see that C = but A < Thus, we improvement of a recent result by Tang(2011) t (1 t)1+ =2 , obtain an (!1) 22 CONCLUSIONS Results In this thesis, we obtained the following results: q; d 1) Finding out the norm p-adic weighted Hardy-Cesaro operators on L ! (Q p), _ q; d q; d B ! q; Q p and CM O ! Q p Wep;bgive a necessary condition and a su _ q; d CMO! d Q is bounded on B! Qp cient condition for (t) to U ;s p with symbol in 2) Finding out the norm of the p-adic weighted multilinear Hardy-Cesaro q d q; operators on the product of the L !(Q p), L d ! (Q p) and B _ q; ! d Q p We give a necessary condition and a su cient condition for (t) to their commutators _ q; d q; d are bounded on the product of the B ! Q p with symbols in CMO ! Q p 3) Given a necessary condition and a su cient condition for (t) to the weighted multilinear Hardy-Cesaro operators are bounded on product of _ ; _ ;p the MK p;q (!), K q (!) Moreover , we give a necessary condition for _ (t); to their d commutators are bounded on product of the MK p;q (!) with symbol in Lip (R ) Recommendations Besides the results achieved in the thesis, some related issues need to be further studied: Study the norm of the weighted multilinear Hardy-Cesaro operators and their commutators on product of the space of Herz types Study the norm of the p-adic weighted multilinear Hardy-Cesaro operators and their commutators on product of the space of Herz types AUTHOR'S WORKS RELATED TO THE THESIS THAT HAVE BEEN PUBLISHED 1) Nguyen Minh Chuong, Ha Duy Hung, Nguyen Thi Hong (2016), Bounds of p adic weighted Hardy-Cesaro operators and their commuta-tors on p adic weighted spaces of Morrey types, p-Adic Numbers Ultra-metric Analysis, and Applications, 8(1), 31-44 2) Nguyen Minh Chuong, Nguyen Thi Hong, Ha Duy Hung (2018), Bounds of weighted multilinear Hardy-Cesaro operators in p adic functional spaces, Frontiers of Mathematics in China, 13(1), 1-24 3) Nguyen Minh Chuong, Nguyen Thi Hong, Ha Duy Hung (2017), Multilinear Hardy-Cesaro Operator and Commutator on the product of Morrey-Herz spaces, Analysis Mathematica, 43(4), 547-565 Results of thesis have been reported at: Seminar of Department of Mathematics, Faculty of Mathematics and In-formatics, Hanoi National University of Education; Workshop on PhD student, Faculty of Mathematics and Informatics, Hanoi National University of Education, 2017; Seminar "Pseudo-di erential operators, wave, harmonic analysis on real eld or p-adic eld", Institute of Mathematics, Vietnam Academy of Sci-ence and Technology 9th National Mathematics Congress, Nha Trang, 08/2018 ... operators such as: Hardy operator, maximal Calderon operators, Riemann-Lioville operators on line, in cases one dimension The estimations in form of (1) is called Hardy' s inequality Hardy' s integral... bilinear Hardy- Cesaro operators In p-adic eld, the commutator of operation of Hardy types have researched by Fu, Lu, Wu, Chuong, Hung, We have the commutators of the weighted q d bilinear Hardy- ... is true with T = H ;A Moreover, which is the best constant C in (1)? The rst question has attracted attention of a lot mathematician over the world and list some results of K Andersen, E Li yand,