Splitting of Volterra Integral Operators with Degenerate Kernels
aa r X i v : . [ m a t h . C A ] J un Splitting of Volterra Integral Operators withDegenerate Kernels ∗† Vyacheslav S. Rychkov
Moscow Institute of Physics and Technology
Abstract
Volterra integral operators with non-sign-definite degenerate kernels A ( x, t ) = P nk =0 A k ( x, t ) , A k ( x, t ) = a k ( x ) t k , are studied acting from one weighted L spaceon (0 , + ∞ ) to another. Imposing an integral doubling condition on one of theweights, it is shown that the operator with the kernel A ( x, t ) is bounded if andonly n + 1 operators with kernels A k ( x, t ) are all bounded. We apply this resultto describe spaces of pointwise multipliers in weighted Sobolev spaces on (0 , + ∞ ) . Introduction
There exists a problem of studying weighted estimates of the form Z ∞ (cid:12)(cid:12)(cid:12)(cid:12) v ( x ) Z x A ( x, t ) f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) p dx ≤ C Z ∞ | u ( x ) f ( x ) | p dx (1)from the viewpoint of finding necessary and sufficient conditions on u ( x ) and v ( x ) , underwhich inequality (1) holds for all functions f with the finite r.h.s. of (1) and a constant C independent of f . In the going back to Hardy [1,2] case A ( x, t ) = 1 the correspondingcriterion was found in [3–5].The first significant progress for A ( x, t ) = 1 was achieved in [6–9], which investigatedthe case A ( x, t ) = ( x − t ) α , α > A ( x, t ) , for which it was possible to characterize u ( x ) and v ( x ) in (1).At present the most general results were obtained, it seems, in [10]. This work finds acondition on u ( x ) and v ( x ) , necessary and sufficient for the validity of inequality (1)under the assumption that the kernel A ( x, t ) is nonnegative and satisfy an additionalcondition of the form A ( x, t ) ≍ A ( x, y ) + A ( y, t ) , y ∈ ( t, x ) , allowing to compare kernelvalues in different points. ∗ This is author’s translation of the Russian original published in
Investigations in the theory of dif-ferentiable functions of many variables and its applications. Part 17, Collection of articles , Trudy Mat.Inst. Steklova, 214, Nauka, Moscow, 1997, 267285; editorial translation published in: Proc. Steklov Inst.Math., 214 (1996), 260278] http://mi.mathnet.ru/eng/tm1040 . † Work supported by the Russian Fund for Basic Research (project 96-01-00243). p = 2 and an additional condition on u ( x ) . Thekernels of our class do not satisfy the above-mentioned conditions of applicability ofknown results.In section 2 the results of section 1 are applied to the problem of describing the setof pointwise multipliers in some weighted Sobolev spaces. Sections 3–5 collect auxiliaryresults and proofs.The author is deeply grateful to O.V. Besov and G.A. Kalyabin for valuable remarks,discussions and support. Note added (June 2020)
See [17] for a short presentation of these results withoutproofs, and [18] for generalizations to p = 2 .
1. Weighted estimates of integral operators
For a Volterra integral operator A with a degenerate kernels of the form( A f )( x ) = Z x " n X k =0 a k ( x ) t k f ( t ) dt, n ∈ N , (1 . k v A f k ≤ C k uf k . (1 . k · k = k · k L ( R + ) ; R + = (0 , ∞ ) ; u, v are nonnegative on R + functions (weights);constant C > f .Denote by L ,u the weighted space of functions f on R + with the norm k uf k . That(1.2) holds now means that A : L ,u → L ,v . We represent A as a sum A = P nk =0 A k ,where ( A k f )( x ) = a k ( x ) Z x t k f ( t ) dt. We will say that for the operator A when acting from L ,u into L ,v the splitting takesplace, if A : L ,u → L ,v ⇐⇒ A k : L ,u → L ,v , k = 0 . . . n. By B δ , δ ≥ R + functions w ,satisfying with some constant D ( w ) the integral doubling condition Z ∆ w ( x ) dx ≤ D ( w ) Z ∆ w ( x ) dx for any interval ∆ ⊂ R + of length | ∆ | ≥ δ , where ∆ is a twice smaller interval withthe same center.The following theorem, given in two equivalent formulations, is the central result ofour work. 2 heorem 1.1. Let u − ∈ B δ for some δ ≥ . If δ > , then assume in addition a k v ∈ L (0 , r ) ∀ r > , k = 0 . . . n − . Then for the inequality (1.2) to be satisfied it isnecessary and sufficient that S k = sup r> k a k v k L ( r, ∞ ) · k x k u − k L (0 ,r ) < ∞ , k = 0 . . . n. (1 . Theorem 1.1 ′ . Assume the conditions of Theorem . Then for the operator A acting from L ,u into L ,v the splitting takes place. Remark 1.
Our method of proof of Theorem 1.1 gives the following estimate for thenorm of A (or, which is the same, the smallest constant C in inequality (2)): c n X k =0 S k ≤ kAk L ,u → L ,v ≤ c n X k =0 S k . Constant c here depends on n , δ , D ( u − ) , as well as (if δ > P n − k =0 k a k v k L (0 ,r ) · k x k u − k L (0 ,r ) , where r is determined by n , δ , D ( u − ) . Constant c is universal. Remark 2.
Consider the adjoint operator A ∗ :( A ∗ f )( x ) = Z ∞ x " n X k =0 x k a k ( t ) f ( t ) dt. Since ( L ,u ) ∗ = L ,u − , we have A : L ,u → L ,v ⇐⇒ A ∗ : L ,v − → L ,u − , and so, underthe assumptions of Theorem 1.1, Eq. (1.3) is also necessary and sufficient for having theinequality k u − A ∗ f k ≤ C k v − f k . Condition u − ∈ B δ of Theorem 1.1 allows to include many interesting cases. E.g.weight u ( x ) = (1 + x ) α log β (2 + x ) satisfies this condition with δ = 0 for any α, β ∈ R .Nevertheless, it is natural to ask to what extent this condition is essential for the validityof Theorem 1.1. The rest of this section is devoted to clarifying this question. Theavailable results are closely related with the paper [11].Consider the Riemann-Liouville integral operator:( I ( α ) f )( x ) = 1Γ( α ) Z x ( x − t ) α − f ( t ) dt, α ≥ . For α ∈ N , operator I ( α ) is an operator of the form (1.1). Let us focus on α = 2 andrepresent I (2) as a sum of two operators: I (2) = I (2)0 + I (2)1 , I (2)0 f ( x ) = x Z x f ( t ) dt, I (2)1 f ( x ) = − Z x tf ( t ) dt. ssertion 1.2. For operator I (2) , acting in the space L ,e − x , splitting does not takeplace. Namely, I (2) : L ,e − x → L ,e − x , while I (2) i : L ,e − x L ,e − x , i = 0 , . This result is basically a reformulation of example 1 in [11]. It shows that condition u − ∈ B δ of Theorem 1.1 is important (clearly, u ( x ) = e − x does not satisfy this conditionfor any δ ≥ u − ∈ B δ is not, generally speaking, necessary for splitting.E.g. for operators I ( α ) it can be replaced by a weaker condition: Z r u − dx ≤ D Z r u − dx for all r ≥ δ ≥ . (1 . Assertion 1.3.
Let the weight u satisfy with some constants D , δ condition (1.4) .If δ > , then let in addition x α − v ∈ L (0 , r ) ∀ r > . Then for α ≥ to have theinequality k v I ( α ) f k ≤ c k uf k (1 . it is necessary and sufficient that sup r> k x α − v k L ( r, ∞ ) · k u − k L (0 ,r ) < ∞ . (1 . For α ∈ N this is equivalent to splitting for operator I ( α ) .
2. Pointwise multipliers in weighted Sobolev spaces
Consider on R + the weighted Sobolev space W = W ( l )2 ,u with the norm k f k W = k f k L (0 , + k f ( l ) u k . For this norm, when the norm of function itself is taken only over aninitial interval of R + , all polynomials of degree ≤ l − W . Spaces W ( l )2 ,u wereintroduced and studied in [12], which used an equivalent norm P l − k =0 | f ( k ) (0) | + k f ( l ) u k .Function ϕ is called a (pointwise) multiplier from one Sobolev space W to another W if ϕf ∈ W ∀ f ∈ W . The space of such multipliers is denoted M ( W → W ) .Various aspects of the theory of multipliers in spaces of differentiable functions werestudied e.g. in the book [13].We are considering the problem of describing the space M ( W ( l )2 ,u → W ( m )2 ,v ) , m ≤ l ,denoted for brevity M ( u, l ; v, m ) . In connection with this problem one should mentionthe work [14], which described multipliers in the Sobolev space on R n with the norm k f k L p ( B (0 , + P | α | = l k D ( α ) f k L p ( R n ) for the case p > n .The first 3 assertions of this section are slight generalizations of author’s results pub-lished in [15]. 4 emma 2.1. Let function g on R + be such that g ( k ) (0) = 0 , k = 0 . . . l − . (2 . Then the following two equations hold: ( ϕg ) ( m ) ( x ) = 1( l − l − X k =0 C kl − (cid:0) ϕx k (cid:1) ( m ) Z x ( − t ) l − k − g ( l ) ( t ) dt, m < l, (2 . ϕg ) ( l ) ( x ) = ϕ ( x ) g ( l ) ( x ) + 1( l − l − X k =0 C kl − (cid:0) ϕx k (cid:1) ( l ) Z x ( − t ) l − k − g ( l ) ( t ) dt. (2 . m = l . Lemma 2.2.
Let u − , v − ∈ L (0 , r ) ∀ r > . Then k ϕvu − k L ∞ ( R + ) < ∞ for anyfunction ϕ ∈ M ( u, l ; v, l ) . From Lemmas 2.1 and 2.2 one easily derives
Theorem 2.3.
Let (1 + x l − ) u − ∈ L ( R + ) , v − ∈ L (0 , r ) ∀ r > . Then the space M ( u, l ; v, m ) , m ≤ l , consists of those and only those ϕ which satisfy the conditions k ( ϕx k ) ( m ) v k < ∞ , k = 0 . . . l − , (2 . and in the case m = l additionally k ϕvu − k L ∞ ( R + ) < ∞ . (2 . Remark.
Theorem 2.3 states, roughly speaking, that (in its conditions) to check whether ϕ belongs to M ( u, l ; v, m ) one should see how multiplication by ϕ acts on polynomialsin W ( l )2 ,u . In other words, we have a weight effect: the growth of u at ∞ implied by thecondition (1 + x l − ) u − ∈ L leads to the fact that the functions of the space W ( l )2 ,u “differlittle” from polynomials. Note in this regard a result from [12]: for (1 + x l − ε ) u − ∈ L , ε > f ∈ W ( l )2 ,u there exists a polynomial P of degree l − x →∞ ( f ( x ) − P ( x )) ( k ) = 0 , k = 0 . . . l − Theorem 2.4.
Let u − ∈ B δ , v − ∈ L (0 , r ) ∀ r > . Then the space M ( u, l ; v, m ) , m ≤ l , consists of those and only those ϕ which satisfy the conditions k ( ϕx k ) ( m ) v k < ∞ , k = 0 . . . l − , (2 . r> k ( ϕx k ) ( m ) v k L ( r, ∞ ) · k x l − k − u − k L (0 ,r ) < ∞ ,k = 0 . . . l − , (2 . nd in the case m = l additionally k ϕvu − k L ∞ ( R + ) < ∞ . (2 . u ( x ) = (1 + x ) α for α < l − / u ( x ) = e x : in this case it is Theorem 2.4 which is not applicable, and wemust use Theorem 2.3. Finally note that u ( x ) = e − x is not covered by any of thesetheorems; describing the corresponding multiplier spaces is a problem for the future.6 . Auxiliary results By ∆ , ∆ , ∆ ′ . . . we denote intervals in R + , by a ∆ , a > a | ∆ | having the same center as ∆ . (cid:3) denotes the end of proof. Lemma 3.1.
Let w ∈ B δ , and let function ψ ≥ satisfy the condition sup ∆ ψ ≤ c inf ∆ ψ, if | ∆ | ≥ δ. Then ψw ∈ B δ . In particular, x γ w ∈ B δ ∀ γ > . Proof is obvious. (cid:3)
The r.h.s. inequality of the next lemma is not surprising, while the l.h.s. one showsthat the class B δ is more narrow than it could seem from the first glance. Lemma 3.2.
Let w ∈ B δ . Then there exist constants α, β, A, B > such that A (cid:18) | ∆ || ∆ | (cid:19) α ≤ R ∆ w dx R ∆ w dx ≤ B (cid:18) | ∆ || ∆ | (cid:19) β , if ∆ ⊂ ∆ , | ∆ | ≥ δ. (3 . Proof.
Let us first prove the estimate Z ∩ R + w dx ≤ E Z ∆ w dx, | ∆ | ≥ δ, with the constant E = D ( w ) . Obviously we only need to consider 2∆ R + . Let∆ = [ a, a + 2 ε ] , a < ε . Then 2∆ = [ a − ε, a + 3 ε ] , 2∆ ∩ R + = [0 , a + 3 ε ] . Now14 (2∆ ∩ R + ) = (cid:20)
38 ( a + 3 ε ) ,
58 ( a + 3 ε ) (cid:21) ⊂ ∆ , and therefore Z ∩ R + w dx ≤ D ( w ) Z (2∆ ∩ R + ) w dx ≤ D ( w ) Z ∆ w dx. Now, to prove the r.h.s. inequality in (3.1) note that | ∩ R + | ≥ | ∆ | for arbitrary ∆ .Therefore if we choose an integer N from the condition (3 / N − < | ∆ | / | ∆ | ≤ (3 / N ,then after applying to ∆ N consecutive operations ∆ ∩ R + we will get aninterval covering ∆ . this implies that R ∆ w dx R ∆ w dx ≤ E N ≤ E ( | ∆ | / | ∆ | ) β for β = log / E , proving the r.h.s. inequality in (3.1).7o prove the l.h.s. inequality in (3.1), consider first the case when ∆ and ∆ havethe same left endpoint. Let ∆ = [ a, a + ε ] . Consider the interval e ∆ = [ a + ε, a + 2 ε ] .Then ∆ ⊂ e ∆ , therefore Z ∆ w dx ≤ Z e ∆ w dx ≤ E Z e ∆ w dx = E (cid:20)Z a +2 εa − Z a + εa w dx (cid:21) , and so Z a +2 εa w dx ≥ (cid:18) E (cid:19) Z a + εa w dx. Applying this inequality N times, where 2 N ≤ | ∆ | / | ∆ | < N +1 , we get R ∆ w dx R ∆ w dx ≥ (cid:18) E (cid:19) N ≥ (cid:18) E (cid:19) − ( | ∆ | / | ∆ | ) α , where α = log (1 + 1 /E ) .For the general relative position of ∆ , ∆ consider intervals ∆ ′ and ∆ ′′ such that1) ∆ ′ ∪ ∆ ′′ = ∆ , ∆ ′ ∩ ∆ ′′ = ∆ ; 2) ∆ and ∆ ′ have the same right endpoint; 3) ∆ and ∆ ′′ have the same left endpoint. Then (cid:18) | ∆ || ∆ | (cid:19) α = (cid:18) | ∆ ′ | + | ∆ ′′ | − | ∆ || ∆ | (cid:19) α ≤ α (cid:18)(cid:18) | ∆ ′ || ∆ | (cid:19) α + (cid:18) | ∆ ′′ || ∆ | (cid:19) α (cid:19) ≤≤ α /A R ∆ ′ w dx R ∆ w dx + R ∆ ′′ w dx R ∆ w dx ! = 2 α /A R ∆ ′ w dx + R ∆ ′′ w dx R ∆ w dx ≤ α +1 /A R ∆ w dx R ∆ w dx . (cid:3) Lemma 3.3.
Let P n ( x ) be an arbitrary degree n polynomial, ∆ ⊂ ∆ intervals in R . Then for some constant c = c ( n )max ∆ | P n ( x ) | ≤ c (cid:18) | ∆ || ∆ | (cid:19) n max ∆ | P n ( x ) | . Proof of this undoubtedly known fact is given for completeness. Iterating Markov’sinequality max ∆ | P ′ n ( x ) | ≤ n | ∆ | max ∆ | P n ( x ) | , we get an estimate for the derivative of order k :max ∆ | P ( k ) n ( x ) | ≤ c ( n )( | ∆ | ) k max ∆ | P n ( x ) | , k ≤ n. (3 . x ∈ ∆ and write the Taylor expansion of P n ( x ) around x : P n ( x ) = n X k =0 ( x − x ) k k ! P ( k ) n ( x ) . ∆ | P n ( x ) | ≤ c ( n ) n X k =0 (cid:18) | ∆ || ∆ | (cid:19) k max ∆ | P n ( x ) |≤ c ( n ) (cid:18) | ∆ || ∆ | (cid:19) n max ∆ | P n ( x ) | . (cid:3) The next lemma is of a mostly technical character and describes some properties ofpolynomials orthogonal with weight B δ . Part (i) asserts, roughly speaking, that roots ofsuch polynomials cannot get too close to each other. Part (ii) is more particular and itwill play an important role in the proof of Lemma 3.5. Lemma 3.4.
Let v , w ∈ B δ . Let P n,r be the n -th polynomial of the orthogonalpolynomial system with weight v on the interval [0 , r ] , i.e. Z r t k P n,r ( t ) v ( t ) dt = 0 , k = 0 . . . n − . (3 . Let < t r, < . . . < t r,n < r be the roots P n,r , breaking up [0 , r ] into ( n + 1) intervals ∆ r, , ∆ r, , . . . , ∆ r,n . There exists a constant ε = ε ( D ( w ) , δ, n ) > such that forall r ≥ r = δ/ε | ∆ r,j | ≥ εr, j = 0 . . . n. (ii) Let us impose normalization P n,r (0) = 1 . Consider the polynomial Q n,r ( t ) = 1 − P n,r ( t ) . There exist constants β ∈ (0 , , γ > /β − , r ≥ for δ = 0) such thatfor each r ≥ r one can select a set A r ⊂ [0 , r ] , for which Z A r w dt = β Z r w dt, (3 . Z [0 ,r ] \ A r Q n,r ( t ) w ( t ) dt ≥ γ Z A r Q n,r ( t ) w ( t ) dt. (3 . Constants γ, β, r depend only on D ( v ) , D ( w ) , δ, n . Proof. (i) We will be omitting the lower index r from the notation. That the roots of P n are simple, real and located on (0 , r ) is not an additional requirement, but followsfrom (3.3), as is shown in the theory of orthogonal polynomials. Furthermore, for each r > j ∈ { , . . . , n } , that | ∆ j | ≥ rn + 1 . (3 . j = j let us consider the polynomial R ( t ) = Q ( t − t k ) , where the product istaken over k ∈ { , ..., n }\{ j, j + 1 } , and let us use orthogonality of R and P n . Let fordefiniteness j = 1 , j = 1 . Then Z r ( t − t )( t − t ) n Y k =3 ( t − t k ) w ( t ) dt = 0 , Z ∆ ( t − t )( t − t ) n Y k =3 ( t − t k ) w ( t ) dt ≥≥ Z ∆ j ( t − t )( t − t ) n Y k =3 ( t − t k ) w ( t ) dt. By (3.6), the polynomial under the last integral sign is not less than c ( n ) r n − on ∆ j ,while on ∆ it does not exceed r n − , therefore Z ∆ w dt ≥ c ( n ) Z ∆ j w dt. (3 . r = 2( n + 1) δ , then | ∆ j | ≥ δ for r ≥ r , and we can continue inequality (3.7)with the help of the r.h.s. inequality from (3.1) (Lemma 3.2): Z ∆ w dt ≥ c Z r w dt. (3 . r = δ/ε ≥ r , where ε ∈ (0 , (2 n + 2) − ) is taken so small, that for r ≥ r and∆ ⊂ [0 , r ] , | ∆ | = εr , the l.h.s. inequality from (3.1) gives Z r w dt ≥ /c Z ∆ w dt. From here and from (3.8) for r ≥ r we get Z ∆ w dt ≥ Z ∆ w dt, if ∆ ⊂ [0 , r ] , | ∆ | = εr. Therefore, | ∆ | ≥ εr , Q.E.D.(ii) On each interval ∆ j (see (i)) let us find the (clearly unique) point z j where | P n ( t ) | attains the maximum on this interval. Clearly, z = 0 , z n = r , P ′ n ( z j ) = Q ′ n ( z j ) = 0 , j = 1 . . . n − [0 ,r ] | P n ( t ) | ≤ c ( n ) ε n max ∆ j | P n ( t ) | , r ≥ r , from where | P n ( z j ) | = max ∆ j | P n ( t ) | ≥ c , r ≥ r . (3 . F j = [ z j − , z j ] , j = 1 . . . n , and consider the intervals I j ( α ) = { t ∈ F j : | Q n ( t ) | ≤ min F j | Q n ( · ) | + α } . Choose α j so that Z I j ( α j ) w dt = β Z F j w dt β ∈ (0 ,
1) , and put A r = ∪ j I j ( α j ) . (3 . β can be chosen so that the inequality R F j \ I j ( α j ) Q n ( t ) w ( t ) dt R I j ( α j ) Q n ( t ) w ( t ) dt ≥ γ (3 . j with a constant γ > /β − F j | Q n | = 0 and 2) min F j | Q n | > j = 1 ) for α j < ≥ R F j \ I j (1) w dtα j R I j ( α j ) w dt = 1 / ( α j β ) R F j \ I j (1) w dt R F j w dt . The closure of the set F j \ I j (1) contains the interval F j = { t ∈ F j : P n ( t ) ≤ } . From | ∆ j | ≥ εr it is easy to deduce an analogous inequality for the intervals F j . Let usincrease if necessary r so that for r ≥ r we have | F j | ≥ δ . Then by Lemma 3.2 wehave R F j \ I j (1) w dt . R F j w dt ≥ c , and inequality (3.11) is satisfied with γ = c/ ( α j β ) .Note that α j = O ( | I j ( α j ) | ) = O ( β ) . [The first equation uses the following from Markov’sinequality and Lemma 3.3 estimates Q ′ n ( x ) = O (1 /r ) on ∆ ⇒ O (1) on [0 , r ] ; thesecond equation follows from Lemma 3.2.] Now it’s clear that for sufficiently small β > γ > /β − Q n varies monotonically and doesnot vanish on F j . For definiteness assume that it’s positive and increasing. From (3.9) itfollows that 0 < Q n ( z j − ) = min F j Q n ≤ − c . Furthermore we can find a ξ ∈ ( z j − , z j )such that Z F j Q n w dt = Q n ( ξ ) Z F j w dt, From the simple estimates Z F j Q n w dt ≥ (min F j Q n ) Z F j \ F j w dt + Z F j w dt == (min F j Q n ) Z F j w dt + = 1 − (min F j Q n ) ) Z F j w dt ≥≥ (cid:20) (min F j Q n ) + c (1 − (1 − c ) ) (cid:21) Z F j w dt
11t follows that Q n ( ξ ) ≥ min F j Q n + θ , θ > η the right endpoint of I j ( α j ) ,i.e. I j ( α j ) = [ z j − , η ] . Monotonicity of Q n and the definition of ξ imply the inequality Z z j η Q n w dt ≥ Q n ( ξ ) Z z j η w dt. [For the proof one considers separately the cases η < ξ and η ≥ ξ .] Using this inequalityand the estimate α j = O ( β ) , we havel . h . s . of (3 . ≥ Q n ( ξ ) R z j η w dt (min F j Q n + α j ) R ηz j − w dt ≥ (cid:18) min F j Q n + θ min F j Q n + Kβ (cid:19) (1 /β − . We let A = (1 − c + θ ) / (1 − c + θ/ > β = θ/ K the l.h.s. of(3.11) ≥ γ = A (1 /β −
1) . (cid:3)
The next lemma will be used in an optimization procedure when proving Lemma 3.6.The values of parameters β and γ will then be taken from Lemma 3.4(ii). Lemma 3.5.
For < a ≤ / (1 + γ ) < β ≤ set F ( a ; α , α ) = aα + 1 − aα , α , α > . There exists a constant M ( β, γ ) < such that min βα +(1 − β ) α =1 F ( a ; α , α ) ≤ M ( β, γ ) . (3 . Proof.
The Lagrange function has the form L ( a ; α , α , λ ) = aα + 1 − aα + λ ( βα + (1 − β ) α − . From ∂L/∂α = − a/α + λβ = 0 , ∂L/∂α = − (1 − a ) /α + λ (1 − β ) = 0 , βα +(1 − β ) α =1 we have α ∗ = (cid:18) aλβ (cid:19) / , α ∗ = (cid:18) − aλ (1 − β ) (cid:19) / ,λ = (cid:0) ( aβ ) / + ((1 − a )(1 − β )) / (cid:1) . By algebraic transformations we find the minimum aα ∗ + 1 − aα ∗ = λ / (cid:0) ( aβ ) / + ((1 − a )(1 − β )) / (cid:1) == 1 − (cid:0) ((1 − a ) β ) / − ( a (1 − β )) / (cid:1) = m ( β, a ) ≤ m (cid:18) β,
11 + γ (cid:19) < . Therefore (3.12) holds with M ( β, γ ) = m ( β, / (1 + γ )) . (cid:3) ε, r in Lemmas3.6 and 3.6 ′ depend only on D ( u − ) , δ, n . Lemma 3.6.
Let u − ∈ B δ , n ∈ N . Then there exist constants ε > , r ≥ for δ = 0) such that for each r ≥ r one can choose a function f r on [0 , r ] satisfyingthe conditions k f r u k L (0 ,r ) < ∞ , (3 . Z r t k f r ( t ) dt = 0 , k = 1 . . . n, (3 . Z r f r ( t ) dt ≥ ε k f r u k L (0 ,r ) · k u − k L (0 ,r ) . (3 . Proof.
We will look for f r in the form u − g r , where g r ∈ L (0 , r ) . Conditions(3.14),(3.15) then take the form: Z r g r ( t ) t k u ( t ) − dt = 0 , k = 1 . . . n, (3 . Z r g r ( t ) u ( t ) − dt ≥ ε k g r k L (0 ,r ) · k u − k L (0 ,r ) . (3 . ϕ r = Q n u − , Q n ( t ) = P nk =1 b k t k , the projection in L (0 , r ) of the func-tion u − on the linear span E r of the set of functions { t k u − : k = 1 . . . n } . Let g r = u − − ϕ r = P n u − , P n = 1 − Q n . Condition (3.16) will be clearly satisfied, while(3.17) will be equivalent to the condition ∃ c < Z r ϕ r ( t ) u − ( t ) dt ≤ c k ϕ r k L (0 ,r ) · k u − k L (0 ,r ) . (3 . L (0 , r ) conditions (3.16),(3.17) meanthat the angle between the vectors u − and g r , g r ⊥ E r , is uniformly in r ≥ r “small”(separated from π/ u − and E r is uniformly“large” (separated from 0). Clearly these are equivalent statements.Therefore, we will be proving (3.18). Condition (3.16) shows that P n is the n -thepolynomial of the orthogonal system of polynomials with weight tu − on the interval[0 , r ] . Since u − ∈ B δ , we have tu − ∈ B δ (Lemma 3.1). By Lemma 3.6 for w = u − , v = tu − we can find constants β ∈ (0 ,
1) , γ > /β − r ≥ r there is a set A r ⊂ [0 , r ] with the properties Z A r u − dt = β Z r u − dt, (3 . Z [0 ,r ] \ A r Q n ( t ) u − ( t ) dt ≥ γ Z A r Q n ( t ) u − ( t ) dt. (3 . Z [0 ,r ] \ A r | ϕ r ( t ) | dt ≥ γ Z A r | ϕ r ( t ) | dt. (3 . , r ] the function α ( t ) = ( α , t ∈ A r ,α , t ∈ [0 , r ] \ A r ,α , α > , βα + (1 − β ) α = 1 . (3 . (cid:18)Z r ϕ r ( t ) u − ( t ) dt (cid:19) = (cid:18)Z r ϕ r ( t ) α / ( t ) u − ( t ) α / ( t ) dt (cid:19) ≤≤ Z r | ϕ r ( t ) | α ( t ) dt Z r u − ( t ) α ( t ) dt == (cid:18) α Z A r | ϕ r ( t ) | dt + 1 α Z [0 ,r ] \ A r | ϕ r ( t ) | dt (cid:19) Z r u − ( t ) dt. Let us minimize the last expression over all α , α satisfying (3.22). Applying Lemma3.5 with a = R A r | ϕ r ( t ) | dt . R r | ϕ r ( t ) | dt [condition a ≤ / (1 + γ ) follows from (3.21)],we have min βα +(1 − β ) α =1 α Z A r | ϕ r ( t ) | dt + 1 α Z [0 ,r ] \ A r | ϕ r ( t ) | dt ≤≤ M ( β, γ ) Z r | ϕ r ( t ) | dt, M ( β, γ ) < , from where (3.18) follows with c = ( M ( β, γ )) / . (cid:3) The next statement is essentially a reformulation of Lemma 3.6 and easily follows fromthe remark made after (3.18). Nevertheless we believe that it may be of independentinterest.
Lemma 3.6 ′ . Let u − ∈ B δ , n ∈ N , and χ r = χ (0 ,r ) be the characteristic function ofthe interval. Then there exist constants ε > , r ≥ for δ = 0) such that for any r ≥ r G ( u − χ r , xu − χ r , . . . , x n u − χ r ) ≥≥ ε k u − χ r k · k xu − χ r k · · · k x n u − χ r k . Here G is the Gram determinant of a system of functions in L . The claim, therefore, isthat the parallelepiped with edges u − χ r , . . . , x n u − χ r is uniformly in r non-degenerate.14 . Proofs of results from Section 1Lemma 4.1. Let u − ∈ B δ . If δ > , assume in addition a v ∈ L (0 , r ) ∀ r > .Then inequality (1.2) implies that S = sup r> k a v k L ( r, ∞ ) · k u − k L (0 ,r ) < ∞ . Proof.
Let us apply Lemma 3.6. For a function f r , r ≥ r , satisfying condition (3.13)–(3.15) and extended by zero on [ r, ∞ ) , we will consecutively have k v A f r k ≤ C k uf r k ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) v ( x ) n X k =0 a k ( x ) Z r t k f r ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( r, ∞ ) ≤ C k uf r k L (0 ,r )(3 . , (3 . = ⇒ k a v k L ( r, ∞ ) · k u − k L (0 ,r ) ≤ C/ε, r ≥ r . It remains to note that r = 0 for δ = 0 , while in the case δ > r < r we have k a v k L ( r, ∞ ) · k u − k L (0 ,r ) ≤≤ ( k a v k L (0 ,r ) + k a v k L ( r , ∞ ) ) k u − k L (0 ,r ) ≤≤ k a v k L (0 ,r ) · k u − k L (0 ,r ) + C/ε. (cid:3)
Lemma 4.2.
For the inequality (cid:13)(cid:13)(cid:13)(cid:13) v ( x ) Z x f ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ c k uf k to hold with a constant c > independent of function f , it is necessary and sufficientthat sup r> k v k L ( r, ∞ ) · k u − k L (0 ,r ) < ∞ . This is the known criterion of the weighted Hardy inequality, obtained in [3–5]
Proof of Theorem 1.1.
By Lemma 4.2 the condition S k < ∞ is necessary andsufficient for the inequality (cid:13)(cid:13)(cid:13)(cid:13) v ( x ) a k ( x ) Z x f ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ c k x − k u ( x ) f ( x ) k , which by replacing ˜ f ( x ) = x − k f ( x ) becomes (cid:13)(cid:13)(cid:13)(cid:13) v ( x ) a k ( x ) Z x t k ˜ f ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ c k u ( x ) ˜ f ( x ) k . A k : L ,u → L ,v . We conclude the equivalence ofTheorems 1.1 and 1.1 ′ and sufficiency of conditions (1.3) of Theorem 1.1.To prove that conditions (1.3) are necessary we will use the induction on n . For n = 0the statement of the theorem follows from Lemma 4.2. Assume the theorem is proved for n ≤ n , and for the operator( A f )( x ) = Z x " n +1 X k =0 a k ( x ) t k f ( t ) dt. inequality (1.2) holds. Then Lemma 4.1 implies S < ∞ , which is equivalent to theinequality (cid:13)(cid:13)(cid:13)(cid:13) va Z x f dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ c k uf k . (4 . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) v ( x ) Z x " n +1 X k =1 a k ( x ) t k f ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C ′ k uf k , which by substituting ˜ f ( t ) = tf ( t ) reduces to the form (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) v ( x ) Z x " n X k =0 a k +1 ( x ) t k ˜ f ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C ′ k t − u ( t ) ˜ f ( t ) k . Since u − ∈ B δ , then ( t − u ) − = t u − ∈ B δ (Lemma 3.1), and by the inductive hypoth-esis we obtain finiteness of the other constants S k ( k = 1 . . . n + 1) . (cid:3) Remark.
For n = 1 Lemmas 3.6, 4.1, and therefore Theorem 1.1 as well, remain trueif the condition u − ∈ B δ is replaced in their formulation by the weaker condition (1.4).In this case Lemma 3.4 used in the proof of Lemma 3.6 is replaced by the followinganalogously proven result. Lemma 3.4 ′ . Let function w ≥ satisfy with some constants D, δ the condition Z r w dt ≤ D Z r w dt, r ≥ δ ≥ . Then there exists such constants β ∈ (0 , , γ > /β − , r ≥ in the case δ = 0) ,that for each r ≥ r Z rr ∗ t w ( t ) dt ≥ γ Z r ∗ t w ( t ) dt, where r ∗ ∈ (0 , r ) is determined by the condition Z r ∗ w dt = β Z r w dt. emma 4.3. Let α ≥ . To have the inequality k v I ( α ) f k ≤ c k uf k it is necessaryand sufficient that the two conditions hold: sup r> k ( x − r ) α − v k L ( r, ∞ ) · k u − k L (0 ,r ) < ∞ , sup r> k v k L ( r, ∞ ) · k ( r − x ) α − u − k L (0 ,r ) < ∞ . This is the criterion of boundedness of the Riemann-Liouville operators in weightedspaces, obtained in [6–8].
Proof of Assertion 1.2.
Apply Lemma 4.3. (cid:3)
Proof of Assertion 1.3.
By Lemma 4.2 condition (1.6) is necessary and sufficient forthe inequality (cid:13)(cid:13)(cid:13)(cid:13) v ( x ) x α − Z x f ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ c k uf k . Since x α − ≥ ( x − t ) α − , x ≥ t ≥ k x α − v k L ( r, ∞ ) · k u − k L (0 ,r ) ≤≤ α − k ( x − r/ α − v k L ( r/ , ∞ ) · k u − k L (0 ,r ) ≤≤ α − D / k ( x − r/ α − v k L ( r/ , ∞ ) · k u − k L (0 ,r/ shows that necessity of (1.6) can be obtained with the help of Lemma 4.3. Such argumentswere used in [11]. (cid:3)
5. Proofs of results from section 2
Proof of Lemma 2.1. m < l : ( ϕg ) ( m ) ( x ) = m X k =0 C km ϕ ( k ) ( x ) g ( m − k ) ( x ) == m X k =0 C km ϕ ( k ) ( x ) Z x ( x − t ) l − m + k − ( l − m + k − g ( l ) ( t ) dt == 1( l − Z x " m X k =0 C km ϕ ( k ) ( x ) ( l − l − m + k − x − t ) l − m + k − g ( l ) ( t ) dt == 1( l − Z x d m dx m ( ϕ ( x )( x − t ) l − ) g ( l ) ( t ) dt. (5 . ϕg ) ( l ) ( x ) = ϕ ( x ) g ( l ) ( x ) + l X k =1 C kl ϕ ( k ) ( x ) g ( l − k ) ( x ) =17 ϕ ( x ) g ( l ) ( x ) + l X k =1 ϕ ( k ) ( x ) Z x ( x − t ) k − ( k − g ( l ) ( t ) dt == ϕ ( x ) g ( l ) ( x ) + 1( l − Z x " l X k =1 C kl ϕ ( k ) ( x ) ( l − k − x − t ) k − g ( l ) ( t ) dt == ϕ ( x ) g ( l ) ( x ) + 1( l − Z x d l dx l ( ϕ ( x )( x − t ) l − ) g ( l ) ( t ) dt. (5 . x − t ) l − in (5.1) and (5.2) by the binomialformula. (cid:3) Lemma 5.1.
For any set h , ..., h l of functions integrable on [ a, b ] , one can find afunction σ with | σ ( x ) | = 1 on [ a, b ] such that Z ba h k ( x ) σ ( x ) dx = 0 , k = 1 . . . l. Proof of this statement can be found in the book [16, p.267]. (cid:3)
Proof of lemma 2.2.
Define the norm k ϕ k M in the space of multipliers as the normof the corresponding operator multiplying by ϕ , acting from W ( l )2 ,u into W ( l )2 ,v .From the conditions u − , v − ∈ L (0 , r ) ∀ r > W –spaces. In this case from Banach’s closed graph theorem it is easy to getthat k ϕ k M < ∞ . Therefore it suffices to prove the inequality k ϕvu − k L ∞ ( R + ) ≤ c k ϕ k M . (5 . g ∈ W ( l )2 ,u and satisfies (2.1). By Lemma 2.1 k ϕg k W ( l )2 ,v ≥ k ( ϕg ) ( l ) v k ≥≥ k ϕg ( l ) v k − c l − X k =0 (cid:13)(cid:13)(cid:13)(cid:13) ( ϕx k ) ( l ) v Z x t l − k − g ( l ) ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) . (5 . A α = { x : | ϕvu − ( x ) | ≥ α } . Let mes A α > ∀ ε > ε , | ∆ ε | = ε , such that mes ∆ ε ∩ A α > g l such that | g l ( t ) | = ( , t / ∈ ∆ ε ∩ A α ,u − ( t ) , t ∈ ∆ ε ∩ A α , and Z ∆ ε t l − k − g l ( t ) dt = 0 , k = 0 . . . l − .
18e put g ( x ) = Z x ( x − t ) l − ( l − g l ( t ) dt. Then g ( l ) = g l , (2.1) is satisfied, while (5.4) gives: k ϕg k W ( l )2 ,v ≥ α k g ( l ) u k L (∆ ε ) −− c l − X k =0 k ( ϕx k ) ( l ) v k L (∆ ε ) · k g ( l ) u k L (∆ ε ) · k x l − k − u − k L (∆ ε ) = (5 . α − c l − X k =0 k ( ϕx k ) ( l ) v k L (∆ ε ) · k x l − k − u − k L (∆ ε ) ! k g ( l ) u k L (∆ ε ) . Note that k g k L (0 , ≤ k g ( l ) u k · k u − k L (0 , , therefore k g k W ( l )2 ,u ≤ c k g ( l ) u k . Since k g ( l ) u k > k ϕ k M ≥ α/c from (5.5) by tending ε to zero. Tending now α to ess sup x> | ϕvu − ( x ) | − (cid:3) Proof of Theorems 2.3 and 2.4.
Consider the case m = l . The case m < l isconsidered analogously Necessity.
Let ϕ ∈ M ( u, l ; v, l ) . Then (2.5) and (2.8) follow from Lemma 2.2, while(2.4) and (2.6) follow from x k ∈ W ( l )2 ,u , k = 0 . . . l − k ( ϕg ) ( l ) v k ≤ c k g ( l ) u k considered on functions satisfying (2.1), Lemma 2.1 and (2.8) we get the inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l − X k =0 C kl − ( ϕx k ) ( l ) v Z x ( − t ) l − k − g ( l ) ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ c k g ( l ) u k . Now (2.7) follows from Theorem 1.1.
Sufficiency.
Let us show that if (2.6)–(2.8) hold and f ∈ W ( l )2 ,u , then ϕf ∈ W ( l )2 ,v .This will show sufficiency of conditions of both Theorem 2.4 and Theorem 2.3, since inthe case (1 + x l − ) u − ∈ L conditions (2.6),(2.8) imply (2.7).Representing f in the form f ( x ) = l − X k =0 x k k ! f ( k ) (0) + g ( x ) , where g clearly satisfies (2.1), we will have k ϕf k W ( l )2 ,v ≤ c l − X k =0 k ϕx k k W ( l )2 ,v + k ϕg k W ( l )2 ,v =19 c l − X k =0 (cid:0) k ϕx k k L (0 , + k ( ϕx k ) ( l ) v k (cid:1) + k ϕg k L (0 , + k ( ϕg ) ( l ) v k . With the help of Lemmas 2.1 and 4.2, (2.7) and (2.8) imply the inequality k ( ϕg ) ( l ) v k ≤ c k g ( l ) u k , providing an estimate for the last term. Taking into account (2.6) it remains to show that k ϕx k k L (0 , , k ϕg k L (0 , < ∞ . But this is obvious, since k ϕ ( l ) v k , k g ( l ) u k < ∞ imply the continuity of ϕ and g on R + . (cid:3) References [1] G. H. Hardy,
Note on a theorem of Hilbert , Math. Zeitschr. vol.6 (1920), 314-317.[2] G. H. Hardy, J.E. Littlewood, G. P´olya
Inequalities , Cambridge University Press,1952[3] G. Talenti,
Osservazioni sopra una classe di disuguaglianze , Rend. Sem. Mat. Fis.Milano vol.39 (1969), 171-185.[4] G. Tomaselli,
A class of inequalities , Bull. Un. Mat. Ital. vol.2 N.4 (1969), 622-631.[5] B. Muckenhoupt,
Hardy’s inequality with weights , Stud. Math. vol.44 N.1 (1972),31-38.[6] V. D. Stepanov,
Two-weighted estimates for Riemann-Liouville integrals , ReportNo 39, Math. Inst., Czechoslovak Academy of Sciences, 1988, pp. 28.[7] V. D. Stepanov,
On one weighted inequality of Hardy type for higher order derivatives ,Proceedings of the Steklov Institute of Mathematics, 1990, 187, 205220[8] V. D. Stepanov,
Weighted inequalities of Hardy type for Riemann-Liouville fractionalintegrals , Siberian Math. J. (1990), 513-522. MR1084772[9] J. F. Martin-Reyes and E. Sawyer, Weighted inequalities for Riemann-Liouville frac-tional integrals of order one and greater , Proc. Amer. Math. Soc. vol.106 N.2 (1989),727-733.[10] R. Oinarov, Two-sided norm estimates for certain classes of integral opera-tors, in
Investigations in the theory of differentiable functions of many variablesand its applications. Part 16 , Trudy Mat. Inst. Steklov., 204, Nauka, Moscow,1993, 240250; [Translated in: Proc. Steklov Inst. Math., 204 (1994), 205214] http://mi.mathnet.ru/eng/tm1271
Some remarks concerning Hardy inequality , Teubner Texte zur Math. , Leipzig:Teubner, 1993, pp.290-294.[12] L. D. Kudryavtsev,
On norms in weighted spaces of functions given on infinite in-tervals , Analysis Mathematica vol.12 N.4 (1986), 269-282.[13] V. G. Maz’ya and T. O. Shaposhnikova,
Theory of multipliers in spaces of differen-tiable functions , Monographs and Studies in Mathematics, vol. 23, Pitman PublishingCo., Brooklyn, New York, 1985[14] G.A. Kalyabin, Pointwise multipliers in some Sobolev spaces containing unboundedfunctions, in
Investigations in the theory of differentiable functions of many vari-ables and its applications. Part 16 , Trudy Mat. Inst. Steklov., 204, Nauka, Moscow,1993, 160165; [Translated in: Proc. Steklov Inst. Math., 204 (1994), 137141] http://mi.mathnet.ru/tm1266 [15] V. S. Rychkov, Pointwise multiplicators in weighted Sobolev spaces on a half-line,Math. Notes, 56:1 (1994), 704710 https://doi.org/10.1007/BF02110561 [16] B.S.Kashin and A.A.Saakyan,
Orthogonal series , American Mathematical Soc., 2005[17] V.S. Rychkov, “On weighted estimates for a class of Volterra integraloperators,” Doklady Ros. Akad. Nauk (1997), vol. 357, p. 455; trans-lated in Doklady Mathematics (1997), vol. 56, no. 3 , p. 906-908. . arXiv:2005.11574 [math.FA][18] V.S. Rychkov, “Some Weighted Hardy-Type Inequalities and Applications,”Proc. of A. Razmadze Georgian Math. Inst. (1997), vol. 112. p. 113-129