Optimal Calderón Spaces for generalized Bessel potentials
aa r X i v : . [ m a t h . F A ] J u l Optimal Calder´on Spaces for generalized Besselpotentials
Elza Bakhtigareeva, Mikhail L. Goldman, and Dorothee D. HaroskeJuly 17, 2020
Abstract
In the paper we investigate the properties of spaces with generalizedsmoothness, such as Calder´on spaces that include the classical Nikolskii-Besov spaces and many of their generalizations, and describe differentialproperties of generalized Bessel potentials that include classical Bessel po-tentials and Sobolev spaces. Kernels of potentials may have non-powersingularity at the origin. With the help of order-sharp estimates for mod-uli of continuity of potentials, we establish the criteria of embeddings ofpotentials into Calder´on spaces, and describe the optimal spaces for suchembeddings.
The paper is devoted to generalized Bessel potentials constructed by the con-volutions of generalized Bessel-McDonald kernels with functions from the basicrearrangement invariant space. If the criterion is satisfied for the embedding ofpotentials into the space of bounded continuous functions, we state the equiva-lent description for the cones of moduli of continuity of potentials in the uniformnorm. This gives the opportunity to obtain the criterion for the embedding ofpotentials into the Calder´on space. We develop here the results of [GH13a].Some results presented here we announced in papers [GH14, GH15]. In the caseof generalized Bessel potentials constructed over the basic weighted Lorentzspace we describe explicitly the optimal Calder´on space for such an embedding.The results of Sections 3, 4 are based on an application of some results obtainedin [BG14].The paper is organized as follows. In Section 2 notation, essential conceptsand definitions are presented. We present the results concerning equivalentdescriptions for the cone of moduli of continuity for generalized Bessel potentialsin uniform norm (Theorem 2.6), and prove two-sided estimates for a variant ofthe continuity envelope function for the space of potentials (Theorem 2.12).The criterion of embedding for the space of potentials into the Calder´on spaceis presented in Theorem 3.5. Sections 4-6 are devoted to the description ofthe optimal Calder´on space for such embedding. Theorem 4.3 gives an explicit1escription in the case when the basic space for the potential is a weightedLorentz space. The proofs of the main results of Section 4 we give in Sections 5and 6. Section 7 contains some explicit descriptions of the optimal Calder´onspace.
First we fix some notation. By N we denote the set of natural numbers, by N the set N ∪ { } . For two positive real sequences { α k } k ∈ N and { β k } k ∈ N we meanby α k ∼ β k that there exist constants c , c > c α k ≤ β k ≤ c α k for all k ∈ N ; similarly for positive functions. Usually B r = { y ∈ R n : | y | < r } stands for a ball in R n centred at the origin and with radius r >
0. We denoteby µ n the Lebesgue measure on R n , n ∈ N , and by V n the volume of the n -dimensional unit ball, that is, V n = µ n ( B ). For a set A we denote by χ A itscharacteristic function.Given two (quasi-) Banach spaces X and Y , we write X ֒ → Y if X ⊂ Y andthe natural embedding of X in Y is continuous.All unimportant positive constants will be denoted by c , occasionally withsubscripts. We use some notation and general facts from the theory of Banach functionspaces and rearrangement-invariant spaces; for general background material werefer to [BS88]. As usual we call f ∗ the decreasing rearrangement of the function f , i.e., 0 ≤ f ∗ is a decreasing, right-continuous function on R + = (0 , ∞ ), equi-measurable with f , µ n ( { x ∈ R n : | f ( x ) | > s } ) = µ ( { t ∈ R + : f ∗ ( t ) > s } ) , s > . (2.1) Definition 2.1. (i)
A linear space X = X (0 , T ) of measurable functions on(0 , T ), equipped with the norm k · k X , is called a Banach function space ,shortly:
BFS , if the following conditions are satisfied: (P1) k f k X = 0 ⇐⇒ f = 0 µ -a.e. on (0 , T ); (P2) | f | ≤ g, g ∈ X implies f ∈ X, k f k X ≤ k g k X ; (P3) If for some functions f n ∈ X , f n ≥ n ∈ N , and f n monotonicallyincreasing to f , then k f n k X → k f k X , i.e. k f k X = lim n →∞ k f n k X . (2.2) (P4) For every measurable set B ⊂ (0 , T ) with µ ( B ) >
0, there exists some c B >
0, such that for all f ∈ X , Z B | f | d µ ≤ c B k f k X . P5)
For every measurable set B ⊂ (0 , T ) with µ ( B ) > k χ B k X < ∞ . (ii) A BFS E is called a rearrangement-invariant space , shortly: RIS , if itsnorm is monotone with respect to rearrangements, f ∗ ≤ g ∗ , g ∈ E implies f ∈ E, k f k E ≤ k g k E . (2.3) Remark . Note that the limit on the right-hand side of (2.2) always exists(finite or infinite), because the sequence of norms increases. The finiteness ofthis limit is a criterion for f ∈ X .For an RIS E ( R n ) its associate space E ′ ( R n ) is an RIS again, equipped withthe norm k g k E ′ ( R n ) = sup k f k E ( R n ) ≤ ∞ Z f ∗ ( s ) g ∗ ( s ) d s, g ∈ E ′ ( R n ) , see [BS88, Ch. 2] for further details. Remark . The following Luxemburg representation formula is known: Foran RIS E ( R n ) there exists a unique RIS e E ( R + ) such that k f k E ( R n ) = k f ∗ k e E ( R + ) . (2.4)Likewise, let e E ′ ( R + ) be the Luxemburg representation for E ′ ( R n ), with k h k e E ′ ( R + ) = sup (cid:26)Z ∞ h ∗ ( t ) f ∗ ( t ) d t : f ∈ e E ( R + ) , k f k e E ( R + ) ≤ (cid:27) . Examples 2.4.
Let us mention some examples of RIS such as L p , classicalLorentz and Marcinkiewicz spaces Λ pq and M p , Orlicz spaces and other. Recallthat the norms in generalized weighted Lorentz and Marcinkiewicz spaces Λ q ( v )and M ( v ) with a weight v are given by k f k Λ q ( v ) = (cid:18)Z ∞ f ∗ ( t ) q v ( t ) d t (cid:19) /q , ≤ q < ∞ ; (2.5) k f k M ( v ) = sup { f ∗∗ ( t ) v ( t ) : t ∈ R + } , (2.6)where f ∗∗ ( t ) = 1 t Z t f ∗ ( s ) d s (2.7)is the well-known maximal function of f ∗ , that is, f ∗ ≤ f ∗∗ , f ∗∗ is monotoni-cally decreasing, whereas tf ∗∗ ( t ) is monotonically increasing.For special weights we obtain a variety of Lorentz and Marcinkiewicz spaces:For instance, v pq ( t ) = t qp − , 1 ≤ p, q < ∞ , yields the classical Lorentz spacesΛ q ( v pq ) = Λ pq , in particular, Λ q ( v qq ) = Λ q (1) = L q , whereas the choice v b ( t ) = [ t γ b ( t )] q t − , γ ∈ R , and b ( t ) a slowly varying function of logarithmic type, leads to the so-calledLorentz-Karamata spaces, see for example [GNO04, Nev02].3 .2 Space of Bessel potentials The space of Bessel potentials is introduced by an integral representation over anRIS E = E ( R n ). Here we need the notion of the generalized Bessel-McDonaldkernel G = G Φ , G ( x ) = Φ( | x | ) , x ∈ R n \ { } , (2.8)where we make the following assumptions on the function Φ : R + → [0 , ∞ )everywhere in the sequel: Φ is continuous, monotonically decreasing, and0 < Z ∞ Φ( z ) z n − d z < ∞ . (2.9)We denote by ϕ ( τ ) = Φ (cid:16) ( τ /V n ) /n (cid:17) , τ > , (2.10)so that ϕ is a positive continuous decreasing function such that0 < Z ∞ ϕ ( τ ) d τ < ∞ . Definition 2.5.
Let E = E ( R n ) be an RIS, G the generalized Bessel-McDonaldkernel as above. Then H GE ( R n ) ≡ H GE = { u = G ∗ f : f ∈ E ( R n ) } , (2.11)equipped with the norm k u k H GE := inf {k f k E : f ∈ E ( R n ); G ∗ f = u } . Here u ( x ) = ( G ∗ f ) ( x ) = Z R n G ( x − y ) f ( y ) d y. Let C ( R n ) be the space of all complex-valued bounded uniformly continuousfunctions on R n , equipped with the sup-norm as usual. The following estimateholds for u = G ∗ f , with fixed T ∈ R + : k u k C := sup x ∈ R n | u ( x ) | ≤ c Z T ϕ ( τ ) f ∗ ( τ ) d τ, c = 1+ (cid:18)Z ∞ T ϕ d τ (cid:19) Z T ϕ d τ ! − , see [GM13a, (1.3)]. Thus, if we require that ϕ ∈ e E ′ (0 , T ) , (2.12)then for every f ∈ E ( R n ) such that G ∗ f = u , we have the inequality k u k C ≤ c k ϕ k e E ′ (0 ,T ) k f ∗ k e E ( R + ) = c k ϕ k e E ′ (0 ,T ) k f k E ( R n ) . (2.13)4e take the infimum over such functions f and obtain k u k C ≤ c k ϕ k e E ′ (0 ,T ) k u k H GE ( R n ) . (2.14)Under the additional conditions (2.25), (2.26) we have the embedding H GE ( R n ) ֒ → C ( R n ) into the space of continuous uniformly bounded functions (see Remark2.8 below).For later use, let us further recall the definition of differences of functions.If f is an arbitrary function on R n , h ∈ R n and k ∈ N , then(∆ kh f )( x ) := k X j =0 (cid:18) kj (cid:19) ( − k − j f ( x + jh ) , x ∈ R n . (2.15)Note that ∆ kh can also be defined iteratively via(∆ h f )( x ) = f ( x + h ) − f ( x ) and (∆ k +1 h f )( x ) = ∆ h (∆ kh f )( x ) , k ∈ N . For convenience we may write ∆ h instead of ∆ h . Accordingly, the k -th modulusof smoothness of a function f ∈ C ( R n ) is defined by ω k ( f ; t ) = sup | h |≤ t (cid:13)(cid:13) ∆ kh f (cid:13)(cid:13) C ( R n ) , t > , (2.16)such that ω ( f ; t ) = ω ( f ; t ). Recall that (2.15) immediately gives ω k ( f ; λt ) ≤ (1 + λ ) k ω k ( f ; t ) , λ > . (2.17) Let k ∈ N , T >
0, and H GE ( R n ) as in Definition 2.5. We introduce the followingcone of moduli of continuity of potentials, M = n h : R + → R + : h ( t ) = ω k (cid:16) u ; t /n (cid:17) , t ∈ (0 , T ) for some u ∈ H GE ( R n ) o , (2.18)equipped with the functional ̺ M ( h ) = inf n k u k H GE : u ∈ H GE ( R n ) , ω k (cid:16) u ; t /n (cid:17) = h ( t ) , t ∈ (0 , T ) o , h ∈ M. (2.19)Plainly, M = M ( k, E, G, T ), but we shall usually write M for convenience.Our next aim is to characterize the above cone by a simpler expression. Forthis purpose we define the notion of covering and equivalence of cones. We claimthat the cone M = M (0 , T ) with ̺ M is covered by the cone K = K (0 , T ) with ̺ K , with covering constant c ∈ (0 , ∞ ), written as M c ≺ K , if for any function h ∈ M there exists some function g ∈ K , such that ̺ K ( g ) ≤ c̺ M ( h ) and g ( t ) ≥ h ( t ) , t ∈ (0 , T ) . (2.20)5e write M ≺ K if there exists c ∈ (0 , ∞ ), such that M c ≺ K . We denote as c ( M ≺ K ) the best constant of covering M ≺ K , that is c ( M ≺ K ) = inf n c ∈ R + : M c ≺ K o . In case of mutual covering we call it equivalence of cones , that is M ≈ K ⇐⇒ M ≺ K ≺ M. (2.21)We need some further notation. For ϕ as above, letΩ ϕ ( t, τ ) = ϕ ( τ )1 + (cid:0) τt (cid:1) k/n , t, τ > , (2.22)and e E (0 , T ) = n σ ∈ e E (0 , T ) : σ ≥ , σ monotonically decreasing o . Then the new cone K = K ( E, ϕ, T ) is given by K = ( h : R + → R + : h ( t ) = Z T Ω ϕ ( t, τ ) σ ( τ ) d τ for some σ ∈ e E (0 , T ) ) , (2.23)equipped with the functional ̺ K ( h ) = inf ( k σ k e E (0 ,T ) : σ ∈ e E (0 , T ) , Z T Ω ϕ ( t, τ ) σ ( τ ) d τ = h ( t ) ) . (2.24)Our first main result reads as follows. Theorem 2.6.
Let Φ be a standard function with (2.9) , ϕ , given by (2.10) ,satisfy (2.12) . Assume, in addition, that Φ ∈ C k ( R + ) , k ∈ N , and Φ satisfies the followingestimates for some positive real numbers a , a , z , max ≤ j ≤ k (cid:0) z j | Φ j ( z ) | (cid:1) ≤ a Φ( z ) , z ∈ (0 , z ] , (2.25)max ≤ j ≤ k (cid:0) z j | Φ j ( z ) | (cid:1) ≤ a z k Φ( z ) , z > z , (2.26) where Φ j ( z ) = (cid:18) z dd z (cid:19) j Φ( z ) . Then, for the cones M given by (2.18) and K given by (2.23) we have thecovering M ≺ K . . If Φ additionally satisfies the following estimate for some positive real num-ber δ : ( − k z k Φ ( k ) ( z ) ≥ δ Φ( z ) , z ∈ (0 , z ] , (2.27) then M and K are equivalent. The constants in the condition of mutualcoverings (2.21) depend on k , n , T , a , a , z , δ and on the norm of theembedding operator (2.14) .Remark . The proof is based on the following crucial estimates. Under the assumptions of Theorem 2.6, Part 1, for any u ∈ H GE ( R n ), thatis u = G ∗ f, f ∈ E ( R n ) , the estimate holds ω k ( u, t /n ) ≤ c Z T Ω ϕ ( t, τ ) f ∗ ( τ ) d τ, t ∈ (0 , T ) , (2.28)see [GM13a], with c = c ( k, n, z , a , a , T, k id k ) ∈ R + , where k id k refersto the norm of embedding operator in (2.14). Under the assumptions of Theorem 2.6, Part 2, for σ ∈ e E (0 , T ) we denote σ ( t ) = σ ( t ) , t ∈ (0 , T ) , σ ( t ) = 0 , t ≥ T . Then, there exists f ∈ E ( R n ),such that f ∗ ( t ) ≤ σ ( t ), t ∈ R + , and for u = G ∗ f ∈ H GE ( R n ) ω k ( u, t /n ) ≥ c Z T Ω ϕ ( t, τ ) σ ( τ ) d τ, t ∈ (0 , T ) , (2.29)with c = c ( k, n, z , a , δ , T ) >
0, see [GM13b, GH14]. Moreover, for thebest constants of coverings in Theorem 2.6 we have c ( M ≺ K ) ≤ c ; c ( K ≺ M ) ≥ c − . (2.30) Proof. Step 1 . We first show that under the assumptions (2.25), (2.26) there isthe covering M c ≺ K for the cones given by (2.18) and (2.23) with any constantof covering c > c . Let c = (1 + ε ) c , ε ∈ (0 , . For h ∈ M there exists u ε ∈ H GE ( R n ) such that ω k ( u ε ; t /n ) = h ( t ) , t ∈ (0 , T ) , k u ε k H GE ≤ (1 + ε ) ̺ M ( h ) . Then, for u ε ∈ H GE ( R n ), we find f ε ∈ E ( R n ), such that u ε = G ∗ f ε , k f ε k E ≤ (1 + ε ) k u ε k H GE ≤ (1 + ε ) ̺ M ( h ) . By (2.28) for u = u ε we have the estimate h ( t ) ≤ g ε ( t ) := c Z T Ω ϕ ( t, τ ) f ∗ ε ( τ ) d τ = Z T Ω ϕ ( t, τ ) σ ε, ( τ ) d τ, t ∈ (0 , T ) . σ ε, = c f ∗ ε ∈ e E (0 , T ); k σ ε, k e E (0 ,T ) ≤ c k f ∗ ε k e E ( R + ) = c k f ε k E . We see that h ≤ g ε ∈ K ; ̺ K ( g ε ) ≤ k σ ε, k e E (0 ,T ) ≤ c k f ε k E ≤ c (1 + ε ) ̺ M ( h ) = c̺ M ( h ) . These estimates show that M c ≺ K for all c > c implies c ( M ≺ K ) ≤ c . Step 2 . We will show that under assumptions of Theorem 2.6, Part 2, thereis the covering K c ≺ M for all c > c − , which implies c ( K ≺ M ) ≤ c − . (2.31)Let c = (1 + ε ) c − , ε ∈ (0 , g ∈ K there exists σ ε, ∈ e E (0 , T ), such that g ( t ) = Z T Ω ϕ ( t, τ ) σ ε, ( τ ) d τ, t ∈ (0 , T ); k σ ε, k e E (0 , ∞ ) ≤ (1 + ε ) ̺ K ( g ) . Now, let σ ε ( t ) = ( σ ε, ( t ) , t ∈ (0 , T ) , , t ≥ T. Then, k σ ε k e E (0 , ∞ ) = k σ ε, k e E (0 ,T ) ≤ (1 + ε ) ̺ K ( g ) . According to (2.29) with σ = σ ε, we find f ε ∈ E ( R n ), such that f ∗ ε ≤ σ ε and for u ε = G ∗ f ε the estimate holds ω k ( u ε , t /n ) ≥ c g ( t ) , t ∈ (0 , T ). So, wedenote h ε ( t ) := ω k ( c − u ε , t /n ) = c − ω k ( u ε , t /n ) ≥ g ( t ) , t ∈ (0 , T ) . Moreover, h ε ∈ M , and ̺ M ( h ε ) ≤ (cid:13)(cid:13) c − u ε (cid:13)(cid:13) H GE ≤ k c − f ε k E = c − k f ∗ ε k e E (0 , ∞ ) ≤ c − k σ ε k e E (0 , ∞ ) ≤ (1 + ε ) c − ̺ K ( g ) . These estimates show that K c ≺ M for all c > c − which implies c ( K ≺ M ) ≤ c − . emark . For Ω ϕ ( t, τ ), see (2.22), ϕ ∈ e E ′ (0 , T ), see (2.12), and σ ∈ e E (0 , T )the following assertions holdΩ ϕ ( t, τ ) σ ( τ ) ≤ ϕ ( τ ) σ ( τ ) ∈ L (0 , T ); Ω ϕ ( t, τ ) σ ( τ ) → t → +0) . Therefore, by Lebesgue’s dominated convergence theorem (2.23) implies h ∈ K ⇒ h ( t ) → t → +0) . Together with the covering M ≺ K, see (2.20), it leads to h ∈ M ⇒ h ( t ) → t → +0) . It means that under the assumptions (2.25), (2.26) ω k ( u, t ) → t → +0) ⇒ H GE ( R n ) ֒ → C ( R n ) . Examples 2.9.
For the classical Bessel potentials the corresponding Bessel-McDonald kernels are determined by (2.8) withΦ ν ( x ) = H ν ( x ) , x ∈ R + , ν = n − α , < α < n, (2.32)where H ν ( x ) = x − ν K ν ( x ), x >
0, and K ν is the modified Bessel function, K ν ( ̺ ) = 12 (cid:16) ̺ (cid:17) ν ∞ Z ξ − ν − e − ξ − ̺ / ξ d ξ (2.33)cf. [Gol10, GH13a, Nik77]. From the well-known properties of these functions,it follows easily that conditions (2.25)–(2.26) are satisfied and thatΦ ν ( y ) ≃ ( y − ν , y ∈ (0 , y ] ,y − ν − e − y , y > y , (2.34)for some appropriate y > ν and y ). Note that (2.10) reads as ϕ ( τ ) ≃ τ α/n − , τ ∈ (0 , T ] , (2.35)in this case which implies that (2.14) is true. In order to apply Theorem 2.6 weneed to verify (2.12) and have found in this case that(2.12) if, and only if, τ α/n − ∈ e E ′ (0 , T ) , T ∈ R + . Recall that e E ′ (0 , T ) is the restriction of e E ′ ( R + ) to (0 , T ). Examples 2.10.
Let Φ ∈ C k ( R + ) satisfy assumptions (2.9)–(2.12) and (2.26)for some z ∈ R + . Assume T = V n z n , andΦ( z ) = z α − n Λ( z ) , < α < n, z ∈ (0 , z ] . (2.36)9ere Λ ∈ C k (0 , z ] is a positive function, with z j Λ ( j ) ( z ) = ε j ( z )Λ( z ) , with ε j ( z ) −−−−→ z → , j = 1 , . . . , k. (2.37)Then Λ is a slowly varying function on (0 , z ], i.e., for all γ > z γ Λ( z ) is monotonically increasing, z − γ Λ( z ) is monotonically decreasing . (2.38)We further conclude that ϕ ( τ ) = τ α/n − λ ( τ ) , λ ( τ ) = Λ (cid:18)(cid:16) τV n (cid:17) /n (cid:19) , τ ∈ (0 , T ] . (2.39)The function λ is slowly varying as well on (0 , T ], and (2.14) is satisfied. Oneverifies that Φ satisfies conditions (2.25) and (2.27) such that Theorem 2.6 isapplicable whenever (2.12) is true, where ϕ is given by (2.39). Proof.
We apply the Leibniz formula to Φ given by (2.36) and getΦ ( k ) ( z ) = ( α − n ) · · · ( α − n − ( k − z α − n − k Λ( z )+ k X j =1 C k,j ( α − n ) · · · ( α − n − ( k − j − z α − n − ( k − j ) Λ ( j ) ( z ) , where C k,j are the binomial coefficients. Therefore, by (2.37), z k Φ ( k ) ( z ) = Φ( z ) h ( − k ( n + k − − α ) · · · ( n − α )+ k X j =1 C k,j ( α − n ) · · · ( α − n − ( k − j − ε j ( z ) i . This assertion implies (2.27) for sufficiently small z > ε j ( z ) → z → j = 1 , . . . , k . Note that (cid:18) z − dd z (cid:19) m z α − n = ( α − n ) · · · ( α − n − m − z α − n − m . (2.40)Furthermore, condition (2.37) implies for m = 1 , . . . , k , (cid:18) z − dd z (cid:19) m Λ( z ) = δ m ( z ) z − m Λ( z ) , (2.41)where δ m ( z ) → z → (cid:18) z − dd z (cid:19) l Φ( z )= l X j =0 C l,j "(cid:18) z − dd z (cid:19) l − j z α − n z − dd z (cid:19) j Λ( z ) = ( α − n ) · · · ( α − n − l − z α − n − l Λ( z )+ l X j =1 C l,j ( α − n ) · · · ( α − n − l − j − z α − n − l − j ) δ j ( z ) z − j Λ( z )= Φ( z ) z − l F ( z ) , where F ( z ) = ( α − n ) · · · ( α − n − l − l X j =1 C l,j ( α − n ) · · · ( α − n − l − j − δ j ( z ) . Since the term F ( z ) is bounded on (0 , z ], we obtain (2.25). Thus Theorem 2.6can be applied. Finally ϕ is determined by (2.39) with 0 < α < n , and λ beingslowly varying on (0 , T ], and we conclude Z t ϕ ( τ ) d τ = Z t λ ( τ ) τ α/n − d τ ≃ λ ( t ) t α/n = ϕ ( t ) t, (2.42)where the involved constants do not depend on t ∈ (0 , T ), see also Remark 2.11below. Thus condition (2.14) is satisfied and we have the equivalence that H GE ( R n ) ֒ → C ( R n ) holds if, and only if, τ α/n − λ ( τ ) ∈ e E ′ (0 , T ) . Remark . In (2.42) we used some well-known properties of slowly varyingfunctions λ , which are positive on (0 , T ): for any γ > Z t τ γ − λ ( τ ) d τ ≃ t γ λ ( t ) , (2.43) Z Tt τ − γ − λ ( τ ) d τ ≤ c γ t − γ λ ( t ) , (2.44) λ ( t ) = o Z Tt τ − λ ( τ ) d τ ! for t → . (2.45)Now we describe some important characteristic of the smoothness of func-tions from H GE , the uniform majorant for moduli of continuity Ω kEG ( t /n ) , t ∈ (0 , T ) , namely,Ω kEG ( t /n ) = sup n ω k ( u, t /n ) : u ∈ H GE ( R n ); k u k H GE ≤ o . (2.46)11n case of k = 1 this is a variant of the continuity envelope function studied in[Har07, Tri01] in general, and in [GH13a] for Bessel potentials. Theorem 2.12.
Let Φ be a standard function with (2.9) , ϕ , given by (2.10) ,satisfy (2.12) . Under the assumptions of Theorem 2.6, Part 1, the following estimateholds: Ω kEG ( t /n ) ≤ c k Ω ϕ ( t, · ) k e E ′ (0 ,T ) , t ∈ (0 , T ) . (2.47) Under the assumptions of Theorem 2.6, Part 2, we have both estimates: (2.47) and Ω kEG ( t /n ) ≥ c k Ω ϕ ( t, · ) k e E ′ (0 ,T ) , t ∈ (0 , T ) . (2.48) The constants c , c are the same as in Theorem 2.6.Proof. Step 1 . Let u ∈ H GE ( R n ). For any f ∈ E ( R n ) such that G ∗ f = u wehave the estimate, see (2.28), ω k ( u, t /n ) ≤ c Z T Ω ϕ ( t, τ ) f ∗ ( τ ) d τ ≤ c k Ω ϕ ( t, · ) k e E ′ (0 ,T ) k f ∗ k e E (0 ,T ) ≤ c k Ω ϕ ( t, · ) k e E ′ (0 ,T ) k f k E ( R n ) . Therefore, ω k ( u, t /n ) ≤ c k Ω ϕ ( t, · ) k e E ′ (0 ,T ) k u k H GE ( R n ) , ∀ u ∈ H GE ( R n ) . This yields (2.47).
Step 2 . Note that Ω ϕ ( t, τ ) ≥ τ ∈ (0 , T ) , so thatwe have by the well-known formula for an associated norm in the RIS e E (0 , T ) k Ω ϕ ( t, · ) k e E ′ (0 ,T ) = sup (Z T Ω ϕ ( t, τ ) σ ( τ ) d τ : σ ∈ e E (0 , T ); k σ k e E (0 ,T ) ≤ ) . Thus, for every ε ∈ (0 ,
1) there exists σ ,ε ∈ e E (0 , T ) such that k σ ,ε k e E (0 ,T ) ≤ Z T Ω ϕ ( t, τ ) σ ,ε ( τ ) d τ ≥ (1 − ε ) k Ω ϕ ( t, · ) k e E ′ (0 ,T ) . Let σ ε ∈ e E ( R + ) be the extension of σ ,ε by zero from (0 , T ) onto R + . Then,according to (2.29), there exists f ε ∈ E ( R n ) such that f ∗ ε ≤ σ ε , G ∗ f ε = u ε ∈ H GE ( R n ), ω k ( u ε , t /n ) ≥ c Z T Ω ϕ ( t, τ ) σ ,ε ( τ ) d τ ≥ (1 − ε ) c k Ω ϕ ( t, · ) k e E ′ (0 ,T ) . (2.49)12oreover, k u k H GE ( R n ) ≤ k f ε k E ( R n ) = k f ∗ ε k e E ( R + ) ≤ k σ ε k e E ( R + ) = k σ ,ε k e E (0 ,T ) ≤ . (2.50)Consequently, by (2.46), (2.49), and (2.50)Ω kEG ( t /n ) ≥ c k Ω ϕ ( t, · ) k e E ′ (0 ,T ) . This leads to (2.29) by passing to the limit for ε → . We need some generalization of the notion of Banach function spaces (BFS),see Definition 2.1. Let µ denote the Lebesgue measure on (0 , T ), T ∈ (0 , ∞ ]. Definition 3.1.
A linear space X = X (0 , T ) of measurable functions on (0 , T ),equipped with the norm k · k X , is called a generalized Banach function space ,shortly: GBFS , if the following conditions are satisfied: (P1) k f k X = 0 ⇐⇒ f = 0 µ -a.e. on (0 , T ); (P2) | f | ≤ g, g ∈ X implies f ∈ X, k f k X ≤ k g k X ; (P3) If for some functions f n ∈ X , f n ≥ n ∈ N , and f n monotonicallyincreasing to f , then k f n k X → k f k X for n → ∞ . (P4) For every measurable set B ⊂ (0 , T ) with µ ( B ) >
0, there exists some h B > µ -a.e. in B , and some c B >
0, such that for all f ∈ X , Z B h B | f | d µ ≤ c B k f k X . (P5) For every measurable set B ⊂ (0 , T ) with µ ( B ) >
0, there exists some f B ∈ X such that f B > µ -a.e. in B . Remark . Note that in [BS88] for a BFS it is required that h B = f B = χ B in (P4), (P5), respectively, where χ B is the characteristic function of B . Notethat, if X = X (0 , T ) is a BFS, and the function ν is µ -measurable such that0 < ν < ∞ µ -a.e. in (0 , T ), then X ν = { f : f ν ∈ X, k f k X ν = k f ν k X } is a GBFS. Moreover, a GBFS is in fact a Banach space, as well as its associ-ated space. The duality principle holds for GBFS (the twice associated spacecoincides with the initial space, cf. [BGZ14]).13et K = K (0 , T ) be some cone of non-negative µ -measurable functions on(0 , T ) equipped with the positively homogeneous functional ̺ K . Recall that forthe GBFS X = X (0 , T ) the embedding K X means that K ⊂ X and ∃ c = c K > k h k X ≤ c K ̺ K ( h ) for all h ∈ K. (3.1) Definition 3.3.
A GBFS X = X (0 , T ) is called optimal for the embedding K X , if (i) K X , (ii) whenever K Y , where Y is a GBFS, then this implies X ⊂ Y . Remark . Let K and M be some cones of non-negative µ -measurable func-tions on (0 , T ) equipped with the functionals ̺ K and ̺ M . If K ≈ M , then(1) for every GBFS X = X (0 , T ) we have K X if, and only if, M X ,and the ratio of the constants c K /c M in (3.1) depends on the constants ofthe mutual coverings of the cones only, see (2.20), (2.21),(2) a GBFS X = X (0 , T ) is optimal for both the embeddings K X as wellas M X .This can be seen as follows. Let us show that whenever M ≺ K and K X ,then M X .For every h ∈ M we can find h ∈ K such that h ≤ h on (0 , T ) and ̺ K ( h ) ≤ c ̺ M ( h ) . Now K X implies h ∈ X and, by property (P2) of Definition 3.1, h ∈ X with k h k X ≤ k h k X ≤ c K ̺ K ( h ). This finally leads to k h k X ≤ c K c ̺ M ( h ) for all h ∈ M. But this is nothing else than M X . Consequently, the equivalence M ≈ K implies the equivalence K X ⇐⇒ M X. Thus the same GBFS X = X (0 , T ) is optimal for both embeddings K X and M X .Let X = X (0 , T ) be a GBFS and k ∈ N . We introduce the Calder´on spaceΛ k ( C, X ) (see for example [Gol88], a more special version was considered in[GNO11]) as follows:Λ k ( C ; X ) = n u ∈ C ( R n ) : ω k ( u ; t /n ) ∈ X (0 , T ) o , (3.2) k u k Λ k ( C,X ) = k u k C + k ω k ( u ; t /n ) k X (0 ,T ) . (3.3)14he following non-trivial conditions hold:Λ k ( C ; X ) = { } ⇐⇒ (cid:13)(cid:13)(cid:13) t k/n (cid:13)(cid:13)(cid:13) X (0 ,T ) < ∞ , (3.4)Λ k ( C ; X ) = C ( R n ) ⇐⇒ k k X (0 ,T ) = ∞ . (3.5)Moreover, it is obvious that X (0 , T ) ֒ → X (0 , T ) implies Λ k ( C ; X ) ֒ → Λ k ( C ; X ) . (3.6)Now we are able to formulate a criterion for the embedding H GE ( R n ) ֒ → Λ k ( C ; X ). Theorem 3.5.
Let the conditions of Theorem 2.6 be satisfied, and let K be thecone given by (2.23) with (2.22) . Then H GE ( R n ) ֒ → Λ k ( C ; X ) , (3.7) if, and only if, K X. (3.8) The norm of the embedding operator in (3.7) depends only on k , n , T , a , a , z , δ and on the norms of the embedding operators in (2.14) and (3.1) .Proof. First we show that under the condition (2.12) we have the equivalence(3.7) ⇐⇒ M X (0 , T ) , (3.9)where M is the cone in (2.18). Indeed, in this case k u k C ≤ c k u k H GE for all u ∈ H GE ( R n ) , and (3.7) is equivalent to k ω k ( u ; t /n ) k X (0 ,T ) ≤ c k u k H GE , u ∈ H GE ( R n ) . This means that for h ∈ M , k h k X (0 ,T ) ≤ c k u k H GE , for every u ∈ H GE ( R n ) such that ω k ( u ; t /n ) = h ( t ), t ∈ (0 , T ). Therefore, for h ∈ M , k h k X (0 ,T ) ≤ c inf n k u k H GE : u ∈ H GE ( R n ) , ω k ( u ; t /n ) = h ( t ) o , that is, k h k X (0 ,T ) ≤ c ̺ M ( h ) , h ∈ M. This is equivalent to the embedding M X (0 , T ).Now the equivalence of (3.7) and (3.8) follows from (3.9), (2.21) due toRemark 3.4. 15 orollary 3.6. Let X = X (0 , T ) be an optimal GBFS for the embedding (3.8) , where K is again the cone given by (2.23) – (2.22) . Then Λ k ( C ; X ) is anoptimal Calder´on space for the embedding (3.7) , that is, H GE ( R n ) ֒ → Λ k ( C ; X ) , and (3.7) implies Λ k ( C ; X ) ֒ → Λ k ( C ; X ) . (3.10) Proof.
For the GBFS X = X (0 , T ) we have K X = ⇒ H GE ( R n ) ֒ → Λ k ( C ; X ) , (3.11)by Theorem 3.5. Assume that the embedding (3.7) is true. Then K X implies X ֒ → X by (3.8), and by the definition of the optimal GBFS for theembedding K X . Now we apply (3.6) and obtain both the assertions in(3.10). We shall exemplify the results of Sections 2 and 3 for the case when the basicRIS E ( R n ) coincides with a weighted Lorentz space, E ( R n ) = Λ q ( v ), 1 ≤ q < ∞ , where the weight v > q ( v ), 1 ≤ q < ∞ , is equipped with the functional k f k Λ q ( v ) = (cid:18)Z ∞ f ∗ ( t ) q v ( t ) d t (cid:19) /q , ≤ q < ∞ . (4.1)General properties of Lorentz spaces can be found, for instance, in [CS93,CPSS01]. Recall thatΛ q ( v ) = { } ⇐⇒ V ( t ) = Z t v ( τ ) d τ < ∞ , t > . (4.2)Expression (4.1) is equivalent to some norm if q = 1 and t − V ( t ) almost de-creases, or, in case q >
1, if there exists some c >
0, such that t q Z ∞ t τ − q v ( τ ) d τ ≤ cV ( t ) , t > . (4.3)We shall assume in the sequel that these conditions are satisfied. We furtherneed the following notation, where T > t ∈ (0 , T ], W ( t ) = V ( t ) − Z t ϕ ( τ ) d τ, (4.4)Ψ q ( t ) = sup τ ∈ (0 ,t ] W ( τ ) , q = 1 , (cid:16)R t W q ′ ( τ ) v ( τ ) d τ (cid:17) /q ′ , < q < ∞ , (4.5)where q ′ is defined as usual, q + q ′ = 1, 1 < q < ∞ .16 emma 4.1. Let
T > , ≤ q < ∞ . Using the above notation, (2.12) is true if, and only if, Ψ q ( T ) < ∞ . (4.6) Proof.
Let us consider the case of basic RIS E ( R n ) = Λ q ( v ) , ≤ q < ∞ , (4.7)see (4.1), (4.2). We define for t >
0, 1 < q < ∞ , w ( t ) = V ( t ) − q ′ v ( t ) . (4.8)Note that Z ∞ T w ( t ) d t = 1 q ′ − (cid:16) V ( T ) − q ′ − lim t →∞ V ( t ) − q ′ (cid:17) . (4.9)We use the well-known description of the associated RIS for Lorentz spaces(4.7) in the Luxemburg representation: k ϕ k e E ′ ( R + ) = sup τ ∈ R + V ( τ ) R τ ϕ ∗ ( s ) d s, q = 1 , (cid:16)R ∞ (cid:0)R τ ϕ ∗ ( s ) d s (cid:1) q ′ w ( τ ) d τ (cid:17) /q ′ , q > , where ϕ ∗ is the decreasing rearrangement of the function ϕ : R + → [0 , ∞ ]. For ϕ given by (2.10) we set ϕ ( τ ) = ( ϕ ( τ ) , τ ∈ (0 , T ) , , τ ≥ T. Since ϕ ≥ ϕ ∗ ( s ) = ϕ ( s ) χ (0 ,T ) ( s ) , and thus Z τ ϕ ∗ ( s ) d s = (cid:18)Z τ ϕ ( s ) d s (cid:19) χ (0 ,T ) ( τ ) + Z T ϕ ( s ) d s ! χ [ T, ∞ ) ( τ ) . In view of k ϕ k e E ′ (0 ,T ) = k ϕ k e E ′ ( R + ) this leads to k ϕ k e E ′ (0 ,T ) = max ( sup τ ∈ (0 ,T ) W ( τ ) , Z T ϕ ( s ) d s ! sup τ ≥ T V ( τ ) − ) = Ψ ( T )in case of q = 1, see (4.4) and (4.5). In case of q >
1, using that q ′ = qq − , weconclude from (4.4), (4.5), (4.8) and (4.9) that k ϕ k e E ′ (0 ,T ) ≃ Z T W ( τ ) q ′ v ( τ ) d τ ! /q ′ + Z T ϕ ( s ) d s ! (cid:18)Z ∞ T w ( τ ) d τ (cid:19) /q ′ = Ψ q ( T ) + Z T ϕ ( s ) d s ! q ′ − /q ′ (cid:16) V ( T ) − q ′ − lim t →∞ V ( t ) − q ′ (cid:17) /q ′ . ϕ and v . For that reason we com-plement our above notation (4.4), (4.5) as follows: f W ( t ) = V ( t ) − t − kn ϕ ( t ) , (4.10) U q ( t ) = sup τ ∈ [ t,T ] f W ( τ ) , q = 1 , (cid:16)R Tt f W q ′ ( τ ) v ( τ ) d τ (cid:17) /q ′ , < q < ∞ , (4.11)where again t ∈ (0 , T ] is assumed. Note that f W is a continuous bounded functionon [ t, T ] for any t ∈ (0 , T ], such that the expressions in (4.11) are well-defined.Now we can formulate the alternative assumptions. (A) There exists a constant d > t ∈ (0 , T ], Z Tt τ − kn ϕ ( τ ) d τ ≤ d t − kn Z t ϕ ( τ ) d τ, (4.12)and, in addition, ∃ ε > t ε V ( t ) − is monotonically decreasing for t ∈ (0 , T ] . (4.13) (B) There exists a constant d > t ∈ (0 , T ] Z t τ − kn ϕ ( τ ) d τ ≤ d t − kn ϕ ( t ) , (4.14)and, in addition, ∃ ε > t ε U q ( t ) is monotonically decreasing for t ∈ (0 , T ] . (4.15) Remark . Let λ > , T ],0 < α < n, ϕ ( t ) = t αn − λ ( t ) , t ∈ (0 , T ] , (4.16)similar to (2.39). Recall that λ ≡ α < k , whereas (4.14) holds, if, and onlyif, α > k .This can be seen as follows. Let ϕ be given by (4.16) and denote A ( t ) = Z Tt τ − k/n ϕ ( τ ) d τ = Z Tt τ α − kn − λ ( τ ) d τ, (4.17) B ( t ) = t − k/n Z t ϕ ( τ ) d τ = t − k/n Z t τ α/n − λ ( τ ) d τ. (4.18)18ccording to (2.43) for α > B ( t ) ≃ t α − kn λ ( t ) , t ∈ (0 , T ) , (4.19)and by (2.44) for 0 < α < k , A ( t ) ≤ c t α − kn λ ( t ) ≃ B ( t ) , t ∈ (0 , T ) , such that (4.12) follows. If α = k , then (4.17), (4.19) and (2.45) show that B ( t ) ≃ λ ( t ) ≃ o ( A ( t )) , t → , such that (4.12) fails. If α > k , then (4.19) and (2.43) show that B (0+) = 0 , A (0+) = Z T τ α − kn − λ ( τ ) d τ ≃ T α − kn λ ( T ) > , so that (4.12) fails. Therefore (4.12) holds if, and only if, α < k .Next we show that (4.14) holds if, and only if, α > k . For a function ϕ givenby (4.16) we denote by C ( t ) = Z t τ − kn ϕ ( τ ) d τ = Z t τ α − kn − λ ( τ ) d τ. If α < k , then C ( t ) = ∞ , t ∈ (0 , T ), for every slowly varying function λ > α = k , then t − kn ϕ ( t ) = λ ( t ) = o (cid:18)Z t λ ( τ ) d ττ (cid:19) for t → λ >
0. Hence (4.14) fails as well. Finally, inthe remaining case α > k , we have by (2.43) that C ( t ) ≃ t α − kn λ ( t ) = t − kn ϕ ( t ) , t ∈ (0 , T ) , and (4.14) holds.Now we present one of our main results. Its proof however has to be post-poned to Section 6 as we shall need some detailed preparation. But here we wantto formulate the corresponding result first and collect some further consequencesand examples below.We introduce the following notation, k f k ˚ X = Z T (cid:18) k f k L ∞ (0 ,t ) Ψ q ( t ) (cid:19) q dΨ q ( t )Ψ q ( t ) ! /q , (4.20)with Ψ q defined by (4.5). Let T ∈ (0 , T ) be such that Ψ q ( T ) = Ψ q ( T ).19 heorem 4.3. Let ≤ q < ∞ , T > , and assume that Ψ q ( T ) < ∞ , where Ψ q is given by (4.5) . Let the cone K be given by (2.23) with (2.22) . Assumethat at least one of the above conditions (A) or (B) , given by (4.12) – (4.15) , issatisfied. Then the optimal GBFS X = X (0 , T ) for the embedding K X has the following norm: (i) if q = 1 and Ψ (0+) = lim t ↓ Ψ ( t ) > , then k f k X = k f k L ∞ (0 ,T ) , (4.21) (ii) if q = 1 and Ψ (0+) = lim t ↓ Ψ ( t ) = 0 , or < q < ∞ , then k f k X = k f k ˚ X + Ψ q ( T ) − k f k L ∞ ( T ,T ) . (4.22) Let the assumptions of Theorem 2.6 be satisfied, and K the cone given by(2.23) with (2.22). We will show that the results of [BGZ14, Thm. 3.1, Rem. 3.2,Ex. 3.3] are applicable here. Note that here A = D = (0 , T ), µ = ν are Lebesguemeasures, Ω ϕ ( t, τ ) is determined by (2.22), so that for t, τ ∈ (0 , T ),Ω ϕ ( t, τ ) ≃ ( ϕ ( τ ) , τ ∈ (0 , t ] ,t k/n τ − k/n ϕ ( τ ) , τ > t. (5.1)To apply [BGZ14, Thm. 3.1] it is sufficient to verify that c = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T Ω ϕ ( t, · ) d t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e E ′ (0 ,T ) < ∞ , (5.2)and the existence of some σ ∈ e E (0 , T ) such that Z T Ω ϕ ( t, τ ) σ ( τ ) d τ > , t ∈ (0 , T ) , (5.3)as these conditions imply [BGZ14, (3.7), (3.8)]. According to (5.1), for every g ∈ M + (0 , T ), Z T Ω ϕ ( ξ, τ ) g ( ξ ) d ξ ≃ τ − k/n ϕ ( τ ) Z τ ξ k/n g ( ξ ) d ξ + ϕ ( τ ) Z Tτ g ( ξ ) d ξ. (5.4)Let g ( t ) ≡ t ∈ (0 , T ), then for τ ∈ (0 , T ), Z T Ω ϕ ( t, τ ) d t ≃ τ ϕ ( τ ) + ( T − τ ) ϕ ( τ ) = T ϕ ( τ ) ,
20o that c = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T Ω ϕ ( t, · ) d t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e E ′ (0 ,T ) ≃ k ϕ k e E ′ (0 ,T ) < ∞ , because of assumption (2.12). Furthermore, with σ = χ (0 ,T ) ∈ e E (0 , T ) , we obtain in view of (5.1) for every t ∈ (0 , T ), Z T Ω ϕ ( t, τ ) σ ( τ ) d τ = Z T Ω ϕ ( t, τ ) d τ ≥ c Z t ϕ ( τ ) d τ > , which implies (5.3). Finally, an application of [BGZ14, Thm. 3.1, Rem. 3.2,Ex. 3.3] shows that the associated norm to the optimal one for the embedding K X coincides with ̺ ( g ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T Ω ϕ ( t, · ) g ( t ) d t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e E ′ (0 ,T ) , g ∈ M + (0 , T ) . (5.5) E = Λ ( v ) Lemma 5.1.
Let the assumptions (2.9) , (2.10) and (4.1) - (4.3) be satisfied with q = 1 . Then the following estimate holds for the norm (5.5) , ̺ ( g ) ≃ e ̺ ( g ) + ̺ ( g ) , g ∈ M + (0 , T ) , (5.6) where e ̺ ( g ) = sup t ∈ (0 ,T ) ( V ( t ) − (cid:18)Z t ϕ ( τ ) d τ (cid:19) Z Tt g ( ξ ) d ξ !) , (5.7) ̺ ( g ) = sup t ∈ (0 ,T ) (cid:26) V ( t ) − (cid:18)Z t Φ k ( ξ, t ) g ( ξ ) d ξ (cid:19)(cid:27) , (5.8) where Φ k ( ξ, t ) = Z ξ ϕ ( τ ) d τ + ξ k/n Z tξ τ − k/n ϕ ( τ ) d τ. (5.9) Proof.
For g ∈ M + (0 , T ) we defineΨ ( g, τ ) = (R T Ω ϕ ( ξ, τ ) g ( ξ ) d ξ, τ ∈ (0 , T ) , , τ ≥ T. (5.10)Then, according to (5.5), ̺ ( g ) = k Ψ ( g ) k e E ′ ( R + ) . (5.11)21n our setting, e E ( R + ) = Λ ( v ) implies e E ′ ( R + ) = M V ( R + ) , (5.12)where M V is the Marcinkiewicz space normed by k f k M V = sup t> V ( t ) − Z t f ∗ ( τ ) d τ , recall (2.6) in Example 2.4. and f ∗ denotes the decreasing rearrangement of f ,as usual. Since 0 ≤ f ( τ ) = Ψ ( g, τ ) is decreasing and right-continuous, hence f ∗ ( τ ) = Ψ ( g, τ ). Thus (5.11) implies ̺ ( g ) = sup t> V ( t ) − Z t Ψ ( g, τ ) d τ, (5.13)which in view of (5.10) leads to ̺ ( g ) = max ( sup t ∈ (0 ,T ) (cid:18) V ( t ) − Z t Ψ ( g, τ ) d τ (cid:19) ; Z T Ψ ( g, τ ) d τ ! sup t ≥ T V ( t ) − ) = sup t ∈ (0 ,T ] (cid:18) V ( t ) − Z t Ψ ( g, τ ) d τ (cid:19) ≃ sup t ∈ (0 ,T ] V ( t ) − Z t Z T Ω ϕ ( ξ, τ ) g ( ξ ) d ξ d τ ! . We substitute (5.4) into the last formula and obtain ̺ ( g ) ≃ e ̺ ( g ) + ̺ ( g ) , where e ̺ ( g ) is determined by (5.7) and ̺ ( g ) = sup t ∈ (0 ,T ] (cid:18) V ( t ) − Z t (cid:18)Z tτ g ( ξ ) d ξ + τ − k/n Z τ ξ k/n g ( ξ ) d ξ (cid:19) ϕ ( τ ) d τ (cid:19) . (5.14)Changing the order of integration, equality (5.14) gives (5.8). Remark . Let the conditions of Lemma 5.1 be satisfied andsup t ∈ (0 ,T ) V ( t ) − Z t ϕ ( τ ) d τ = ∞ . Then g ∈ M + (0 , T ) with ̺ ( g ) < ∞ implies g = 0 a.e. on (0 , T ). This is aconsequence of (5.7). Corollary 5.3.
Let the conditions of Lemma 5.1 be satisfied and the followingestimate be valid for ξ ∈ (0 , T ) , ξ k/n Z Tξ τ − k/n ϕ ( τ ) d τ ≤ d Z ξ ϕ ( τ ) d τ, (5.15)22 or some d ∈ R + not depending on ξ . Then ̺ ( g ) ≃ b ̺ ( g ) = sup t ∈ (0 ,T ] ( V ( t ) − Z t Z Tτ g ( ξ ) d ξ ! ϕ ( τ ) d τ !) . (5.16) Proof.
Note that in this case (5.8) and (5.15) yield ̺ ( g ) ≃ sup t ∈ (0 ,T ] ( V ( t ) − Z t Z ξ ϕ ( τ ) d τ ! g ( ξ ) d ξ ) = sup t ∈ (0 ,T ] (cid:26) V ( t ) − Z t (cid:18)Z tτ g ( ξ ) d ξ (cid:19) ϕ ( τ ) d τ (cid:27) . (5.17)We insert this identity in (5.6), take into account (5.7) and arrive at (5.16). Remark . If k > n , then the estimate (5.15) is valid for every positive function ϕ which decreases on (0 , T ). Indeed, in this case ξ k/n Z Tξ τ − k/n ϕ ( τ ) d τ ≤ ξ k/n ϕ ( ξ ) Z ∞ ξ τ − k/n d τ = ξϕ ( ξ ) (cid:18) kn − (cid:19) − ≤ Z ξ ϕ ( τ ) d τ ! (cid:18) kn − (cid:19) − . Corollary 5.5.
If the conditions of Lemma 5.1 are satisfied, and the followingestimate takes place for ξ ∈ (0 , T ) , Z ξ τ − k/n ϕ ( τ ) d τ ≤ d ξ − k/n ϕ ( ξ ) , (5.18) with d ∈ R + not depending on ξ , then ̺ ( g ) ≃ e ̺ ( g ) + b ̺ ( g ) , g ∈ M + (0 , T ) , (5.19) where e ̺ is given by (5.7) and b ̺ ( g ) = sup t ∈ (0 ,T ] (cid:18) V ( t ) − t − k/n ϕ ( t ) Z t ξ k/n g ( ξ ) d ξ (cid:19) = sup t ∈ (0 ,T ] (cid:18) U ( t ) Z t ξ k/n g ( ξ ) d ξ (cid:19) , (5.20) and U is defined by (4.11) .Proof. In this case, by (5.18), Z ξ ϕ ( τ ) d τ ≤ ξ k/n Z ξ τ − k/n ϕ ( τ ) d τ ≃ ξϕ ( ξ ) ≤ Z ξ ϕ ( τ ) d τ,
23o that ξϕ ( ξ ) ≃ Z ξ ϕ ( τ ) d τ ≃ ξ k/n Z ξ τ − k/n ϕ ( τ ) d τ, (5.21)and Z ξ ϕ ( τ ) d τ + ξ k/n Z tξ τ − k/n ϕ ( τ ) d τ ≃ ξ k/n Z t τ − k/n ϕ ( τ ) d τ ≃ ξ k/n t − k/n ϕ ( t ) . (5.22)From here, and from (5.8)-(5.9) it follows that ̺ ( g ) ≃ sup t ∈ (0 ,T ] (cid:18) V ( t ) − t − k/n ϕ ( t ) (cid:18)Z t ξ k/n g ( ξ ) d ξ (cid:19)(cid:19) = b ̺ ( g ) . (5.23)Therefore, (5.6) implies (5.19). The second equality in (5.20) with U from(4.11) is a well-known consequence of the fact that0 ≤ σ k ( t ) = Z t ξ k/n g ( ξ ) d ξ (5.24)increases in t ∈ (0 , T ]. Remark . For a positive decreasing function ϕ the estimate (5.18) is possibleonly when k < n . Otherwise the integral diverges. For such a function ϕ theinverse estimate is evident, such that (5.18) implies Z ξ τ − k/n ϕ ( τ ) d τ ≃ ξ − k/n ϕ ( ξ ) . (5.25) Lemma 5.7.
Let the assumptions of Lemma 5.1 be satisfied and Z t V ( ξ ) ξ − d ξ ≤ cV ( t ) , t ∈ (0 , T ) , (5.26) for some c > independent of t . Then b ̺ ( g ) ≃ e ̺ ( g ) , (5.27) using the notation (5.16) and (5.7) .Proof. For every t ∈ (0 , T ] we havesup ξ ∈ (0 ,t ] (cid:16) V ( ξ ) − Z tξ g ( s ) d s Z ξ ϕ ( τ ) d τ (cid:17) ≤ sup ξ ∈ (0 ,T ] (cid:16) V ( ξ ) − Z Tξ g ( s ) d s Z ξ ϕ ( τ ) d τ (cid:17) = e ̺ ( g ) . Therefore, Z tξ g ( s ) d s Z ξ ϕ ( τ ) d τ ≤ e ̺ ( g ) V ( ξ ) , ξ ∈ (0 , t ] , t ∈ (0 , T ] . Z t ϕ ( ξ ) (cid:18)Z tξ g ( s ) d s (cid:19) d ξ ≤ Z t Z ξ ϕ ( τ ) d τ ! (cid:18)Z tξ g ( s ) d s (cid:19) ξ − d ξ ≤ e ̺ ( g ) Z t V ( ξ ) ξ − d ξ, and, according to (5.26),sup t ∈ (0 ,T ] V ( t ) − Z t ϕ ( ξ ) (cid:18)Z tξ g ( s ) d s (cid:19) d ξ ≤ c e ̺ ( g ) . Consequently, by (5.16), b ̺ ( g ) = sup t ∈ (0 ,T ] V ( t ) − Z t ϕ ( ξ ) Z Tξ g ( s ) d s ! d ξ ≤ sup t ∈ (0 ,T ] V ( t ) − Z t ϕ ( ξ ) (cid:18)Z tξ g ( s ) d s (cid:19) d ξ + sup t ∈ (0 ,T ] V ( t ) − (cid:18)Z t ϕ ( ξ ) d ξ (cid:19) Z Tt g ( s ) d s ! ≤ ( c + 1) e ̺ ( g ) . Since e ̺ ( g ) ≤ b ̺ ( g ), we obtain (5.27). Corollary 5.8.
Let the assumptions of Lemma 5.1 be satisfied and the estimates (5.15) and (5.26) be valid. Then ̺ ( g ) ≃ e ̺ ( g ) , g ∈ M + (0 , T ) , (5.28) with constants not depending on g , recall (5.5) and (5.7) .Proof. Plainly (5.16) and (5.27) imply (5.28).
Lemma 5.9.
Let the assumptions of Lemma 5.1 be satisfied, the estimate (5.18) be valid, and assume that for some ε > the function U ( t ) t ε is decreasing on (0 , T ] , recall (4.10) - (4.11) . Then the estimate (5.28) holds for the norms (5.5) and (5.7) .Proof. Without loss of generality we assume that U ( T ) = 1 and introduce adiscretizing sequence { ν m } m ∈ N by ν m = sup { t ∈ (0 , T ] : U ( t ) = 2 m } , m ∈ N . (5.29)Note that U is a positive and decreasing function on (0 , T ], with U (0+) = ∞ ,such that ν m is well-defined, m ∈ N , and ν = T, < ν m +1 < ν m , m ∈ N , lim m →∞ ν m = 0 . (5.30)25y assumption, U ( t ) t ε decreases which leads to ν m +1 < ν m ≤ /ε ν m +1 such that ν m +1 ≃ ν m , m ∈ N , (5.31)for fixed ε . For convenience we use the notation∆ m = ( ν m +1 , ν m ] , m ∈ N . (5.32)The discretized version of (5.20) then yields b ̺ ( g ) = sup m ∈ N sup t ∈ ∆ m U ( t ) Z t ξ k/n g ( ξ ) d ξ ≃ sup m ∈ N m sup t ∈ ∆ m Z t ξ k/n g ( ξ ) d ξ = sup m ∈ N m Z ν m ξ k/n g ( ξ ) d ξ = sup m ∈ N m X j ≥ m Z ∆ j ξ k/n g ( ξ ) d ξ. (5.33)Here we used the assertion U ( t ) ≃ m , t ∈ ∆ m , m ∈ N . (5.34)Now we apply some well-known estimate for non-negative sequences { α m } m ∈ N and positive sequences { β m } m ∈ N which satisfy, in addition, β m +1 /β m ≥ B > B . Then for 0 < p ≤ ∞ and 1 ≤ r ≤ ∞ , X m ∈ N β m X j ≥ m α rj r p p ≤ c ( B, p ) X m ∈ N ( β m α m ) p ! p , (5.35)where c ( B, p ) is a positive constant (and the usual modification for p = ∞ or r = ∞ ). Since the inequality inverse to (5.35) is valid (with c = 1), the estimate(5.35) is in fact an equivalence. Now we use (5.35) with β m = 2 m , r = 1 , p = ∞ , α j = Z ∆ j ξ k/n g ( ξ ) d ξ, insert it in (5.33) and conclude b ̺ ( g ) ≤ c sup m ∈ N m Z ∆ m ξ k/n g ( ξ ) d ξ ≤ c sup m ∈ N m ν k/nm Z ∆ m g ( ξ ) d ξ, (5.36)where we used in the latter estimate that ξ ≃ ν m for ξ ∈ ∆ m , m ∈ N . Itremains to estimate e ̺ ( g ) given by (5.7) from below. By similar discretization26rguments as above we observe that e ̺ ( g ) ≥ sup m ∈ N sup t ∈ ∆ m V ( t ) − tϕ ( t ) Z Tt g ( ξ ) d ξ ≥ sup m ∈ N Z ∆ m − g ( ξ ) d ξ ! sup t ∈ ∆ m V ( t ) − tϕ ( t ) ≃ sup m ∈ N ν k/nm − Z ∆ m − g ( ξ ) d ξ sup t ∈ ∆ m V ( t ) − t − k/n ϕ ( t ) , where we used that ϕ decreases and the obvious estimate Z Tt g ( ξ ) d ξ ≥ Z Tν m g ( ξ ) d ξ ≥ Z ∆ m − g ( ξ ) d ξ, t ∈ ∆ m , m ∈ N . Thus (5.29) implies2 m +1 = U ( ν m +1 ) = max (cid:26) U ( ν m ) , sup t ∈ ∆ m V ( t ) − t − k/n ϕ ( t ) (cid:27) = sup t ∈ ∆ m V ( t ) − t − k/n ϕ ( t ) . Finally this leads to e ̺ ( g ) ≥ c sup m ∈ N ν k/nm − m +1 Z ∆ m − g ( ξ ) d ξ = 4 c sup m ∈ N ν k/nm m Z ∆ m g ( ξ ) d ξ. Together with (5.36) this results in e ̺ ( g ) ≥ c b ̺ ( g ) , g ∈ M + (0 , T ) , where c > g . In view of (5.19) this yields (5.28) asdesired. Remark . In Lemma 5.9 we considered the alternative situation to Corol-lary 5.8, where the estimate (5.15) is replaced by (5.18). In this more delicatecase a more flexible method of discretization was needed for the proof. E = Λ q ( v ) , < q < ∞ Now we deal with Lorentz spaces Λ q ( v ) for 1 < q < ∞ , recall Example 2.4and (4.1). Our main aim is to obtain some counterparts of Corollary 5.8 andLemma 5.9 from the preceding section, but now corresponding to the case q > Lemma 5.11.
Let the assumptions (2.9) , (2.10) , and (4.1) - (4.3) be satisfiedwith < q < ∞ . Then the following estimate holds for the norm (5.5) , ̺ ( g ) ≃ e ̺ ( g ) + ̺ ( g ) + ̺ ( g ) , g ∈ M + (0 , T ) , (5.37)27 here e ̺ ( g ) = Z T (cid:18)Z t ϕ ( τ ) d τ (cid:19) Z Tt g ( s ) d s !! q ′ w ( t ) d t /q ′ , (5.38) ̺ ( g ) = Z T (cid:18)Z t Φ k ( ξ, t ) g ( ξ ) d ξ (cid:19) q ′ w ( t ) d t ! /q ′ , (5.39) ̺ ( g ) = Z T Φ k ( ξ, T ) g ( ξ ) d ξ ! (cid:18)Z ∞ T w ( t ) d t (cid:19) /q ′ , (5.40) where Φ k is defined by (5.9) and w by (4.8) .Proof. Note that the line of arguments is similar to those strengthened in theproof of Lemma 5.1. Recall that for an RIS e E = Λ q ( v ) the associated RIS isthe space e E ′ = Γ q ′ ( w ) with the norm k f k e E ′ ( R + ) = Z ∞ (cid:18)Z t f ∗ ( τ ) d τ (cid:19) q ′ w ( t ) d t ! /q ′ . Since in our case f ( τ ) = Ψ ( g, τ ) = f ∗ ( τ ), see (5.10), thus ̺ ( g ) = k Ψ ( g ) k e E ′ ( R + ) = Z ∞ (cid:18)Z t Ψ ( g, τ ) d τ (cid:19) q ′ w ( t ) d τ ! /q ′ . (5.41)Substitution of (5.10) into (5.41) gives ̺ ( g ) ≃ b ̺ ( g ) + ̺ ( g ) , (5.42)where b ̺ ( g ) = Z T (cid:18)Z t Ψ ( g, τ ) d τ (cid:19) q ′ w ( t ) d t ! /q ′ , (5.43) ̺ ( g ) = Z T Ψ ( g, τ ) d τ ! (cid:18)Z ∞ T w ( t ) d t (cid:19) /q ′ . (5.44)Now we introduce some function G k for t ∈ (0 , T ], and τ ∈ (0 , t ], G k ( t, τ ) = Z tτ g ( ξ ) d ξ + τ − k/n Z τ ξ k/n g ( ξ ) d ξ. (5.45)Then (5.4) and (5.10) imply that Z t Ψ ( g, τ ) d τ = Z t G k ( T, τ ) ϕ ( τ ) d τ = Z t ϕ ( τ ) d τ Z Tt g ( ξ ) d ξ + Z t G k ( t, τ ) ϕ ( τ ) d τ. (5.46)28fter some change of the order of integration we obtain Z t G k ( t, τ ) ϕ ( τ ) d τ = Z t Φ k ( ξ, t ) g ( ξ ) d ξ, (5.47)recall (5.45) and (5.9). As a special case we get Z T Ψ ( g, τ ) d τ = Z T G k ( T, τ ) ϕ ( τ ) d τ = Z T Φ k ( ξ, T ) g ( ξ ) d ξ, (5.48)which means the coincidence of (5.44) and (5.40). Moreover, substituting (5.46)into (5.43), we arrive at b ̺ ( g ) ≃ e ̺ ( g ) + ̺ ( g ) , g ∈ M + (0 , T ) , (5.49)where e ̺ ( g ) is given by (5.38) and ̺ ( g ) = Z T (cid:18)Z t G k ( t, τ ) ϕ ( τ ) d τ (cid:19) q ′ w ( t ) d t ! /q ′ . (5.50)Now in view of (5.47), equality (5.39) coincides with (5.50), and (5.42) and(5.49) imply (5.37). Lemma 5.12.
Let < q < ∞ , v > be a measurable function on (0 , T ) , where T ∈ (0 , ∞ ] , and V and w be given on (0 , T ) by (4.2) and (4.8) , respectively.Let ε > , δ ∈ [0 , ε/q ) , and V ( t ) t − ε increasing on (0 , T ) . (5.51) Then, for t ∈ (0 , T ) , Z T t w ( τ ) τ δq ′ d τ ! /q ′ ≤ (cid:18) εqq ′ ( ε − δq ) (cid:19) /q ′ t δ V ( t ) − /q , (5.52) (cid:18)Z t w ( τ ) − q/q ′ τ − ( δ +1) q d τ (cid:19) /q ≤ t − δ V ( t ) /q ε /q ′ ( ε − δq ) /q . (5.53) Proof.
We start proving (5.52). Here we apply (4.8) and (5.51) to conclude Z T t w ( τ ) τ δq ′ d τ ≤ (cid:0) t ε V ( t ) − (cid:1) δq ′ /ε Z T t V ( τ ) q ′ ( δ/ε − v ( τ ) d τ = t δq ′ V ( t ) − δq ′ /ε ( q ′ ( δ/ε −
1) + 1) − (cid:16) V ( τ ) q ′ ( δ/ε − (cid:17) (cid:12)(cid:12)(cid:12) T τ = t ≤ t δq ′ (cid:18) q ′ (cid:18) q − δε (cid:19)(cid:19) − V ( t ) − q ′ , which implies (5.52). 29t remains to verify (5.53). Property (5.51) yields that v ( τ ) = V ′ ( τ ) ≥ ετ − V ( τ ) , τ ∈ (0 , T ) . Therefore, Z t w ( τ ) − q/q ′ τ − ( δ +1) q d τ = Z t V ( τ ) q v ( τ ) − q/q ′ τ − ( δ +1) q d τ ≤ ε − q/q ′ Z t V ( τ ) τ − δq − d τ. Now (5.51) implies that Z t V ( τ ) τ − δq − d τ ≤ V ( t ) t − ε Z t τ ε − δq − d τ = V ( t ) t − δq ( ε − δq ) − . These estimates conclude the proof.We formulate an immediate consequence of the estimates (5.52) and (5.53).
Corollary 5.13.
Let the assumptions of Lemma 5.12 be satisfied. Then (cid:18)Z t w ( τ ) − q/q ′ τ − ( δ +1) q d τ (cid:19) /q Z T t w ( τ ) τ δq ′ d τ ! /q ′ ≤ ( q/q ′ ) /q ′ ε − δq . (5.54) Lemma 5.14.
Let the assumptions of Lemma 5.11 be satisfied, and assume theconditions (5.15) and (5.51) to hold. Then there is the equivalence ̺ ( g ) ≃ e ̺ ( g ) , g ∈ M + (0 , T ) , (5.55) where the norm ̺ is given by (5.5) and e ̺ ( g ) by (5.38) .Proof. Step 1 . Lemma 5.11 implies the estimates (5.37)–(5.40). Next we apply(5.15) to Φ k ( ξ, t ), given by (5.9), thatΦ k ( ξ, t ) ≃ Z ξ ϕ ( τ ) d τ, t ∈ (0 , T ] , and thus ̺ ( g ) = Z T Z t Z ξ ϕ ( τ ) d τ ! g ( ξ ) d ξ ! q ′ w ( t ) d t /q ′ = Z T (cid:18)Z t ϕ ( τ ) (cid:18)Z tτ g ( ξ ) d ξ (cid:19) d τ (cid:19) q ′ w ( t ) d t ! /q ′ . ̺ ( g ) ≤ c Z T Z t ϕ ( τ ) Z Tτ g ( ξ ) d ξ ! d τ ! q ′ w ( t ) d t /q ′ . (5.56)Similarly, ̺ ( g ) ≃ Z T ϕ ( τ ) Z Tτ g ( ξ ) d ξ ! d τ ! (cid:18)Z ∞ T w ( t ) d t (cid:19) /q ′ . (5.57)Therefore it is enough to prove that ̺ ( g ) ≤ c e ̺ ( g ) , (5.58) ̺ ( g ) ≤ c e ̺ ( g ) , (5.59)where c , c are positive constants independent of g ∈ M + (0 , T ). Step 2 . We verify (5.58). We apply Hardy’s inequality [Maz85, Thm. 2,p. 41] in adapted notation, that is, (cid:16) Z T Z t ϕ ( τ ) Z Tτ g ( ξ ) d ξ ! d τ ! q ′ w ( t ) d t (cid:17) /q ′ ≤ c Z T tϕ ( t ) Z Tt g ( ξ ) d ξ ! q ′ w ( t ) d t /q ′ (5.60)if, and only if, B = sup t ∈ (0 ,T ) Z Tt w ( τ ) d τ ! /q ′ (cid:18)Z t w ( τ ) − q/q ′ τ − q d τ (cid:19) /q < ∞ . (5.61)Moreover, for the best possible constant c in (5.60) we have B ≤ c ≤ B (cid:18) q ′ q ′ − (cid:19) q ′− q ′ ( q ′ ) q ′ = B q q ( q ′ ) q ′ . (5.62)Corollary 5.13 with δ = 0, T = T , implies that B ≤ ε − ( q ′ − − /q ′ , (5.63)and consequently c ≤ qε . (5.64)Now (5.58) follows from (5.56), (5.60), (5.38) and from the obvious estimate tϕ ( t ) ≤ R t ϕ ( τ ) d τ due to the monotonicity of ϕ .31 tep 3 . It remains to verify (5.59) which is much simpler. By H¨older’sinequality we get from (5.57) that ̺ ( g ) ≤ c Z T τ ϕ ( τ ) Z Tτ g ( ξ ) d ξ ! q ′ w ( τ ) d τ /q ′ ×× Z T τ − q w ( τ ) − q/q ′ d τ ! /q (cid:18)Z ∞ T w ( t ) d t (cid:19) /q ′ ≤ c e ̺ ( g ) Z T τ − q w ( τ ) − q/q ′ d τ ! /q (cid:18)Z ∞ T w ( t ) d t (cid:19) /q ′ . Now application of (5.54) with T = ∞ , δ = 0, t = T gives ̺ ( g ) ≤ c ε (cid:18) qq ′ (cid:19) /q ′ e ̺ ( g ) . Thus we have finally shown (5.58) and (5.59) which together with (5.37) imply(5.55).The last preparatory lemma we need is the following.
Lemma 5.15.
Let the assumption of Lemma 5.11 be satisfied. If the estimate (5.18) holds, and for the function U q , given by (4.11) for q > , there is some ε > such that t ε U q ( t ) decreases on (0 , T ) , (5.65) then the assertion (5.55) holds with e ̺ ( g ) given by (5.38) . All the constantsappearing in (5.55) are positive, finite, and independent of g ∈ M + (0 , T ) .Proof. We strengthen a similar line of arguments like in the proof of Lemma 5.9and use the method of discretization again.
Step 1.
According to (5.9) and (5.21)-(5.22) we haveΦ k ( ξ, t ) ≃ ξ k/n t − k/n ϕ ( t ) . (5.66)Hence (5.37)-(5.40) read as ̺ ( g ) ≃ Z T (cid:18)(cid:18)Z t ξ k/n g ( ξ ) d ξ (cid:19) t − k/n ϕ ( t ) (cid:19) q ′ w ( t ) d t ! /q ′ (5.67) ̺ ( g ) ≃ e ̺ ( g ) = Z T ξ k/n g ( ξ ) d ξ ! T − k/n ϕ ( T ) (cid:18)Z ∞ T w ( t ) d t (cid:19) /q ′ . (5.68)32 tep 2. First we estimate ̺ ( g ). We substitute formulas (4.8) and (4.10)into (5.67) and obtain ̺ ( g ) ≃ e ̺ ( g ) = Z T (cid:18)Z t ξ k/n g ( ξ ) d ξ (cid:19) q ′ f W ( t ) q ′ v ( t ) d t ! /q ′ . (5.69)Note that U q ( T ) = 0, U q (0+) = ∞ , and U q ( t ) t ε is monotonically decreasing.We introduce the discretizing sequence { δ m } m ∈ Z by δ m = sup { τ ∈ (0 , T ) : U q ( τ ) = 2 m } , m ∈ Z . (5.70)Thus we observe δ ∈ (0 , T ), δ m → m → + ∞ , δ m → T for m → −∞ , and δ m +1 < δ m ≤ δ m +1 /ε , m ∈ Z . (5.71)We use the notation e ∆ m = ( δ m +1 , δ m ] , m ∈ Z . (5.72)Therefore, e ̺ ( g ) q ′ = X m ∈ Z Z e ∆ m (cid:18)Z t ξ k/n g ( ξ ) d ξ (cid:19) q ′ f W ( t ) q ′ v ( t ) d t. In view of (5.70) this can be continued by e ̺ ( g ) q ′ ≤ X m ∈ Z Z δ m ξ k/n g ( ξ ) d ξ ! q ′ Z e ∆ m f W ( t ) q ′ v ( t ) d t = X m ∈ Z Z δ m ξ k/n g ( ξ ) d ξ ! q ′ (cid:16) U q ( δ m +1 ) q ′ − U q ( δ m ) q ′ (cid:17) = (2 q ′ − X m ∈ Z mq ′ X j ≥ m Z e ∆ j ξ k/n g ( ξ ) d ξ q ′ . Now we apply in appropriately adapted notation estimate (5.35) again andobtain e ̺ ( g ) ≤ c ( q ′ ) X m ∈ Z mq ′ (cid:18)Z e ∆ m ξ k/n g ( ξ ) d ξ (cid:19) q ′ ! /q ′ . Using (5.71) and (5.72) we observe ξ ≃ δ m , ξ ∈ e ∆ m , m ∈ Z , such that finally e ̺ ( g ) ≤ c ( q, ε ) X m ∈ Z mq ′ δ kq ′ /nm (cid:18)Z e ∆ m g ( ξ ) d ξ (cid:19) q ′ ! /q ′ . (5.73)33 tep 3. We deal with ̺ ( g ) and (5.68). We apply (5.52) with δ = 0 andobtain e ̺ ( g ) ≤ c ( q ) Z T ξ k/n g ( ξ ) d ξ ! T − k/n ϕ ( T ) V ( T ) − /q . (5.74)Moreover, Z δ ξ k/n g ( ξ ) d ξ = X m ∈ N Z e ∆ m ξ k/n g ( ξ ) d ξ ≃ X m ∈ N δ k/nm Z e ∆ m g ( ξ ) d ξ, such that H¨older’s inequality leads to Z δ ξ k/n g ( ξ ) d ξ ≤ c X m ∈ N − mq ! q X m ∈ N mq ′ δ q ′ k/nm (cid:18)Z e ∆ m g ( ξ ) d ξ (cid:19) q ′ ! q ′ , such that Z T ξ k/n g ( ξ ) d ξ ≤ c ( q, ε ) X m ∈ N mq ′ δ q ′ k/nm (cid:18)Z e ∆ m g ( ξ ) d ξ (cid:19) q ′ ! q ′ + T k/n Z Tδ g ( ξ ) d ξ . Together with (5.73), (5.74) this leads to e ̺ ( g )+ e ̺ ( g ) ≤ c ( q, ε, T ) X m ∈ N mq ′ δ q ′ k/nm (cid:18)Z e ∆ m g ( ξ ) d ξ (cid:19) q ′ ! q ′ + Z Tδ g ( ξ ) d ξ . (5.75) Step 4.
We estimate e ̺ ( g ) in (5.38) from below. First of all, e ̺ ( g ) ≥ Z δ (cid:18)Z t ϕ ( τ ) d τ (cid:19) Z Tt g ( ξ ) d ξ !! q ′ w ( t ) d t q ′ ≥ Z Tδ g ( ξ ) d ξ ! Z δ ( tϕ ( t )) q ′ w ( t ) d t ! q ′ , hence Z Tδ g ( ξ ) d ξ ≤ c ( δ , q ) e ̺ ( g ) , g ∈ M + (0 , T ) . (5.76)34urthermore, according to (5.70)-(5.71), e ̺ ( g ) q ′ = X m ∈ Z Z e ∆ m (cid:18)Z t ϕ ( τ ) d τ (cid:19) Z Tt g ( ξ ) d ξ !! q ′ w ( t ) d t ≥ X m ∈ Z Z e ∆ m − g ( ξ ) d ξ ! q ′ Z e ∆ m ( tϕ ( t )) q ′ w ( t ) d t ≃ X m ∈ Z Z e ∆ m − g ( ξ ) d ξ ! q ′ δ kq ′ /nm − Z e ∆ m f W ( t ) q ′ v ( t ) d t. (5.77)For t ∈ e ∆ m we have t ≤ δ m < δ m − < T , such that R Tt g ( ξ ) d ξ ≥ R e ∆ m − g ( ξ ) d ξ .In addition, we know that t ≃ t − /k/n δ k/nm − and R t ϕ ( τ ) d τ ≥ tϕ ( t ), recall alsonotation (4.11) and (4.8).Now by similar arguments as presented in the proof above following (5.72)we obtain Z e ∆ m f W ( t ) q ′ v ( t ) d t = (2 q ′ − mq ′ = 2 q ′ (2 q ′ − ( m − q ′ . Substituting this into (5.77) and an index shift lead to e ̺ ( g ) ≥ c ( ε, q ) X m ∈ Z mq ′ δ kq ′ /nm (cid:18)Z e ∆ m g ( ξ ) d ξ (cid:19) q ′ ! /q ′ . (5.78)Thus the estimates (5.75), (5.76) and (5.78) result in e ̺ ( g ) + e ̺ ( g ) ≤ c ( ε, δ , q, T ) e ̺ ( g ) , g ∈ M + (0 , T ) . Then the last estimate, together with (5.69), (5.68) and (5.37) yields (5.55).Recall that our aim is to prove Theorem 4.3 above. For that reason wesummarize our preceding results in the following theorem.
Theorem 5.16.
Let the conditions of Theorem, 4.3 be satisfied. Then theassociated GBFS X ′ = X ′ (0 , T ) to the optimal space X = X (0 , T ) is generatedby the function norm ̺ such that ̺ ( g ) ≃ e ̺ ( g ) , g ∈ M + (0 , T ) , (5.79) where for q = 1 , e ̺ ( g ) = sup t ∈ (0 ,T ] V ( t ) − (cid:18)Z t ϕ ( τ ) d τ (cid:19) Z Tt g ( s ) d s !! , (5.80)35 nd for < q < ∞ , e ̺ ( g ) = Z T (cid:18)Z t ϕ ( τ ) d τ (cid:19) Z Tt g ( s ) d s !! q ′ w ( t ) d t /q ′ , (5.81) where q + q ′ = 1 , as usual. Here again V ( t ) = Z t v ( τ ) d τ, w ( t ) = V ( t ) − q ′ v ( t ) . (5.82) Proof.
First we assume that condition (A) (see (4.12) and (4.13)) is satisfied.Then for q = 1 we can apply Corollary 5.8, and find that (5.15) coincides with(4.12), and (4.13) implies (5.26) (as in the last estimate before Lemma 5.14 with δ = 0). Thus (5.79) is just (5.28) in this case. If 1 < q < ∞ , we receive (5.55)by Lemma 5.14.Secondly we consider the situation when (B) holds, that is, (4.14) and (4.15).For q = 1 we can apply Lemma 5.9, since (5.18) coincides with (4.14) and (4.15)provides the required property of U . This yields (5.28) which coincides with(5.79). Finally, if 1 < q < ∞ , then (5.79) is a consequence of Lemma 5.15. Recall that we already stated in Section 4 above one of our main results, The-orem 4.3. Now we are ready to present its proof.
Proof of Theorem 4.3.
Theorem 5.16 above shows that (under the given as-sumptions) the associated norm is optimal. So what is left to verify are theexplicit representations for the optimal norm k · k X in (4.21) and (4.22), re-spectively, with (4.20). This norm is associated to the norm e ̺ presented in(5.80), (5.81). We benefit from the paper [BG14] and an application of [BG14,Thm. 1.2] (in appropriately adapted notation) concludes the argument.The combination of Theorem 3.5 and 4.3 now yields the following result. Theorem 6.1.
Let the assumptions of Theorems 2.6 and 4.3 be satisfied. Let q = 1 and Ψ (0+) = 0 or < q < ∞ . Then the optimal Calder´on space for theembedding (3.7) has the following norm k u k Λ k ( C ; X ) = k u k C + Z T (cid:18) ω k ( u ; t /n )Ψ q ( t ) (cid:19) q dΨ q ( t )Ψ q ( t ) ! /q . (6.1) Proof.
Theorem 4.3 states that the GBFS X = X (0 , T ) is optimal for theembedding K X , where K is the cone described by (2.23) with (2.22). Then36orollary 3.6 shows that the corresponding Calder´on space Λ k ( C ; X ) is optimalfor the embedding (3.7). Thus k u k Λ k ( C ; X ) = k u k C + (cid:13)(cid:13)(cid:13) ω k ( u ; τ /n ) (cid:13)(cid:13)(cid:13) X (0 ,T ) . (6.2)We substitute (4.22) into (6.2) and arrive at k u k Λ k ( C ; X ) ≃k u k C + Ψ q ( T ) − (cid:13)(cid:13)(cid:13) ω k ( u ; τ /n ) (cid:13)(cid:13)(cid:13) L ∞ ( T ,T ) + Z T (cid:13)(cid:13) ω k ( u ; τ /n ) (cid:13)(cid:13) L ∞ (0 ,t ) Ψ q ( t ) ! q dΨ q ( t )Ψ q ( t ) ! q . (6.3)But ω k ( u ; τ /n ) increases with respect to τ , hence (cid:13)(cid:13)(cid:13) ω k ( u ; τ /n (cid:13)(cid:13)(cid:13) L ∞ ( T ,T ) ≤ ω k ( u ; T /n ) ≤ k k u k C , (cid:13)(cid:13)(cid:13) ω k ( u ; τ /n ) (cid:13)(cid:13)(cid:13) L ∞ (0 ,t ) ≤ ω k ( u ; t /n ) , t ∈ (0 , T ) . Together with (6.3) this implies (6.1).
Remark . In the case q = 1 and Ψ (0+) >
0, the embedding (3.7) takes place‘on the limit of the smoothness’, and we obtain Λ k ( C ; X ) = C ( R n ). Accordingto the results in [GH13a] in this case there exist functions u ∈ H GE ( R n ) suchthat ω k ( u ; t /n ) → t →
0+ arbitrarily slowly. Note that in this case X (0 , T ) = L ∞ (0 , T ) by (4.21), such that the above norm k u k Λ k ( C ; X ) = k u k C + (cid:13)(cid:13)(cid:13) ω k ( u ; τ /n ) (cid:13)(cid:13)(cid:13) L ∞ (0 ,T ) ≃ k u k C . In that case we cannot say anything else than u ∈ C ( R n ). Remark . We return to Examples 2.9, 2.10. If α = k , then Theorems 4.3,6.1 can be applied. In the limiting case α = k some special care is needed.This follows from estimate (2.35) in Example 2.9. In Example 2.10 we have theequality (2.39) when λ is slowly varying on (0 , T ]. So in some sense (2.22) canbe understood as a special case of (2.39) with λ ≡
1. Thus Remark 4.2 appliesand implies that we can use Theorems 4.3, 6.1 in case of α = k .Before we can state our next main result, Theorem 6.8 below, which alsocovers the delicate limiting case α = k , we need some further preparation. Lemma 6.4.
Let λ > be a continuous function on (0 , T ] such that for some δ ∈ (0 , the function λ ( t ) t − δ decreases. Then λ ( s ) + Z ts λ ( τ ) τ − d τ ≤ λ ( s ) δ (cid:18) ts (cid:19) δ , s ∈ (0 , t ] , t ∈ (0 , T ] . roof. We use the assumed monotonicity and argue as follows, Z ts λ ( τ ) τ − d τ ≤ λ ( s ) s − δ Z ts τ δ − d τ = λ ( s ) s − δ δ (cid:0) t δ − s δ (cid:1) . Hence λ ( s ) + Z ts λ ( τ ) τ − d τ ≤ λ ( s ) δ (cid:18) ts (cid:19) δ . Corollary 6.5.
Let ϕ be given by (4.16) with α = k , and λ > be slowlyvarying on (0 , T ] . Then for every δ ∈ (0 , the function Φ k given by (5.9) canbe estimated by < Φ k ( ξ, t ) ≤ c t δ ξ k/n − δ λ ( ξ ) , ξ ∈ (0 , t ] , t ∈ (0 , T ] , (6.4) where c = c ( δ, k, n ) > .Proof. If ϕ is given by (4.16) with α = k , then Z ξ ϕ ( τ ) d τ = Z ξ τ k/n − λ ( τ ) d τ ≃ ξ k/n λ ( ξ ) , (6.5)recall (2.42). HenceΦ k ( ξ, t ) = Z ξ ϕ ( τ ) d τ + ξ k/n Z tξ τ − k/n ϕ ( τ ) d τ ≃ ξ k/n (cid:18) λ ( ξ ) + Z tξ λ ( τ ) τ − d τ (cid:19) . Application of Lemma 6.4 leads to (6.4).
Corollary 6.6.
Let the assumptions of Corollary 6.5 be satisfied, and let ̺ be given by (5.8) - (5.9) . Then for any δ ∈ (0 , there is some positive constant c = c ( δ, k, n ) , such that ̺ ( g ) ≤ c sup t ∈ (0 ,T ] (cid:18) V ( t ) − t δ Z t s kn − δ λ ( s ) g ( s ) d s (cid:19) , g ∈ M + (0 , T ) . (6.6) Proof.
This follows immediately by substituting (6.4) into (5.9).
Corollary 6.7.
Let the assumptions of Corollary 6.5 be satisfied, and let ̺ begiven by (5.8) - (5.9) . Then for any δ ∈ (0 , min { , k/n } ) there is some positiveconstant c = c ( δ, k, n ) , such that ̺ ( g ) ≤ c sup t ∈ (0 ,T ] (cid:18) V ( t ) − t δ Z t τ kn − δ − λ ( τ ) (cid:18)Z tτ g ( s ) d s (cid:19) d τ (cid:19) , g ∈ M + (0 , T ) . (6.7)38 roof. Recall that when λ > < δ < k/n that s kn − δ λ ( s ) ≃ Z s τ kn − δ. − λ ( τ ) d τ, s ∈ (0 , T ] . (6.8)Substituting this into (6.6) and changing the order of integration we arrive at(6.7).Now we come to our next essential result. Theorem 6.8.
Let ϕ be determined by (4.16) . Assume that α ≤ k and (4.13) is satisfied, or α > k and (4.15) is satisfied with f W ( t ) = V ( t ) − t α − kn λ ( t ) . Then the formulas (4.20) - (4.22) , (6.1) hold with Ψ q , given by (4.5) , where W ( t ) = V ( t ) − t α/n λ ( t ) , t ∈ (0 , T ] . (6.9) Proof.
Note first that it is left to consider the case α = k only since the othercases are already covered by Theorem 4.3, in particular using condition (A) if α < k , and condition (B) if α > k . So we may assume in the following that α = k . Step 1 . First we deal with the case q = 1. For ϕ given by (4.16) we have(6.5). Thus e ̺ given by (5.7) can be estimated by e ̺ ( g ) ≃ sup τ ∈ (0 ,T ] V ( τ ) − τ k/n λ ( τ ) Z Tτ g ( s ) d s !! , g ∈ M + (0 , T ) , such that τ k/n λ ( τ ) Z tτ g ( s ) d s ≤ c e ̺ ( g ) V ( τ ) , τ ∈ (0 , t ] , t ∈ (0 , T ] . In view of this we can continue (6.7) here by ̺ ( g ) ≤ c sup t ∈ (0 ,T ] (cid:18) V ( t ) − t δ (cid:18)Z t τ − δ − V ( τ ) d τ (cid:19)(cid:19) e ̺ ( g ) , for any δ ∈ (0 , min { , k/n } ). Recall that V ( τ ) τ − ε is monotonically increasingby (4.13). Assume that δ < ε . Then Z t τ − δ − V ( τ ) d τ ≤ V ( t ) t − ε Z t τ ε − δ − d τ = V ( t ) t − δ ( ε − δ ) − . Consequently, ̺ ( g ) ≤ c e ̺ ( g ) , g ∈ M + (0 , T ) , c = c ( δ, ε, k, n ) >
0. Together with (5.6) this shows that (5.55) is validfor the norm ̺ given by (5.5) which is associated to the optimal one. In otherwords, we can apply the results given in (4.20)-(4.22) as explained in the proofof Theorem 4.3. Step 2 . Assume now 1 < q < ∞ . First we show that assertion (5.79) isvalid with e ̺ ( g ) from (5.81) where ϕ is given by (4.16) with α = k. Applying(6.5) implies e ̺ ( g ) ≃ Z T t k/n λ ( t ) Z Tt g ( s ) d s !! q ′ w ( t ) d t /q ′ . (6.10)Further, (6.4), (5.39), and (5.40) yield ̺ ( g ) ≤ c Z T (cid:18)Z t s k/n − δ λ ( s ) g ( s ) d s (cid:19) q ′ t δq ′ w ( t ) d t ! /q ′ ,̺ ( g ) ≤ c Z T s k/n − δ λ ( s ) g ( s ) d s ! T δ (cid:18)Z ∞ T w ( t ) d t (cid:19) /q ′ . Now, substituting (6.8) into these formulas and some change of order ofintegration lead to ̺ ( g ) ≤ ˜ c Z T Z t τ k/n − δ − λ ( τ ) Z Tτ g ( s ) d s ! d τ ! q ′ t δq ′ w ( t ) d t /q ′ , (6.11) ̺ ( g ) ≤ ˜ c Z T τ k/n − δ − λ ( τ ) Z Tτ g ( s ) d s ! d τ ! T δ (cid:18)Z ∞ T w ( t ) d t (cid:19) /q ′ , (6.12)in view of the obvious estimate R tτ g ( s ) d s ≤ R Tτ g ( s ) d s, τ < t ≤ T. Now we apply in appropriately adapted notation Hardy’s inequality (see[Maz85, Thm. 2, Ch. 1]) to estimate the right-hand side in (6.11) and obtainthat Z T Z t τ k/n − δ − λ ( τ ) Z Tτ g ( s ) d s ! d τ ! q ′ t δq ′ w ( t ) d t /q ′ ≤ c Z T t k/n λ ( t ) Z Tτ g ( s ) d s ! q ′ w ( t ) d t /q ′ (6.13)40olds if, and only if, B δ := sup t ∈ (0 ,T ) Z Tt τ δq ′ w ( τ ) d τ ! /q ′ (cid:18)Z t τ − ( δ +1) q w ( τ ) − q/q ′ d τ (cid:19) /q < ∞ . This condition is satisfied in view of Corollary 5.13: if 0 ≤ δ < ε/q, then B δ ≤ ( q/q ′ ) /q ′ ε − δq . (6.14)Moreover, for the best possible constant c in (6.13) we have B δ ≤ c ≤ B δ q /q ( q ′ ) /q ′ ≤ qε − δq . (6.15)Estimates (6.11),(6.13), and (6.10) imply that ̺ ( g ) ≤ c e ̺ ( g ) , g ∈ M + (0 , T ) , (6.16)where c > g .Similarly, by H¨older’s inequality we get from (6.12) in view of (6.10) ̺ ( g ) ≤ c e ̺ ( g ) Z T τ − ( δ +1) q w ( τ ) − q/q ′ d τ ! /q T δ (cid:18)Z ∞ T w ( τ ) d τ (cid:19) /q ′ . But T δ (cid:18)Z ∞ T w ( τ ) d τ (cid:19) /q ′ ≤ (cid:18)Z ∞ T τ δq ′ w ( τ ) d τ (cid:19) /q ′ , and estimating (5.54) with t = T, T = ∞ , yields ̺ ( g ) ≤ c e ̺ ( g ) , g ∈ M + (0 , T ) , (6.17)where c > g . The assertions (5.37), (6.16), and (6.17) imply(5.79) with e ̺ ( g ) from (5.81).Recall that here ϕ is given by (4.16), where α = k, such that (5.81) coincideswith (6.10).It remains to describe the optimal norm k · k X which is associated to thenorm e ̺ ( g ) in (6.10). This description is given by the formulas (4.20) - (4.22)where we have to consider ϕ given by (4.16) with α = k. Thus, formula (4.5) isvalid where for given ϕ the function W (4.4) has the equivalent form (6.9). Here we present a more detailed consideration in the case of classical Besselpotentials, see Example 2.9. Note that in Example 7.3 below we extend some41receding results in [GNO11]. The results presented here were announced inour paper [GH14].We start with a preparatory lemma which we shall need in the argumentsbelow.
Lemma 7.1.
Let < T ≤ ∞ , β > , λ be a positive slowly varying functionon (0 , T ) , and A ( t ) = Z Tt τ − β − λ ( τ ) d τ, t ∈ (0 , T ) . Then there exists some ε > such that t ε A ( t ) is monotonically decreasing.Proof. We have the equality[ t ε A ( t )] ′ = t ε − " ε Z Tt τ − β − λ ( τ ) d τ − t − β λ ( t ) . Applying estimate (2.44) yields[ t ε A ( t )] ′ ≤ t ε − (cid:2) t − β λ ( t )( εc β − (cid:3) < , if we choose ε ∈ (0 , c − β ) . Examples 7.2.
Let 0 < α < n , 1 ≤ q < ∞ , v = 1 in (4.1), such that E = L q ( R n ) in Example 2.9. For 0 < α < n , q = 1, the space H GE ( R n ) is notembedded into C ( R n ). If q >
1, the criterion for the embedding into C ( R n )reads as (2.13) is true ⇐⇒ α > nq . If nq < α < min (cid:16) n, k + nq (cid:17) , then the optimal Calder´on space for the embedding(3.7) has the norm k u k Λ k ( C ; X ) = k u k C + Z T (cid:18) ω k ( u ; t /n ) t α/n − /q (cid:19) q d tt ! /q . (7.1)In particular, this means that Λ k ( C ; X ) coincides with the classical Besov space B α − n/q ∞ ,q ( R n ), cf. [Nik77, Tri83] for further details on Besov spaces. Proof. Step 1 . In the case considered here we have equivalence (2.35). Withoutloss of generality we can assume that0 < α < n ; ϕ ( t ) = t α/n − , t ∈ (0 , T ] . (7.2)For the basic RIS E ( R n ) = L q ( R n ) , ≤ q < ∞ , the criterion of the embedding(2.13) has the form (2.12) , such that(2.13) ⇐⇒ τ α/n − ∈ L q ′ (0 , T ) ⇐⇒ α > n/q. (7.3)42t means that in case of q = 1 the embedding (2.13) is impossible for this Ex-ample. Step 2 . Now let 1 < q < ∞ , n/q < α < n , and α ≤ k . We have here v ( t ) =1 , V ( t ) = t ; condition (4.13) is satisfied. So we may apply the correspondingresults of Theorem 6.8. According to (4.4), and (4.5) with ϕ given by (7.2), and n/q < α < n, we get W ( t ) = nα t α/n − , Ψ q ( t ) = ct α/n − , t ∈ (0 , T ] , (7.4)where c = c ( α, n, q ) > . Therefore, formula (6.1) leads to (7.1).
Step 3 . Now we consider the case 1 < q < ∞ ; max( n/q, k ) < α < min( n, k + n/q ) . Condition (4.14) is satisfied, we may apply Theorem 4.3 (case (B) ). Here,according to (4.10), and (4.11) with ϕ given by (7.2) we get f W ( t ) = t α − kn − , (7.5) U q ( t ) = Z Tt τ ( α − kn − q ′ d τ ! /q ′ . (7.6)We apply Lemma 7.1 to the function A ( t ) = U q ( t ) q ′ which corresponds to thespecial case λ ( t ) = 1; β = (cid:2) − α − kn (cid:3) q ′ − . For α < k + n/q we have β > , such that in view of Lemma 7.1 there exists some ε > t εq ′ A ( t ) decreases monotonically ⇐⇒ t ε U q ( t ) decreases monotonically . It shows that (4.15) is valid. Therefore, we may apply Theorem 4.3 (case (B) )) here, as well as Theorem 6.1. We arrive at the descriptions (4.22), (6.1)for 1 < q < ∞ , and Ψ q given by (7.4). It leads to (7.1). Examples 7.3.
Let 0 < α < n , 1 ≤ q < ∞ , 1 < p < ∞ , E = Λ q ( v ), recall(4.1), where v is given by v ( t ) = t q/p − b q ( t ) , t ∈ (0 , T ) , (7.7)where b is a positive slowly varying continuous function on (0 , T ). In otherwords, E ( R n ) is a so-called Lorentz-Karamata space, cf. [GNO11]. Now weexplicate Example 2.9. Note that in this caseΨ q ( t ) = sup τ ∈ (0 ,t ] τ αn − p b ( τ ) − , q = 1 , (cid:16)R t τ q ′ ( αn − p ) b ( τ ) − q ′ d ττ (cid:17) /q ′ , q > , (7.8)for t ∈ (0 , T ). (i) If 0 < α < np , then H GE ( R n ) is not embedded into C ( R n ).43 ii) If α = np , then we have to distinguish further between q = 1 and q >
1: incase of q = 1, we require also Ψ (0+) = 0, and(2.13) holds ⇐⇒ Ψ ( t ) = sup τ ∈ (0 ,t ] b ( τ ) − < ∞ , t ∈ (0 , T ] . In case of q >
1, we arrive at(2.13) holds ⇐⇒ Ψ q ( t ) = (cid:18)Z t b ( τ ) − q ′ d ττ (cid:19) /q ′ < ∞ , t ∈ (0 , T ] . In that case the optimal Calder´on space for the embedding (3.7) has thenorm (4.20), where in case of q > q ( t )Ψ q ( t ) ≃ b ( t ) − q ′ R t b ( τ ) − q ′ d ττ d tt . (iii) In case of np < α < min (cid:16) n, k + np (cid:17) , the optimal Calder´on space for theembedding (3.7) has the norm (4.20), where we conclude from (7.8) thatΨ q ( t ) ≃ ( t αn − p b ( t ) − , q = 1 ,t αn − p b ( t ) − , q > , hence dΨ q ( t )Ψ q ( t ) ≃ d tt . Proof. Step 1 . Here we consider the case of ϕ given by (7.2). For the basicLorentz-Karamata space E = Λ q ( v ) with 1 ≤ q < ∞ , < p < ∞ , and v ( t ) isgiven by (7.7). Applying (2.43) in appropriately adapted notation leads to V ( t ) = Z t v ( τ ) d τ ≃ t q/p b ( t ) q , t ∈ (0 , T ) , (7.9)such that in view of (4.4) W ( t ) ≃ t αn − qp b ( t ) − q , t ∈ (0 , T ) . Thus, for q = 1 we have by (4.5) thatΨ ( t ) ≃ t αn − p b ( t ) − , α > n/p,B ( t ) , α = n/p, ∞ , α < n/p, (7.10)where B ( t ) = sup (cid:8) b ( τ ) − : τ ∈ (0 , t ] (cid:9) . (7.11)For 1 < q < ∞ we get by (4.5)Ψ q ( t ) ≃ (cid:18)Z t τ q ′ ( αn − p ) b ( τ ) − q ′ τ − d τ (cid:19) /q ′ , (7.12)44uch that in view of (2.43), for t ∈ (0 , T ),Ψ q ( t ) ≃ t αn − p b ( t ) − , α > n/p,B q ( t ) , α = n/p, ∞ , α < n/p, (7.13)where B q ( t ) = (cid:26)Z t b ( τ ) − q ′ τ − d τ (cid:27) /q ′ . (7.14)According to Lemma 4.1,(2.13) ⇐⇒ (2.12) ⇐⇒ Ψ q ( T ) < ∞ . We see that (2.13) is impossible for α < n/p ; (2.13) is valid for α > n/p ;and the validity of (2.13) for α = n/p depends on B q : (2.13) ⇐⇒ B q ( T ) < ∞ , where B q is given by (7.11), (7.13). Step 2 . Assume that the conditions of embedding (2.13) are satisfied, inparticular, for α ≥ n/p. If α ≤ k we may apply Theorem 6.8 with λ ( t ) = 1 . Condition (4.13) is valid for function V given by (7.9) for any ε ∈ (0 , q/p ) . In-deed, in this case V ( t ) t − ε = t q/p − ε b ( t ) q increases monotonically, because b ( t ) q is slowly varying. Then, by Theorem 6.8 we get the descriptions (4.20)-(4.22)with Ψ q from (7.10) in case of q = 1, or from (7.13) in case of 1 < q < ∞ . Step 3 . Finally, let α ≥ n/p , k < α < min( n, k + n/p ). We need to verifycondition (4.15). According to (4.10) with ϕ given by (7.2), and V given by(7.9) we obtain f W ( t ) = t α − kn − qp b ( t ) − q , t ∈ (0 , T ) . Here, α − kn − p < , such that for q = 1 the function f W ( t ) decreases mono-tonically, and U ( t ) = f W ( t ) = t α − kn − p b ( t ) − , (7.15)see (4.11). For 1 < q < ∞ we obtain by (4.11) and (7.7) that U q ( t ) = Z Tt τ q ′ ( α − kn − p ) b ( τ ) − q ′ τ − d τ ! /q ′ . (7.16)We see that for α < k + n/p condition (4.15) is valid. For U given by (7.15) thisis obvious because b ( t ) − is slowly varying. For U q given by (7.16) we define β = (cid:20) p − α − kn (cid:21) q ′ > λ ( t ) = b ( t ) − q ′ ; A ( t ) = U q ( t ) q ′ . Then, by Lemma 7.1 there exists some ε > A ( t ) t εq ′ decreases monotonically ⇐⇒ U q ( t ) t ε decreases monotonically . (7.17)45onsequently, we may apply Theorem 4.3 (case (B) ) and get descriptions (4.20)-(4.22), (6.1) with Ψ q determined by (7.10) for q = 1 , or by (7.13) for 1 < q < ∞ . Remark . Note that for 1 < q < ∞ and the function Ψ q defined by integral(7.12) we have in view of estimate (2.43) thatdΨ q ( t )Ψ q ( t ) ≃ d (cid:16) Ψ q ′ q ( t ) (cid:17)(cid:16) Ψ q ′ q ( t ) (cid:17) ≃ d tt . (7.18)For the function Ψ q = B q given by integral (7.14) we getdΨ q ( t )Ψ q ( t ) ≃ b ( t ) − q ′ t − d t R t b ( τ ) − q ′ τ − d τ . (7.19)These assertions are useful when we apply formulas (4.20), (6.1). Acknowledgements
The work of Elza Bakhtigareeva and Mikhail L. Goldman is supported by theRussian Science Foundation under grant no.19-11-00087 and performed in theSteklov Mathematical Institute of the Russian Academy of Sciences.
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