A simpler description of the $\kappa$-topologies on the spaces $\mathscr{D}_{L^p}$, $L^p$, $\mathscr{M}^1$

The $\kappa$-topologies on the spaces $\mathscr{D}_{L^p}$, $L^p$ and $\mathscr{M}^1$ are defined by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre-)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi-norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of semi-norms generating the $\kappa$-topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence-space representation of the spaces $\mathscr{D}_{L^p}$ equipped with the $\kappa$-topology, which complements a result of J.~Bonet and M.~Maestre. As a byproduct, we give a characterisation of the compact subsets of the spaces $\mathscr{D}'_{L^p}$, $L^p$ and $\mathscr{M}^1$.


Introduction
In the context of the convolution of distributions, L. Schwartz introduced the spaces D L p of C 8 -functions whose derivatives are contained in L p and the spaces D 1 L q of finite sums of derivatives of L q -functions, 1 ď p, q ď 8. The topology of D L p is defined by the sequence of (semi-)norms D L p Ñ R`, ϕ Þ Ñ p m pϕq :" sup |α|ďm B α ϕ p , whereas D 1 L q " pD L p q 1 for 1 {p`1{q " 1 if p ă 8 and D 1 L 1 " p 9 Bq 1 , carry the strong dual topology.
Equivalently, for 1 ď p ă 8, by barrelledness of these spaces, the topology of D L p is also the topology βpD L p , D 1 L q q of uniform convergence on the bounded sets of D 1 L q . In the case of p " 1, the duality relation p 9 B 1 q 1 " D L 1 , see [3,Proposition 7,p. 13], provides that D L 1 also has the topology of uniform convergence on bounded sets of 9 B 1 . Following L. Schwartz in [26], 9 B 1 denotes the closure of E 1 in B 1 " D 1 L 8 which is not the dual space of 9 B but in some sense is its analogon for distributions instead of smooth functions. For p " 8, D L 8 carries the topology βpD L 8 , D 1 L 1 q due to pD 1 L 1 q 1 " D L 8 [26, p. 200]. By definition, the topology of D 1 L q is the topology of uniform convergence on the bounded sets of D L p and 9 B for q ą 1 and q " 1, respectively.
In [25], the spaces D L p ,c and D 1 L q ,c are considered, where the index c designates the topologies κpD L p , D 1 L q q and κpD 1 L q , D L p q of uniform convergence on compact sets of D 1 L q and D L p , respectively. Let us mention three reasons why the spaces L q c and D 1 L q ,c are of interest: (1) Let E and F be distribution spaces. If E has the ε-property, a kernel distribution Kpx, yq P D 1 x pF y q belongs to the space E x pF y q if it does so "scalarly", i.e., Kpx, yq P E x pF y q ðñ @f P F 1 : xKpx, yq, f pyqy P E x .
The spaces L q c , 1 ă q ď 8, and D 1 L q ,c , 1 ď q ď 8, have the ε-property [25, Proposition 16, p. 59] whereas for 1 ď p ď 8, L p and D 1 L p do not. For L p , 1 ď p ă 8, this can be seen by checking that Kpx, kq " pk 1{p {p1`k 2 x 2 qq P D 1 pRq p bc 0 satisfies xKpx, kq, ay P L p for all a P ℓ 1 but is not contained in L p p b ε c 0 ; in the case of p " 8 one takes Kpx, kq " Y px´kq instead. For D 1 L q a similar argument with Kpx, kq " δpx´kq can be used.
(2) The kernel δpx´yq of the identity mapping E x Ñ E y of a distribution space E belongs to E 1 c,y ε E x . Thus, e.g., the equation where the H k denote the Hermite functions, due to G. Arfken in [2, p. 529] and P. Hirvonen in [15], is valid in the spaces L q c, [22], or, classically, in S 1 (3) The classical Fourier transform is well-defined and continuous by the Hausdorff-Young theorem. By means of the kernel e´i xy of F we can express the Hausdorff-Young theorem by if p ą 1, and by e´i xy P L 8 c,x pC 0,y q if p " 1. Obvious generalizations are e´i xy P D L q ,c,x p 8 ď k"0 pL q q k,y q, 2 ă q ă 8, and e´i xy P D L 8 ,c,x pO 0 c,y q.
Whereas the topology of D L p , 1 ď p ď 8, can be described either by the seminorms p m or, equivalently, by the topology of D L p ,c only is described by An analogue statement holds for L p c , 1 ă p ď 8 and M 1 c .
Thus, our task is the description of the topologies κpD L p , D 1 L q q, κpD L 1 , 9 B 1 q, κpL p , L q q and κpM 1 , C 0 q (for 1 ă p ď 8) by seminorms involving functions and not sets (Propositions 4,8,16 and 18). As a byproduct, the compact sets in D 1 L q and L q for 1 ď q ă 8 are characterised in Proposition 3 and in Proposition 17, respectively. In addition, we also give characterisations of the compact sets of 9 B 1 and M 1 .
The notation generally adopted is the one from [24][25][26][27][28] and [17]. However, we deviate from these references by defining the Fourier transform as We follow [26, p. 36] in denoting by Y the Heaviside-function. The translate of a function f by a vector h is denoted by pτ h f qpxq :" f px´hq. Besides the spaces L p pR n q " L p , D L p pR n q " D L p , 1 ď p ď 8, we use the space M 1 pR n q " M 1 of integrable measures which is the strong dual of the space C 0 of continuous functions vanishing at infinity. In measure theory the measures in M 1 usually are called bounded measures whereas J. Horváth, in analogy to the integrable distributions, calls them integrable measures. Here we follow J. Horváth's naming convention.
The topologies on Hausdorff locally convex spaces E, F we use are βpE, F q -the topology (on E) of uniform convergence on bounded sets of F , κpE, F q -the topology (on E) of uniform convergence on absolutely convex compact subsets of F , see [17, p. 235].
In addition, we consider the following sequence spaces. By s we denote the space of rapidly decreasing sequences, by s 1 its dual, the space of slowly increasing sequences. Moreover we consider the space c 0 of null sequences and its weighted variant pc 0 q´k " tx P C N : lim jÑ8 j´kxpjq " 0u.
The uniquely determined temperate fundamental solution of the iterated metaharmonic operator p1´∆ n q k , where ∆ n is the n-dimensional Laplacean, is given by wherein the distribution Particular cases of this formula are: For s ą 0, L s ą 0 and L s decreases exponentially at infinity. Moreover, L s P L 1 for Re s ą 0. In contrast to [29, p. 131] and [36], we maintain the original notation L s for the Bessel kernels and we write L s˚i nstead of J s . The spaces D 1 L q can be described as the inductive limit see, e.g., [26, p. 205]. The space 9 B 1 of distributions vanishing at infinity has the similar representation by [26, p. 205]. If we equip p1´∆ n q m C 0 with the final topology with respect to p1´∆ n q m an application of de Wilde's closed graph theorem provides the topological equality since by [3, Proposition 7, p. 65] 9 B 1 is ultrabornological and lim Ý Ñm p1´∆ n q m C 0 has a completing web since it is a Hausdorff inductive limit of Banach spaces.

"Function"-seminorms in
In order to describe the topology of D L p ,c p1 ă p ă 8) by "function"seminorms it is necessary to characterise the compact sets of the dual space and endowed with the strong topology βpD 1 L q , D L p q. The description of D L 8 ,c is already well-known [10].
Due to the definition of the space D L p , 1 ď p ă 8, as the countable projective limit Ş 8 m"0 H 2m,p of the Banach spaces H 2m,p , which are called "potential spaces" in [29, p. 135], we conclude that the strong dual D 1 L q coincides with the countable inductive limit Ť 8 m"0 H´2 m,q . Note that the topological identity follows from the ultrabornologicity of pD 1 L q , βpD 1 L q , D L p qq, which follows for example by the sequence-space representation D 1 L ps 1 p b ℓ q given independently by D. Vogt in [31] and by M. Valdivia in [30], by means of Grothendiecks Théorème B [14, p. 17]. The completeness of D 1 L q implies the regularity of the inductive limit Ť 8 m"0 H´2 m,q [6, p. 77]. An alternative proof of the representation of D 1 L q as the inductive limit of the potential spaces above can be given using [4,Theorem 5] and the fact that 1´∆ n is a densely defined and invertible, closed operator on L q .
We first show that the (LB)-space D 1 L q , 1 ď q ă 8 is compactly regular [6, 6. Definition (c), p. 100]: L q is compactly regular. @m P N 0 Dk ą m @ε ą 0 @ℓ ą k DC ą 0 : By definition, But this inequality follows from Ehrling's inequality [36], which states that for 1 ď q ă 8 and 0 ă s ă t, By density of S in H´2 pm´ℓq,q this implies the validity of (Q).  The next proposition characterises compact sets in D 1 L q .
Proof. "ð": The compactness of L 2m˚C in L q implies its compactness in D 1 L q and, hence, The following proposition generalizes the description of the topology of the space B c " pD L 8 , κpD L 8 , D 1 L 1 qq by the "function"-seminorms Proposition 4. Let 1 ă p ď 8 and 1 {p`1{q " 1. The topology κpD L p , D 1 L q q of D L p ,c is generated by the seminorms D L p Ñ R`, ϕ Þ Ñ p g,m pϕq :" gp1´∆ n q m ϕ p , g P C 0 , m P N 0 , or equivalently by ϕ Þ Ñ sup |α|ďm gB α ϕ p , g P C 0 , m P N 0 .
Proof. Due to [10, (3.5) Cor., p. 71] it suffices to assume 1 ă p ă 8. We denote the topology on D L p generated by tp g,m : g P C 0 , m P N 0 u by t. Moreover, B 1,p shall denote the unit ball in L p .
(a) t Ď κpD L p , D 1 L q q: If U g,m :" tϕ P D L p : p g,m pϕq ď 1u is a neighborhood of 0 in t we have U g,m " ppU g,m q˝q˝by the theorem on bipolars. We show that Ug ,m is a compact set in D 1 L q . We have ϕ P U g,m ðñ gpL´2 m˚ϕ q P B 1,p ðñ sup Hence, Ug ,m " L´2 m˚p gB 1,q q Ď D 1 L q . By Proposition 3, Ug ,m is compact in (i) Because ℓ ą m, µ :" L 2pℓ´mq P L 1 and hence, for ϕ P B 1,q , i.e., C is bounded in L q .
(ii) The set C has to be small at infinity: for ϕ P B 1,q , Hence, lim RÑ8 Y p|.|´Rqpµ˚pgϕqq q " 0 uniformly for ϕ P B 1,q .
L q q-neighborhood of 0 with C a compact set in D 1 L q then, by Proposition 3, there exists m P N 0 such that the set L 2m˚C is compact in L q . By means of Lemma 5 below there is a function g P C 0 such that L 2m˚C Ď gB 1,q , hence C Ď L´2 m˚p gB 1,q q Ď Ug ,m . Thus, C˝Ě U g,m , i.e., C˝is a neighborhood in t.
Lemma 5. Let 1 ď q ă 8. If K Ď L q is compact then there exists g P C 0 such that K Ď gB 1,q .
Proof. Apply the Cohen-Hewitt factorization theorem [11, (17.1), p. 114] to the bounded subset K of the (left) Banach module L q with respect to the Banach algebra C 0 having (left) approximate identity te´k 2 |x| 2 : k ą 0u.

Remark 6.
Denoting by τ pB, D 1 L 1 q the Mackey-topology on B " D L 8 we even have for p " 8, B " D L 8 : 3 The case p " 1 Using the sequence-space representation 9 B 1s 1 p b c 0 given in [3, Theorem 3, p. 13], the compact regularity of the (LB)-space 9 B 1 can be shown similarly to the proof of Proposition 12. Moreover, one has the following characterisation of the compact sets of 9 B 1 .

Proposition 7.
A set C Ď 9 B 1 is compact if and only if for some m P N 0 , L 2m˚C is compact in C 0 .
Proof. The proof is completely analogous to the one of Proposition 3. Proposition 8. The topology κpD L 1 , 9 B 1 q of D L 1 ,c is generated by the seminorms D L 1 Ñ R`, ϕ Þ Ñ p g,m pϕq :" gp1´∆ n q m ϕ 1 , g P C 0 , m P N 0 , or equivalently by ϕ Þ Ñ sup |α|ďm gB α ϕ 1 , g P C 0 , m P N 0 .
Proof. We first show that the topology t generated by the above seminorms is finer than the topology of uniform convergence on compact subsets of 9 B 1 . Similarly to the proof of Proposition 4, we have to show that the set L 2pℓ´mq˚p gB 1,C 0 q is a relatively compact subset of 9 B 1 , where B 1,C 0 denotes the unit ball of C 0 . We pick ℓ ą m and, by the compact regularity of 9 B 1 and the Arzela-Ascoli theorem, we have to show that L 2pℓ´mq˚p gB 1,C 0 q is bounded as a subset of C 0 and equicontinous as a set of functions on the Alexandroff compactification of R n . Since ℓ ą m, we have that L 2pℓ´mq P L 1 . For every ϕ P B 1,C 0 , Young's convolution inequality implies Since a translation of a convolution product can be computed by applying the translation to one of the factors, we can again use Young's convolution inequality to obtain for all ϕ in the unit ball of C 0 . From this inequality we may conclude that L 2pℓ´mq˚p gB 1,C 0 q is equicontinuous at all points in R n . Therefore we are left to show that it is also equicontinous at infinity. In order to do this, first observe that |g| P C 0 and |L 2pℓ´mq | " L 2pℓ´mq . Moreover, Lebesgue's theorem on dominated convergence implies that the convolution of a function in C 0 and an L 1 -function is contained in C 0 . Finally, by the above reasoning the inequality |pL 2pℓ´mq˚p gϕqqpxq| ď ż R n |gpx´ξq||L 2pℓ´mq pξq| dξ " pL 2pℓ´mq˚| g|qpxq shows that L 2pℓ´mq˚p gB 1,C 0 q is equicontinuous at infinity.
The proof that κpD L 1 , 9 B 1 q is finer than t is completely analogous to the corresponding part of Proposition 4 if we can show that every compact subset of C 0 is contained in gB 1,C 0 for some g P C 0 . Let C Ď C 0 be a compact set. By the Arzela-Ascoli theorem, it is equicontinuous at infinity, i.e., for every k P N there is an R k such that |hpxq| ď 1{k for h P C and all |x| ą R k . This condition implies the existence of the required function g P C 0 with the above property.

Properties of the spaces D L p ,c
In [25, p. 127], L. Schwartz proves that the spaces B c " D L 8 ,c and B " D L 8 have the same bounded sets, and that on these sets the topology κpB, D 1 L 1 q equals the topology induced by E " C 8 . Moreover, κpB, D 1 L 1 q is the finest locally convex topology with this property. By an identical reasoning we obtain: (a) The spaces D L p and D L p ,c have the same bounded sets. These sets are relatively κpD L p , D 1 L q q-compact and relatively κpD L 1 , 9 B 1 q-compact for 1 ă p ă 8 and p " 1, respectively.
(b) The topology κpD L p , D 1 L q q of D L p ,c is the finest locally convex topology on D L p which induces on bounded sets the topologies of E or D 1 or D 1 L p .
In the following proposition we collect some further properties of the spaces D L p : (2) D L p ,c is quasinormable since its dual D 1 L q is boundedly retractive (see the argument in [10, p. 73] and use [13, Def. 4, p. 106]) which, by [6, 7. Prop., p. 101] is equivalent with its compact regularity (Proposition 1).
(3) Since bounded and relative compact sets coincide in D L p ,c it is a semi-Montel space.
(4) Infrabarrelledness and the Montel-property would imply that D L p ,c is Montel which in turn implies the coincidence of the topologies κpD L p , D 1 L q q and βpD L p , D 1 L q q. This is a contradiction, since together with the compact regularity of the inductive limit representation D 1 L q " Ť H´2 m,q this would imply that for some m the unit ball of H´2 m,q is compact which is impossible for an infinite dimensional Banach space. (6) Using the sequence-space representation D L p ,cℓ p c p b s, we can conclude by [14, Ch. II §3 n°2, Prop. 13, p. 76] that D L p ,c is nuclear if and only if ℓ p c is nuclear. We first consider the case p ą 1. The nuclearity of ℓ p c would imply where ℓ 1 tℓ p c u and ℓ 1 xℓ p c y is the space of absolutely summable and of unconditionally summable sequences in ℓ p c , respectively, see [18, pp. 341, 359]. We now proceed by giving an example of an element of the space at the very right which is not contained in the space at the very left. Fix ε ą 0 small enough. Using Hölder's inequality, we observe that k"1 P ℓ q which together with the characterisation of unconditional convergence in [33,Theorem 1.15] and the condition that compact subsets of ℓ q are small at infinity implies that the sequence is unconditionally convergent. On the other hand taking pk´αq 8 k"1 P c 0 with α " 1´1`ε p yields from which we may conclude that pδ jk k´p 1`εq{p q j,k is not an absolutely summable sequence in ℓ p c . For the case p " 1 we use the Grothendieck-Pietsch criterion, see [18, p. 497], and observe that Λpc 0,`q " ℓ 1 c . Choosing α " p1{kq 8 k"1 provides the necessary sequence with pα k {β k q 8 k"1 R ℓ 1 for every β P c 0,`w ith β ě α.
Remark 11. The above proof actually shows that ℓ p c , for 1 ď p ď 8 is not nuclear. From this we may also conclude that D 1 L p ,c , 1 ď p ď 8, is not nuclear.
Analogous to the table with properties of the spaces D F , D 1F (defined in [17, p. 172,173]) in [5, p. 19], we list properties of D L p and D L p ,c in the following table (1 ď p ď 8): property Sequence space representations of the spaces D L p ,c , D 1 L q ,c and the compact regularity of D 1 L q P. and S. Dierolf conjectured in [10, p. 74] the isomorphy where ℓ 8 c " pℓ 8 , κpℓ 8 , ℓ 1 qq " pℓ 8 , τ pℓ 8 , ℓ 1 qq. This conjecture is proven in [7,1. Theorem,p. 293]. More generally, we obtain: Proof.
(1) By means of M. Valdivia's isomorphy (see Remark 2 (a)) D 1 L qℓ q p b s 1 it follows, for q ă 8, by [8, 4.  (2) The second Theorem on duality of H. Buchwalter [20, (5), p. 302] yields for two Fréchet spaces E, F : and hence, for E " ℓ p and F " s, E p b F " ℓ p p b s which implies An alternative proof for (2) can also by given using [8, 4. The compact regularity of D 1 L q as a countable inductive limit of Banach spaces is proven in Proposition 1. By means of the sequence space representation which has been presented in Remark 2 (a) we can give a second proof: Proposition 14. Let E be a Banach space.
(a) The inductive limit representations (b) The inductive limit lim Ý Ñk pc 0 pEqq´k is compactly regular. @m P N 0 Dk ą m @ε ą 0 @l ą k DC ą 0 @x " px j q j P pc 0 pEqq´m : For the sequences px j q j P pc 0 pEqq´m the sequence p x j q j is contained in s 1 " lim Ý Ñk pc 0 q´k. Thus, (Q) is fulfilled because s 1 is an (LS)-space.
6 "Function"-seminorms in L p c and M 1 c .
Motivated by the description of the topology κpℓ p , ℓ q q of the sequence space ℓ p c by the seminorms g P c 0 , see I. Mack's master thesis [21] for the case p " 8, we also investigate the description of the topologies κpL p , L q q and κpM 1 , C 0 q by means of "function"-seminorms.
Denoting the space of bounded and continuous functions by B 0 (see [24, p. 99]) we recall that B 0 c " pB 0 , κpB 0 , M 1 qq has the topology of R. C. Buck.
Proposition 16. Let 1 ă p ď 8. The topology κpL p , L q q of L p c is generated by the family of seminorms L p Ñ R`, f Þ Ñ p g,h pf q " h˚pgf q p , g P C 0 , h P L 1 .
Proof. (a) t Ď κpL p , L q q where t denotes the topology generated by the seminorms p g,h above and let U g,h :" tf P L p : p g,h pf q ď 1u be a neighborhood of 0 in t. We show that (i) U g,h " pgpȟ˚B 1,q qq˝, (ii) gpȟ˚B 1,q q is compact in L q . ad (i): If S P gpȟ˚B 1,q q Ď L q there exists ψ P B 1,q with S " gpȟ˚ψq. For f P L p with p g,h pf q ď 1 we obtain |xf, Sy| "ˇˇxf, gpȟ˚ψqyˇˇ" |xh˚pgf q, ψy| and sup SPgpȟ˚B 1,q q |xf, Sy| " h˚pgf q p " p g,h pf q, so (i) follows.
ad ( (a) The set gpȟ˚B 1,q q is bounded in L q .
If s tends to 0, τ s g´g 8 Ñ 0 because g is uniformly continuous, and τ sȟ´ȟ 1 Ñ 0 because the L 1 -modulus of continuity is continuous.
(b) t Ě κpL p , L q q: For a compact set C Ď L q we have to show that there exist g P C 0 , h P L 1 such that p g,h pf q " gph˚f q p ě sup sPC |xf, Sy| for f P L p . By applying the factorization theorem [11, (17.1). p. 114] twice, there exist g P C 0 , h P L 1 such that C Ď gpȟ˚B 1,q q. More precisely, take L 1 as the Banach algebra with respect to convolution and C 0 as the Banach algebra with respect to pointwise multiplication, L p as the Banach module. As approximate units we use e´k 2 |x| 2 , k ą 0, in the case of C 0 and p4πtq´n {2 e´| x| 2 {4t , t ą 0, in the convolution algebra L 1 . Then, sup SPC |xf, Sy| ď sup ψPB 1,qˇx f, gpȟ˚ψqyˇˇ" sup ψPB 1,q |xh˚pgf q, ψy| " h˚pgf q p , which finishes the proof.
An exactly analogous reasoning as for Proposition 16 yields Proposition 18. The topology κpM 1 , C 0 q of the space M 1 c is generated by the seminorms M 1 Ñ R`, µ Þ Ñ p g,h pµq :" h˚pgµq 1 .