On solvability of integro-differential equations

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper.


Introduction
We consider the equation on H T = [0, T ] × R d for a given T > 0, with initial condition u(0, x) = ψ(x) for x ∈ R d , where A is an integro-differential operator of the form A = L + M + N + R, with a "zero-order" linear operator R, a second order differential operator and linear operators M and N defined by M(t)ϕ(x) = Z (ϕ(x + η t,z (x)) − ϕ(x) − η t,z (x)∇ϕ(x)) µ(dz), (1.2) for a suitable class of real-valued functions ϕ(x) on R d . Here a ij , b i and c are real-valued bounded functions defined on H T , µ and ν are σ-finite measures on a measurable space (Z, Z). The functions η and ξ are R d -valued mapping defined on H T × Z. Under "zeroorder operators" we mean bounded linear operators R mapping the Sobolev spaces W k p into themselves for k = 0, 1, 2, .., n for some n. Examples include integral operators R(t) defined by with appropriate functions ζ on H T × Z and finite measures λ on Z.
Our aim is to investigate the solvability of equation (1.1) in Bessel potential spaces H m p and Sobolev-Slobodeckij spaces W m p for p ≥ 2 and m ∈ [1, ∞). Such kind of equations arise, for example, as Kolmogorov equations for Markov processes given by stochastic differential equations, driven by Wiener processes and Poisson random measures, see e.g., [1], [2], [11], [12] and [15]. They play important roles in studying random phenomena modelled by Markov processes with jumps, in physics, biology, engineering and finance, see e.g., [3], [8], [31], [36] and the references therein. There is a huge literature on the solvability of these equations, but in most of the publications some kind of non-degeneracy, conditions on the equations, or specific assumptions on the measures µ and ν are assumed. Results in this direction can be found, for example, in [11], [12], [15], [25], [27], [28], [30] and [37], and for nonlinear equations of the type (1.1), arising in the theory of stochastic control of random processes with jumps, we refer to [12] and [38]. Extensions of the L p -theory of Krylov [16] to stochastic equations and systems of stochastic equations with integral operators of the type M and N above are developed in [7], [6], [17], [18] and [29].
In this paper we are interested in the solvability of equation (1.1) when it can degenerate, and besides some integrability conditions, no specific conditions on the measures µ and ν are assumed. An L 2 -theory of degenerate linear elliptic and parabolic PDEs is developed in [32], [33], [34] and [35]. The solvability in L 2 -spaces of linear degenerate stochastic PDEs of parabolic type were first studied in [21] (see also [39]). The first existence and uniqueness theorem on solvability of these equations in W m p spaces, for integers m ≥ 1 and any p ≥ 2, is presented in [22]. A gap in the proof of a crucial L p -estimate in [22] is filled in, and the existence and uniqueness theorem is substantially improved in [14]. The solvability of degenerate stochastic integro-differential equations, which include the type of equations (1.1), are studied in [9], [23] and [24]. Existence and uniqueness theorems are obtained in Hölder spaces in [23], and in L 2 -spaces in [9] and [24]. Our main result, Theorem 2.1 below, is an existence and uniqueness theorem in L p -spaces, which generalises the corresponding results in [9] and [24], but instead of stochastic integro-differential equations here we consider only the deterministic equation (1.1). A generalisation of Theorem 2.1 to stochastic integro-differential equations will be presented in a forthcoming paper.
In conclusion we introduce some notations used throughout the paper. For vectors v = (v i ) and w = (w i ) in R d we use the notation vw = m i=1 v i w i and |v| 2 = i |v i | 2 . For realvalued Lebesgue measurable functions f and g defined on R d the notation (f, g) means the integral of the product f g over R d with respect to the Lebesgue measure on R d . A finite list α = α 1 α 2 , ..., α n of numbers α i ∈ {1, 2, ..., d} is called a multi-number of length |α| := n, and the notation D α := D α 1 D α 2 ...D αn is used for integers n ≥ 1, where .., d}. We use also the multi-number of length 0, and agree that D means the identity operator. For an integer n ≥ 0 and functions v on R d , whose partial derivatives up to order n are functions, we use the notation D n v for the collection {D α v : |α| = n}, and define |D n v| 2 = |α|=n |D α v| 2 .
For differentiable functions v = (v 1 , ..., v d ) : R d → R d the notation Dv means the Jacobian matrix whose j-th entry in the i-th row is D j v i .
For a separable Banach space V we use the notation , which are continuos with respect to the strong topology and with respect to the weak topology, respectively, on V . For m ∈ R and p ∈ (1, ∞) we use the notation H m p for the Bessel potential space with exponent p and order m, defined as the space of generalised functions ϕ on R d such that For p ∈ [1, ∞) and integers m ≥ 0 the notation W m p means the Sobolev space defined as the completion of C ∞ 0 , the space of smooth functions with compact support on R d , in the norm For integers m ≥ 0 the space W m ∞ is the completion of C ∞ b , the space of bounded functions on R d with bounded smooth derivatives, in the norm |ϕ| W m ∞ := |α|≤m ess sup |D α ϕ|.
One knows that H m p and W m p are the same as vector spaces, and their norms are equivalent for p ∈ (1, ∞) and integers m ≥ 0. When m > 0 is not an integer, then W m p denotes space of functions f ∈ W m p such that for every multi-index α of length m , where m is the largest integer smaller than m, and {m} = m − m . When m > 0 is not an integer, then W m p with the norm is a Banach space, called Slobodeckij space. Derivatives are understood in the generalised sense unless otherwise noted. The summation convention with respect to repeated indices is used thorough the paper, where it is not indicated otherwise. For basic notions and results on solvability of parabolic PDEs in Sobolev spaces we refer to [19]. The paper is organised as follows. The formulation of the problem and the main result, Theorem 2.1, is in Section 2. Some technical tools and the crucial L p estimates are collected in Sections 3 and 4, respectively. The proof of Theorem 2.1 is given in the last section, Section 5.

Formulation of the main results
Let K be a constant and letη andξ be nonnegative Z-measurable functions on Z such that Let p ∈ [2, ∞) and m ≥ 0 be real numbers, and let m denote the smallest integer which is greater than or equal to m. We make the following assumptions. for all (t, x, z) ∈ H T × Z, and for every v ∈ C ∞ 0 , where to ease notation we do not write the arguments t and z and write v k instead of D k v for functions v. Due to Assumption 2.2 these equations extend to v ∈ W 1 p for p ≥ 2 as well. Hence after changing the order of integrals, by integration by parts we obtain where for the sake of short notation the arguments t, z of η and η k have been omitted, and its derivatives in x ∈ R d up to order max{ m , 2} exist and are continuous in x such that |ξ| ≤ξ, |D k ξ| ≤ξ ∧ K, k = 1, 2, ..., max( m , 2) =: m ξ for all (t, x, z) ∈ H T × Z, and Let V s p denote H s p or W s p for every s ≥ 0. Assumption 2.5. We have ψ ∈ V m p and f ∈ L p ([0, T ], V m p ). Using Remark 2.1 we define the notion of generalised solutions to (1.1) as follows.
Observe that, if Assumptions 2.2 and 2.3 hold, then there is a constant N such that If Assumptions 2.1 through 2.5 hold with m = 0, then there is at most one generalised solution.

preliminaries
First we present some lemmas which are probably well-known from textbooks in analysis. Recall that we use multi-numbers α = α 1 . . . α n , where α j ∈ {1, . . . , d}, to denote higher order derivatives. For a multi-number α = α 1 ....α k of length k and a subset κ ofk := {1, 2, ..., k} we use the notation α(κ) for the multi-number α l 1 ...α ln , where l 1 ,...,l n are the elements of κ, listed in increasing order. For short we use the notation v α := D α v for functions v of x ∈ R d . We write κ 1 · · · κ n =k for the partition ofk := {1, 2, .., k} into n nonempty disjoint sets κ 1 ,...,κ n . Two partitions are considered different if one of the sets in one of the partitions is different from each set in the other partition. Using the above notation the chain rule for (u(ρ)) α := D α (u(ρ)) for functions u : R d → R and ρ : R d → R d can be formulated as follows.
where the second summation on the right-hand side means summation over the different partitions ofl := {1, 2, ..., l}, and for each l and each partition ofl there is also a summation with respect to the repeated indices i j ∈ {1, 2, ..., d} for j = 1, 2, ..., n.
Proof. One can prove this lemma by induction on l, and it is left for the reader as an easy exercise.
A one-to-one function, mapping R d onto R d , is called a C k (R d )-diffeomorphism on R d for an integer k ≥ 1, if the derivatives up to order k of the function and its inverse are continuous. If ρ is a C k (R d ) diffeomorphism such that M ≤ | det(Dρ)| and |D i ρ| ≤ N for i = 1, 2, ..., k. (3.2) for some positive constants M and N , then Lemma 3.1 can be extended to u ∈ W k p for any p ∈ [1, ∞).
Lemma 3.2. Let ρ be a C k (R d )-diffeomorphism for some k ≥ 1 such that (3.2) holds. Then the following statements hold.
(i) There is a constant Proof. We prove (3.3) by induction on l, assuming that u ∈ W k p , v ∈ W k ∞ are smooth functions and p = ∞. For l = 0 by the change of variable ρ(x) = y and by the first inequality in (3.2) we have which proves (3.3) for l = 0. Let l ≥ 1 and assume that statement (i) is true for l − 1 in place of l. By the Leibniz rule and the chain rule .., d. Hence by the induction hypothesis and the second inequality in (3.2) we have which finishes the induction proof. When p = ∞ and l = 0 then (3.3) is obvious, and by induction on l we get the result as before. Clearly, the condition given by the first inequality in (3.2) is not needed in this case. Since C ∞ 0 is dense in W l p when p = ∞ and C ∞ b is dense in W l p , we can finish the proof of (ii) by a standard approximation argument. Making use of (ii) we can get (i) also by approximating u by C ∞ 0 functions when p = ∞ and by C ∞ b functions when p = ∞. Proof. It follows from the second estimate in (3.2 which proves the first estimate in (3.2) for g = ρ −1 in place of ρ. To estimate |Dg| notice that To estimate |D i g| for 1 ≤ i ≤ k and k > 1, we claim that for every multi-number α of length i < k each entry B rl (α) of the matrix B(α) := D α Dg is a linear combination of products of at most k + 2 functions, with multiplicity, taken from the set This gives B rl (α) = −g r j ρ j pi (g)g i α g p l for r, l = 1, 2, .., d, which proves the claim for k = 2, and our claim follows by induction on k. Hence also by induction on k we immediately obtain that |D i g| ≤ N for 1 ≤ i ≤ k with a constant N = N (N, M, d, k), since we have already proved this statement for k = 1 above.
In Section 5 we will approximate equation (1.1) by mollifying the data ψ and f , the coefficients of L and the functions η and ξ in the variable x ∈ R d . It is easy to see that the mollifications of the data and the coefficients of L by a nonnegative C ∞ 0 kernel of unit integral satisfy Assumptions 2.5 and 2.1. It is less clear, however, that mollifications of η and ξ satisfy Assumptions 2.2 and 2.3. We clarify this by the help of some lemmas below. In the rest of the paper for ε > 0 and locally integrable functions v defined on R d we use the Proof. We show first that | det Dρ (ε) | is separated away from zero for sufficiently small ε > 0.
To this end observe that if v = (v 1 , v 2 , ..., v d ) is a Lipschitz function on R d with Lipschitz constant L, and in magnitude it is bounded by a constant K, then for every ε > 0 By virtue of this observation, taking into account that D i ρ l is bounded by N and it is Lipschitz continuous with a Lipschitz constant N , we get for all x, y ∈ R d and ε > 0. Observe that which implies lim |x|→∞ |ρ (ε) (x)| = ∞, i.e., that under ρ (ε) the pre-image of any compact set is a compact set for ε ∈ (0, ε ). A continuous function with this property is called a proper function, and by Theorem 1 in . Now we can complete the proof of the lemma by noting that since Recall the definition τ θη by (2.4). Similarly, for each t ∈ [0, T ], θ ∈ [0, 1] and z ∈ Z we use the notation τ θξ for the R d valued function on R d , defined by for x ∈ R d . To ease notation we will often omit the variables t and z of η and ξ. We can apply the above lemmas to τ θη and τ θξ by virtue of the following proposition. Proof. By the inverse function theorem τ θη and τ θξ are local C 1 (R d )-diffeomorphisms for each t, θ and z. Since Hence τ θη and τ θξ are global C 1 -diffeomorphisms by Theorem 1 in [10] for each t ∈ [0, T ], z ∈ Z and θ ∈ [0, 1]. Note that by the formula on the derivative of inverse functions a C 1 (R d )-diffeomorphism and its inverse have continuous derivatives up to the same order. This observation finishes the proof of the proposition.
To prove (i) notice that by the definition of generalised derivatives and by Fubini's theorem and (3.9) follows.
For each t ∈ [0, T ] and z ∈ Z define the operators T t,z , I = I t,z and J = J t,z by for ϕ ∈ C ∞ 0 (R d ). By Taylor's formula we have for t ∈ [0, T ], z ∈ Z and p ∈ [1, ∞), where N is a constant depending only on d, K, m, p.
Corollary 3.9. Let Assumptions 2.2 and 2.3 hold. Then for every t, z the operators T t,z , I t,z and J t,z extend to bounded linear operators from W k p to W k p , from W k+1 p to W k p and from W k+2 p to W k p , respectively, for k = 0, 1, 2, ..., m, such that T t,z ϕ, I t,z f and J t,z g are Proof. It is sufficient to prove the proposition for v ∈ C ∞ 0 (R d ). Then clearly, the statement on L with a constant N = N (d, K, T, m, p) is obvious. By Taylor's formula Hence, due to Assumption 2.2, by Lemma 3.2 for k ∈ [2, m] we get which proves (3.16) for M when k ≥ 2. For every ϕ by integration by parts we have (Mv, ϕ) = I 1 + I 2 + I 3 with Proceeding as before, using Assumption 2.3 we get (3.16) for N . Estimates (3.17) can be proved similarly.
Lemma 3.11. Let Assumptions 2.2 and 2.3 hold with m = 0. Then for ϕ ∈ W 1 1 and φ ∈ W 2 1 with a constant N = N (K, d). Proof. The proof of (3.19) is given in [9]. For the convenience of the reader we prove both estimates here. We may assume that t,z,θ and τ −1 t,z,θ denote the inverse of the functions x → x + θξ t,z (x) and x → x + θη t,z (x), respectively. Using (3.12) and (3.13) by change of variables we have Next we present a special case of Theorem 2.1 from [20] on the L p -norm of semimartingales with values in Sobolev spaces, where we use the notation D * α = −D k for α = k = 1, 2, ..., d, and D * 0 = D 0 stands for the identity operator.
The following lemma is a vector-valued version of a special case of Lemma 5.1 from [20]. Its proof is a simple exercise left for the reader.

L p estimates
In this section we prove estimate (2.6) for p = 2 k and m = n for integers k ≥ 1, and n ≥ 0, provided Assumptions 2.1, 2.2 and 2.3 hold with sufficiently high m. To this end we use the formula for W n+2 p -valued solutions u to equation (1.1), which we obtain by an application of Lemma 3.13 with ψ α , D α u and D α Au + D α f in place of ψ α , u α and f α , respectively.
To estimate the right-hand side of (4.1), the crucial result is the following. We prove this theorem after some lemmas. Proof. This lemma can be obtained from general estimates given in [14]. Here we give a direct proof of it. For functions g and h on R d we write g ∼ h if they have identical integrals over R d , and we write g h if g ∼ h +h such that the integral ofh over R d can be estimated by the right-hand side of (4.2). Consider first the case n = 0. It is easy to see that where, and later on, we use the notation g α := D α g for functions g over R d and multinumbers α = α 1 . . . α n . This by virtue of Assumption 2.1 proves (4.2) when n = 0. Let us now estimate Q when n ≥ 1. Then it is easy to see that where α(l) denotes the l-th element of multi-number α, andᾱ(l) is the multi-number we get from α by leaving out its l-th element. Notice that Hence integrating by parts and using Assumption 2.1, with c p = p(p − 2)/4 ≥ 0 we have and p|D n v| p−2 We estimate B by using the simple inequality for every ε > 0 and multi-number α, to get It is well-known, see e.g. [35], that for symmetric matrices V ∈ R d×d and functions a = (a ij ) mapping R d into the space of symmetric non-negative definite d × d matrices, such that the second order derivatives of a are bounded by a constant L, the inequality holds for any l ∈ {1, 2, ..., d}, where N is a constant depending only on L and d. Using this with V ij := D ij vᾱ (l) for each l = 1, 2, ..., n and multi-number α of length n, we get N (d, K, n). Thus, choosing ε sufficiently small in (4.7), from (4.6) we obtain A 0, which proves the lemma.
For the following lemmas recall the definition of the operators I = I t,z and J = J t,z by (3.11), and notice that the identities Proof. Consider first the case n = 0. Then by identity (4.8) Hence integrating over R d , by (3.18) we have Assume now that n ≥ 1 and let α be a multi-number of length n. Let T denote the operator defined by T g(x) = g(x + ξ(x)) on functions g = g(x) of x ∈ R d . Then (Recall that we use the notation g α = D α g for multi-numbers α.) Hence, by induction on the length n of the multi-number of α, we obtain (Iv) α = Iv α + 1≤|β|≤n q α,β T v β , with some polynomial q α,β of {ξ i γ : 1 ≤ |γ| ≤ n, i = 1, ..., d} for each multi-number β of length between 1 and n. The degree of these polynomials is not greater than n, their constant term is zero, and the other coefficients are nonnegative integers. Hence where the repeated multi-numbers α mean summation over |α| = n. By using the same calculation as in (4.11) we have with constants N and N depending only on m, d and p. Integrating here over R d we get (4.10). Proof. Consider first the case n = 0. Then using identity (4.9) and proceeding with the proof in the same way as in the proof of the previous lemma we get Integrating here over R d by (3.19) we have Assume now that n ≥ 1 and let α be a multi-number of length n. Let T g denote the operator defined by T g(x) := g(x + η(x)). Then for (T v) k := D k (T v), (I η v) k := D k (I η v) and (Jv) k := D k (Jv) we have for every k = 1, ..., d, where, and later on within the proof we write I in place of I η to ease notation. Hence by induction on the length of α we get with some polynomials p α,β and q α,β of {η i γ : 1 ≤ |γ| ≤ n, i = 1, ..., d}. The degree of these polynomials is not greater than n, their constant term is zero, the coefficients of each first order term in the polynomials q α,β is also zero, all the other coefficients in p α,β and q α,β are nonnegative integers. Hence we get where repeated α means summation over the multi-numbers α of length n.

Proof of Theorem 4.1. By the definition of
where I and J are defined in (4.10) and (4.13), respectively. Due to Assumption 2.4 by the Cauchy-Schwarz and Hölder inequalities we obtain for v ∈ C ∞ 0 with constants N 1 = N 1 (d, p, K), N 2 (d, p, K, K ξ ) and N 3 (d, p, K, K η ). We show below that these estimates hold also for v ∈ W 1 p . To this end we use the following lemma from [20]. Lemma 5.1. Let (S, S, ν) be a measure space, and let {v n } n∈N be a sequence of real-valued S-measurable functions defined on S such that such that v n → v in the measure ν, and |v n | r dν → |v| r dν.
for some r > 0. Then |v n − v| r dν → 0 as n → ∞. Proof. Let {v n } ∞ n=1 be a sequence of C ∞ 0 functions, which converges in the W 1 p norm to some v ∈ W 1 p as n → ∞. We claim that since D k , J k and J 0 are bounded linear operators from W 1 2 into L 2 , and the inner product (ϕ, φ) in L 2 is continuous in ϕ, φ ∈ L 2 . Assume now that p = 2 k for k ≥ 1. By choosing subsequences we may assume that v n → v also almost surely. Clearly, On the one hand, by Hölder's inequality, Since v n → v in W 1 p , it is easy to see that B for v ∈ W 1 p . Due to Assumption 2.4 we have |Q (4) (v)| ≤ K|v| p Lp . Hence, by taking into account Proposition 5.2 we get a constant N such that which proves u = v.

Existence of a generalised solution.
In the whole subsection we assume that Assumptions 2.1 through 2.5 hold with m ≥ 1 and p ≥ 2. We prove the existence of a solution to equation (1.1) with initial condition u(0) = ψ in several steps below. In the first three steps we assume that p = 2 k for some integer k ≥ 1 and that m is an integer. We construct a solution u in L p ([0, T ], W m p ) by approximation procedures, and estimate its norm in L p ([0, T ], W s p ) for integers s = 0, 1, ..., m (for p = 2 k ) by the right-hand side of (2.6). Hence, using standard results from interpolation theory we prove the existence of a generalised solution u ∈ L p ([0, T ], V m p ) when p ≥ 2 and m ≥ 1 are any real numbers. Moreover, we show that u ∈ C([0, T ], V s p ) ∩ C w ([0, T ], V m p ) for every s < m, and obtain also the estimate (2.6). We note that similar interpolation arguments are used in [13] to obtain estimates in L p -spaces for solutions of stochastic finite difference schemes.
Step 1. First, in addition to Assumptions 2.1, 2.2, 2.3 and 2.5, we assume that ψ and f are compactly supported, and that µ(Z) < ∞ and ν(Z) < ∞. Under these assumptions we approximate the Cauchy problem (1.1) with initial condition u(0) = ψ by smoothing the data and the coefficients in the problem. Recall that for ε > 0 and functions v on R d the notation v (ε) means the mollification v (ε) = S ε v of v defined in (3.5). We consider the Cauchy problem for ε ∈ (0, ε 0 ), where ε 0 is given in Corollary 3.6, and with operator R ε = S ε R and operators L ε , M ε and N ε , defined by for ϕ ∈ C ∞ 0 (R d ). (Recall that I denotes the d × d unit matrix.) Since ψ (ε) and f (ε) are compactly supported, they belong to W n 2 for every n ≥ 0. By standard results of the L 2 -theory of parabolic PDEs, (5.4)-(5.5) has a unique solution u ε , which is a continuous W n 2 -valued function of t ∈ [0, T ] for every n ≥ 0 (see, e.g., [21] or [39]). Thus for any ϕ ∈ C ∞ 0 we have (u ε (t), ϕ) = (ψ (ε) , ϕ) for t ∈ [0, T ], where J i ε and J 0 ε are defined as J i and J 0 , respectively in (2.3), but with η k(ε) and η l(ε) k in place of η k and η l k , respectively, for k, l = 1, 2, ..., d. Notice that (5.6) can be rewritten as and, equivalently, as for all multi-numbers α of length n. By Sobolev embedding u ε is a continuous W n p -valued function for every n ≥ 0 and p ≥ 2. Hence by Lemma 3.13 we have for p = 2 k , which by Theorem 4.1, known properties of mollifications and Young's inequality gives for every n ≥ 0 and p = 2 k , for integers k ≥ 1, with a constant N = N (T, p, d, n, K, K ξ , K η ). For r > 1 and p ≥ 2 we denote by W n p,r the space of W n p -valued functions v of t ∈ [0, T ] such that We use also the notation W n p and L p for W n p,p and W 0 p,p , respectively. Observe that with this norm W n p,r is a reflexive Banach space, and from (5.7) we have for all ε ∈ (0, ε 0 ), p = 2 k , r > 1 and n = 0, 1, 2, ..., m, with a constant N depending only on T , p, d, m, K, K ξ and K η . Hence there exists a sequence of positive numbers {ε k } k∈N such that ε k → 0 for k → ∞, and u ε k converges weakly to a function u in W n p,r for every n = 0, 1, . . . , m and integers r > 1. From (5.8) we get Our aim now is to pass to the limit in equation (5.6) along ε k → 0. To this end we take a real-valued bounded Borel function h of t ∈ [0, T ], multiply both sides of equation (5.6) with h(t) and then integrate it against dt over [0, T ]. Thus for a fixed ϕ ∈ C ∞ 0 and taking ε k in place of ε, we obtain where F and F i k are functionals defined for v ∈ W 1 p by For each i define also the functional F i in the same way as F i k is defined above, but with a, b, c, J i , J 0 , N and R in place of a ε k , b (ε k ) , c (ε k ) , J i ε k , J 0 ε k and N ε k , R ε k , respectively. Clearly, due to the boundedness of h we have a constant C such that for all v ∈ W 1 where q = p/(p − 1). This means F ∈ W 1 * p , the Banach space of bounded linear functionals on W 1 p . To take the limit k → ∞ in equation (5.9) we show below that F i k and F i are in W 1 * p , and F i k → F i strongly in (W 1 p ) * , for every i as k → ∞. Since the functions h, a ε ,b (ε) and c (ε) are in magnitude bounded by a constant, by Hölder's inequality we have In the same way we get F 1 ∈ W 1 * p . Since |h| is bounded by a constant, by simple estimates and using Hölder's inequality we have and C is a constant, independent of v. By the change of variable and taking into account that by virtue of Corollary 3.6 |det D(τ (ε) θη ) −1 | is bounded by a constant, uniformly in t, θ, z, we get A ≤ Cµ 1/p (Z)|v| W 1 p with a constant C independent of v. Consequently, there is a constantC such that for all v ∈ W 1 p , i.e., F 2 k ∈ W 1 * p for every k ≥ 1. We can prove in the same way that F 2 , F i k ∈ W 1 * p and F i ∈ W 1 * p for i = 3, 4 and k ≥ 1. It is easy to see that F 5 k ∈ W 1 * p and F 5 ∈ W 1 * p . To prove F 1 k → F 1 notice that since h is bounded by a constant N , we have for all k ≥ 1 with By Hölder's inequality for i = 1, 2, 3, by virtue of Lebesgue's theorem on dominated convergence, which proves that F 1 k → F 1 strongly in W 1 * p as k → ∞. Next notice that by using the boundedness of h and by changing variables we have ) µ(dz) dθ ds, and a constant C, independent of v and k, where , and ϕ i := D i ϕ. By Hölder's inequality where, and later on, we write τ instead of τ θη to ease notation. Hence which implies that for every s, θ, z Hence, taking into account (5.13), we have Thus for every s, θ, z and x Hence, using also (5.13) we have θη (x)| 1−q dx ds, (5.14) By virtue of Corollary 3.6 we can use here Lebesgue's theorem on dominated convergence to get lim k→∞ |γ k θ,z | q Lq = |γ θ,z | q Lq for each θ ∈ (0, 1) and z ∈ Z. Thus by Lemma 5.1 we have lim k→∞ |γ k θ,z − γ θ,z | q Lq = 0 for every (θ, z).
Notice that by (5.14)-(5.15) and by virtue of Corollary 3.6 the function |γ k θ,z − γ θ,z | q Lq of (θ, z) can be estimated by a constant timesη q , which has finite integral with respect to µ. Therefore letting k → ∞ in (5.12) we obtain by Lebesgue's theorem on dominated convergence. We get in the same way Consequently, letting k → ∞ in (5.11) we get which means F 2 k → F 2 strongly in W 1 * p . We can prove similarly that F i k → F i strongly in W 1 * p for i = 3, 4. It is easy to see that this holds also for i = 5. Thus due to the convergence of u ε k to u weakly in W 1 p , we have Clearly, Thus taking k → ∞ in equation (5.9) we obtain This means for every bounded real function h the function u : [0, T ] → W 1 p satisfies the equation . Thus for each ϕ ∈ C ∞ 0 equation (2.5) holds for dt-almost every t ∈ [0, T ]. Hence taking into account that u ∈ L p ([0, T ], W 1 p ), by Lemma 3.12 u has a modification, denoted also by u, which is continuous as an L p -valued function and it is the solution of equation (1.1) with initial value ψ.
Step 2. We are going to dispense with the additional assumption that µ and ν are finite measures, i.e., we assume now that Assumptions 2.1 through 2.5 hold with m ≥ 1 and f (t, x) and ψ(x) vanish for |x| ≥ R for some R > 0.
Since µ and ν are σ-finite, there is a sequence (Z n ) ∞ n=1 of sets Z n ∈ Z such that Z i ⊂ Z i+1 for i ≥ 1, Z = ∪ ∞ n=1 Z n , and µ(Z n ) < ∞ and ν(Z n ) < ∞ for all n. For each n define the measures µ n and ν n by and consider the equation with initial condition u(0) = ψ, where M n and N n are defined as M and N in (1.2) and in (1.3), but with µ n and ν n n place of µ and ν, respectively. By virtue of Step 1 for each n there is a solution u n to this problem in the sense that t 0 (f (s), ϕ)h(t) ds dt (5.18) holds with arbitrary ϕ ∈ C ∞ 0 and h ∈ L ∞ ([0, T ], R), where the functional F is defined by (5.10), as before, and for v ∈ W 1 p . Here J i n and J 0 n are defined as J i and J 0 respectively in (2.3), but with µ n in place of µ. Just as for F 1 k before, we can see that Φ 1 ∈ W 1 * p . Let Φ j be defined as Φ j n for j = 2, 3, 4 above, but with J i and N in place of J i n and N n for i = 0, 1, ..., d in their definition. By Step 1 u n ∈ W j p,r for j = 0, 1, ..., m, p = 2 k for integers k ≥ 1, r ∈ (1, ∞), and for all n |u n | p with a constant N = N (d, p, T, K, m, K η , K ξ ). By virtue of this estimate there is a subsequence of integers n → ∞, such that u n converges weakly to some u in each W j p,r for j = 0, 1, 2, ..., m, p = 2 k for integers k ≥ 1 and integers r > 1. Due to the weak convergence estimate (5.19) remains valid also for u with the same constant N , i.e. we have Observe that due to the boundedness of h there is a constant C, independent of v ∈ W 1 p and n, such that Hence, taking into account that with a constant N  Proof. Since S is dense in [0, T ], for a given t ∈ [0, T ] there is a sequence {t n } ∞ n=1 with elements in S such that t n → t. Due to sup n∈N |f (t n )| V < ∞ and the reflexivity of V there is a subsequence {t n k } such that f (t n k ) converges weakly in V to some element v ∈ V . Since f is weakly continuous in U , for every continuous linear functional ϕ over U we have lim k→∞ ϕ(f (t n k )) = ϕ(f (t)). Since the restriction of ϕ in V is a continuous functional over V we have lim k→∞ ϕ(f (t n k )) = ϕ(v). Hence f (t) = v, which proves that f is a V -valued function. Moreover, by taking into account that Let φ be a continuous linear functional over V . Due to the reflexivity of V , the dual U * of the space U is densely embedded into V * , the dual of V . Thus for φ ∈ V * and ε > 0 there is φ ε ∈ U * such that |φ − φ ε | V * ≤ ε. Hence ≤ |φ ε (f (t) − f (t n ))| + ε|f (t) − f (t n )| V ≤ |φ ε (f (t) − f (t n ))| + 2εK. Letting here n → ∞ and then ε → 0, we get Indeed, since we know 2 ∈ U, a repeated application of this proposition gives U = [2, ∞), i.e., the uniqueness holds for every p ≥ 2 when R = 0. Now we show that hence the uniqueness of the solution also in the general case. To this end consider the full equation To prove the proposition we use for every R ≥ 1 a smooth cutting function χ R , defined by χ R (x) = χ(x/R), x ∈ R d , where χ is a nonnegative smooth function on R d such that χ(x) = 1 for |x| ≤ 1 and it vanishes for |x| ≥ 2. We introduce also linear operators F R = F R (t) and G R = G R (t) defined on W 1 p for each t ∈ [0, T ] as follows: where T ξ ϕ(x) = ϕ(x + ξ t,z (x)), I ξ ϕ(x) = ϕ(x + ξ t,z (x)) and I η ϕ(x) = ϕ(x + η t,z (x)) for x ∈ R d and each t ∈ [0, T ], z ∈ Z, for functions ϕ on R d . We will prove Proposition 5.4 by the following lemma.
Proof of Proposition 5.4. Let q ∈ U, p ∈ [q, q + 1/d], assume that Assumptions 2.1 through 2.4 hold with p, and let u i ∈ W 1 p be generalised solution to equation (1.1) (with R = 0) with initial condition u i (0) = ψ for i = 1, 2. Then v := u 1 − u 2 is a generalised solution of (1.1) (with R = 0), v(0) = 0 and f = 0. We want to get an equation for v R := vχ R . To this end notice first that for w ∈ W 2 p we have where the operators F R and G R are defined in (5.26). Clearly, v R ∈ W 1 q . Hence, by using test functions χ R ϕ instead of ϕ in (2.5), it is not difficult to see that v R is a generalised solution of dv R (t) = (Av R (t) + f R (t)) dt, v R (0) = 0, which by the previous lemma belongs to L q = L q ([0, T ], L q ). Since q ∈ U, this solution is unique. Consequently, we can apply the estimate we have for the solutions constructed in the existence proof, according to which we have |v R | Lq ≤ N |f R | Lq with a constant N independent of R. Using Lemma 5.5 it is easy to see that there is a constant N such that |f R | Lq ≤ N R r−1 |v| Lp , for all R ≥ 1. Hence |v R | Lq ≤ N N R r−1 |v| Lp (5.28) for all R ≥ 1. Notice that for p ∈ [q, q + d −1 ] we have Consequently, letting R → ∞ in (5.28) we get |v| Lq = 0, which finishes the proof of the proposition.