1. Introduction We are going to investigate the equation ut(t, x ... - ICMS

THE HEAT EQUATION IN L q ((0,T),L p )-SPACES WITH WEIGHTSN.V. KRYLOVSIAM J. on Math. Anal., Vol. 32, No. 5 (2001), 1117-1141.1. IntroductionWe are going to investigate the equationu t (t, x) =a ij (t)u xi xj(t, x)+f(t, x) (1.1)in the domainsR×R d = {(t, x) :t ∈ R,x∈ R d }, R×R d + =(R×R d )∩(x 1 > 0), R + ×R d , R + ×R d +,where a ij are bounded measurable functions of t ∈ R satisfying the uniform ellipticitycondition:K|λ| 2 ≥ a ij (t)λ i λ j ≥ δ|λ| 2 . (1.2)Equation (1.1) is understood in the sense of generalized functions only with respectto x. In other words, say in the case of the whole R × R d by a solution of (1.1) we1

2mean a function u(t), t ∈ R, taking values in the set of generalized functions on R dsuch that, for any t, s ∈ R satisfying t ≥ s and test function ϕ ∈ C ∞ 0 (R d ),wehave(u(t),ϕ)=(u(s),ϕ)+∫ ts[a ij (r)(u(r),ϕ xi xj)+(f(r),ϕ)] dr.The emphasis is on proving solvability in function spaces of Sobolev type with differentpowers of summability p and q with respect to x and t. This issue arose from the theoryof stochastic partial differential equations in domains in Sobolev spaces with weightsand it turns out that, in this theory, the spaces with weights are the only reasonableones where to look for solutions to equations in domains. To illustrate this necessityconsideru t = u xx in R 2 + with u(0,x)=0,u(t, 0) = g(t),where g is a bounded nondifferentiable function. Thenu xx cannot be continuous sinceotherwise0=u(t, 0) − u(0, 0) = g(t) − g(0) =∫ t0u xx (s, 0) ds.This equation arrises from the SPDEu(t, x) − u(0,x)=∫ t0u xx (s, x) ds + w t ,where w t is a one-dimensional Wiener process. Hence comes the need of weights.

Next, equation (1.1) will be considered in L q ((0,T),L p )-spaces, so that the powersof summability with respect to x and t are different.Surprisingly enough, to the best of my knowledge, the case q ≠ p was never addressed3before even for the heat equation in R × R d without weights.We could only findreferences [1] and [12], where different powers of summability can be related to theCauchy problem in {t >0} for f =0. It turns out that in R × R d the result we needcan be obtained quite easily on the basis of a Banach space version of the Calderón-Zygmund theorem (see Sec. 2), which allows one to pass from q = p to q ≠ p.For equation (1.1) in R×R d + we impose zero boundary condition and look for solutionsin weighted Sobolev spaces with weights allowing the spatial derivatives of solutions toblow up near the boundary x 1 =0. Our results in this setting extend the correspondingresults in [5], where q = p.This time we again use the Calderón-Zygmund theorem starting with the results validfor q = p. However, in order to check the conditions of this theorem, we need somenontrivial properties of the heat semigroup in weighted spaces, which we prove in Sec. 4.In Sec. 5, this allow us to get the result for R×R d +, but only if a 1j ≡ 0 for j ≥ 2. Usually,if one proves a sufficiently strong result for the heat equation, the same or very close

4result can be proved for equations with variable continuous coefficients. However, inthe main application, which we have in mind, to stochastic partial differential equationsfrom filtering theory, the regularity of a ij in time is hard to control. Therefore, wealways deal only with measurable coefficients and, in Sec. 6 after some additional work,we prove our main result for equations in R × R d +in full generality.The arguments of Sec. 6 are based on Lemma 2.5, which also allows us to give adifferent proof of the result in R × R d by using the Marcinkiewicz interpolation theoremrather than the Calderón-Zygmund theorem.The proofs of part of the results are based on a general theorem saying, roughlyspeaking that, whatever estimate is true for the heat equation, it is also true for parabolicequations with coefficients depending only on time. This theorem is equally applicableto Sobolev and Hölder spaces.Finally, it is worth noting that we also give results for the initial value problems. Togive the reader a flavor of our results in R×R d + we state a particular case of Theorem 5.2.

Theorem 1.1. Let p, q ∈ (1, ∞), −1

6This allows us to reduce estimating the L np (L p )-norm of u(t, x) to estimating L p -normof the function u(t, x 1 ) · ... · u(t, x n ) which happens to satisfy the heat equation in ahigher dimension space. It turns out that this device works equally well for stochasticpartial differential equations.DefineLu = a ij u xi x j − u t, (2.1)L p = L p (R d ), H γ p =(1− ∆)−γ/2 L p , H γ,qp = L q (R,H γ p ),H γ,qp (T )=L q((0,T),H γ p ), Lq p = H0,q p , Lq p(T )=H0,q p(T ).It is well known that, for any s ∈ Randf ∈ C ∞ 0 (R d ), there exists a unique boundedcontinuous function u(t, x) on [s, ∞) × R d satisfying Lu(t) =0for t>swith initialcondition u(s) =f. Wedenoteu(t) =T s,t fand recall that, for each s and t, theoperatorT s,t is written as the convolution of fwith a Gaussian density. In particular, T s,t f is infinitely differentiable in x. Finally, forf ∈ C ∞ 0 (R × R d ),letRf(t) :=∫ t−∞T s,t f(s) ds,Af := D 2 Rf.

7Also remember that, for f ∈ C ∞ 0 (R × Rd ), the function Rf satisfiesLRf = −f.Theorem 2.1. Let q, p ∈ (1, ∞), γ ∈ R.Then the operator A is uniquely extendibleto a bounded operator acting in H γ,qp . If we keep the same notation for theextension, then||D 2 Rf|| Hγ,qp≤ N(δ, d, q, p)||f|| Hγ,qp . (2.2)Let us emphasize that the constant N in (2.2) is independent of K (see (1.2)).The following is a corollary of Theorem 2.1.Theorem 2.2. Let q, p ∈ (1, ∞), T ∈ (0, ∞), and γ ∈ R. Take ε>0, f ∈ H γ,qp (T )and u 0 ∈ H γ+2−2/q+εp. Then in H γ+2,q (T ) there is a unique solution of (1.1) withthe initial condition u(0) = u 0 . For this solutionp||u xx || Hγ,qp(T ) ≤ N(||f|| Hγ,qp(T ) + ||u 0 || γ+2−2/q+εH), (2.3)pwhere N = N(δ, d, q, p, T, ε), and if u 0 =0, then N is independent of T . Finally,if q = p, one can take ε =0.

8The following corollary of Theorem 2.2 is obtained by odd continuation of the functionsinvolved.Corollary 2.3. All assertions of Theorem 2.2 hold true for γ =0if we replace R dwith R d + everywhere, assume that a 1j ≡ 0 for j =2, ..., d, and supplement equation(1.1) with zero boundary condition at x 1 =0.To prove Theorem 2.1, we use the following Banach space version of the Calderón-Zygmund theorem. This a standard result which is discussed, for instance, in Chapter1 of [10] and can be extracted from more general results of [1]. For a Hilbert spaceversion of this theorem in the form of multipliers along with a version of Theorem 2.1for q = p and different operators L we refer the reader to [9].Theorem 2.4. Let F and G be Banach spaces, p ∈ (1, ∞), and A : L p (R n ,F) →L p (R n ,G) be a linear bounded operator. Assume that if a bounded strongly measurableF -valued function f has compact support Γ, then, for almost any x ∉ Γ, wehave∫Af(x) = K(x, y)f(y) dy,R n

where K(x, y) is a bounded operator from F into G, defined for x ≠ y, stronglymeasurable with respect to y with norm bounded in y outside any neighborhood of x.Also assume that K(x, y) is strongly measurable with respect to x and there existsa constant N such that∫|K(x, y) − K(x, z)| dx ≤ N|x−y|>2|y−z|for any y and z, which holds, for instance, if K(x, y) is weakly differentiable in yand|∇ y K(x, y)| ≤N|x − y| −d−1 .9Then the operator A is uniquely extendible to a bounded operator from L q (R n ,F)to L q (R n ,G) for any q ∈ (1,p] and A is of weak-type (1, 1) on bounded functionswith compact support.The first proof of Theorem 2.1.In [3] a general theorem is proved whichroughly speaking says that, whatever estimate is true for the heat equation in translationinvariant spaces, it is also true with the same constant for equation (1.1) with thecoefficients depending only on t provided(a ij (t)) ≥ (δ ij ).

10Therefore we may and will assume that a ij ≡ δ ij . Also assuming γ =0does notrestrict generality.Now we are ready to use Theorem 2.4. From Sec. 4.3 of [8] we know that A isuniquely defined and is bounded as an operator acting in L p (R,L p ). We are going tocheck that A satisfies the assumptions of Theorem 2.4 with F = G = L p .Observe the simple fact that, for t>0, k =1, 2, ..., andf ∈ L p ,wehave∂T t f/∂t =∆T t f and ||∂ k T t f/∂t k || Lp ≤ Nt −k ||f|| Lp ,where N depends only on d and k. Fort>0 introduce the operatorK(t) =∆T t : L p → L pwith norm bounded by Nt −1 ,whereN is independent of t. Fort ≤ 0, letK(t) =0.Since a ij = δ ij ,wehaveRf(t, x) =∫ t−∞T t−s f(s, ·)(x) ds.In addition, if t is at a distance from the support of f, then differentiating the aboveformula presents no difficulties and we findAf(t, x) =∫ t−∞∆T t−s f(s, ·) dx =∫RK(t − s)f(s)(x) ds.

In order to prove that the assumptions of Theorem 2.4 are satisfied, it only remains touse11||∂K(t − s)f/∂s|| Lp = ||∂ 2 T t−s f/∂s 2 || Lp I t>s ≤ N|t − s| −2 ||f|| Lp .By Theorem 2.4, A is well defined and bounded as an operator from L q (R,L p ) intoitself for 1

12by using self similarity. In almost the same form as stated this lemma is proved inAppendix in [11].In the next lemma we do the first step to considering the power of summability in tequal to multiples of p.Lemma 2.6. Let T ≤∞, p ∈ (1, ∞), n =1, 2, .... For k =1, ..., n, let u k∈H 2,pp (T ) be solutions of the equationu k t = aij u k x i x j + f kwith zero initial data and with f k ∈ L p p (T ). Then∫ T0n∏||∆u k (t)|| p L pdt ≤ Nk=1n∑∫ Tk=10∏||f k (t)|| p L p||∆u j (t)|| p L pdt, (2.4)j≠kwhere N = N(n, d, p, δ).Proof. Define v k =∆u k .ForX =(x 1 , ..., x n ) ∈ R nd with x i ∈ R d ,defineV (t, X) =v 1 (t, x 1 ) · ... · v n (t, x n ).Observe thatV t (t, X) =LV (t, X)+F (t, X),

where LV = a rs (V xr1 x s 1 + ... + V x r nx s n ),13F (t, X) =∆ xi G i (t, X),G i (t, X) =f i (t, x i ) ∏ j≠iv j (t, x j ).Hence by Lemma 2.5||V || Lp ((0,T )×R nd ) ≤ N ∑ i||G i || Lp ((0,T )×R nd )and this is exactly (2.4). The lemma is proved.The second proof of Theorem 2.1.As in the first proof, we only have toconsider the case a ij ≡ δ ij and γ =0. Also obviously it suffices to prove (2.2) forf ∈ C ∞ 0 (R × Rd ). Without loss of generality we assume that f(t) =0for t ≤ 0.Let u = Rf. Thenu is a classical solution ofu t =∆u + ffor t>0 with zero initial condition and, even more than that, u(s) =0for s ≤ 0. Inaddition, it is easy to check that u ∈ L p ((0,T),H 2 p) for each T

14which by Hölder’s inequality yields ||u xx || Lnp ((0,T ),L p ) ≤ N||f|| Lnp ((0,T ),L p ). By lettingT →∞we obtain (2.2).To treat general q ≥ p, it suffices to use the Marcinkiewicz interpolation theorem.As in the first proof, the case q ≤ p is considered by duality. The theorem is proved.3. Sobolev spaces with weightsM α is the operator of multiplying by (x 1 ) α , M = M 1 .D(R d +) the space of all distributions on R d +.L p,θ := H 0 p,θ = L p (R d +, (x 1 ) θ−d dx).If γ is a nonnegative integer, the space H γ p,θ is{u : u, x 1 u x ,... ,(x 1 ) |α| D α u ∈ L p,θ ∀α : |α| ≤γ}.For negative γ one may use duality and for all other γ the complex interpolation.One may think that it is natural to replace (x 1 ) |α| with (x 1 ) α 1 ...

Definition 3.1. Take and fix a nonnegative function ζ ∈ C ∞ 0 (R + ) such15that∞∑n=−∞ζ p (e x−n ) ≥ 1 ∀x ∈ R.For γ,θ ∈ R, and p ∈ (1, ∞) let H γ p,θbe the set of all distributions uon R d + such that||u|| p H γ p,θ∞∑:= e nθ ||u(e n ·)ζ|| p H < ∞.p γn=−∞Independence of ζ...The space C ∞ 0 (Rd + ) is dense in Hγ p,θ .Theorem 3.2. (i) For any a>0 and α ∈ R,||u(a ·)|| p H γ p,θ≤ a −θ N||u|| p H γ p,θ≤ N||u(a ·)|| p H γ p,θ,||M α u|| Hγp,θ ≤ N||u|| H γ p,θ+pα ≤ N||M α u|| Hγp,θ ,||MDu|| Hγ + ||DMu|| p,θH γ ≤ N||u|| p,θ H γ+1 .p,θ(ii) If M −1 u ∈ H γ p,θand θ ≠ d − 1,d− 1+p, then||MD 2 u|| Hγ−2p,θ≤ N||M −1 u|| Hγ ≤ N||MD2 u||p,θ Hγ−2. (3.1)p,θ

16(iii) Let µ ≤ γ and q ≥ p be such thatγ − d/p = µ − d/q.Denoteτ = θq/p(so that τ/q = θ/p). Then for any u ∈ H γ p,θwe haveu ∈ H µ q,τ, ||u|| Hµq,τ≤ N||u|| Hγp,θ .(iv) Assume γp > d and represent γ − d/p as k + ε, where k is aninteger and ε ∈ (0, 1]. Leti, j be multi–indices such that |i| ≤k, |j| = k.Then for any u ∈ H γ p,θ ,wehaveM |i|+θ/p D i u ∈ C(R d +), M k+ε+θ/p D j u ∈C ε loc(R d +),||M |i|+θ/p D i u|| C(Rd+ ) ≤ N||u|| γ,p,θ ,[M k+ε+θ/p D j u] Cε (R d +) ≤ N||u|| γ,p,θ ,where C ε (R d + ) is the Zygmund space (coinciding with Cε (R d + ) if ε ∈ (0, 1)).

Theorem 3.3. (i) We have17(H γ p,θ )′ = H γ′p ′ ,θ ′ , γ ′ = −γ, 1/p +1/p ′ =1, θ/p+ θ ′ /p ′ = d.(ii) For κ ∈ (0, 1), p ∈ (1, ∞), γ i ,θ i ∈ R, i =0, 1,θ = κθ 1 +(1− κ)θ 0 , γ = κγ 1 +(1− κ)γ 0 ,we have [H γ 0p,θ 0,H γ 1p,θ 1] κ = H γ p,θ .For constant b, c defineL b,c = M 2 ∆+bMD 1 − c.It turns out that the fractional powers of the degenerate operator −L b,c can be definedandH γ p,θ =(−L b,c) −γ/2 L p,θ .

18Theorem 3.4. For any γ,ν,p,θ,b, there exist a constant c 0 > 0 suchthat, for any c ≥ c 0 , the operator(−L b,c ) γ : H 2γ+νp,θ→ H ν p,θis bounded and, for any u ∈ H γ+νp,θ,wehave||u|| 2γ+ν,p,θ ≤ N||(−L b,c ) γ u|| ν,p,θ ≤ N||u|| 2γ+ν,p,θ . (3.2)The main difficulty of the theory is that ∆ and L b,c do not commute.Theorem 3.5. For any b, γ, p, and θ, the operator L b,0 is the generatorof an analytic semigroup T bt acting in H γ p,θ .Notice that, if d =1, the theorem is quite simple. Indeed,(D 2 +(b − 1)D)(v(e x )) = (L b,0 v)(e x ),so that the properties of the semigroup related to L b,0 can be easily obtained from thewell-known properties of the semigroup related to the operator D 2 +(b − 1)D withconstant coefficients. However, for d ≥ 2, we do not know any easy way to deal withL b,0 .

194. Some smoothing properties of the heat semigroup in spaces H γ p,θLetD 2 = D 2lx 1D2−2l x ′be a second order derivative operator, where l ∈{0, 1}, D 2lx 1is the operator of taking2l derivative in x 1 ,andD 2−2lx ′ is an (2 − 2l)th derivative with respect to x ′ .Denote by ˜T t the semigroup associated with the operator ∆ in R d + with zero boundarycondition on {x 1 =0}.The following is a corollary of a general theorem, which shows that the solutions ofthe Cauchy problem for equation (1.1) are “naturally smoother” than the initial data.This result interesting in its own right plays a central role in Sec. 5 in proving solvabilityof parabolic equations in weighted spaces.For t ≥ s introduce the operator ˜T s,t so that, for f ∈ C ∞ 0 (Rd + ), ˜T s,t f is the solutionof the equationu t (t, x) =a ij (t)u xi xj(t, x)for t>sand x ∈ R d + satisfying u(s, x) =f(x) with zero boundary condition at x 1 =0.One has a very well known representation of the kernel of ˜T s,t as the difference of certainGaussian densities.

22in our range of θ.theorem forTherefore, instead of consideringĀ = M∆ ˜RM −1 f.Ã we may and will prove theNotice that, if f ∈ C ∞ 0 (R × Rd ) and t is not in the support of f(s) as a function of s,then ˜T s,t M −1 f(s) is infinitely differentiable in x andĀf(t) =M∆ ˜RM −1 f(t) =∫ t−∞M∆ ˜T s,t M −1 f(s) ds. (5.3)Moreover, the boundedness of Ā as an operator from L p (R,H γ p,θ ) to L p(R,H γ p,θ ) andpoinwise estimates of the operator M∆ ˜T s,t M −1 show that (5.3) holds for almost all toutside the support of f if f is a bounded H γ p,θ-valued function with compact support.In other words, for those t,∫Āf(t) =RK(t, s)f(s) ds,where the operator K(t, s) is defined by the formulaK(t, s)h = I t>s M∆ ˜T s,t M −1 h.Observe that by a general theorem, K(t, s) is a bounded operator from H γ p,θinto itselfwith norm less than N|t − s| −1 .

Now we claim that Ā is a bounded operator from L q (R,H γ p,θ ) to L q(R,H γ p,θ ) for1

24Theorem 5.2. Let p, q ∈ (1, ∞), T ∈ (0, ∞), γ ∈ R,d − 1

Proof. Clearly (6.1) becomes stronger if ν decreases. Therefore we may assume thatν = γ − k, wherek is an integer, and bearing in mind an obvious induction, we seethat, without loss of generality, we may let ν = γ − 1.Now notice that25||M −1 u|| npH γ,npp,θ∫=R∫||M −1 u(t)|| np dt ≤ N ||u(t)|| np dtH γ p,θH γ p,θ−pR∞∑∫ n∏= Ne (θ−p)¯m ||u(t, e mi ·)ζ|| p Hp γm 1 ,...,m n =−∞Ri=1dt, (6.2)with ¯m := m 1 + ... + m n .Here||u(t, e m ·)ζ|| p H γ p≤ N||∆[u(t, em ·)ζ]|| p H γ−2p= ||(1 − ∆) γ/2−1 ∆[u(t, e m ·)ζ]|| p L p= e m(pγ−d) ||(λ m − ∆) γ/2−1 ∆[u(t)ζ m ]|| p L p,where λ m = e −2m and ζ m (x) =ζ(e −m x). Furthermore,L(uζ m )= ¯f m , where,¯f m = fζ m +2a ij ζ mx iu xj + ua 11 ζ mx1 x 1,and similarly to the above computation||(λ m − ∆) γ/2−1 ¯fm (t)|| p L p= e −m(pγ−2p−d) || ¯f m (t, e m ·)|| p .Hpγ−2

26Therefore, by Lemma 2.6, which is obviously valid for R in place of (0,T), foranym 1 , ..., m n ,wehave∫n∏||u(t, e mi ·)ζ|| p H γ p dt ≤ NRi=1∫Rn∑i=1e 2m ip || ¯f mi (t, e mi ·)|| p H γ−2p∏||u(t, e mj ·)ζ|| p H dt.p γj≠iComing back to (6.2), we conclude||M −1 u|| npH γ,npp,θ∫≤ NRF (t)||u(t)|| (n−1)pH γ p,θ−pdt,whereF (t) :=∞∑m=−∞e m(θ+p) || ¯f m (t, e m ·)|| p .Hpγ−2Next we use (see [5]) that the operator M β is a bounded operator from H γ p,θ to Hγ p,θ+βpand that M∇ is a bounded operator from H γ p,θto Hγ−1p,θ. Then we find+N∞∑m=−∞F (t) ≤ N∞∑m=−∞e mθ ||u x (t, e m ·)ζ ′ || p H γ−2p≤ N(||Mf(t)|| p H γ−2p,θe m(θ+p) ||f(t, e m ·)ζ|| p H γ−2p+ N∞∑m=−∞+ ||Mu x (t)|| p H γ−2p,θ−pe m(θ−p) ||u(t, e m ·)ζ ′′ || p H γ−2p+ ||M −1 u(t)|| p )H γ−2p,θ≤ N(||Mf(t)|| p H γ−2p,θ+ ||u(t)|| p )H γ−1p,θ−p

27≤ N(||Mf(t)|| p H γ−2p,θ+ ||M −1 u(t)|| p ).H γ−1p,θThus||M −1 u|| npH γ,npp,θ∫≤ NER(||Mf(t)|| p H γ−2p,θ+ ||M −1 u(t)|| p )||M −1 u(t)|| (n−1)p dt,H γ−1H γ p,θp,θand, to get (6.1) for ν = γ − 1, it only remains to use Hölder’s inequality. The lemmais proved.Lemma 6.2. Let p, q ∈ (1, ∞), d−1

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