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Potential Analysis 20: 237–263, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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Singular Integral Operators, Morrey Spaces and Fine Regularity of Solutions to PDE’s DIAN K. PALAGACHEV1 and LUBOMIRA G. SOFTOVA2

1 Dipartimento di Matematica, Politecnico di Bari, Via E. Orabona, 4, 70125 Bari, Italy (e-mail: [email protected]) 2 Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, ‘Acad. G. Bonchev’ Str., bl. 8, 1113 Sofia, Bulgaria (e-mail: [email protected])

(Received: 30 September 2002; accepted: 4 April 2003) Abstract. Boundedness in Morrey spaces is studied for singular integral operators with kernels of mixed homogeneity and their commutators with multiplication by a BMO-function. The results are applied in obtaining fine (Morrey and Hölder) regularity of strong solutions to higher-order elliptic and parabolic equations with VMO coefficients. Mathematics Subject Classifications (2000): 35B45, 31B10, 42B20, 35B65, 35R05, 46E35. Key words: a’priori estimates, PDE’s with discontinuous coefficients, singular integral operators, fine regularity.

Introduction The article deals with continuity properties in Morrey spaces of singular integral operators and commutators, and their applications to the regularity problem for solutions of linear PDE’s with discontinuous coefficients. Precisely, we are going to study operators of the type k(x; x − y)f (y) dy Kf (x) = P.V. Rn

and commutators K(af ) − aKf , of K and multiplication by a function a(x) k(x; x − y)[a(y) − a(x)]f (y) dy. C[a, f ](x) = P.V. Rn

The kernel k(x; ξ ) is measurable in x ∈ Rn and infinitely smooth in ξ ∈ Rn \{0}, its integral mean over the unit sphere Sn−1 = {ξ ∈ Rn : |ξ | = 1} equals zero and it has suitable homogeneity properties with respect to ξ . Singular integrals of such in the general theory of linear kind arise naturally β a (x)D be a linear partial differential operaPDE’s. Let L(x, D) = |β|=m β tor of order m 2. The regularity problem for a solution u of L, generally distributional, formulates as follows: if Lu belongs to some functional class F,

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DIAN K. PALAGACHEV AND LUBOMIRA G. SOFTOVA

can we say that the highest-order distributional derivatives D m u are locally in F? In the case of elliptic/parabolic operators L with constant coefficients there is an affirmative answer in broad scale of functional classes. To attack variable operators, a classical approach of perturbation about the constant coefficient case applies. Thus, freezing the coefficients at a point x0 leads to the operator L(x0 , D) with constant coefficients which admits at least one fundamental solution by Malgrange–Ehrenprise’s theorem. In the case of uniformly elliptic/parabolic operators, has a singularity at a single point, say zero, and = (x0 ; ξ ). Further, a local solution u(x) of L(x0 , D) supported in the ball Br of radius r > 0 (Rn is supposed to be endowed with a suitable metric) can be expressed as a Newtonian/Gaussian potential with kernel (x0 ; x − y) and density L(y, D)u(y) + (L(x0 , D) − L(y, D))u(y) (see Gilbarg and Trudinger [19] and Ladyzhenskaya, Solonnikov and Ural’tseva [24]). At this point one applies a classical approach due to Calderón and Zygmund based on taking the m-order derivatives of the potential and then unfreezing the aβ (i.e., set x = x0 ). If (x0 ; ξ ) has ‘good’ properties with respect to ξ , which happens in the elliptic/parabolic case, this procedure leads to an expression of D m u(x) in terms of the singular operator K acting on the known term L(y, D)u and the commutator C[aβ , D m u] acting on the very same derivatives D mu. The kernel in both cases is of the form k(x; ξ ) = Dξm (x; ξ ). (Note that uniform ellipticity/parabolicity implies aβ ∈ L∞ and therefore the commutator C is well defined.) Now suppose we want to use that representation formula in order to get a’priori estimates of D m u in F. The simplest case is that of an operator L(D) with constant coefficients aβ = C(β). Then the commutator vanishes and the regularity properties of u(x) in various functional scales (Hölder, Sobolev, Morrey, etc.) depend on these of L(D)u through K(L(D)u). The situation becomes more complicate when dealing with operators of variable coefficients. It is clear that we have to make the norm of C[aβ , D m u] small in order to move

D m u F on the left-hand side of an eventual a’priori inequality, and to realize that aim, boundedness of the aβ is not enough. The presence of [aβ (x) − aβ (y)] in the kernel of C[aβ , D m u], however, suggests a way to impose suitable continuity requirements on the coefficients of L. For example, the case of Hölder continuous coefficients aβ leads to well-known Schauder estimates for classical solutions to elliptic/parabolic equations (see [17, 19, 24]) in the framework of Hölder spaces. If F ≡ Lp , the solution of L(x, D) belongs to a suitable Sobolev space (strong solution) and it turns out that L(x, D)u ∈ Lp implies D m u ∈ Lp for any p in the range (1, +∞) provided the coefficients aβ (x) are only uniformly continuous. This is the essence of Calderón–Zygmund’s theory ([5, 6]) of linear elliptic operators, obtained from the Lp -boundedness of K and C[aβ , ·] established by these authors. The appearance of aβ (x) − aβ (y) in the commutator C[aβ , D m u] suggests replacing the assumption of continuity by continuity in an average sense in order to generalize regularity assumptions on the coefficients. A good basis for such an attempt is the deep result of Coifman, Rochberg and Weiss [12] claiming Lp boundedness of the commutator with ‘constant’ kernel k(x − y), if a(x) belongs

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to John–Nirenberg’s class of functions with bounded mean oscillation (BMO). Moreover, the norm of the commutator is less than the BMO-seminorm a ∗ of a(x). Thus, apart from the fact that in general k depends also on the space variable x, the commutator norm becomes small if a ∗ vanishes as r → 0, and this leads to the Sarason [30] famous class of functions with vanishing mean oscillation (VMO). Second-order elliptic equations with VMO principal coefficients were first studied in the Lp -framework by Chiarenza, Frasca and Longo [10, 11]. Using precise results on Lp -boundedness of commutators with variable kernels k(x; x − y) and a ∈ VMO, these authors gave an affirmative answer of the regularity question for second-order elliptic operators with VMO coefficients for all p ∈ (1, +∞). Applying the same technique, Bramanti and Cerutti [3] obtained similar results concerning uniformly parabolic operators. Indeed, for suitable values of p > p0 (n), the respective strong solutions will have Hölder continuous gradients with norms estimated in terms of the Lp -norm of L(x, D)u, and the utility of such an a’priori estimate is a matter of enormous concern in the theory of nonlinear equations with discontinuous ingredients (cf. Palagachev [27], Maugeri, Palagachev and Softova [25]). Regarding the value of the critical exponent p0 (n), it can be decreased if the highest-order derivatives of u belong to the Morrey space Lp,λ instead of Lp . In fact, Lp,λ possesses better embedding properties into Hölder’s spaces than Lp while, since Lp,λ is a proper subspace of Lp , the already cited results remain valid also if L(x, D)u ∈ Lp,λ . Anyway, a positive answer of the regularity problem in the framework of Morrey spaces requires detailed study of the boundedness properties of K and C in Lp,λ . For what concerns secondorder elliptic operators with VMO coefficients, such results of local character were obtained in Di Fazio and Ragusa [15] by adapting the approach from [10] to the new situation, and in Taylor [31] by the powerful tools of microlocal analysis. We refer the reader also to Di Fazio, Palagachev and Ragusa [16] and Palagachev, Ragusa and Softova [28] where global Morrey regularity was derived for Dirichlet and oblique derivative problems for second-order elliptic operators. The possibility of applying the above procedure depends essentially on the homogeneity properties of the kernel k(x; ξ ) with respect to ξ = (ξ1 , . . . , ξn ). In the case of elliptic operators, k(x; ξ ) is the m-order derivative Dξm (x; ξ ) and has the same homogeneity −1 in any of the components ξi . These are the classical Calderón–Zygmund kernels ([5, 6]). In the case of parabolic operators, however, the respective derivatives of the fundamental solution have the same homogeneity −1 in the space variables but −2 with respect to time. This phenomena caused Fabes and Riviére to introduce in [18] the notion of kernels k(x; ξ ) with mixed homogeneity with respect to the components of ξ and thus to extend Calderón– Zygmund’s results on Lp -boundedness of the operator K to the new case. As a product, Lp -regularity follows for the highest-order derivatives of solutions to parabolic equations with continuous coefficients. Concerning the commutator C with a ∈ VMO and constant kernel k(x − y), its continuity in Lp ∀p ∈ (1, +∞),

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has been derived in [3] whence regularity theory for parabolic equations with VMO-coefficients does follow. The principal object of the present paper is twofold. First of all, we are interested in the continuity properties in Morrey spaces Lp,λ of the singular integrals K and C[a, ·], a ∈ VMO, with variable kernels k(x; ξ ) having mixed homogeneity in ξ . Bearing in mind the above considerations, k depends on x through the coefficients of a differential operator, and if we want to deal with operators with discontinuous ingredients, no more than simple measurability in x would be expected. This lack of regularity in x forces us to employ many deep results (see Torchinsky [32]) concerning singular integral operators. On the other hand, however, the regularity and homogeneity in ξ enable us to apply Calderón–Zygmund’s approach of expansion into spherical harmonics with respect to ξ , arriving thus at expansions of K and C into singular integrals of similar type but with constant kernels. Precise estimates of these yield the desired boundedness of K and C in Morrey spaces. Our second goal is to apply the results obtained in the study of fine (Morrey and Hölder) regularity of local solutions to elliptic and parabolic equations with VMO principal coefficients. To be more precise, given a linear uniformly elliptic operator L(z, D) of order 2m 2 with VMO principal coefficients, we prove that L(x, D)u ∈ Lp,λ implies D 2mu ∈ Lp,λ , generalizing thus the results in [15] and [31]. The approach applies, without essential modifications, also to systems which are elliptic in the sense of Petrovskii. In the particular case λ ∈ (n−p, n) our result yields Hölder continuity of the derivatives D 2m−1 u. Another application regards local regularity in Morrey spaces of solutions to second-order parabolic equations with VMO principal coefficients. The respective results are of the same kind as in the elliptic case and for particular values of p and λ we get local Hölder continuity of the solution or its spatial gradient. Applications to another classes of PDE’s will be proposed in a forthcoming paper.

1. Definitions and Auxiliary Results We are going to study continuity in Morrey spaces of singular integral operators with kernels of Calderón–Zygmund type, but having mixed homogeneity with respect to the respective variables. Let α1 , . . . , αn be real numbers, αj 1 and define α = ni=1 αi . Set Sn−1 for the unit sphere {x ∈ Rn , |x| = 1} in Rn . The kind of kernel under consideration is specified in the following DEFINITION 1.1. homogeneity if:

A function k : Rn \{0} → R is said to be a kernel of mixed

(i) k ∈ C ∞ (Rn \{0}); α1 x1 , . . . , µαn xn ) = µ−α (ii) k(µ k(x) for each µ > 0; (iii) Sn−1 |k(x)| dσx < ∞ and Sn−1 k(x) dσx = 0.


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Let us note that, in the special case αi = 1 and thus α = n, Definition 1.1 gives rise to the classical Calderón–Zygmund kernels (cf. [5, 6]). The mixed homogeneity condition suggests to endow Rn with a metric that takes into account (ii). Thus, following Fabes and Riviére ([18]), the function F (x, ρ) = n 2 2αi , considered for a fixed x ∈ Rn , is a decreasing function of ρ > 0 i=1 xi /ρ and therefore there is a unique solution ρ(x) of the equation F (x, ρ) = 1. It is a simple matter to check that ρ(x − y) defines a distance between any two points x, y ∈ Rn . Thus Rn endowed with the metric ρ is a metric space ([18, Remark 1]). Further, for any x ∈ Rn \{0} one has x1 xn x ∈ Sn−1 . := ,..., x¯ := ρ(x) ρ(x)α1 ρ(x)αn The balls with respect to ρ(x), centered at the origin and of radius r are simply the ellipsoids x12 xn2 n Er (0) = x ∈ R : 2α + · · · + 2α < 1 r 1 r n which are dilations of the n-dimensional Euclidean unit ball with respect to ρ(x). Indeed, E1 (0) and Sn−1 are the unit ball and its surface, respectively, with respect to both ρ(x) and the standard Euclidean metric. One more concept is needed – that of variable kernels. DEFINITION 1.2. We say that a function k(x; y) : Rn × (Rn \{0}) → R is a variable kernel of mixed homogeneity, if: (i) k(x; ·) is a kernel in the sense of Definition 1.1 for almost all fixed x ∈ Rn ; (ii) supρ(y)=1 |(∂/∂y)β k(x; y)| C(β) for every multiindex β, independently of x. For the sake of completeness we shall recall here the definitions and some properties of the spaces we are going to use. DEFINITION 1.3. For a measurable and locally integrable function f : Rn → R set 1 |f (y) − fEr | dy for every R > 0, ηf (R) = sup rR |Er | Er where Er is any ellipsoid in Rn of radius r, fEr is the average |Er |−1 Er f (y) dy and |Er | stands for the measure of Er , comparable to r α . Then: • f ∈ BMO(Rn ) (bounded mean oscillation, see John and Nirenberg [22]) if

f ∗ := supR ηf (R) < +∞. f ∗ is a norm in BMO modulo constant functions under which BMO is a Banach space. • f ∈ VMO(Rn ) (vanishing mean oscillation, cf. Sarason [30]) if f ∈ BMO and limR→0 ηf (R) = 0. The quantity ηf (R) is referred to as a VMO-modulus of f .

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For a given domain % ⊂ Rn , the spaces BMO(%) and VMO(%) are defined in the same fashion, just taking Er ∩ % instead of Er above. DEFINITION 1.4. A measurable function f ∈ Lp (Rn ), p ∈ (1, +∞), belongs to the Morrey space Lp,λ (Rn ) with λ ∈ (0, α), if the following norm is finite 1/p 1 p |f (y)| dy , (1.1)

f p,λ = sup λ r>0 r Er where Er stands for any ellipsoid of radius r. Similarly, the space Lp,λ (%) and the norm f p,λ,% are defined by taking Er ∩ % in (1.1). When λ = 0, Lp,λ coincides with the Lebesgue space Lp and (1.1) gives rise to the norm · p := · Lp . These are Banach spaces and we refer the reader to Morrey [26], Da Prato [14] and Campanato [7] for various embedding properties of Lp,λ (Rn ). For a given measurable function f ∈ L1loc (Rn ), define the Hardy–Littlewood maximal operator 1 |f (y)| dy for almost all x ∈ Rn , Mf (x) = sup E(x) |E(x)| E(x) where the supremum is taken over all ellipsoids E(x) centered at the point x. A variant of Mf is the sharp maximal operator 1 ' |f (y) − fE(x) | dy for almost all x ∈ Rn . f (x) = sup E(x) |E(x)| E(x) The next results will be employed in the forthcoming considerations. LEMMA 1.5 (Maximal inequality). Let p ∈ (1, +∞), λ ∈ (0, α) and f ∈ Lp,λ (Rn ). Then there exists a constant C, independent of f and such that

Mf p,λ C f p,λ . LEMMA 1.6 (Sharp inequality). There exists a constant C, independent of f ∈ Lp,λ (Rn ), such that

f p,λ C f ' p,λ . LEMMA 1.7 (John–Nirenberg type lemma). Let f ∈ BMO(Rn ) and p ∈ [1, +∞). Then 1/p 1 p |f (y) − fE | dy C(p) f ∗ |E| E for any ellipsoid E.


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Lemmas 1.5 and 1.6 in the Lp -settings, and Lemma 1.7 are proved in [2] and [13] in the framework of homogeneous spaces (let us note that (Rn , ρ, dx), dx = ρ α−1 dρ dσ, dσ = the element of area of Sn−1 , is a homogeneous space). The proof of Lemma 1.6 is provided in [9] in the case of Rn endowed with the standard Euclidean metric. To prove Lemmas 1.5 and 1.6 in our situation (Rn , ρ(x)), it suffices to repeat the arguments of [9] making use of the metric ρ(x) and corre+∞ n sponding to it dyadic partition R = 2E ∪ { k=1 2k+1 E\2k E}, where E is a fixed ellipsoid of radius r centered at some point x ∈ Rn , and 2k E stands for the ellipsoid with the same center and of radius 2k r. The details are left to the reader. 2. Singular Integral Estimates in Morrey Spaces Let k(x; y) be a variable kernel of mixed homogeneity, f ∈ Lp,λ (Rn ) with p ∈ (1, +∞) and λ ∈ (0, α), and a ∈ BMO(Rn ). For ε > 0 define the operator Kε and the commutator Cε [a, f ] of Kε and multiplication by a(x), by the setting k(x; x − y)f (y) dy, Kε f (x) := ρ(x−y)>ε

Cε [a, f ](x) := Kε (af )(x) − a(x)Kε f (x) k(x; x − y)[a(y) − a(x)]f (y) dy. = ρ(x−y)>ε

We are going to prove below that Kε and Cε [a, ·] are bounded from Lp,λ (Rn ) into itself, uniformly in ε. This and properties of the kernel k(x; y) will enable us to let ε → 0 obtaining as limits in the Lp,λ (Rn )-topology the singular integrals k(x; x − y)f (y) dy = lim Kε f (x), Kf (x) := P.V. ε→0 Rn k(x; x − y)[a(y) − a(x)]f (y) dy C[a, f ](x) := P.V. Rn

= lim Cε [a, f ](x). ε→0

Moreover, these will result bounded in Lp,λ (Rn ). Let us note that, assuming f ∈ Lp (Rn ) Fabes and Riviére [18] proved existence of Kf as the strong Lp -limit limε→0 Kε f . Moreover, the operator K : Lp (Rn ) → Lp (Rn ) is continuous and this leads to continuity in Lp (Rn ) of C[a, f ] as well, if a(x) is essentially bounded. The case a ∈ BMO(Rn ) is more delicate and further careful analysis of the kernel is needed. Thus, a ∈ BMO and the John–Nirenberg p Lemma 1.7 give a ∈ Lloc (Rn ) for any p ∈ [1, +∞) whence C[a, f ] is well defined p and lies in Lloc (Rn ). However, concerning the commutator C[a, f ], we are going to derive a result similar to that of Coifman, Rochberg and Weiss [12]. That is, C[a, f ] will be bounded from Lp,λ (Rn ) (and therefore on Lp also) into itself in terms of a ∗ . In the framework of homogeneous spaces and constant kernel k, we dispose of [4, Theorem 2.5] which asserts Lp -boundedness of C[a, f ].

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To realize our aim, we will exploit a well-known technique, based on expansion of the variable kernel k(x; y) into spherical harmonics depending on y (cf. [5, 6, 10]). This will allow us to expand Kε f and Cε [a, f ] in Fourier series with respect to integrals and commutators with constant kernels. Due to the properties of the kernels, these operators will turn out to be bounded in Lp,λ uniformly in ε > 0. Thus Kf and C[a, f ] will be defined as limits in Lp,λ of the series expansions of Kε f and Cε [a, f ] as ε → 0. To begin, consider any homogeneous polynomial P : Rn → R of degree m that satisfies *P (x) = 0 (* = the Laplacian). The restriction P (x)|Sn−1 is called an ndimensional spherical harmonic of degree m. Denote by ϒm the linear space of all n-dimensional spherical harmonics of degree m. Indeed, ϒ0 ≡ R, while ϒ1 is the space of all linear functions ni=1 ci xi with ci ∈ R. In general, gm = dim ϒm < ∞ and m+n−1 m+n−3 − C(n)mn−2 (2.1) gm = n−1 n−1 with the second binomial coefficient to be set equal to 0 when m = 0, 1, i.e., gm be an orthonormal base of ϒm . Then g0 = 1, g1 = n. Further, let {Yms (x)}s=1 gm +∞ {Yms }s=1m=0 is a complete orthonormal system in L2 (Sn−1 ) and β ∂ Yms (x) C(n)m|β|+(n−2)/2 , m = 1, 2, . . . . (2.2) sup ∂x x∈Sn−1 If, for instance, φ ∈ C ∞ (Sn−1 ) and m,s bms Yms (x) is the Fourier series expansion gm of φ(x) with respect to {Yms } ( m,s is a substitute for +∞ s=1 ), then m=0 φ(y)Yms (y) dσ, bms = Sn−1 β ∂ −2l sup φ(y) (2.3) |bms | C(n, l)m ∂y |β|=2l y∈Sn−1

for any integer l. In particular, the expansion of φ into spherical harmonics converges uniformly to φ. We are in a position now to formulate our main result concerning existence and boundedness of singular integral operators. THEOREM 2.1. Let k(x; y) be a variable kernel of mixed homogeneity, p ∈ (1, +∞), λ ∈ (0, α) and a ∈ BMO(Rn ). For any f ∈ Lp,λ (Rn ) the singular integrals Kf , C[a, f ] ∈ Lp,λ (Rn ) exist as limits in Lp,λ (Rn ) of Kε f and Cε [a, f ], respectively, as ε → 0. Moreover, the operators K, C[a, ·] : Lp,λ (Rn ) → Lp,λ (Rn ) are bounded

Kf p,λ C f p,λ ,

C[a, f ] p,λ C a ∗ f p,λ

(2.4)

with C = C(n, p, λ, α, k). (The dependence on k is through the constant C(β) in Definition 1.2(ii) for suitable β.)

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Proof. Let x, y ∈ Rn and y¯ = y/ρ(y) ∈ Sn−1 . Keeping in mind the homogeneity properties of the variable kernel k(x; y) and k(x; ·) ∈ C ∞ (Sn−1 ), we get

¯ = bms (x)Yms (y) ¯ ρ(y)α k(x; y) = k(x; y) m,s g

+∞

m in L2 (Sn−1 ), whence by the completeness of {Yms }s=1m=0

−α bms (x)Yms (y), ¯ bms (x) = k(x; y) = ρ(y)

Sn−1

m,s

k(x; y)Y ¯ ms (y) ¯ dσy¯ .

(Note that the summation in m below runs from 1 to +∞ since bs0 = 0 because of Definition 1.1(iii) and ϒ0 ≡ R.) These equalities, Definition 1.2(ii) and (2.3) imply

bms ∞ := bms L∞ (Rn ) C(n, l)m−2l

(2.5)

for any integer l > 1 and we fix hereafter l > (3n − 2)/4. Adopting the expansion of the kernel k(x; y), we get

bms (x)Hms (x − y)f (y) dy, Kε f (x) = ρ(x−y)>ε m,s

Cε [a, f ](x) =

bms (x)Hms (x − y)[a(y) − a(x)]f (y) dy

ρ(x−y)>ε m,s

with Hms (x − y) standing for the kernel Yms (x − y)ρ(x − y)−α . It is easy to check that Hms (·) are kernels in the sense of Definition 1.1. Indeed, (i) and (ii) are trivial while (iii) follows from the fact that Yms (x) is a harmonic and m-homogeneous polynomial, Yms (0) = 0 and the mean value property on spheres for the harmonic functions. In order to get series expansions of Kε f and Cε [a, f ], we let x ∈ Rn and y ∈ Rn be such that ρ(x − y) > ε. Then (2.1), (2.2) and (2.5) yield N g +∞ m

Yms (x − y) |f (y)| −2l+n/2−1+n−2 bms (x) f (y) C(n) m α α ρ(x − y) ρ(x − y) m=1 s=1 m=1 and the series converges since l > (3n − 2)/4 > (3n − 4)/4. Similar inequality holds also for the partial sums of the integrand of Cε [a, f ](x). Thus, the dominated convergence theorem and f (·)/ρ(x − ·)α ∈ L1 (Rn ) for almost all x ∈ Rn yield

bms (x)Kms,ε f (x), (2.6) Kε f (x) = m,s

Cε [a, f ](x) =

m,s

bms (x)Cms,ε [a, f ](x),

(2.7)

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with

Kms,ε f (x) :=

Hms (x − y)f (y) dy, ρ(x−y)>ε

Cms,ε [a, f ](x) :=

Hms (x − y)[a(y) − a(x)]f (y) dy. ρ(x−y)>ε

The crucial step in deriving existence and boundedness of Kf and C[a, f ] consists of letting ε → 0 in the series expansions (2.6) and (2.7). For this goal we have to study continuity in Lp,λ (Rn ) of the singular integral Hms (x − y)f (y) dy Kms f (x) := Rn

with constant kernel Hms (x − y) and of its commutator Cms [a, f ](x) := Hms (x − y)[a(y) − a(x)]f (y) dy. Rn

For boundedness of Kms in Lp (Rn ) we use [18, Theorem II.1] and this implies, through [4, Theorem 2.5], boundedness in Lp (Rn ) of Cms [a, ·] as well, since a ∈ BMO(Rn ). The cited Fabes–Riviére result, however, requires the kernel to have some ‘integral continuity’ expressed in terms of a bound, commonly called Hörmander’s integral condition. It turns out that the kernels Hms (x) satisfy an even stronger condition as we show in the following lemma. LEMMA 2.2 (Pointwise Hörmander’s condition). Let E and 2E be ellipsoids centered at x0 and of radius r and 2r, respectively. Then the kernel Hms (x) satisfies |Hms (x − y) − Hms (x0 − y)| C(n, α)mn/2

ρ(x0 − x) ρ(x0 − y)α+1

(2.8)

for each x ∈ E and y ∈ / 2E. Proof. We will apply the mean value theorem to Hms and therefore a decay estimate for the gradient ∇Hms (x) is needed. Let x ∈ Rn \{0} be an arbitrary point. The implicit function theorem applied to the equation F (x, ρ(x)) = 1 gives an expression for the the gradient ∇ρ(x) and straightforward calculations imply ∂Hms (x) ∂xi

xi ∂Yms 1 ¯ n + (x) ¯ − −αYms (x) = 2 −2α α+α α j i i ρ(x) ∂ x¯i ρ(x) j =1 αj xj ρ(x) n

xi xk ∂Yms αk (x) ¯ . − n 2 −2αj αi ρ(x)αk ∂ x ¯ ρ(x) α x ρ(x) k j j =1 j k=1

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Remembering x¯ ∈ Sn−1 and taking into account (2.2), xi ρ(x)αi and min αi n 2 −2αj max αi , we get j =1 αj xj ρ(x) n/2 ∂Hms C(n, α) m (x) ∂x ρ(x)α+αi i

∀x ∈ Rn \{0}.

(2.9)

Now, taking x ∈ E and y ∈ / 2E, the left-hand side of (2.8) becomes Hms (x − y) − Hms (x0 − y) =

n

∂Hms i=1

∂xi

(ξ )(x − x0 )i

(2.10)

with ξ = t (x − y) + (1 − t)(x0 − y) and some t ∈ (0, 1). Bearing in mind ρ(x − y) r and ρ(x0 − y) 2r, we have ρ(x0 − y) ρ(ξ ) + ρ((x0 − y) − ξ ) = ρ(ξ ) + ρ(t (x0 − y) − t (x − y)) = ρ(ξ ) + tρ(x0 − x) whence ρ(ξ ) > r and 2ρ(ξ ) > ρ(x0 − y). Since (x − x0 )i ρ(x − x0 )αi , (2.9) and (2.10) yield |Hms (x − y) − Hms (x0 − y)| n

ρ(x0 − x)αi C(n, α)mn/2 ρ(ξ )α+αi i=1 C(n, α)m

n/2

n ρ(x0 − x) ρ(ξ ) ρ(x0 − x)αi −1 . ρ(x0 − y)α+1 i=1 ρ(ξ )αi

In order to derive (2.8), it remains to note ρ(x0 − x) r ρ(ξ ).

✷

REMARK 2.3. The right-hand side of (2.8) decays as ρ(x0 −y)−α−1 and therefore the kernel Hms satisfies Hörmander’s integral condition |Hms (x − y) − Hms (y)| dy C {y∈Rn : ρ(y)4ρ(x)}

with a constant C independent of x (see (1.1) in [18]). In view of Remark 2.3, [18, Theorem II.1] and [4, Theorem 2.5], there exist Kms f , Cms [a, f ] ∈ Lp (Rn ) such that lim Kms,ε f − Kms f Lp (Rn ) = lim Cms,ε [a, f ] − Cms [a, f ] Lp (Rn ) = 0.

ε→0

ε→0

The remaining part of the proof of Theorem 2.1 is broken up into a number of lemmas.

248


LEMMA 2.4. The singular integrals Kms f and their commutators Cms [a, f ] satisfy (Kms f )' (x) Cmn/2 (M(|f |p )(x))1/p , (Cms [a, f ])' (x) C a ∗ (M(|Kms f |p )(x))1/p + mn/2 (M(|f |p )(x))1/p ,

(2.11) (2.12)

where the constant depends on n, p, λ, α and k, but not on f . Proof. Fix x0 ∈ Rn and set E for the ellipsoid centered at x0 and of radius r. Then I (x0 , E) 1 |Kms f (y) − (Kms f )E | dy := |E| E 1 |Kms f (y) − Kms,2r f (x0 ) + Kms,2r f (x0 ) − (Kms f )E | dy = |E| E 2 |Kms f (y) − Kms,2r f (x0 )| dy. |E| E Set (2E)c = Rn \2E and write f = fχ2E + fχ(2E)c = f1 + f2 with χ being the characteristic function of the respective set. Hence C C |Kms f1 (y)| dy + |Kms f2 (y) − Kms,2r f (x0 )| dy I (x0 , E) |E| E |E| E =: I1 (x0 , E) + I2 (x0 , E). Remark 2.3 asserts that Kms is a bounded operator from Lp (Rn ) into itself as follows from [18, Theorem II.1] and therefore 1−1/p 1/p C C p dy |Kms f1 (y)| dy

Kms f1 p I1 (x0 , E) |E| E |E|1/p E C

f1 p C(M(|f |p )(x0 ))1/p |E|1/p Cmn/2 (M(|f |p )(x0 ))1/p . Regarding I2 (x0 , E), we have C |Hms (y − ξ ) − Hms (x0 − ξ )||f (ξ )| dξ dy I2 (x0 , E) |E| E (2E)c ρ(x0 − y) n/2 1 |f (ξ )| dξ dy Cm |E| E (2E)c ρ(x0 − ξ )α+1 +∞

|f (ξ )| dξ Cmn/2 r α+1 k+1 E\2k E ρ(x0 − ξ ) 2 k=1

249


Cm

n/2

Cm

n/2

r

+∞

1 k (2 r)α+1

k=1 +∞

1 rα

k=1

1 2k(α+1)

1−1/p

1/p

dξ 2k+1 E

2k+1 E

|2

k+1

E|

|f (ξ )| dξ p

1 |2k+1 E|

1/p

|f (ξ )| dξ p

2k+1 E

Cmn/2 (M(|f |p )(x0 ))1/p , after applying Lemma 2.2 for the ellipsoid E centered at x0 and containing y, while ξ ∈ (2E)c . Taking supE I (x0 , E) and keeping in mind the arbitrariness of x0 , we obtain (2.11) for almost all x ∈ Rn . As concerns the commutator, we shall employ an idea of Stromberg (see [32]) which consists of expressing Cms [a, f ] as a sum of integral operators and estimating their sharp functions. Precisely, Cms [a, f ](x) = Kms (a − aE )f (x) − (a(x) − aE )Kms f (x) = Kms (a − aE )fχ2E (x) + Kms (a − aE )fχ(2E)c (x) − − (a(x) − aE )Kms f (x) =: J1 (x) + J2 (x) + J3 (x). Before proceeding further, let us point out the inequalities |a2E − aE | C(n) a ∗ ∀a ∈ BMO(Rn ), |a2k E − aE | C(n)k a ∗ .

(2.13) (2.14)

Indeed, (2.13) is obvious, while (2.14) follows from (2.13) by running induction. Now, for arbitrary p > 1 and q ∈ (1, p), we have 1 2 |J1 (x) − (J1 )E | dx |Kms (a − aE )fχ2E (x)| dx G1 := |E| E |E| E 1/q (q−1)/q C |Kms (a − aE )fχ2E (x)|q dx dx |E| E E 1/p (p−q)/pq C|E|−1/q |f (y)|p dy |a(y) − aE |pq/(p−q) dy 2E

2E

as follows from [18, Theorem II.1]. Further, (2.13) and Lemma 1.7 yield |a(y) − aE |pq/(p−q) dy 2E pq/(p−q) pq/(p−q) |a(y) − a2E | dy + |a2E − aE | dy C(p, q) 2E 2E 1 pq/(p−q) pq/(p−q) |a(y) − a2E | dy + |2E|C(n) a ∗ C(p, q) |2E| |2E| 2E . C(n, p, q)|2E| a pq/(p−q) ∗

250


Therefore,

G1 C a ∗

1 |2E|

1/p

|f (y)| dy p

C a ∗ (M(|f |p )(x0 ))1/p .

2E

To estimate J2' (x), we proceed in the same way as for I2 (x0 , E). We get 1 2 |J2 (x) − (J2 )E | dx |J2 (x) − J2 (x0 )| dx G2 := |E| E |E| E and the integrand satisfies |J2 (x) − J2 (x0 )| = |Kms (a − aE )fχ(2E)c (x) − Kms (a − aE )fχ(2E)c (x0 )| |Hms (x − y) − Hms (x0 − y)||a(y) − aE ||f (y)| dy (2E)c |a(y) − aE ||f (y)| n/2 Cm ρ(x0 − x) dy ρ(x0 − y)α+1 (2E)c 1/p 1−1/p |f (y)|p |a(y) − aE |p/(p−1) n/2 dy dy Cm r α+1 ρ(x0 − y)α+1 (2E)c ρ(x0 − y) (2E)c as consequence of Lemma 2.2 since x ∈ E, y ∈ (2E)c . Later on, |f (y)|p dy α+1 (2E)c ρ(x0 − y) +∞

|f (y)|p dy = k+1 k ρ(x0 − y)α+1 k=1 2 E\2 E +∞

2α (2k+1 r)α 1 p M(|f |p )(x0 ), |f (y)| dy k r)α+1 |2k+1 E| k+1 (2 r 2 E k=1 while (2.14) and Lemma 1.7 imply |a(y) − aE |p/(p−1) dy ρ(x0 − y)α+1 (2E)c +∞

|a(y) − aE |p/(p−1) dy = k+1 k ρ(x0 − y)α+1 k=1 2 E\2 E +∞

1 |a(y) − aE |p/(p−1) dy k r)α+1 k+1 (2 2 E k=1 C(n, p)

+∞

k=1

= C

1 (2k r)α+1

k p/(p−1)|2k+1 E| a p/(p−1) ∗

+∞ p/(p−1) p/(p−1)

2

a ∗ k

a p/(p−1) . = C(n, α, p) ∗ k r 2 r k=1 α

251


Hence G2 C(n, α, p)mn/2 a ∗ (M(|f |p )(x0 ))1/p . Finally,

1 2 |J3 (x) − (J3 )E | dx |a(x) − aE ||Kms f (x)| dx G3 := |E| E |E| E 1−1/p 1/p 1 1 p/(p−1) p |a(x) − aE | dx |Kms f (x)| dx 2 |E| E |E| E C(p) a ∗ (M(|Kms f |p )(x0 ))1/p .

Summing up G1 , G2 and G3 and taking supE , we get (2.12).

✷

The result just proved and the sharp inequality (Lemma 1.6) yield Lp,λ -boundedness of the integral operator Kms f and its commutator. LEMMA 2.5. The singular operators Kms and their commutators Cms [a, ·] are bounded from Lp,λ (Rn ) into itself, that is,

Kms f p,λ Cmn/2 f p,λ ,

Cms [a, f ] p,λ Cmn/2 a ∗ f p,λ

(2.15) (2.16)

with C = C(n, p, λ, α, k). Proof. We are going to estimate the Lp,λ -norms of the sharp functions of the corresponding operators. Let E be an arbitrary ellipsoid of radius r and q ∈ (1, p). Keeping in mind (2.11), Lemma 1.5 asserts |(Kms f )' (x)|p dx E p/q pn/2 |M(|f |q )(x)|p/q dx Cmpn/2 r λ M(|f |q ) p/q,λ Cm E

p/q

p

Cmpn/2 r λ |f |q p/q,λ Cmpn/2 r λ f p,λ . Dividing by r λ and taking supr>0 , we arrive at

(Kms f )' p,λ Cmn/2 f p,λ which implies (2.15) through Lemma 1.6. The Lp,λ -estimate (2.16) for the commutator follows in the same manner, making use of (2.12) and (2.15), and is left to the reader. ✷ LEMMA 2.6. The nonsingular integral operators Kms,ε and Cms,ε [a, ·] are bounded from Lp,λ (Rn ) into itself, that is,

Kms,ε f p,λ C(n, p, α)mn/2 f p,λ ,

Cms,ε [a, f ] p,λ C(n, p, α)mn/2 a ∗ f p,λ for any f ∈ Lp,λ (Rn ).

(2.17) (2.18)

252


Proof. Let E denote the ellipsoid centered at x ∈ Rn and of radius ε. Then 1 |Kms,ε f (x)| dy Kms,ε f (x) = −1 |2 E| 2−1 E 1 −1 |Kms f (y)| dy + |2 E| 2−1 E 1 |Kms fχE (y)| dy + + −1 |2 E| 2−1 E 1 |Kms fχE c f (y) − Kms,ε f (x)| dy + −1 |2 E| 2−1 E

C M(Kms f )(x) + I1 (x, 2−1 E) + I2 (x, 2−1 E) , where I1 and I2 stand for the terms introduced in the proof of Lemma 2.4, and the same arguments as therein lead to

|Kms,ε f (x)| C M(Kms f )(x) + mn/2 (M(|f |q )(x))1/q for any q ∈ (1, p). It remains to take the Lp,λ -norms of the both sides above and to apply Lemma 1.5 and Lemma 2.5 in order to get (2.17). The commutator estimate (2.18) follows in the same fashion. ✷ Returning to the series expansions (2.6) and (2.7), we are in a position now to complete the proof of Theorem 2.1. First of all, note that gm +∞

m=1 s=1 gm +∞

bms Kms,ε f p,λ C f p,λ

+∞

m−2l+n−2+n/2 ,

m=1

bms Cms,ε [a, f ] p,λ C a ∗ f p,λ

m=1 s=1

+∞

m−2l+n−2+n/2

m=1

as follows from (2.5), (2.1), Lemma 2.6 and l > (3n − 2)/4. Therefore, the series (2.6) and (2.7) are absolutely convergent in Lp,λ (Rn ) uniformly in ε > 0, whence

Kε f p,λ C f p,λ ,

Cε [a, f ] p,λ C a ∗ f p,λ

with C = C(n, p, λ, α, k). Setting

bms (x)Kms f (x), Kf (x) :=

C[a, f ](x) :=

m,s

bms (x)Cms [a, f ](x)

m,s

and applying Lemma 2.5 we obtain as above

Kf p,λ C f p,λ ,

C[a, f ] p,λ C a ∗ f p,λ .

253


Finally, the absolute convergence in Lp,λ (Rn ) of the series expansions (2.6) and (2.7), uniformly in ε > 0, gives

lim Kε f = bms (x) lim Kms,ε f (x) = Kf, ε→0

m,s

lim Cε [a, f ] =

ε→0

ε→0

bms (x) lim Cms,ε [a, f ](x) = C[a, f ] ε→0

m,s

in Lp,λ (Rn ) and this completes the proof of Theorem 2.1.

✷

The result of Theorem 2.1 can immediately be localized. COROLLARY 2.7. Let % be an open subset of Rn and k(x; y) : % × (Rn \{0}) → R a variable kernel of mixed homogeneity, a ∈ BMO(%), p ∈ (1, +∞), λ ∈ (0, α). Then, for any f ∈ Lp,λ (%) and almost all x ∈ %, there exist in Lp,λ (%) singular integral Kf (x) = P.V. k(x; x − y)f (y) dy %

and its commutator

k(x; x − y)[a(y) − a(x)]f (y) dy.

C[a, f ](x) = P.V. %

Moreover, there is a constant C = C(n, p, λ, α, %, k) such that

Kf p,λ,% C f p,λ,% ,

C[a, f ] p,λ,% C a ∗ f p,λ,% .

(2.19)

Corollary 2.7 follows from Theorem 2.1 by defining k(·; y) and f (·) to be zero outside %. One more detail necessary is to extend a in BMO(Rn ) in such way that the BMO norm of the resulting extension is bounded by a constant times a ∗ which can be done using arguments of Jones [23] and Acquistapace [1]. Another outgrowth of Theorem 2.1 is the ‘good behaviour’ of the commutator for VMO-densities a. COROLLARY 2.8. In addition to the assumptions of Corollary 2.7, let a ∈ VMO(%) with VMO-modulus ηa . Then, for each ε > 0 there is r0 = (ε, ηa ) > 0 such that for any r ∈ (0, r0 ) and any ellipsoid Er ⊂ % one has

C[a, f ] p,λ,Er Cε f p,λ,Er

∀f ∈ Lp,λ (Er ).

(2.20)

We refer the reader to [10, Theorem 2.13] for details regarding how Corollary 2.7 implies Corollary 2.8. REMARK 2.9. All the above results remain valid also in the case λ = 0, i.e., in the framework of Lp (= Lp,0 )-spaces. Indeed, continuity of K in Lp (Rn ) is already proved in [18], while Theorem 2.1 with λ = 0 extends to the variablekernel-case the results of [4] on Lp -boundedness of the commutator C[a, f ] with constant kernel.

254


3. Fine Regularity of Solutions to PDE’s We will apply now the results obtained to the study of regularity problem for strong solutions to elliptic and parabolic partial differential equations. ELLIPTIC OPERATORS OF ORDER

2m WITH VMO COEFFICIENTS

Let % be a domain in Rn and consider the following elliptic equation with real measurable coefficients

aβ (x)D β u = f (x), (3.1) L(x, D)u := |β|=2m

where β = (β1 , . . . , βn ), |β| = β1 + · · · + βn , D β = ∂ |β| /∂x1 1 · · · ∂xn n . A function p u(x) is called a local strong solution to (3.1) in % provided u ∈ Lloc (%), p > 1, to2m,p gether with all its weak derivatives up to order 2m (i.e., u(x) ∈ Wloc (%)) and sat2m,p,λ 2m,p isfies (3.1) for almost all x ∈ %. To continue, set Wloc (%) := {u ∈ Wloc (%) : p,λ D 2mu ∈ Lloc (%)} with u W 2m,p,λ (% ) = u W 2m,p (% ) + D 2m u p,λ,% for % %. β

β

REMARK 3.1. It should be noted that Lp,λ (% ) ⊂ Lq,µ (% ) iff 1 q p and (n − µ)/q (n − λ)/p (cf. [29]). Further, assuming ∂% to be sufficiently smooth, Sobolev’s imbedding theorem and Hölder’s inequality assert W 1,p (% ) ⊂ Lp,p (% )

if p < n

while w ∈ W 1,p (% ), Dw ∈ Lp,λ (% ) and the Poincaré inequality imply that w belongs to the Campanato space Lp,p+λ (% ) ([7, Theorem 2.2, Lemma 3.III]) whence w ∈ Lp,p+λ (% ) if p + λ < n. In general, w ∈ W 2m,p (% ), D 2mw ∈ Lp,λ (% ) imply D β w ∈ Lp,λ (% ) for each β, 0 |β| < 2m (see [7]). Indeed, ellipticity of (3.1) means absence of real characteristic surfaces, so

aβ (x)ξ β = 0 for almost all x ∈ %, ∀ξ ∈ Rn \{0}, 0 there is r0 = r0 (ε, ηaβ ) > 0 such that if r < r0 one has

β β

D v p,λ,Br C ε

D v p,λ,Br + L(·, D)v p,λ,Br |β|=2m

|β|=2m

with C = C(n, m, =, p, λ, ηaβ ). Thus, choosing ε such that Cε < 1/2, we get

D β v p,λ,Br C L(·, D)v p,λ,Br (3.4)

D 2m v p,λ,Br := |β|=2m

256

DIAN K. PALAGACHEV AND LUBOMIRA G. SOFTOVA 2m,p

for any v ∈ W0 (Br ) ∩ W 2m,p,λ (Br ), if r < r0 = r0 (n, m, p, λ, =, ηaβ ). p,λ Now, suppose aβ ∈ VMO(%) ∩ L∞ (%), f ∈ Lloc (%) with p ∈ (1, +∞) 2m,p and λ ∈ (0, n), and let u ∈ Wloc (%) be a strong solution to (3.1) such that p,λ p,λ D 2mu ∈ Lloc (%). In order to derive an interior a’priori estimate for the Lloc (%)norm of D 2mu, we choose arbitrary % % % and fix the point x0 ∈ % . Let r < min{dist(% , ∂% ), r0 } be fixed and for θ ∈ (0, 1) consider a cut-off function ϕ ∈ C0∞ (Br ), Br = Br (x0 ), such that 0 ϕ 1, ϕ(x) = 1 if x ∈ Bθr , ϕ(x) = 0 if x ∈ / Bθ r with θ = (1 + θ)/2 > θ, and |D α ϕ| C0 (1 − θ)−|α| r −|α| . Indeed, 2m,p v(x) := u(x)ϕ(x) ∈ W0 (Br ) ∩ W 2m,p,λ (Br ) and thus (3.4) yields

D 2m u p,λ,Bθ r = D 2m v p,λ,Bθ r C L(·, D)v p,λ,Br 2m

2m−k k C ϕL(·, D)u p,λ,Bθ r +

D u p,λ,Bθ r D ϕ L∞ (Br ) k=1

C f p,λ,Br +

2m−1

k=1

+

1

D 2m−k u p,λ,Bθ r + (1 − θ)k r k

1

u p,λ,Bθ r (1 − θ)2m r 2m

with a constant C independent of u. Defining the weighted Morrey seminorms Bk = sup (1 − θ)k r k D k u p,λ,Bθ r ,

k = 0, 1, . . . , 2m,

θ∈(0,1)

the last inequality becomes B2m C r

2m

f p,λ,Br +

2m−1

Bk + B0 .

(3.5)

k=1

To proceed further, we need the following version of the interpolation inequality for the seminorms Bk . PROPOSITION 3.2. For each k = 1, 2, . . . , 2m − 1, there exists a constant C = C (n, m, k, p) such that Bk εB2m +

C ε k/(2m−k)

B0

for any ε > 0. Proof. There is a θ0 ∈ (0, 1) such that Bk 2(1 − θ0 )k r k D k u p,λ,Bθ0 r ,

257


while [19, Theorem 7.28] (see also [24, Chapter II, Lemma 3.3]) imply

D k u p,λ,Bθ0 r µ D 2m u p,λ,Bθ0 r +

C µk/(2m−k)

u p,λ,Bθ0 r ,

µ>0

by means of simple rescaling argument. This way, C k k 2m Bk 2(1 − θ0 ) r µ D u p,λ,Bθ0 r + k/(2m−k) u p,λ,Bθ0 r µ C ε(1 − θ0 )2mr 2m D 2m u p,λ,Bθ0 r + k/(2m−k) u p,λ,Bθ0 r ε C εB2m + k/(2m−k) B0 ε by taking µ = 2ε (1 − θ0 )2m−k r 2m−k .

✷

Returning to (3.5), we interpolate the intermediate seminorms Bk , k = 1, . . . , 2m − 1, in order to get

B2m C r 2m f p,λ,Br + B0 whence

D 2m u p,λ,Bθ r

2m C r f p,λ,Br + u p,λ,Br 2m 2m (1 − θ) r

with θ ∈ (0, 1). Finally, fixing θ = 1/2 and covering % with a finite number of balls each of radius r/2, r < min{dist(% , ∂% ), r0 }, we get

D 2m u p,λ,% C( u p,λ,% + f p,λ,% )

(3.6)

for any % % % with C = C(n, m, p, λ, =, ηaβ , % , dist(% , ∂% )). The representation formula (3.3) allows us also to derive the regularizing property in Morrey spaces of the operator L(x, D) with VMO-coefficients, that is, 2m,q

p,λ

p,λ

u ∈ Wloc (%), q ∈ (1, p], L(x, D)u ∈ Lloc (%) ⇒ D 2mu ∈ Lloc (%).

(3.7)

For, following the ideas in [10], let Br % with radius r smaller than r0 given above, s ∈ (1, +∞), µ ∈ [0, n) and for any couple of multiindices β and γ with |β| = |γ | = 2m, and g ∈ Ls,µ(Br ) set β Dξ (x; x − y)[aγ (x) − aγ (y)]g(y) dy. Sβγ g(x) := P.V. Br s,µ s,µ It follows from Corollary 2.7 that the operators Sβγ : L (Br ) → L (Br ) are well-defined, while Corollary 2.8 ensures β,γ Sβγ < 1 if r < r0 is small enough.

258


Let ϕ ∈ C0∞ (Br ) be a cut-off function such that ϕ ≡ 1 on Br/2 and set v := 2m,q uϕ ∈ W0 (Br ). It follows L(x, D)v = ϕ(x)L(x, D)u + L (x, D)u, where L is a linear differential operator of order 2m − 1. We have Lu ∈ Lp,λ (Br ) by hypothesis. Further, L u ∈ W 1,q (Br ) ⊂ Lq ,λ (Br ) for some q > q and λ > 0. In fact, if q < n one has W 1,q ⊂ Lnq/(n−q) ≡ Lnq/(n−q),0 ⊂ Lq ,λ by virtue of Remark 3.1, with q = (q + nq/(n − q))/2, λ = q/2. If q = n, then L u ∈ W 1,n ⊂ W 1,s ⊂ Lns/(n−s),0 ⊂ Lq ,λ with q = n+1 if s ∈ [n(n + 1)/(2n + 1 − λ), n), while q > n implies W 1,q ⊂ C 0,1−n/q ⊂ C 0 ⊂ Lq ,λ ∀q > 1, ∀λ ∈ (0, n). Therefore, Lv ∈ Lq1 ,λ1 (Br ) with q1 = min{p, q } ∈ [q, p], λ1 = min{λ, λ } ∈ (0, λ], and we are going to show v ∈ W 2m,q1 ,λ1 (Br ). For, Corollary 2.7 yields β Dξ (x; x − y)L(y, D)v(y) dy + Tβ (x) := P.V. Br

+ bβ (x)L(x, D)v(x) ∈ Lq1 ,λ1 (Br )

is the cardinality of the set for any β, |β| = 2m. Further, note that N = 2m+n−1 n−1 B = {β = (β1 , . . . , βn ) : |β| = 2m}. Take arbitrary s ∈ [q, q1 ] and µ ∈ [0, λ1 ], and consider the product space (Ls,µ(Br ))N . Its elements are the vector-valued functions w of entries wβ ∈ Ls,µ (Br ) with β’s running into B. Now define the operator U : (Ls,µ(Br ))N → (Ls,µ(Br ))N by setting U(w) = (Uβ (w))β∈B with

Sβγ (wγ ), β ∈ B, w = (wβ )β∈B . Uβ (w) = Tβ + γ ∈B

Indeed, U isa contraction mapping on (Ls,µ(Br ))N ∀s ∈ [q, q1 ] and ∀µ ∈ [0, λ1 ], ¯ of U in all because of β,γ Sβγ < 1, and thus there is a unique fixed point w s,µ N the spaces (L (Br )) ∀s ∈ [q, q1 ], ∀µ ∈ [0, λ1 ]. On the other hand, (3.3) and Remark 2.9 show that also v := (D β v)β∈B ∈ (Lq,0 (Br ))N is a fixed point of U and since Lq1 ,λ1 (Br ) ⊂ Ls,µ(Br ) ⊂ Lq,0 (Br ), we get D β v = wβ ∈ Lq1 ,λ1 (Br ) for any β ∈ B. Thus, the properties of the cut-off function ϕ imply D β u ∈ Lq1 ,λ1 (Br/2). If q1 < p or λ1 < λ, we have to repeat the above arguments finitely many times in order to get (3.7). Let us recall that all results above were obtained assuming n to be odd. In case ˜ D) = the space dimension n is even, it suffices to consider the operator L(x, 2m 2m L(x, D) + ∂ /∂xn+1 , where xn+1 is a new variable and extend all the functions making them independent of xn+1 (see Douglis and Nirenberg [17] for details). Finally, the known properties of Morrey spaces (cf. [26, 14, 7]) imply Hölder continuity of the derivatives D 2m−1 u for suitable values of p and λ. The results obtained can be summarized as follows. THEOREM 3.3. Let % be a domain of Rn , aβ ∈ VMO(%) ∩ L∞ (%) satisfy (3.2) p,λ and f ∈ Lloc (%) with p ∈ (1, +∞), λ ∈ (0, n). Suppose further that u ∈

259

SINGULAR INTEGRAL OPERATORS 2m,q

2m,p,λ

Wloc (%) is a local strong solution of (3.1) with q ∈ (1, p]. Then u ∈ Wloc (%) and (3.6) holds true. Moreover, if λ ∈ (n − p, n), then the derivatives D 2m−1 u are locally Hölder continuous in % with exponent 1 − (n − λ)/p and sup

x,y∈% ,x=y

|D 2m−1 u(x) − D 2m−1 u(y)| C( u p,λ,% + f p,λ,% ) |x − y|1−(n−λ)/p

for any % % % with C = C(n, m, p, λ, =, ηaβ , dist(% , ∂% )). REMARK 3.4. Without essential difficulties, the above results extend also to systems which are elliptic in the sense of Petrovskii. The case of uniformly elliptic operators L(x, D) of second order provides good advantages and is better studied. Thus, the result of Theorem 3.3 (with m = 1) has been derived in [15] by ‘weighted Lp -approach’, and improved in [31] through microlocal analysis tools. Moreover, the fact that second-order elliptic operators satisfy maximum principle of Aleksandrov–Bakel’man–Pucci type allows one to obtain global (up-to the boundary) Morrey regularity for strong solutions of boundary value problems. The interested reader is referred to [16] for Dirichlet’s problem and to [28] for the case of the regular oblique derivative problem.

SECOND - ORDER PARABOLIC OPERATORS WITH VMO COEFFICIENTS

Due to its general character, Theorem 2.1 is applicable to parabolic equations with discontinuous coefficients as well. In fact, as was already mentioned, kernels of mixed homogeneity were introduced and studied by Fabes and Riviére [18] just to extend Calderón–Zygmund’s results to parabolic operators. Let QT = % × (0, T ) be a cylinder in Rn , of base % ⊂ Rn−1 and height , t) ∈ QT and consider the uniformly T > 0. Set x = (x , t) = (x1 , . . . , xn−1 n−1 ij parabolic operator P (x, D) := Dt − i,j =1 a (x)Dxi xj with real and measurable coefficients. Indeed, uniform parabolicity means existence of a positive constant = such that =−1 |ξ |2

n−1

a ij (x)ξi ξj =|ξ |2

(3.8)

i,j =1

for almost all x ∈ QT , ∀ξ ∈ Rn−1 , and this is certainly satisfied if a ij ∈ L∞ (QT ) and the matrix a(x) = {a ij (x)}n−1 i,j =1 is strictly positive definite. Fix the coefficients at a point x0 = (x0 , t0 ) ∈ QT . Then the fundamental solution of the operator P0 = P (x0 , D) with constant coefficients a ij (x0 ) is given by  Aij (x0 )ξi ξj  (4π τ )(1−n)/2 exp − , τ > 0, (x0 ; θ) = (x0 , t0 ; ξ, τ ) = ij (x )} 4τ det{a 0  0, τ 0. Moreover, the theory of second-order parabolic operators (see [24]) implies C ∞ -regularity of ij (x; θ) with respect to θ ∈ Rn \{0} and also their vanishing property over Sn−1 . In other words, ij (x; θ) are kernels of mixed homogeneity in the sense of Definition 1.2 with α1 √= · · · = αn−1 = 1, |x |2 + |x |4 +4t 2 . According αn = 2, α = n + 1 and therefore ρ(x) = ρ(x , t) = 2 n to what was shown in Section 1, the balls in R with respect to the metric ρ(x) are the ellipsoids (xn−1 )2 (x1 )2 t2 n + ··· + + 4 0. Suppose further a ij ∈ VMO(QT ) ∩ L∞ (QT ) satisfy (3.8) and f ∈ Lp,λ (QT ), p ∈ (1, +∞), λ ∈ (0, n + 1). Let u ∈ Wq2,1 (QT ) be a strong solution of (3.9) in QT with q ∈ (1, p], u(x , 0) = 0. Then u ∈ Wp2,1 (QT ) and Dx2 u, Dt u ∈ Lp,λ (QT ). Moreover, there is a constant C = C(n, p, λ, =, T , % , dist(% , ∂% )), such that

(3.11)

Dx2 u p,λ,QT , Dt u p,λ,QT C u p,λ,QT + f p,λ,QT . For appropriate values of p and λ, we get from Theorem 3.5 Hölder continuity of u and Dx u with respect to the metric ρ(x). In fact, Dx u, Dt u ∈ Lp,λ (QT ) and [14, Theorem 4.1] imply Hölder continuity of u(x) in QT with exponent 1/n + λ/p − (n + 1)/p if n + 1 − p/n < λ < n + 1. It is not so simple to show Hölder regularity of the spatial gradient Dx u because of the lack of derivatives Dt t u and Dx t u. Anyway, a standard approach (cf. [8], [25,

262


Chapter 3]) consisting of passage through Campanato spaces leads to the goal. Thus, for any ellipsoid Er we have

2 p p p |Dx u − (Dx u)Er | dx C(n)r |Dx u| + |Dt u|p dx Er Er

2 p 1 |Dx u| + |Dt u|p dx Cr p+λ λ r E

2r p p p+λ

Dx u p,λ,Q + Dt u p,λ,Q Cr T

T

as a consequence of [8, Lemma 2. II]. By virtue of (3.11), this means that the spatial gradient Dx u belongs to Campanato’s space Lp,p+λ (QT ) and [14, Theorem 3.1] yields Hölder continuity of Dx u with exponent 1 + λ/p − (n + 1)/p if n + 1 − p < λ < n + 1. Thus, remembering the equivalence of the metrics ρ(x) and ρ(x), ˜ we have COROLLARY 3.6. Under the assumptions of Theorem 3.5, let λ ∈ (n + 1 − p, n + 1). Then there is a constant C = C(n, p, λ, =, T , % , dist(% , ∂% )), such that

|Dx u(x) − Dx u(y)| + f p,λ,Q sup C

u p,λ,Q T T ˜ − y)1+λ/p−(n+1)/p x,y∈QT ,x=y ρ(x √ with ρ(x ˜ − y) = max{|x − y |, |t − s|} for any couple x = (x , t), y = (y , s). If moreover, λ ∈ (n + 1 − p/n, n + 1) then sup

x,y∈QT ,x=y

|u(x) − u(y)| C u p,λ,QT + f p,λ,QT . 1/n+λ/p−(n+1)/p ρ(x ˜ − y)

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