# Boundedness of Singular Integral Operators with Operator-Valued Kernels and Maximal Regularity of Sectorial Operators in Variable Lebesgue Spaces

Authors: Qinghua Zhang; Yueping Zhu; Feng Wang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This paper is devoted to the maximal regularity of sectorial operators in Lebesgue spacesLp⋅ with a variable exponent. By extending the boundedness of singular integral operators in variable Lebesgue spaces from scalar type to abstract-valued type, the maximal Lp⋅−regularity of sectorial operators is established. This paper also investigates the trace of the maximal regularity space E01,p⋅I, together with the imbedding property of E01,p⋅I into the range-varying function space C−I,X1−1/p⋅,p⋅. Finally, a type of semilinear evolution equations with domain-varying nonlinearities is taken into account.


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## Body

## 1. Introduction

MaximalLp−regularity of sectorial operators is an important theory, which brings a powerful tool in investigating the evolution equations in Lp−spaces. Let X be a Banach space and A be a closed operator defined in X with the dense domain DA and dense range RA, and let X1=DA endowed with the graph norm. A is called a sectorial operator, if there are constants M0>0 and 0<ω<π/2, such that the sector(1)Σω=λ∈\ℂ0:argλ≤π−ω,is contained in ϱ−A, and the inequality(2)λI+A−1≤Mλ,holds for all λ∈Σω. Recall that for a sectorial operator A, its negative −A generates an analytic C0−semigroup e−tA (refer to [1], Section 2.5).LetI=0,b with 0<b<∞ or I=0,∞, and consider the abstract differential equation(3)u′t+Aut=ft,0<t≤b.We say thatA satisfies the maximal Lp−regularity on I, or A∈ℳℛpI in symbol, if for all f∈LpI,X, there is a unique solution u∈W1,pI,X∪L1,pI,X1 of equation (3) with the initial value u0=0. Using the interpolation method for the convolution operators with singular kernels, we know that (see [2, 3], etc.) if A∈ℳℛqI for some 1<q<∞, then A∈ℳℛpI for all 1<p<∞.In [4, 5], the authors gave a general introduction on the Lp−regularity of sectorial operators and [6–9] investigated the maximal Lp−regularity of the second order elliptic and Stokes operators, made some Lp− or Lp−Lq−estimates for the parabolic evolution and nonstationary Navier–Stokes equations. Maximal regularity of sectorial operators in weighted Lp−spaces was established in [10] and applied in quasilinear equations in [11, 12]. During the same period, Chill and Fiorenza [13] dealt with the maximal regularity of sectorial operators in Orlicz spaces of rearrangement invariant Banach functions.In some concrete situations, the nonlinear termf attached to (3) may be lying in Lp⋅I,X, so it is natural to consider the maximal regularity in such spaces. Since p⋅ is a variable exponent, the interpolation method used in [2, 3] is not suitable anymore. Because of lacking of translation invariance, space Lp⋅I,X is not arrangement invariant, hence tools developed in [13] are not applicable directly yet. In order to establish the maximal Lp⋅−regularity of A, a recently developed method for the maximal operator and singular integral operators can be employed. This method is associated with the maximal operator M, the sharp maximal operator M# and the singular integral operator T attached with A in Lp−spaces with variable exponents (refer to [14, 15]). By employing this method, with the aid of the estimate obtained in [13], in this paper, we will prove that if A∈ℳℛqI for some 1<q<∞, then A∈ℳℛp⋅I for all log-Hölder continuous exponents p⋅ with 1<p−<p+<∞, where p+ and p− denote the supremum and infimum of p⋅ on the interval I, respectively.In order to apply the maximalLp⋅−regularity theory to the quasilinear evolution equations, in this paper, we also make some investigations on the trace of the maximal regularity space W1,p⋅I,X∩L1,p⋅I,X1. As we know that, for all subintervals J of I, W1,pJ−J,X∩L1,pJ−J,X1 can be imbedded into the space CJ,X1−1/pJ−,pJ− (refer to [5, 10]), where pJ−=inft∈Jpt, and X1−1/pJ−,pJ−=X,X11−1/pJ−,pJ− is the real interpolation space between X and X1, a question arises naturally, that is, for arbitrary t∈I, whether or not the trace space of W1,p⋅I,X∩L1,p⋅I,X1 is X1−1/pt,pt exactly. This question was raised in [16] and has had not an answer until now. The main obstacle is that the imbedding bounds of W1,pJ−J,X∪L1,pJ−J,X1 ↪ CJ,X1−1/pJ−,pJ− depend on the length of J, and it could not be controlled as the interval J shrinks to the point t. Here, by using the properties of the log-Hölder function, together with the theory of range-varying function spaces developed in [16, 17], we give this question an affirmative answer. We will show that, in case that −A generates a exponentially decaying semigroup and p⋅ is a log-Hölder function, then the homogeneous maximal regularity space W01,p⋅I,X∩L1,p⋅I,X1 can be imbedded in C−J,X1−1/p⋅,p⋅, a range-varying function space established on the regular Banach space net Xα,pα:α∈0,1. This gives an affirmative answer to the question about the trace of the homogeneous maximal regularity space.This paper is organized as follows. As preliminaries, in this and the next sections, we make a brief review on the maximalLp−regularity of sectorial operators and the Xθ⋅−valued function spaces. In Section 3, the main results on singular integral operators with operator-valued kernels with application to maximal regularity in Lp⋅I,X and time-varying trace of the maximal regularity space are derived. All the results will be applied to a semilinear evolution equation with the time-dependent nonlinearity at the end of the paper. This example implies the wide application of our work in the study of parabolic partial differential equations with nonstandard growth.


## 2. Preliminaries

Given a Banach spaceX and a sectorial operator A which is densely defined in X. Let X1=DA endowed with the graph norm as above, and let I=0,b or I=0,∞.Given1<q<∞, define the maximal regularity space(4)E1,qI=W1,qI,X∩LqI,X1,endowed with the norm uE1,qI=uW1,qI,X+uLqI,X1 and the homogeneous subspace(5)E01,qI=u∈E1,qI,u0=0.Under present situations,E1,qI ↪ CI,X1−1/q,q for I=0,∞ and E01,qI ↪ CI,X1−1/q,q for I=0,b with the imbedding bounds independent of b>0 (refer to [18], Section 3.4.10).By the inverse operator theorem of the closed operators, we can assert that, ifA∈ℳℛqI, then there is a constant Cq>0 such that(6)uE1,qI≤CqfLqI,X,where f∈LqI,X and u∈E01,qI is the solution of equation (3). Furthermore, if A∈ℳℛqℝ+, then Cq is independent of the length of I, and −A generates an exponentially decaying analytic semigroups e−tA, i.e., there are constants M0≥1 and ω>0 such that(7)maxe−tAℒX0,tAe−tAℒX0≤M0e−ωt,for all t≥0. In this case, the real interpolation space X1−1/q,q=X,X11−1/q,q has an equivalent norm (cf. [19], Section 5.1)(8)xX1−1/q,q=∫01Ae−sAxq1/q.It is well known that (cf. [4, 5, 13]) A has the maximal Lq−regularity on the interval I if and only if the singular integral operator T defined through(9)Tft=∫0tAe−t−sAfsds,f∈C0∞I,Xis well defined and can be extended onto LqI,X as a bounded linear operator.As preparations for the discussions on the trace of the spaceE01,p⋅I, let us recall the definition and construction of the abstract-valued function space of the range-varying type. For the detailed discussions, please refer to [16, 17].Suppose thatA is an ordered topological space with the order ≺, in which every order-bounded subset has the order supremum and order infimum. Suppose also A is totally order-bounded, i.e., there are α± in another order space containing A such that α−≺α≺α+ for all α∈A. Under present situation, A is called a totally bounded lattice. Let αk⊆A and α∈A, we say that αk is approaching α, we mean that αk≺β for all k∈ℕ and limk⟶∞αk=β at the same time.LetXα:α∈A be a family of Banach spaces attached to A. We say it is a regular Banach space net, provided the hypotheses are both fulfilled:(1)
Ifα≺β, then Xβ ↪ Xα, and there is a constant C>0 independent of α,β such that xα≤Cxβ for all x∈Xβ.(2)
Ifαk approaches β, then limk⟶∞xαk=xβ for all x∈Xβ. Moreover, if x∈Xαk for all k∈ℕ and C=supk⟶∞xαi<∞, then x∈Xβ and xβ≤C.LetI be an interval as above and ΛI be the collection of all bounded subintervals of I. Consider the map θ:I⟶A. When we say θ is order-continuous, we mean that for any nest of intervals Jk∈ΛI:k=1,2,… shrinking to t, the limit(10)limk⟶∞θJk−=limk⟶∞θJk+=θt,always holds, where θJ− and θJ+ denote the order infimum and supremum of θ on J, respectively.Define(11)L0I,Xθ⋅=f∈L0I,X:fJ∈L0J,XθJ−for allJ∈ΛI,andft∈Xθtfor a.e.t∈I.This is a linear space according to the addition and scalar multiplication of functions. Moreover, for allf∈L0I,Xθ⋅, the composite function t⟼ftθt is measurable.There are two types of range-varying function spaces derived fromL0I,Xθ⋅, one is of continuous type defined through(12)C−I,Xθ⋅=f∈L0I,Xθ⋅:fJ∈CJ,XθJ−for allJ∈ΛI,andsupt∈Iftθt<∞,which is a Banach space equipped with the norm supt∈Iftθt or equivalently supJ∈ΛIfJCJ,XθJ−. And the other is of an integral type defined through(13)Lp⋅I,Xθ⋅=f∈L0I,Xθ⋅:f⋅θ⋅∈Lp⋅I,with the Luxemburg norm(14)fLp⋅I,Xθ⋅=infλ>0:∫Iftθtλptdt≤1,where p:I⟶1,∞ is a measurable variable exponent. If θt≡0, then we obtain the familiar Lebesgue–Bochner space of variable exponent type Lp⋅I,X0.Discussions in [16] tell us that, if we take 0,1 as the totally bounded lattice, then Xα≔Xα,1/1−α:α∈0,1,andX0=X is a regular Banach space net. Hence, for the continuous exponent p:I⟶1,∞, we obtain the linear space L0I,X1−1/p⋅,p⋅ and the Banach space C−I,X1−1/p⋅,p⋅. We can also construct the maximal regularity space with variable exponent E1,p⋅I=W1,p⋅I,X∩Lp⋅I,X1 with the norm uE01,p⋅I=u′Lp⋅I,X+uLp⋅I,X1 and homogeneous subspace E01,p⋅I=W01,p⋅I,X∩Lp⋅I,X1. All of them will be applied in the coming arguments.


## 3. Main Results and Proofs

We firstly focus on boundedness of the singular integral operator with operator-valued kernel onLp⋅ℝN,X.LetX and Y be two Banach spaces, ϒ=x,y:x,y∈ℝN,x≠y, and let k:ϒ⟶ℒY,X is a locally integrable function. Define a linear operator T as follows:(15)Tfx=∫ℝNkx,yfydy,f∈C0∞ℝN,Y,x∈¯suppf.T is called a singular integral operator of strong q,q type, provided it can be extended onto LqℝN,X to LqℝN,Y for some 1<q<∞, and there is a C1>0 such that(16)TfLqℝN,X≤C1fLqℝN,Y,for all f∈LqℝN,X.If there are constantsC2>0, δ>0 such that(17)kx,yℒY,X≤C2x−yN,x≠y,(18)kx,y−kz,yℒY,X≤C2x−zδx−y−N−δ,x−z≤x−y2,(19)ky,x−ky,zℒY,X≤C2x−zδx−y−N−δ,x−z≤x−y2,then k is called a standard kernel. Here assumption (17) tells us that kx,y is a singular kernel, and (18) and (19) together imply that kx,y is locally Hölder continuous in some way. All of them are connected to the strong q,q boundedness of T in case that k is a scalar kernel. And under the strong q,q assumption of T, we only use (18) to deal with the strong p⋅,p⋅ property of T for the operator-valued kernel.We sayk satisfies the Hörmander’s integral condition, if there is another constant C3>0 such that for every cube Q with sides parallel to the coordinate axes and all y,z∈Q, we have(20)∫ℝN\2Qkx,y−kx,zdx≤C3,where 2Q represents the cube with the same center and double sides of Q.Similar to the scalar case, for the operator-valued kernel, we have [13].Lemma 1.
Under Hörmander’s integral condition (20), a singular integral operator T of strong q,q type is also of weak 1,1 type in the sense that(21)mx∈ℝN:TfxX>λ≤C4λfL1ℝN,Y,for all f∈LqℝN,Y∩L1ℝN,Y and some constant C4=CN,q,C1,C3>0.The following lemma is a natural extension of [20, 21] of the standard kernel from the scalar type to the operator-value type. For the convenience of the reader, we state it here and give it a complete proof.Lemma 2.
LetT be an operator defined through (15) with the standard kernel k. Suppose that T can be extended as a weak 1,1 type operator as above and 0<s<1, then for all f∈C0∞ℝN;Y, the scalar function Tf⋅Xs lies in the space BMOℝN, and there is a constant C5=CN,s,δ,C4>0 such that(22)M#TfXsx≤C5Mfsx≕C5MfYsx,for all x∈ℝN.Proof.
Take anyf∈C0∞ℝN;Y and x0∈ℝN. Without loss of generality, assume that Mfx0>0. Let Q be a cube containing x0 with sides parallel to the coordinate axes. Consider the split f=f1+f2, f1=fχ2Q. For the first part f1, we have(23)1Q∫QTf1xXsdx=sQ∫0∞λs−1mx∈Q:Tf1xX>λdλ≤sQ∫0tλs−1Qdλ+∫t∞λs−1C4f1L1ℝN;Yλdλ=ts+C4ts−11−s1Q∫2QfyYdy≤ts+2NC41−sMfx0ts−1.
Taket=Mfx0, and we obtain(24)1Q∫QTf1Ysdx≤1+2nC11−sMfsx0.
For the second partf2, we have(25)1Q∫QTf2xX−Tf2x0Xsdx≤1Q∫QTf2x−Tf2x0Xdxs.
Notice that for allx∈Q, by (18),(26)Tf2x−Tf2x0X≤∫ℝN\2Qkx,y−kx0,yℒY,XfyYdy≤∑j=1∞∫2j+1Q∖2jQx−x0δx−yN+δfyYdy≤∑j=1∞2j−1δ∫2j+1Q∖2jQfyY2jrNdy≤2N∑j=1∞2j−1δ⋅12j+1Q∫2j+1QfyYdy≤2N1−2δMfx0,where r denotes the radius of Q, we obtain(27)1Q∫QTf2xX−Tf2x0Xsdx≤2N1−2δsMfsx0.
Putting the above two estimates together, we obtain(28)1Q∫QTfxX−Tf2x0Xsdx≤CMfsx0,for some constant C5=CN,s,δ,C4, which means that TfYs∈BMOℝN, and estimate (22) holds.
Given a variable exponentp:ℝN⟶1,∞. We say p is log-Hölder continuous, or symbolically p∈PlogℝN, if there are constants C0>0 and p∞≥1 such that(29)px−py≤C0loge+x−y−1,px−p∞≤C0loge+x,for all x,y∈ℝN.Remark 1.
IfΩ is a bounded domain of ℝN, then p is log-Hölder continuous on Ω, if and only if the first inequality of (29) is satisfied.
Next lemma is an important result in harmonic analysis, and it was first proved in [14] for bounded exponents and later extended to general cases in some literatures. For the complete proof with detail discussions, please refer to [15].Lemma 3.
Assume thatPlogℝN and p−>1, then the maximal operator M is bounded from Lp⋅ℝN to Lp⋅ℝN, i.e., there is a constant C6=CN,p−,C0>0 such that(30)MfLp⋅ℝN≤C6fLp⋅ℝN.Furthermore, under the extra assumptionp+<∞, for the sharp operator M#, there is another constant C7=Cp±,C6>0 for which estimate(31)fLqℝN≤C7M#fLqℝN,for all f∈LqℝN (refer to [22], P.148).Putting all the facts together, we obtain the following.Theorem 1.
LetT be a singular integral operator of strong q,q type for some 1<q<∞ with the standard kernel k satisfying Hörmander’s integral condition and p∈PlogℝN be a variable exponent satisfying 1<p−≤p+<∞. Then, T is bounded from Lp⋅ℝN;Y to Lp⋅ℝN;X with the bounds C8=CN,p±,δ,C1,C2,C3>0.Proof.
Take a constant exponents such that 0<s<1, and then the variable exponent p⋅/s is also log-Hölder continuous the same constant C0, and p⋅/s−=p−/s>1 and p⋅/s+=p+/s<∞. Thus, combining (22), (30), and (31), we can deduce that(32)TfLp⋅ℝN;X=TfXsLp⋅/sℝN1/s≤C7M#TfXsLp⋅/sℝN1/s≤C5C7MsfYLp⋅/sℝN1/s=C5C7MfLp⋅ℝN;Y≤C8fLp⋅ℝN;Y,where the constant C8=C6C5C7=CN,p±,δ,C1,C2,C3.Remark 2.
This conclusion is a natural extension of that in [15], Section 1.6.3 for the singular integral operator from the scalar type to the abstract-valued type. For another treatment of the extension, please refer to [23].
Now we can establish the maximalLp⋅−regularity for the sectorial operator A. Define(33)Kt=Ae−tA,ift>0,0,ift≤0.
Straight calculations show thatkt,s=Kt−s is a standard kernel satisfying Hörmander’s integral condition, and Tf can be expressed by K∗f˜, where f˜(34)f˜t=ft,ift∈I,0,ift∈ℝ\I,is the zero extension of f. In this setting, A∈ℳℛqI is equivalent to say that T is a singular integral operator of strong q,q type.
Given a variable exponentp∈PI satisfying the log-Hölder condition (29) with ℝN replaced by I and 1<p−≤p+<∞. From [15], Section 4.1, we know that p has an extension p˜∈Pℝ with the same constant C0 and p˜±=p±. Analogous to E1,qI and E01,qI, define the maximal Lp⋅−regularity space E1,p⋅I and its closed subspace E01,p⋅I with the norm(35)uE1,p⋅I=uW1,p⋅I,X+uLp⋅I,X1.
Applying Theorem1, we can drive the following.Theorem 2.
Assume thatA∈ℳℛqI for some 1<q<∞ and p∈PI with 1<p−≤p+<∞. Then, A satisfies the maximal Lp⋅−regularity on I, that is, for all f∈Lp⋅I,X, there is a unique function u∈E1,p⋅I solving (3) with u0=0, and satisfying(36)uE1,p⋅I≤CfLp⋅I,X1,with the constant C>0 depending on A,C0,Cq, and p±.In the following paragraphs, we turn our attention to the trace ofE1,p⋅I. Here and after we need assumption (7) for the semigroup e−tA. Denote by γE1,p⋅I the trace space of E1,p⋅I, that is,(37)γE1,p⋅I=u0∈X:∃u∈E1,p⋅Is.t.u0=u0,with the norm(38)u0γE1,p⋅I=infuE1,p⋅I:u∈E1,p⋅I,u0=u0.Proposition 1.
Suppose thatp∈Plogℝ+ with 1<p−≤p+<∞, then(39)γE1,p⋅ℝ+=X1−1/p0,p0,with the equivalent norms.Proof.
Firstly, for allu0∈X1−1/p0,p0 with u0X1−1/p0,p0≤1, we have u0X≤C for some constant C>0 independent of u0, and ∫01e−tAu0p0dt≤1. Consider the split(40)∫0∞Ae−tAu0ptdt=∫01+∫1∞Ae−tAu0ptdt≔I1+I2.
For the second part, we have(41)I2≤∫1∞M0te−ωtu0Xptdt≤1ωp−maxM0Cp+,1.
DefineE=t∈0,1:pt≤p0, let p1t=ptχE+p01−χE and p2t=pt1−χE+p0χE, then both of p1 and p2 are log-Hölder continuous with the same constant C0. Thus, for the first part I1, we have(42)I1≤∫01Ae−tAu0p1tdt+∫01Ae−tAu0p2tdt≔I1,1+I1,2,where(43)I1,1≤∫01Ae−tAu0p0dt+1−p−p0≤2,I1,2≤∫01Ae−tAu0p0Mu0Xtp2t−p0dt≤maxM0Cp+−p−,1∫01Ae−tAu0p0t−C0/loge+1/tdt≤maxM0Cp+−p−,1eC0.
Putting all the parts together, we obtain∫0∞e−tAu0ptdt≤C with the constant C>0 independent of u0. And by scaling arguments, we have e−tAu0E1,p⋅ℝ+≤Cu0X1−1/p0,p0, which in turn yields u0γE1,p⋅ℝ+≤Cu0X1−1/p0,p0.
Conversely, suppose thatu∈E1,p⋅ℝ+ with uE1,p⋅ℝ+≤1 and u0=u0. By the unit ball property, we have ∫01Autptdt≤1 and u′Lp⋅01,X≤1. By imbedding W1,p⋅ℝ+,X ↪ C0,1,X, we get the estimate uC0,1,X≤C for some constant C>0 independent of u. Notice that(44)u0=ut−∫0tu′sds,we have(45)Ae−tAu0Lp00,1,X≤Ae−tAuLp00,1,X+Ae−tA∫0tu′sdsLp00,1,X.
Since(46)∫01Ae−tAutp0dt≤M0p0∫EAutptuttp0−ptdt+M0p0∫0,1\EAutp0dt≤M0p0max1,uC0,1,Xp0−p−∫01Autptt−C0/loge+1/tdt+M0p0∫01Autptdt+1≤M0p+eC0max1,Cp+−p−+2,∫01Ae−tA∫0tu′sdsp0dt≤M0p0∫E1t∫0tu′sdsptut−u0tp0−ptdt+M0p0∫0,1\E1t∫0tu′sdsp0dt≤M0p0max1,2uC0,1,Xp0−p−∫01Mu′tptt−C0/loge+1/tdt+M0p0∫01Mu′tptdt+1≤M0p+max1,2Cp+−p−eC0+1maxC6p+,C6p−+1,where(47)∫01Mu′tptdt≤maxMu′Lp⋅0,1p+,Mu′Lp⋅0,1p−≤maxC6u′Lp⋅0,1p+,C6u′Lp⋅0,1p−≤maxC6p+,C6p−,by the unit ball property of Lp⋅ and boundedness of the maximal operator, we then have ∫01Ae−tAu0p0dt≤C with the constant C>0 independent of u. By the scaling arguments and definition of γE1,p⋅ℝ+, we can also derive that u0X1−1/p0,p0≤Cu0γE1,p⋅ℝ+. Thus, equivalence of γE1,p⋅ℝ+ and X1−1/p0,p0 has been reached.Proposition 2.
Under the same assumptions uponp as above, we have(48)E1,p⋅ℝ+↪C−ℝ+,X1−1/p⋅,p⋅.Proof.
irstly, for every bounded subintervalJ of ℝ+, imbedding(49)E1,p⋅J↪E1,pJ−J↪CJ,X1−1/pJ−,pJ−,holds with the imbedding bounds depending on J, from which we obtain(50)E1,p⋅+sℝ+⊆L−0ℝ+,X1−1/p⋅,p⋅,where the range-varying function space L−0ℝ+,X1−1/p⋅,p⋅ comes from [17].
For eachs>0, consider the translation operator Tsut=ut+s. Obviously, Ts is a bounded linear operator from E1,p⋅ℝ+ to E1,p⋅+sℝ+, and(51)TsuE1,p⋅+sℝ+≤uE1,p⋅ℝ+.
Another fact is that for everys>0, the translated exponent Tsp=p⋅+s is also log-Hölder continuous with the same constant C0 and 1<p−≤Tsp−≤Tsp+≤p+<∞. Thus, for all u∈E1,p⋅ℝ+ and all s>0, using Proposition 1, we have(52)usX1−1/ps,ps≤CTsu0γE1,p⋅+sℝ+≤CTsuE1,p⋅+sℝ+≤CuE1,p⋅ℝ+.
Therefore,u∈C−ℝ+,X1−1/p⋅,p⋅ and(53)uC−ℝ+,X1−1/p⋅,p⋅≤CuE1,p⋅ℝ+,with a constant C>0 independent of u (refer to [17]).Theorem 3.
Suppose thatp∈Plog0,b with 1<p−≤p+<∞, then(54)E01,p⋅0,b↪C−0,b,X1−1/p⋅,p⋅,with the imbedding bounds independent of T>0.Proof.
Let(55)p˜t=pt,0≤t≤b,p2b−t,b≤t≤2b,p0,t≥2b,u˜t=ut,0≤t≤b,u2b−t,b≤t≤2b,0,t≥2b.
It is easy to check thatp˜∈Plogℝ+ with p˜±=p± and p˜∞=p0, u˜∈E01,p⋅ℝ+, and(56)u˜E01,p˜⋅ℝ+≤2uE01,p⋅0,b.
By invoking Proposition2, we know that u˜∈C−ℝ+,X1−1/p˜⋅,p˜⋅, and(57)supt≥0u˜tX1−1/p˜t,p˜t≤Cu˜E01,p˜⋅ℝ+.
Notice thatp˜t=pt and u˜t=ut for all t∈0,b, we obtain(58)supt∈0,butX1−1/pt,pt≤Cu˜E01,p˜⋅ℝ+≤2CuE01,p⋅0,b.
Thus, the proof has been proved.
At the end of the paper, we will use the maximalLp⋅−regularity results to deal with the semilinear evolution equation:(59)u′t+Au=Ft,u,t>0,u0=u0,where A∈ℳℛqℝ+ for some 1<q<∞, 0<b0≤∞, p∈Plog0,b0, and F:0,b0×X⟶X is a nonlinear map fulfilling assumptions HF as follows:(1)
For almost allt∈0,b0, Ft,⋅ can be defined and locally Lipschitz continuous on X1−1/pt,1/pt(2)
For allu∈L00,b0;X1−1/p⋅,1/p⋅, the compound function t⟶Ft,ut is strongly measurable on 0,b0(3)
u0∈X1−1/p0,1/p0, and F⋅,e−⋅Au0∈Lp⋅0,b0;XTheorem 4.
Under present situations, for everyr>0, there is a number 0<b<b0, such that for each u1∈ℬu0,r, and semilinear equation (59) has a unique solution u⋅=u⋅,u1∈E1,p⋅I∩C−I,X1−1/p⋅,p⋅ on the interval I=0,b with the initial condition u0=u1. Here,(60)ℬu0,r=u∈X1−1/p0,p0:u−u0X1−1/p0,p0<r,is a ball in X1−1/p0,p0. Moreover, for every two points ui∈ℬu0,r, i=1,2, the corresponding solutions u⋅,ui, i=1,2, satisfy(61)maxu⋅,u1−u⋅,u2C−I,1−1/p⋅,p⋅,u⋅,u1−u⋅,u2E1,p⋅I≤Cu1−u21−1/p0,p0.Proof of this theorem is much similar to that completed in [16] where embedding E01,p+I↪C−I,X1−1/p⋅,p⋅ is replaced by E01,p⋅I↪C−I,X1−1/p⋅,p⋅.Remark 3.
Using the maximalLp⋅−regularity theory, hypothesis F⋅,e−⋅Au0∈Lp+0,b0;X used in [16] is dropped. Instead, here, a weaker assumption HF3: F⋅,e−⋅Au0∈Lp⋅0,b0;X is applied. In this sense, Theorem 4 is an improvement of that in [16].


## 4. Conclusions and Discussion

In this paper, we study the maximalLp⋅−regularity for the sectorial operators. By extending the boundedness of singular integral operators from the scalar type to abstract-valued type; we see that, if a sectorial operator A lies in ℳℛqI for some 1<q<∞, then it lies in ℳℛp⋅I for every Hölder continuous variable exponent p⋅ with 1<p−<p+<∞. We also prove that if −A generates an exponentially decaying analytic semigroup, then for the maximal regular space E1,p⋅ℝ+, its trace space is exactly X1−1/p0,p0, and the homogeneous maximal regular space E01,p⋅I can be embedded continuously into the range-varying function space C−I,X1−1/p⋅,p⋅ with the embedding bounds independent of the length of the interval I. Different to the constant exponent type, translation series Ts:Lp⋅ℝ+,X⟶Lp⋅+sℝ+,X,s≥0 could not make up a C0 semigroup on Lp⋅ℝ+,X, since Lp⋅ℝ+,X does not have the translation-invariant property. Consequently, whether or not the following estimates:(62)supλ≥0λλ+∂t+A−1ℒLp⋅I,X,Lp⋅I,X<∞,supλ≥0λ+∂t+A−1ℒLp⋅I,X,0E1,p⋅I<∞,still hold for the variable exponents remains unknown. We also wonder that under what situations maximal Lp⋅−regularity can be preserved under time-dependent perturbation Bt.



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*Source: 3672892-2020-03-10.xml*