Explicit improvements for Lp$\mathrm{L}^p$ ‐estimates related to elliptic systems

We give a simple argument to obtain Lp$\mathrm{L}^p$ ‐boundedness for heat semigroups associated to uniformly strongly elliptic systems on Rd$\mathbb {R}^d$ by using Stein interpolation between Gaussian estimates and hypercontractivity. Our results give p$p$ explicitly in terms of ellipticity. It is optimal at the endpoint p=∞$p=\infty$ . We also obtain Lp$\mathrm{L}^p$ ‐estimates for the gradient of the semigroup, where p>2$p>2$ depends on ellipticity but not on dimension.


INTRODUCTION
In dimension  ⩾ 3, we consider uniformly strongly elliptic systems on ℝ  of  ⩾ 1 equations in divergence form with bounded, measurable and complex coefficients, see Section 2 for precise definitions.This gives rise to a contraction semigroup (e − ) >0 in (L 2 )  ∶= L 2 (ℝ  ; ℂ  ).Surprisingly, little is known about explicit L  -estimates when no further regularity on the coefficients is imposed.For systems with minimally smooth coefficients, we refer, for example, to [8].More precisely, consider the set  () ∶= {  ∈ (1, ∞) ∶ e − is bounded in L  , uniformly for  > 0 } .
By complex interpolation, it is an interval around 2, the endpoints of which are often denoted by  ± ().All of our results will be stable under taking adjoints.As  − () = ( + ( * )) ′ , we shall concentrate on the upper endpoint  + ().It is known that  + () > 2 * , where 2 * ∶= 2 ∕(−2) is the Sobolev conjugate of 2, and that the improvement  + () − 2 * can be arbitrarily small even when  = 1 [12, section 2.2].What seems to be missing though, are explicit lower bounds for  + () in terms of the data of , such as ellipticity constants and dimensions, in particular when the improvement is expected to be large or even covers  + () = ∞.Indeed, all results for systems (that we are aware of) are perturbative from the general L 2 -theory and provide small, nonquantifiable improvements [2,3,19].In contrast, we proceed by interpolation from the L ∞ -theory for special systems described further below.Our results are new also for elliptic equations ( = 1).
The number  + () is related to the optimal ranges of various L  -estimates for , such as Riesz transforms, boundary value problems and functional calculus, see the introduction of [4] for a comprehensive account on the literature.Thus, improving lower bounds for  + (), as we shall do here, leads to automatic improvements in all these topics.
All of our results are perturbative from the diagonal Laplacian system corresponding to  =  (ℂ  )  ∶= ( ,  , ) , , but not necessarily on a small scale.This is in the nature of things, because every uniformly strongly elliptic  is an L ∞ -perturbation of  (ℂ  )  of size smaller than 1 up to normalization: The "distance" d() is a well-known measure of ellipticity [14].It can be bounded from above and below in terms of the usual ellipticity constants and when  =  * , there is an easy formula (Lemma 2.1).The dimensional constant () ∶= ( 1 + ( − 2) 2  − 1 will play an important role in this paper.Our main result is as follows.
The result in (ii) is optimal in the sense that for each  > 0 there is some   with d(  ) ⩽ () +  and  + (  ) < ∞.
The dimensional constants in Theorem 1.1 are quite large in small dimensions and we collect some values in Figure 1.
Part (ii) is proved in Section 3 by combining results of Koshelev [14], see also [15], with a characterization of Gaussian estimates and Hölder regularity of the kernel associated with e − due to Auscher-Tchamitchian [5].In fact, in Theorem 3.3 we shall not only prove that d() < () implies  + () = ∞, but that e − has a Hölder regular integral kernel with Gaussian decay.As F I G U R E 1 Approximate constants in Theorem 1.1 in small dimensions.The third column contains the ellipticity ratio () = () ∕Λ() ∈ (0, 1] that is sufficient for having d() = () in the special case  =  * , see Lemma 2.1.
Part (i) is proved in Section 4. The idea is to rewrite d() < 1 as  = ( (ℂ  )  − ), where ‖‖ ∞ = d() and  > 0. We embed  as  1 into an analytic family of elliptic matrices given by where  ⩽ || ⩽  with 0 <  < 1 < .Then, in the spirit of Stein interpolation, we estimate e − = e − 1 by using the generic information  + (  ) ⩾ 2 * on the outer circle || =  and  + (  ) = ∞ on the inner circle || =  provided  is small.This gives a lower bound for  + ( 1 ) that becomes the larger, the closer  = 1 is to the inner circle and the farther away it is from the outer one.Thus, the best bound is achieved when ,  are the largest possible and the optimal choice for  comes from (ii).We believe that this simple analytic perturbation argument is of independent interest and has multiple applications to other types of L  -estimates for divergence form operators.
Writing Theorem 1.1(i) as 1 ∕2 * − 1 ∕ + () ⩾ ln(d()) ∕2 * ln(()) =∶ (, d()), we see that (, d()) → 0 as  → ∞.Inspired by Stein's result [18] on dimensionless bounds for the Riesz transform, we ask whether an improvement can be given independently of .To this end, it will be advantageous to consider where  ≈ 5.69061 is the unique real solution to For curiosity, let us mention that the first bound in Theorem 1.2 produces a larger improvement for  + () compared to Theorem 1.1(i) in dimension  ⩾ 922100.
It would be interesting to know to what extent our results can be extended to more general domains and boundary conditions.In case of Theorem 1.1, we provide an extension to bounded C 1 -domains with Dirichlet boundary conditions in Section 5.

UNIFORMLY STRONGLY ELLIPTIC SYSTEMS
Let  ⩾ 3,  ⩾ 1 and  ∶ ℝ  → ((ℂ  )  ) be measurable.We assume that  is "uniformly strongly elliptic", that is, Let  = − div(∇⋅) be realized as an m-accretive operator in (L For the second claim we take  ∶= 2 ∕(Λ()+()).Then  (ℂ  )  − () is a self-adjoint matrix for almost every  ∈ ℝ  with eigenvalues contained in ] , and thus by the spectral radius formula.□ The next smoothing of the coefficients lemma will be important in Sections 3 and 5 to absorb terms, which are a priori not finite for nonsmooth coefficients.We include the simple proof for convenience.To this end, we let  ∈ C ∞ c ((0, 1)) be nonnegative with ∫ ℝ   d = 1 and put   () ∶=   () for  ∈ ℕ and  ∈ ℝ  .We define the smoothed coefficients   ∶=  *   .Lemma 2.2.Let  ⊆ ℝ  be open and bounded,  ∈ W 1,2 ()  be a weak solution to  = 0 in  and   ∈ W 1,2 ()  be the unique weak solution to Then the following assertions are satisfied.

NEW THOUGHTS ON OLD RESULTS OF KOSHELEV
In a series of articles, culminating in the monograph [14], Koshelev studied qualitative (Hölder) regularity of weak solutions to elliptic systems.In this section, we explain how they lead us to an optimal perturbation result for Gaussian estimates for heat semigroups, when reinterpreted appropriately as quantitative statements.The appropriate setting to study regularity of -harmonic functions turns out to be the following "weighted Morrey spaces" H  ()  ,  ∈ ℝ, which are defined as the spaces of all  ∈ W 1,2 ()  modulo ℂ  for which the norm is finite.For  >  − 2 sufficiently close to  − 2 and  > 0 small enough we have (, , ) ∶= This quantity will play an important role.In fact, (, , ) → () −1 in the limit as  →  − 2 and  → 0. From now on we shall assume d() < ().
We begin by looking at -harmonic functions on the unit ball .Let  > 0. Guided by the perturbation principle in Lemma 2.1, it begins with writing the equation  = 0 in  in the weak sense as Due to technical reasons, we replace  by   and  by   as defined in Lemma 2.2 and call the term on the right-hand side − div(  ).In addition, we choose  > 0 such that d(  ) = ‖ (ℂ  )  −   ‖ ∞ .
Temporarily, fix  0 ∈ 1 4 .To derive optimal Morrey estimates for the solutions   , Koshelev considers two variational integrals where  >  − 2 is as above and   is an ingeniously chosen test function for the equation on 1 4  that they constructs from   using spherical harmonics [14, eq.(2.3.2)].The precise formula for   is not needed here -it suffices to use the estimates below "off-the-shelf".In fact, this specific   dates back to Giaquinta and Nečas [16].Koshelev goes on by proving in [14, Corollary 2.3.1] the bounds .
As   is a test function for the equation for   in 1 4 , we have and the Cauchy-Schwarz inequality along with the bound d(  ) ⩽ d() in Lemma 2.2(i) yields Combining the previous three estimates and recalling the definition of (, , ) leads to .
As d() < (), we can fix  small and  close to  − 2 depending only on dimension and d() such that the first term on the right can be absorbed.This is the key point and the result is After taking the supremum over all  0 ∈ 1 4 , we arrive at Then, Koshelev proves in [14, Theorem 2.1.1]that the left-hand side controls the Hölder seminorm of order  = (−+2) ∕2 on 1 4 .Applying Caccioppoli's inequality on the right-hand side eventually leads to Finally, we invoke Lemma 2.2(ii) in order to deduce [] () for each  0 ∈ ℝ  and  > 0, a scaling argument shows that the outcome of revisiting Koshelev's results is the following proposition.

Proposition 3.2. Suppose d(𝐴) < 𝛿(𝑑).
There are  ∈ (0, 1] and  > 0, both depending only on  and d(), such that we have for all balls  = (, ) ⊆ ℝ  and every or  * -harmonic  in  that The quantitative Hölder estimate in Proposition 3.2 appeared much later in a different context.Namely, Auscher and Tchamitchian [3,5] called it property (H) and proved that it implies that e − has a kernel with pointwise Gaussian bounds.If we combine their Theorem 10 in [5, chapter 1, section 1.4.1] with Proposition 3.2, then we obtain the following result for Gaussian estimates.It can be seen as a perturbation result from the Laplacian.From Young's inequality for convolutions, we obtain: We shall see next that the "radius"  = () is optimal for the conclusion in Corollary 3.4 and hence also for the one in Theorem 3.3.Again this is implicit in Koshelev's work and relies on a counterexample due to De Giorgi.
Our notation should be interpreted as what comes out from the product rule.The only thing that matters is that ∇ vanishes near the origin and hence  ∈ (C ∞ c )  .If we had  + () = ∞, then according to [2, Proposition 5.3] we would get  ∈ (L  )  for every  ∈ (2 * , ∞).However, |()| = || 1− in a neighborhood of 0 cannot belong to L  for  ⩾  ∕(−1).□

THE INTERPOLATION ARGUMENT
We come to the proof of our main result, Theorem 1.1, for the case d() ⩾ ().We will use basic properties of semigroups and vector-valued holomorphic functions.For further background, we refer to [1].
Proof.Fix  > 0 and  ∈ [0, 1].Let us define Φ() ∶= e −  for  ∈ S  .As an ((L 2 )  )-valued map, Φ is bounded by 1, holomorphic on S  and in particular continuous on S.This follows from (iii), see [20].Now, we estimate Φ on the boundary of S. Let  ∈ (L 2 )  .(A) Estimate on the left boundary.Let  ∈ S with Re() = 0. Due to (ii) and Theorem 3.3 we have Gaussian estimates for the kernel ( , ) >0 of (e −  ) >0 at our disposal.By (i), implicit constants depend only on , ,  and Λ. Young's inequality for convolutions yields (B) Estimate on the right boundary.Let  ∈ S with Re() = 1.By holomorphy of the semigroup it follows that e −   ∈ (W In the language of L  − L  -estimates, this means that (e −  ) >0 is L 2 − L Let us show that for  in a strip S  with sufficiently small  > 0 we can define the interpolating operators   = − div(  ∇ ⋅), where   is still elliptic by Lemma 2.1.Indeed, for  > 0 sufficiently small our choice of  delivers This also proves (i) in Proposition 4.1.Part (iii) follows immediately and our choice of  yields which is (ii).Now, pick  ∈ (0, 1) such that 1 = which was used in (B) above.

EXTENSION TO BOUNDED 𝐂 𝟏 -DOMAINS
Let us extend Theorems 1.1 and 3.3 to bounded C 1 -domains with Dirichlet boundary conditions.The divergence form operator  = − div(∇ ⋅) with uniformly strongly elliptic  ∈ L ∞ (Ω; ((ℂ  )  )) is now realized in L 2 (Ω)  as the m-accretive operator associated to the form We fix our geometric setup.
We can choose  arbitrarily small by choosing the neighborhoods small enough.This is exactly the reason, why we assume that the boundary is C 1 and not just Lipschitz.Theorem 5.2.In the setting above suppose that d() < ().Then the kernel of (e − ) >0 is represented by a measurable function (  ) >0 for which there are ,  > 0 and  ∈ (0, 1) such that for all  > 0 and ,  ′ , ,  ′ ∈ Ω.The constants  and  depend only on , d(), (), Λ() and geometry.
In view of d() = d( * ), we stick to the case of -harmonic functions.When  0 ∈ Ω and ( 0 , ) ⊆ Ω, this estimate has already been obtained in Section 3 with  = 1 ∕4.By a case distinction (whether or not ( 0 ,  ∕2) intersects Ω) it suffices to treat in addition the case  0 ∈ Ω.
Transforming back gives (5.1) for some  = () ∈ (0, 1).□ At this point, we are in the same situation as on ℝ  and we can derive the following statement.
Corollary 5.3.In the setup of this section, the following assertions hold true.
(iii) Part (ii) is sharp in the sense that for all bounded C 1 -domains Ω ⊆ ℝ  and for each  > 0 there is some   with d(  ) ⩽ () +  and  + (  ) < ∞.
Proof.The estimates for  + () follow as before, see also Remark 4.3.As for the sharpness of the radius d() = () we can, after translation, assume 0 ∈ Ω.We take the same coefficients  =  DG as in the proof of Proposition 3.5 and localize  to a ball contained in Ω.As before, this produces some  ∈ () with  ∉ (L  )  for  large but  ∈ (C ∞ c )  .Arriving at a contradiction with  + () = ∞ requires a different (and in fact simpler) argument compared to the case Ω = ℝ  .
By ellipticity and Poincaré's inequality, there is some  2 > 0 such that  −  2 is still m-accretive.Hence,  is invertible in (L 2 )  and the semigroup enjoys the exponential bound ‖ e − ‖ 2 ⩽ e − 2  ‖‖ 2 for all  > 0 and  ∈ (L 2 )  .By interpolation with the uniform bound on (L  )  for some  > , we get ‖ e − ‖  ≲ e −   ‖‖  with some   > 0. But then the formula valid in (L  ∩ L 2 )  by the exponential estimate, implies that  −1 maps (L  ∩ L 2 )  into itself, in contradiction with the properties of .□
Inverting Φ leads to which proves the theorem.□

A C K N O W L E D G M E N T S
We would like to thank Salvatore Leonardi for helping us with the literature underlying Section 3 and Oliver Dragičević for helpful exchange about the constant () in Section 6.We are
1−   .Then  =   and Proposition 4.1 implies that  + () ⩾ 2 * ∕.Finally, we notice that Remark 4.3.The proof of Theorem 1.1 reveals that the same results hold for divergence form operators with form domain  on general open sets  ⊆ ℝ  , provided that Theorem 3.3 holds true with implicit constants depending only on geometry, ellipticity and dimension, and that we have additionally □Remark 4.2.In Proposition 4.1, we assume that  ↦   (, ) is holomorphic in a larger strip for convenience to get continuity of  ↦ e −  up to S. If this holds true for any other reason, it is enough to suppose that  ↦   (, ) is holomorphic in S.
( + )  is a weak solution of provided that the right-hand side is finite, and   and   are sufficiently smooth, which is the case thanks to our smoothing procedure.Here, (, , ) is as in(3.1),where  >  − 2 and  > 0 are chosen such that (, , ) is positive and finite.By definition of   and (5.2), we derive the estimate As d() < (), we can pick  > 0,  >  − 2 and  > 0 depending only on  and d() such that the first term on the right can be absorbed in order to obtain ‖  ‖ H − ( 1 4  + ) ≲ ,d() ‖∇  ‖ L 2 ( 1 4  + ) .