Maximal potentials, maximal singular integrals, and the spherical maximal function

We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.


Introduction
Let Ω be a bounded function defined on the unit sphere S n−1 := S n−1 (0, 1) with zero mean value K(x) = Ω(x/|x|) |x| n , x = 0 be the associated homogeneous kernel of degree −n. It is well known and easy to prove that under these assumptions the principal value p.v. K is a tempered distribution. Moreover the convolution with p.v. K defines a singular integral that is bounded in L p for all 1 < p < ∞. Namely the singular integral and also the maximal singular integral are bounded in L p (R n ), 1 < p < n. This follows from [4,Theorems 4.2.7 and 4.2.11] and the fact that the kernel K can be represented as a sum of an odd and an even one. Actually, the result is true under much weaker assumptions about Ω, see [4].
In this note we investigate related results for convolutions with homogeneous kernels of degree −(n − 1). Such operators have smoothing properties in the sense that they increase regularity of functions. More precisely if Ω ∈ C 1 (S n−1 ) andK (x) = Ω(x/|x|) |x| n−1 , x = 0 (we no longer require (1.1)), then the operator is bounded from L p to the homogeneous Sobolev spaceẆ 1,p for all 1 < p < n. This is well known (see Proposition 2.1). Actually we do not have to pass to the limit as the integral converges for a.e. x ∈ R n by the Fractional Integration Theorem, [12,Theorem 2.8.4].
It is natural to consider the associated maximal operator We will call it a maximal potential. Such operators withK(x) = x/|x| s+1 are called maximal s-Riesz transforms and in our case s = n−1. These operators have been studied for example in [2], [11], however, they were investigated from a different perspective than the one discussed in this paper. In Section 4 we will show that the spherical maximal operator of a Sobolev function can be represented as a maximal (n − 1)-Riesz transform.
The question that we investigate in the paper is for what values of p, the operator A * Ω : L p →Ẇ 1,p is bounded. Clearly the operator A * Ω f is bounded by the Riesz potential and the Riesz potential is bounded for all for 1 < p < n. Thus A * Ω f has the growth properties of a function fromẆ 1,p . On the other hand the radius t at which the supremum in (1.4) is attained may depend on x in a very irregular way and this perhaps can result in the lack of smoothness of A * Ω f .
In our main result (Theorem 1.1) we will prove that the operator A * Ω : L p →Ẇ 1,p is bounded for all n/(n − 1) < p < n. We do not know what happens when 1 < p ≤ n/(n − 1), but we will show that the positive answer to the questions about boundedness for 1 < p ≤ n/(n − 1) would imply a positive answer to a question about boundedness of the spherical maximal function in the Sobolev space, [5], [6], see Proposition 1.2. Now we can state our main result.
We could formulate the theorem under weaker assumptions than Ω ∈ C 1 (S n−1 , R m ), but we wanted to keep the presentation simple. After all the C 1 case is the most interesting one.
Kinnunen, [7], proved that the Hardy-Littlewood maximal operator is bounded in the Sobolev space M : W 1,p → W 1,p , 1 < p < ∞. This result initiated the search for the boundedness results for various maximal operators in Sobolev spaces.
It was proved by Bourgain [1] and Stein [9], (see also [4], [10]) that the spherical maximal operator is bounded in L p (R n ) when n ≥ 2 and p > n/(n − 1). Using this result it is easy to prove, see [5], [6], that actually the spherical maximal operator is bounded in the Sobolev spaces S : W 1,p → W 1,p for all n/(n − 1) < p < ∞. In particular S : W 1,p →Ẇ 1,p is bounded for n/(n − 1) < p < n. It was posted as an open problem in [5], [6], whether S : W 1,p →Ẇ 1,p is bounded for 1 < p ≤ n/(n − 1). The case of the spherical maximal function has also been discussed in [8]. As a corollary of Theorem 1.1 we will prove Proposition 1.2. If the claim of Theorem 1.1 is true for 1 < p ≤ n/(n−1), then the spherical maximal operator S : W 1,p →Ẇ 1,p is bounded for all 1 < p ≤ n/(n − 1). The paper is organized as follows. In Section 2 we recall basic facts about Sobolev spaces and potentials and provide a proof that the operator (1.3) is bounded from L p to the homogeneous Sobolev spaceẆ 1,p for all 1 < p < n. In Section 3 we prove Theorem 1.1 and in Section 4 we prove Proposition 1.2.
Notation is pretty standard. By C we denote a generic positive constant that may have different values in different inequalities. The volume of the unit ball is denoted by ω n , so the volume of the unit sphere is nω n . Finally stands for the integral average.

Sobolev spaces and singular integrals
The Sobolev space W 1,p (R n ), 1 ≤ p ≤ ∞ is defined as a class of all f ∈ L p whose distributional partial derivatives of first order also belong to L p , ∇f ∈ L p . W 1,p is a Banach space with respect to the norm It follows from the Sobolev embedding theorem that W 1,p ⊂Ẇ 1,p for 1 < p < n.
If a function f ∈ L p * is absolutely continuous on almost all lines parallel to coordinate axes, then f has partial derivatives defined a.e. If in addition these partial derivatives belong to L p , then f ∈Ẇ 1,p , see [3,Section 4.9]. We will use this characterization ofẆ 1,p in the proof of Theorem 1.1. Now we will show that certain potential type operators are bounded from L p toẆ 1,p , 1 < p < n. The following result seems to be well known, but we could not find a direct reference, so we provide a proof. In particular it implies that the Riesz potential I 1 : L p →Ẇ 1,p , 1 < p < n is bounded.
If Ω ∈ C 1 (S n−1 ) and Proof. First observe that Since by the Fractional Integration Theorem, [12,Theorem 2.8.4], the Riesz potential is bounded from L p to L p * , 1 < p < n, we conclude that A Ω : L p → L p * is bounded.
Lemma 2.2. Under the above assumptions the pointwise gradient ∇K(x) defined for x = 0 is homogeneous of degree −n and Hence the condition (1.1) is satisfied and thus Finally the distributional gradient ∇K satisfies where δ 0 is the Dirac distribution and In other words for ϕ ∈ S n and 1 ≤ j ≤ n we have Proof. The fact that ∇K(x) is homogeneous of degree −n is easy and left to the reader. For 0 < t < r let A(t, r) = {x : t ≤ |x| ≤ r}. The integration by parts yields Indeed, the last two integrals are equal by a simple change of variables and homogeneity ofK. Thus the integral on the left hand side equals 0 independently of r. Hence its derivative with respect to r is also equal zero.
This proves (2.1). Therefore p.v. ∇K ∈ S ′ n is a well defined tempered distribution. We are left with the proof that the distributional gradient ∇K satisfies (2.2). Let ϕ ∈ S n . We have It remains to prove that The last equality follows from a simple change of variables and homogeneity ofK. We have Since the surface area of the sphere {|x| = ε} is nω n ε n−1 , the integral on the right hand side of (2.3) converges to 0 as ε → 0. ✷ For ϕ ∈ S n we have ∇(A Ω ϕ) = cϕ + (p.v. ∇K) * ϕ and hence ∇A Ω ϕ p ≤ C ϕ p , 1 < p < ∞, because the convolution with p.v. ∇K is a singular integral. This and the fact that A Ω : L p → L p * , 1 < p < n, is bounded yields A Ω ϕ Ẇ 1,p ≤ C ϕ p , 1 < p < n. Now the result follows by a density argument. ✷ 3. Proof of Theorem 1.1 Recall that Φ t = (Φ 1 t , . . . , Φ m t ) is a vector valued function and the convolu- We can also write it in an abbreviated form ∇Φ t = ∇Kχ R n \B(0,t) + µ t , µ t = (µ 1 t , . . . , µ m t ). Proof. The lemma is a straightforward consequence of the integration by parts. Indeed, for ϕ ∈ C ∞ 0 (R n ) we have The proof is complete. ✷ Lemma 3.2. The convolution with ∇Φ t is a bounded operator from L p (R n , R m ) to L p (R n ) for all 1 < p < ∞, Proof . For f = (f 1 , . . . , f m ) ∈ L p we have where σ t is the normalized Lebesgue measure on S n−1 (0, t) and I * i f i are maximal singular integrals defined independently of t. Now it suffices to observe that the operators I * i are bounded in L p and |f | * σ t p ≤ f p σ t = f p by Young's inequality. ✷ with the constant C > 0 that does not depend on t. Moreover Proof. Clearly f * Φ t p * ≤ C f p , because |f * Φ t | can be estimated by the Riesz potential of |f |.
If ϕ ∈ C ∞ 0 (R n , R m ), then ∇(ϕ * Φ t ) = ϕ * ∇Φ t and hence Lemma 3.2 yields The lemma follows now from a density argument. ✷ Lemma 3.4. The maximal operator Proof. Observe that |f | * σ t ≤ Sf and hence the estimates from Lemma 3.2 give Thus the result follows from the boundedness of the maximal singular integrals and from the boundedness of the spherical maximal function. This is the only moment where we use the assumption p > n/(n − 1). ✷ We are ready now to complete the proof of the theorem. Clearly for f ∈ L p (R n , R m ) we have |A * Ω f | ≤ CI 1 |f |, so A * Ω : L p (R n , R m ) → L p * (R n ), 1 < p < n is bounded. It suffices to prove now that for f ∈ L p (R n , R m ), p > n/(n−1), A * Ω f is absolutely continuous on almost all lines and pointwise partial derivatives of A * Ω f satisfy (3.1) |∇A * Ω f | ≤ T * f a.e. Indeed, Lemma 3.4 will give the estimate ∇A * Ω f p ≤ C f p for p > n/(n − 1) and the result will follow form the characterization ofẆ 1,p by absolute continuity on lines [3, Section 4.9].
To prove absolute continuity along with (3.1) it suffices to show that for almost all lines ℓ in R n the function A * Ω f satisfies Indeed, since the function T * f is in L p on almost all lines ℓ (Fubini), estimate (3.2) implies absolute continuity of A * Ω f on compact intervals in ℓ with the estimate for the directional derivatives e. In particular partial derivatives of A * Ω f are in L p , so A * Ω f ∈Ẇ 1,p . Now taking the supremum in (3.3) over a countable and dense set of directions ν ∈ S n−1 we obtain |∇A * Ω f | ≤ T * f. Thus we are left with the proof of (3.2).
According to Lemma 3.3, the functions f * Φ t are inẆ 1,p and hence they are absolutely continuous on almost all lines. Thus for all rational t > 0 and almost all lines ℓ, all of the functions f * Φ t are absolutely continuous on ℓ.
Fix x, y ∈ R n such that A * Ω f (x) < ∞, A * Ω f (y) < ∞ and (3.4) is true for all positive rational numbers t.
By symmetry we can assume that A * Ω f (x) ≥ A * Ω f (y). Choose a sequence of positive rational numbers t k > 0 such that Proof. For ϕ ∈ C ∞ 0 (R n ) we have The case of general f ∈ W 1,p follows by the approximation argument with the use of the Hölder inequality: ∇f ∈ L p , 1 ≤ p < n and the function z/|z| n χ R n \B(0,t) belongs to L p ′ . The proof is complete. ✷ According to the lemma the spherical maximal function of f ∈ W 1,p , 1 ≤ p < n can be represented as where Ω(z) = z for z ∈ S n−1 . Thus if the claim of Theorem 1.1 remains true for 1 < p ≤ n/(n − 1), then we immediately obtain that the spherical maximal function S : W 1,p →Ẇ 1,p is bounded for 1 < p ≤ n/(n − 1). ✷