$p$-capacity with Bessel convolution

We define and examine nonlinear potential by Bessel convolution with Bessel kernel. We investigate removable sets with respect to Laplace-Bessel inequality. By studying the maximal and fractional maximal measure, a Wolff type inequality is proved. Finally the relation of B-$p$ capacity and B-Lipschitz mapping, and the B-$p$ capacity and weighted Hausdorff measure and the B-$p$ capacity of Cantor sets are examined.


Introduction
Classical, nonlinear, and Bessel potentials are widespread, have an extensive literature, and are widely applicable, see e.g.[17], [10], [15] and the references therein.Below we introduce and examine nonlinear potential defined by Bessel convolution with Bessel kernel.
Bessel translation was defined by Delsarte [5] and the basic investigation is due to Levitan, [14].In a series of works the authors pointed out that Bessel translation and convolution methods are effective tools to handle Bessel-type partial differential operators, see e.g.[16], [12], [18], [13].It also proved useful for deriving Nikol'skii type inequality, see [4], and for giving compactness criteria in some Banach spaces, see [11].
This leads to examine nonlinear potential and p-capacity with respect to Bessel convolution.The curiosity of the method is that the underlying space of Besselp capacity is automatically weighted.Weighted nonlinear potential was studied already in the '80-s, see e. g. [1], [3].For logarithmic potentials with external field see the monograph [21].In our investigation the Bessel weighted space is a natural consequence of the definition of convolution, and so many of the results are very similar to the ones proved in the unweighted case.
The paper is organized as follows.After the preliminaries, in the third section, applying recent results on Bessel potential, we investigate removable sets for Laplace-Bessel equation.In the fourth section a Wolff type inequality is proved, which is the basis of the study of the last section.This last section contains some "metric" results on Lipschitz type mapping and on capacity of Cantor sets.Since Bessel translation is not a geometric similarity, moreover the underlying space is weighted, we have to introduce a special property (B-Lipschitz mapping), and the notion of weighted Hausdorff measure.
The translation can also be expressed as an integral with respect to a kernel function: (2) where , |x − t| < z < x + t 0, otherwise.

Obviously
T t a f (x) = T x a f (t).T a is a positive operator, and (4) T t a,x f (x) p,a ≤ f p,a , 1 ≤ p ≤ ∞, see e.g.[14].
The generalized convolution with respect to the Bessel translation is a,x f (x)g(x)dλ a (x).
Subsequently if it does not cause any confusion, T t f (x) stands for T t a f (x).For any set H ⊂ R n we denote by H + := H ∩ R n + .The next technical lemma will be useful in the following sections.
There is a c > 0 such that for all Proof.The first two statements are direct consequences of the definition, for ( 6) see [8, p. 321].( where 2 ) sin ϑ 2αi dϑ i is a probability measure on [0, π].
If r < x i + t i ≤ 2r, using (2) and recalling that |x i − t i | ≤ r 2 we have there are positive constants A and B such that Af < g < Bf ) and we have for any x, t.Together with Lemma 1 it implies that c 1 T t χ B+(0,r) (x) ≤ T t χ B+(0,cr) (x) ≤ c 2 T t χ B+(0,r) (x).

Radially decreasing kernels and B-p capacity.
Definition 1.Let g be a non-negative lower semi-continuous, non-increasing function on R + for which Then κ := g(|x|) is a radially decreasing kernel on R n .
The B-p capacity with respect to κ is as follows.
) As usual, the definitions above can be extended to any subsets of R n + as it follows.
In view of (2.1) The last inequality is equivalent with the assumption, and according to Remark 2, it proves the statement.
(3) It is enough to show that λ a (E∩B + (0, r)) = 0 for all r > 0. Let F = E∩B + (0, r) and f ∈ L p+ a such that κ * a f (x) ≥ 1 on F .Then by Fubini's theorem We estimate the second factor.
x a on |x − t| < r.Thus by the assumption we have In the first integral |t| < 2r and |x − t| < r, so the convolution can be estimated as where in spherical coordinates the last inequality is just (8).Thus, I is also bounded by a constant.Taking infimum over appropriate functions f , we have that λ a (F ) ≤ cC p,κ (F ), which implies the statement.

2.3.
Bessel and Riesz kernels.The modified Bessel function of the second kind, K α is defined as follows.
, where J α is the Bessel function, and Considering r > 0, around zero and around infinity (10) K The Bessel kernel is Below we also need the Riesz kernel: (12) In the last section we use Bessel kernel rather than the Riesz kernel, because its behavior at infinity allows wider function classes.On the other hand, around the origin the Riesz kernel, I ν−|a| (x), behaves similarly to the Bessel kernel and is simpler, thus it proved to be a useful tool for computations.
Below we examine B-p capacity, which is defined by generalized convolution referring to the Bessel kernel: C p,a,ν (E) := C p,Ga,ν (E).In view of ( 9), (10) 3. The Laplace-Bessel operator B-elliptic equations are investigated by several authors.For instance fundamental solutions are given, see e.g.[12] and [13].Harmonic analysis associated with Bessel operator is examined, see e.g.[16] and mean-value theorems are proved, see [19].Here we give a simple application of B-p capacity.
We begin this section by introducing some additional notation.According to (13) Thus we define the next Banach space.
∂ ∂x be The Bessel operator.The Laplace-Bessel operator is defined as With this notation we define the Sobolev space W m p,a with m ∈ N as it follows.
Notation.We need the "even" functions from the Schwartz class in R n + .
The next lemma describes the relation of Bessel potential and Sobolev spaces above.
Lemma 2. Let m be a positive integer.Then Comparing this with (14 On the other hand taking into consideration that S e is a dense subset in W m p,a , let f ∈ S e .According to [18,Theorem 4.5] Thus by ( 15) and ( 16) (1) If p = 2 with standard convolution, N is the spectral measure defined by Deny, see [6].
(2)Of course, N p,a,ν can be extended as above and C p,a,ν (E) ≤ N p,a,ν (E).

Notation. Let us introduce the inner product for measurable functions
We denote in the same way the effect of a distribution.
Indeed, let ε > 0 be arbitrary and f = G a,ν * a g such that g p ′ ,a ≤ ε, f ≡ 1 on K.By ( 14) Thus λ a (K) = 0 and so λ(K) = 0. ( Lf, g a = f, Lg a . In one dimension for L = B α it is [16, (2.4)].Similarly to this case if f ∈ C ∞ 0 (intR n + ) and g is smooth enough, then integration by parts implies the result.For general elements of the dual space we extend Lg by the formula above, that is In view of (1) of Remark 5 u is a.e.defined, so it can be handled as a distribution.Since , where c = c(L) depends on the coefficients of L. Applying Lemma 2 and the inversion of Bessel potential (for formulae see e.g.[7, Theorem 1]) we have where c depends on u and L. Since ε was arbitrary, for all g ∈ C ∞ 0 (O) u, Lg a = 0, so in view of (17) u is a weak solution on O.
The fundamental solution for the Laplace-Bessel operator, that is On the other hand we prove that if C p ′ ,a,ν (K) = 0, then N p ′ ,a,2m (K) = 0 too.According to Definition 2 for an ε > 0 there is a nonnegative function Noticing that h(G a,ν * a f ) fulfils the requirements of Definition 3, since ε is arbitrary, the statement is proved.

Maximal measure and a Wolff type inequality
Bessel maximal function was introduced and examined e.g. in cf.e.g.[8], see also the references therein.The boundedness of the maximal operator in some Morrey spaces is studied and applied to prove a Hardy-Littlewood-Sobolev type theorem in [9].The maximal measure presented below has proved useful in formulating a Wolff type inequality which is the main tool of the next section.Wolff type inequalities can be applied in different situations, for instance in martingale theory, see [3] or deducing trace inequalities or characterize the trace measures via Wolff's inequality, see e.g.[24], [25], [23] and the references therein.
Below we define the maximal measure with respect to Bessel convolution.
Since λ a (B + (0, r)) = cr n+|a| , we define the fractional maximal measure as Changing the order of integration we get Notation.
Lemma 4.There is a ̺ > 1 and a b > 0, such that for all s > 0 and ε ∈ (0, 1], By lower semicontinuity we can take Whithney's decomposition of H µ s , i.e. (Dyadic cubes means cubes with side 2 −k , k ∈ Z, whose vertices belong to the lattice {m2 −k : m ∈ Z n }.For Whithney's decomposition see [22,page 16,Theorem 3].)In addition, to prove (24) if diamQ i ≥ 1  8 , then we decompose it to subcubes with diameter is between 1  16 and 1 8 , and we consider this new sequence of cubes.
Let Q be an element of this decomposition.Let x ∈ Q be arbitrary, denote the center of Q by x c and let d := diamQ.Let G := B(x c , 6d), B = B(x, 8d), that is At first we deal with I ν−|a| * a µ 2 .To this we estimate T t χ B+(0,r) (x).We can assume, that r > 11  2 , otherwise suppµ Now we choose ̺ so that I ν−|a| * a µ 2 (x) ≤ ̺s 2 , which implies that (25) If the diameter of Q was originally less then 1  8 , then the whole construction is contained in a ball of radius less than one, so the same chain of ideas leads to If there is no . Then, recalling that r > 11  2 d, we have In view of Lemma 1 B = suppT t χ B+(0,8d)(x0) ⊂ T + (x 0 , 8d).Thus by ( 7) Taking into consideration that λ a (Q) Since diamQ ≤ 1 8 , similarly we have 28) or (29), respectively, is fulfilled, otherwise Q ⊂ K εs µ.Recalling (25) or (26) and adding over all Q ∈ {Q i }, we obtain the required result.
Corollary 2. With the assumptions of Theorem 2 we have 9) and ( 10) ensure the first inequality and According to (5) 1,a M a,ν,δ µ p,a , which, together with Theorem 2, implies the statement.
The corresponding Wolff-function is Remark 6.In view of Lemma 3 and ( 20) we can observe that (31) We are in position to prove the next Wolff type inequality.

Metric properties
Applying the previous section, below we investigate some "metric" properties of B-p capacity.Since the Bessel-translation is not a geometric congruence, we need a special "Lipschitz"-condition.It is also necessary to introduce the notion of "weighted Hausdorff measure", to examine Cantor-type sets.
At the beginning of this section let us recall that for 1 and the B-p capacity is non trivial if 1 < p < n+|a| ν .
5.1.B-Lipschitz mappings.The next Lipschitz type property is corresponding to the Bessel translation.Example.Let f : R + → R + be a Lipschitz function.Let K ⊂ intR n + compact and Φ : K → R n + ; Φ(x) = f (|x|)x.On K Φ also fulfils the Lipschitz condition with constant L(K) and If L > M (K), we have It can be readily seen, that the Hessian is positive (negative) definite if ϑ = 0 (ϑ k = π k = 1, . . ., n), respectively.Thus the Lipschitz property of Φ(x) implies that the B-Lipschitz condition fulfils for all ϑ ∈ [0π) n .
where c depends only on n, p, a, L.

Proof. By standard arguments it is enough to prove for any
where c = c(n, p, a, L).This implies that Indeed, if L ≤ 1, we have immediately the inequality above, if L > 1 we have to consider Remark 7 with δ = 1 L , which leads again to the inequality above.According to Theorem 3 it proves the statement, cf. the definition above.5.2.Coverings.In the next subsections coverings in Bessel-weighted space are introduced.Since Bessel-convolution lives in a weighted space, B-p capacity of a set depends also on the location of the set.As capacity is in close connection with Hausdorff measure, in the next subsection we extend this notion to weighted spaces as well.
) Let A(r) be the minimal number of balls of radius r required to cover K.

B(r) := inf{
j=1 B(u j , r) be a minimal covering of K. Then there is a constant C n such that any point of K belongs to at most C n balls.Indeed, let C n = C n (q) be the minimal number of balls of radius q ≤ 1 2 which covers the unite ball.Suppose that there is a point x ∈ K which belongs to C n (q) + 1 balls.Then B x, r q contains all these balls and it can be covered by C n (q) balls of radius r which contradicts with minimality, cf.e.g.[2, page 145 and {B(u i , r k+2 )} be minimal coverings of K with the corresponding points {m j } A(r) , respectively.Any m j belongs to a ball, B(u i , r k+2 ), and so m a j ≤ M a i .Since at most C n 1 q+2 maximum points (m j ) can be in the same ball, ). Repeating the chain of ideas with r k−1 and q 1 = 2 k−1 r, we have where c = c(n, a).
Proof.Let µ ∈ M(K) and r k = 1 2 k , as above.According to Corollary 2 and (31) we have where the last inequality follows from the monotone convergence theorem.Let K ⊂ ∪ A(r k+1 ) j=1 B(u j , r k+1 ) be a minimal covering.Recalling that χ B+(0,r k ) * a µ(x) = Considering again the support of T t χ B+(0,r k ) , by Hölder's inequality we have Since x ∈ B (u j , r k+1 ), B(x, r k ) ⊃ B(u j , r k+1 ).As T t χ B+(0,r k ) (x) is continuous (actually it belongs to the Lip( 1 2 ) class) on Estimating the last integral we have So we have By Hölder's inequality Taking into consideration Remark 9 we have Comparing it with (33) the proof is finished.

5.3.
Hausdorff measure with Bessel external field.Definition 6.Let h be an increasing function on R + with h(0 where x a i,ri := max{t a : t ∈ B(x i , r i )}.Since Λ ̺ h,a (E) is a decreasing function of ̺, we can define the (finite or infinite) a-Hausdorff measure of E as Remark 10. (1) h(x, r) := x a r h(r) is an increasing function of r, but it depends on x as well, that is the a-Hausdorff measure of E depends also on the location of E.
(2) If K ⊂ intR n + is compact, then there are constants ) is not a ̺-covering of E, there exists an r l > ̺, and because Let us denote by Q k the set of the dyadic cubes in R n + with edge length 1 2 k , k ∈ Z. Theorem 6.With the notation above, let h be an increasing function, E ⊂ R n + and µ ∈ M(E) such that for all balls µ(B(x, r)) ≤ h(r).Then Then there is a constant c depending only on n, k and a and a measure µ ∈ M(E) satisfying that µ(B(x, r)) ≤ h(r) for all balls, such that The first part of the second statement is proved in [2, page 137], namely there are measures µ ll such that suppµ ll = {∪ j Q , where r i = 1 2 i .Moreover µ ll has constant density on each Q j ∈ Q l .Finally µ is defined as a weak accumulation point of {µ ll }.Then suppµ = E and µ where c = c(n, a, k) and the infimum is taken over all finite or denumerable coverings of E. So Taking into consideration that a where c = c(n, a, k).
i=1 q k,i as above, where q k,i are the closed cubes in C n k of edge length l k .Let v a k,i := max x∈q k,i x a .Let us denote by Obviously, h(l k ) > h(l k+1 ).Let h L (r) := h Q,L,a (r) be an increasing function on [0, ∞), h L (0) = 0 and h L (l k ) is given by (37).
Proof.With the notation above C L can be covered by 2 kn balls of radius l k Comparing with Theorem 5 it shows that C a,ν,p (C L ) = 0 if diverges.
On the other hand, considering h L let us construct the measure µ L ensured by Theorem 6.In view of Lemma 1 In view of (33) which proves the converse statement.

5. 4 .
Capacity of Cantor sets with Bessel external field.Let L := {l k } ∞ k=0 be a decreasing sequence such that 0 < 2l k+1 < l k for k ∈ N. Let C 0 be a closed interval of length l 0 .C 1 is obtained by removing an open interval of length