Monatsh Math DOI 10.1007/s00605-014-0688-1

Holomorphic potentials and multipliers for Hardy-Sobolev spaces C. Cascante · J. Fàbrega · J. M. Ortega

Received: 12 February 2014 / Accepted: 3 September 2014 © Springer-Verlag Wien 2014

Abstract We show that the bounded holomorphic potentials are pointwise multipliers for the Hardy-Sobolev spaces. As a consequence, we construct nontrivial examples of such multipliers and we give some applications. Keywords

Nonlinear holomorphic potentials · Multipliers · Hardy-Sobolev spaces

Mathematics Subject Classification

31C15 · 32A35

1 Introduction The main object of this paper is to show that the bounded holomorphic potentials are pointwise multipliers for the Hardy-Sobolev in the unit ball B of Cn . This fact allows us to construct nontrivial examples of such multipliers. p We recall that if 1 ≤ p < ∞ and s ∈ R, then the Hardy-Sobolev space Hs consists  of the holomorphic functions f on B such that if f = f k is its homogeneous k

polynomial expansion, and the fractional radial derivative is defined by (1 + R)s f :=



(1 + k)s f k ,

k

Communicated by G. Teschl. Partially supported by DGICYT Grant MTM2011-27932-C02-01 and Grant 2014SGR289. C. Cascante (B) · J. Fàbrega · J. M. Ortega Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, Barcelona, Spain e-mail: [email protected]

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then  f  Hsp := (1 + R)s f  H p < ∞. Given X a Banach space of holomorphic functions, we denote by Mult (X ) the space of pointwise multipliers from X to itself. p It is well known that any pointwise multiplier of Hs is a function in H ∞ and that p if s ≤ 0, the space of multipliers coincide with H ∞ . If s > n/ p, then the space Hs is a multiplicative algebra and, in consequence, it coincides with the algebra of its multipliers (see, for instance [4]). In what follows, we will assume that 0 < s ≤ n/ p. In this situation, even in the cases where the characterization is known, for instance n − sp < 1 or p = 2, the conditions are difficult to check and it is not easy to construct explicit non trivial examples of multipliers. Our focus of interest in this paper comes out from the following fact in Rn : for a nonlinear potential of a positive measure, it is enough to impose its boundedness to assure that the potential is a pointwise multiplier of the Bessel space L s, p (see the unpublished work [5] and the general theory of multipliers on potential spaces in [10]). We will check, using completely different methods, an analogous result for non isotropic holomorphic potentials on the unit ball. p We recall a representation of the functions in Hs , when 1 < p < ∞ and 0 < s < n, p which will be used in the paper. Namely,  a function f is in Hs if and only if it can h(ζ ) dσ (ζ ), for some h ∈ L p (S), be represented as f (z) = Cs (h)(z) := S (1 − zζ )n−s where dσ is the normalized Lebesgue measure on S. The functions h can be restricted to be boundary values of H p functions. In this case, Cs is a bijective operator from p H p to Hs . These facts allow us to construct different nontrivial examples of multipliers, which are summarized in the following theorem. We prove that the only boundedness is enough both for the holomorphic potentials C2s (μ) and for Us, p,λ (μ) and Vs, p,λ (μ) (see Sect. 2 for the precise definitions) introduced by [8], for being a multiplier for p Hs2 and Hs respectively. Theorem 1.1 Let 1 < p < ∞, 0 < n − sp < λ < 1 and μ a finite positive Borel measure on S. 1. Assume that either the holomorphic potential Us, p,λ (μ) is bounded if p < 2 or the holomorphic potential Vs, p,λ (μ) is bounded if p ≥ 2. p We then have that if p < 2 the function Us, p,λ (μ) is a multiplier for Hs and if p ≥ 2 p the function Vs, p,λ (μ) is a multiplier for Hs .  dμ(ζ ) 2. Assume that C2s (μ) := S (1−zζ is bounded, then C2s (μ) is a multiplier for )n−2s Hs2 .

In particular, if the measure is the capacitary measure associated to compact subsets of S, which will be defined later, we have: Theorem 1.2 Let 1 < p < ∞, 0 < s < n/ p and let E be a compact subset in S with nonisotropic Riesz capacity different from zero. If μ E is the capacitary extremal measure associated to E, we have:

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1. If 1 < p < 2 and in addition n −s < λ < 1, the holomorphic potential Us, p,λ (μ E ) p is a multiplier for Hs . 2. If p ≥ 2, and n−sp < λ < 1, the holomorphic potential Vs, p,λ (μ E ) is a multiplier p for Hs . 3. The potential C2s (μ E ) is a multiplier for Hs2 . Although the first statement is probably true for the more general case n − sp < 1, the techniques used in the proof need to assume that n − s < 1. Finally, we give some applications of the results obtained. We solve a strong Corona p problem for multipliers of Hs with data holomorphic potentials of capacitary measures, and we show that the sets of capacity zero are “weak exceptional sets” for the p multipliers of Hs in a sense that will be specified. The paper is organized as follows: In Sect. 2, we give the definitions and results on trace measures and holomorphic potentials that we will use in the proof of our main results. In Sect. 3 we prove Theorem 1.1, in Sect. 4 we prove Theorem 1.2 and in the last Section, we give some applications. 2 Preliminaries 2.1 Potentials, nonisotropic capacities and trace measures Consider, in Rn , the space of Bessel potentials of functions in L p (Rn ), G s [L p (Rn )], 1 < p < ∞ and 0 < sp ≤ n. If μ is a positive measure on Rn , G s is a Bessel  potential, and the nonlinear potential of ν defined by Vs, p [μ] = G s ∗ (G s ∗ μ) p −1 is bounded, then Vs, p [μ] is a multiplier on G s [L p ] (see [5] and also the general theory  on multipliers in [10]). Hence, the measure (G s ∗ μ) p −1 d x is a trace measure for p the space G s [L ]. We will check that we can state a nonisotropic version of this last assertion. We first recall some definitions and results concerning to nonisotropic potentials, capacities and multipliers that we will need. Throughout the paper, if 0 < s < n, Is will denote the nonisotropic Riesz operator given by  ϕ(ζ ) Is (ϕ)(z) := dσ (ζ ). S |1 − zζ |n−s We recall that if E ⊂ S, 1 < p < ∞ and 0 < s < n, the nonisotropic Riesz capacity of the set E is given by p

Cs, p (E) = inf{|| f || L p ; f ≥ 0, Is ( f ) ≥ 1 on E}. Definition 2.1 If 1 < p < ∞ and 0 < s < n, we say that a finite positive Borel p measure μ on S is a trace measure for the Hardy-Sobolev space Hs if there exists p C > 0 such that for any f ∈ Hs , Mrad [ f ] L p (dμ) ≤ C f  Hsp ,

(2.1)

where Mrad [ f ](ζ ) = supr <1 | f (r ζ )|.

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Definition 2.2 If 1 < p < ∞ and 0 < s < n, we say that a finite positive Borel measure μ on S is a trace measure for the space Is [L p ] if there exists C > 0 such that for any f ∈ L p , (2.2) Is [ f ] L p (dμ) ≤ C f  L p . A proof of the following capacitary characterization of the trace measures can be found in [2]. Proposition 2.3 Let 0 < s < n and 1 < p < ∞. A finite positive Borel measure μ on S is a trace measure for the space Is [L p ] if and only if there exists C > 0 such that for any open set G ⊂ S, μ(G) ≤ CCsp (G). The proof of our main results heavily rely in a nice result obtained in [11], which give a sufficient condition for a measure to be a trace measure for Is [L p ]. This condition involves weighted L p estimates for arbitrary A1 -weights. Proposition 2.4 ([11]) Let g be an integrable function on S such that |g| p dσ is a trace measure for Is [L p ]. Let h be a measurable function on S satisfying that there exists C > 0 such that for any weight w in A1 ,   p |h| wdσ ≤ C |g| p wdσ. (2.3) S

S

We then have that the measure |h| p dσ is a trace measure for Is [L p ]. 

We recall that Is [Is (μ)] p −1 denotes the nonisotropic potential and that the definition of the nonisotropic Wolff-type potential of a measure μ is given by  Ws, p (μ)(ζ ) =

1  μ(B(ζ, 1 − r ))  p  −1

0

(1 − r )n−sp

dr . 1−r

(2.4)

The methods in [7] can be adapted to the nonisotropic case to show the following theorem. Proposition 2.5 Let 1 < p < ∞, 0 < s < n/ p and let μ be a finite positive measure in S. Then the measure dμ1 = W dμ is a trace measure for Is [L p ]. (μ) p−1 s, p

We also recall the nonisotropic versions of Theorem 2.1 and Lemma 3.1 in [11]. Proposition 2.6 Let 1 < p < ∞, 0 < s < n/ p and let μ be a positive finite Borel measure on S. Then the following assertions are equivalent: 1. The measure μ is a trace measure for Is [L p ].  2. The measure (Is (μ)) p dσ is a trace measure for Is [L p ]. We can now state our first result. Theorem 2.7 Let 1 < p < ∞, 0 < s < n and let μ a finite positive Borel measure on S. Assume that the nonisotropic potential Ws, p (μ) is bounded on S. Then μ and  (Is (μ)) p dσ are trace measures for Is [L p ].

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Proof We first observe that the boundedness of the nonisotropic Wolff potential Ws, p (μ) gives that the measure μ is a trace measure for Is [L p ]. Indeed 

 S

Is ( f ) p (ζ )dμ(ζ ) =

S

Is ( f ) p (ζ )Ws, p (μ) p−1 (ζ )

 

S

Is ( f ) p (ζ )

dμ(ζ ) Ws, p (μ) p−1 (ζ )

dμ(ζ ) p   f L p , Ws, p (μ) p−1 (ζ )

(2.5)

where in the last estimate we have used that by Proposition 2.5, the measure dμ is a trace measure for Is [L p ]. By Proposition 2.6, we finally obtain Ws, p (μ) p−1  that the measure (Is (μ)) p dσ is also a trace measure for Is [L p ]. If μ is a positive Borel measure on S, 1 < p < ∞, 0 < s < n and w is an A p -weight. We recall (see for instance [1]) that the nonlinear weighted nonisotropic  potential of the measure μ is defined by Is [(w −1 Is (μ)) p −1 ] . The (s, p)-energy of μ with weight w (see [1]), is defined by      Es, p,w (μ) := Is [(w −1 Is (μ)) p −1 ](ζ )dμ(ζ ) = (Is (μ)(ζ )) p w(ζ )−( p −1) dσ (ζ ). S

S

(2.6)

It is also introduced in [1] a weighted Wolff-type potential of a measure μ as 1  μ(B(ζ, 1 − r ))  p  −1 

 Ws, p,w (μ)(ζ ) = 0

(1 − r )n−sp

\



B(ζ,1−r )

w −( p −1) (η)dσ (η)

dr . 1−r (2.7)

Here   1 \ θ := θ. |B| B B We have the pointwise estimate 

Ws, p,w (μ)(ζ ) ≤ CIs [(w −1 Is (μ)) p −1 ](ζ ).

(2.8)

The weighted version of Wolff’s theorem gives that if w in an A p -weight, the following weighted Wolff-type theorem holds (see [6]):   −1 p  −1 Es, p,w (μ) = Is [(w Is (μ)) ](ζ )dμ(ζ ) Ws, p,w (μ)(ζ )dμ(ζ ). (2.9) S

S

When w ≡ 1 we will just write Ws, p (μ) and Es, p (μ). We also recall an extremal theorem for the nonisotropic Riesz capacities. See, for instance, the books of [2] and [10] and the paper [9]. Proposition 2.8 Let 1 < p < ∞, 0 < s < n/ p and G ⊂ S be an open set. There exists a positive capacitary measure μG such that

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(i) (ii) (iii) (iv) (v)

supp μG ⊂ G. μG (G) = Cs, p (G) = Es, p (μG ). Ws, p (μG )(ζ ) ≥ C, for every ζ ∈ G.  Is ((Is (μG )) p −1 )(ζ ) ≤ C, for any ζ ∈ S. If = min(1, p − 1), 

Cs, p ({ζ ∈ S; Is (Is (μG ) p −1 )(ζ ) ≥ t})  (vi) If 1 < p ≤ 2 −

s n

Cs, p (G) . t

( p−1)n n s n−s or 2 − n < p and 1 < δ < n−sp , then the ))δ is in A1 . Moreover, for any η ∈ S and y > 0,

and 1 < δ <

weight w δ = (Is (Is (μG )

p  −1

w δ (B(η, y))  w δ (η). yn

(2.10)

2.2 Holomorphic potentials If 1 < p < ∞ and 0 < n − sp < 1, [8] introduced two families of holomorphic potentials (one for p > 2 and the other for p ≤ 2) that extend some of the properties of the linear holomorphic potential C2s (μ). Let μ be a finite positive Borel measure on S. For any n − sp < λ < 1, we set the following holomorphic functions on B defined by  Us, p,λ (μ)(z) = 0

1



S

and

μ(B(ζ, 1 − r )) (1 − r )n−sp 

Vs, p,λ (μ)(z) = 0

1 

S

 p −1

(1 − r )λ−n (1 − r zζ )λ

(1 − r )λ+sp−n (1 − r zζ )λ

dσ (ζ )

 p −1 dμ(ζ )

dr , 1−r

dr . 1−r

(2.11)

(2.12)

It is proved in [8] the following proposition: Proposition 2.9 Let 1 < p < ∞, 0 < s and λ such that 0 < n − sp < λ < 1. We then have: 1. If 1 < p < 2 there exist C1 , C2 > 0 such that for any finite positive Borel measure μ on S the following assertions hold: (a) For any η ∈ S, lim inf Re Us, p,λ (μ)(ρη) ≥ C1 Ws, p (μ)(η). ρ→1

p

(b) ||Us, p,λ (μ)|| H p ≤ C2 Es, p (μ). s 2. If p ≥ 2, there exist C1 , C2 > 0 such that for any finite positive Borel measure μ on S the following assertions hold:

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(a) For any η ∈ S, lim inf Re Vs, p,λ (μ)(ρη) ≥ C1 Ws, p (μ)(η). ρ→1

p

(b) ||Vs, p,λ (μ)|| H p ≤ C2 Es, p (μ). s

p

2.3 Multipliers of Hs

We conclude this section stating some basic properties of the pointwise multipliers of p Hs (see Theorem 4.1 in [13]). Theorem 2.10 Let 1 < p < ∞, s ∈ R and k > s. 1. Mult (Hs ) ⊂ H ∞ and if s ≤ 0, then Mult (Hs ) = H ∞ . p p 2. If s > n/ p, then Mult (Hs ) = Hs . p 3. For some (any) positive integer k > s, g ∈ Mult (Hs ) if and only if g ∈ H ∞ and p p k R g ∈ Mult (Hs → Hs−k ). p p p 4. If g ∈ Mult (Hs ), then R j g ∈ Mult (Hs → Hs− j ). p p 5. If t < s, then Mult (Hs ) ⊂ Mult (Ht ). p

p

3 Proof of Theorem 1.1 Proof We begin proving (1). We first observe that by Proposition 2.9, the hypothesis give that the nonisotropic Wolff potential Ws, p (μ) is bounded on S.  By Theorem 2.7, the measure (Is (μ)) p dσ is a trace measure for Is [L p ]. p Case p < 2. We want to check that for any f ∈ Hs , Us, p,λ (μ) f  Hsp   f  Hsp , p

that is, that the function Us, p,λ (μ) is a multiplier for the space Hs . By (3) in Theorem p 2.10, Us, p,λ (μ) is a a multiplier for the space Hs if and only if Us, p,λ (μ) is bounded p p p and for k > s, R k Us, p,λ (μ) ∈ Mult (Hs → Hs−k ), that is, for every f ∈ Hs    1  2 dρ  21      |R k Us, p,λ (μ)(ρζ )|| f (ρζ )|(1 − ρ)k−s  1−ρ   0 

  f  Hsp . L p (S)

(3.1) p,q But, if we denote by Fs is the holomorphic Triebel-Lizorkin space consisting of holomorhic functions f such that for k > s,    1 

q dρ  q1   

k  

R f (ρζ ) (1 − ρ)k−s  1−ρ   0 

< +∞, L p (S)

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p,2

p

it is well known that Fs ⊂ Fs = Hs , (see for instance [12]). Hence, in order to prove (3.1) it is enough to show that if k > s,    

1

|R k Us, p,λ (μ)(ρζ )|| f (ρζ )|(1 − ρ)k−s

0

 dρ     f  Hsp . 1 − ρ  L p (S)

(3.2)

But the left hand side of the above estimate is bounded by    

1

0

|R k Us, p,λ (μ)(ρζ )| | f (ρζ )|(1 − ρ)k−s



≤

S



1

(Mrad ( f )(ζ )) p (

p dρ   1 − ρ  L p (S)

|R k Us, p,λ (μ)(ρζ )|(1 − ρ)k−s

0

dρ p ) dσ (ζ ). 1−ρ

Thus (3.2) will follow if we show that the measure 

1

|R Us, p,λ (μ)(ρζ )|(1 − ρ) k

k−s

0

dρ 1−ρ

p dσ (ζ )

(3.3)

p

is a trace measure for Hs .  Since we have that the measure (Is (μ)) p dσ is a trace measure for Is [L p ], Proposition 2.4 gives that it is enough that we show that for any weight w in A1 ,   S



0 S

p

dρ



w(η)dσ (η)

(1 + R)k Us, p,λ (μ)(ρη) (1 − ρ)k−s 1−ρ

1



(Is (μ)(η)) p w(η)dσ (η).

(3.4)

We have 

1

(1 − ρ)k−s |R k Us, p,λ (ρη)|

0





1

 (1 − ρ)k−s

0

0

1

S



dρ 1−ρ

μ(B(ζ, 1 − r )) (1 − r )n−sp

 p −1

(1 − r )λ−n

dσ (ζ )dr dρ . |1 − rρηζ |λ+k (1 − r )(1 − ρ)

But 

1

0

(1 − ρ)k−s |1 − rρηζ |λ+k

1 dρ  . 1−ρ |1 − r ηζ |λ+s

The preceding estimate, together with Fubini’s theorem give that 

1 0

123

(1 − ρ)(k−s) |R k Us, p,λ (ρη)|

dρ  Φ(η), 1−ρ

(3.5)

Holomorphic potentials and multipliers for Hardy-Sobolev spaces

where 



1

Φ(η) =

S

0

μ(B(ζ, 1 − r )) (1 − r )n−sp

 p −1

(1 − r )λ−n |1 − r ηζ |λ+s

dσ (ζ )

dr . 1−r

We will now follow some of the arguments in [8], page 87. The key point is to apply Hölder’s inequality in such a way that we obtain on one hand μ(B(ζ, 1 − r )) raised to the power 1 and on the other hand an expression where in the denominator we have |1 − r ηζ | raised to some power strictly greater that n. Precisely, let ε > 0 such that p) n − s < ε < λ+s−n(2− . We then have that p−1  p−1 μ(B(ζ, 1 − r )) (1 − r )ε−n (1 − r )n−s |1 − r ζ η|ε S 0    μ(B(ζ, 1 − r )) p − p (1 − r )λ+s−n(2− p)−ε( p−1) dr × dσ (ζ ) . n−s λ+s−ε( p−1) (1 − r ) 1−r |1 − r ηζ | 

Φ(η) =

1



Hölder’s inequality with exponents 1/( p − 1) > 1 and 1/(2 − p) gives Φ(η)  Φ1 (η) p−1 Φ2 (η)2− p , where 

1

Φ1 (η) = 0

μ(B(ζ, 1 − r )) (1 − r )ε−n dr dσ (ζ ) n−s ε (1 − r ) |1 − r ζ η| 1 −r S

and 



1

Φ2 (η) = 0

S

μ(B(ζ, 1−r )) (1−r )n−s

 p

(1 − r )(λ+s−n(2− p)−ε( p−1))/(2− p) |1 − r ηζ |(λ+s−ε( p−1))/(2− p)

dr . 1−r

dσ (ζ )

Since n − s < ε, Fubini’s Theorem gives that Φ1 (η)  Is (μ)(η). Applying again Hölder’s inequality with exponents γ = 1 p(2− p) ,

> 1 and γ  =

we obtain that 

 Φ(η) w(η)dσ (η) 

p

p

S

1 ( p−1)2

S

 1 

Is (μ) (η)w(η)dσ (η)

 1

γ

S

Φ2 (η)w(η)dσ (η)

γ

.

Next, 

 S



1

Φ2 (η)w(η)dσ (η)  0

S

μ(B(ζ, 1 − r )) (1 − r )n−s

 p Ω(ζ, r )dσ (ζ )

dr , 1−r

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where  Ω(ζ, r ) =

(1 − r )(λ+s−n(2− p)−ε( p−1))/(2− p) |1 − r ηζ |(λ+s−ε( p−1))/(2− p)

S

λ+s−ε( p−1) 2− p

We choose 1 < q such that nq < λ+s−ε( p−1) ). 2− p

w(η)dσ (η)

(which is possible since n <

We recall that any w ∈ A1 satisfies a doubling condition of order τ ,

p−1) , by decomfor any τ > n, that is, w(2k B)  2kτ w(B). Choosing τ < λ+s−ε( 2− p posing in coronas in the usual way the integral over S with respect to the variable η, we obtain:

Ω(ζ, r ) 

w(B(ζ, 1 − r )) . (1 − r )n

Altogether, since w ∈ A1 , we have that  S

Φ2 (η)w(η)dσ (η) 

 0

  =



1

S

1



μ(B(ζ, 1 − r )) (1 − r )n−s μ(B(ζ, 1 − r )) (1 − r )n−s

 p  p

w(B(ζ, 1 − r )) dr dσ (ζ ) n (1 − r ) 1−r

dr w(ζ )dσ (ζ ) 1 −r S 0  p −1   1  μ(B(ζ, 1 − r )) B(ζ,1−r ) dμ(η) S

0

(1 − r )n−s

(1 − r )n−s

w(ζ )dσ (ζ )

dr . 1−r

Next, if η ∈ B(ζ, 1 −r ), we have that B(ζ, 1 −r ) ⊂ B(η, 4(1 −r )). This fact together with Fubini’s Theorem gives that the above integral is bounded by 

  μ(B(η, 4(1 − r ))) p −1 1 dr w(B(η, 4(1 − r )))dμ(η) n−s n−s (1 − r ) (1 − r ) 1 −r S 0  p −1  1  μ(B(η, 4(1 − r ))) w(B(η, 4(1 − r ))) dr = dμ(η) n−sp n (1 − r ) (1 − r ) 1−r  0 S  ≈ Ws, p,w−( p−1) (μ)dμ ≈ Is (μ) p (η)w(η)dσ (η), 1



S

S

where in the last estimate we have used the weighted Wolff’s theorem [see (2.9)]. Altogether, we obtain that 

 S

Φ(η) p w(η)dσ (η) 



S

Is (μ) p (η)w(η)dσ (η).

Hence, from (3.5), we have proved (3.4) and then the case p < 2.

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Case p ≥ 2. The same arguments of the preceding case, give that in order to check that p the potential Vs, p,λ (μ) is a multiplier for Hs , it is enough to prove that the estimate (3.3) holds replacing Us, p,λ (μ) by Vs, p,λ (μ), that is, 

p

dρ

k

dσ

R Vs, p,λ (μ)(ρζ ) (1 − ρ)k−s 1−ρ

1

0

p

is a trace measure for Hs . It is proved in [8], page 90, that 

1

|R k Vs, p,λ (μ)(ρζ )|(1 − ρ)k−s

0

1  μ(B(ζ, δ))  p  −1





δ n−s

0

dρ 1−ρ

dδ := Ψ (ζ ). δ

(3.6)

By Proposition 2.4, it will be enough to prove that the measure Ψ p dσ is a trace measure for Is [L p ]. We will partially follow some of the arguments used in that paper to prove an extension of Wolff’s inequality. We first observe that 1  μ(B(ζ, t))  p  −1

 Ψ (ζ ) = p p

t n−s

0

  t 0

μ(B(ζ, y)) y n−s

 p −1

dy y

p−1

dt . t

If we choose 0 < ε < s, Hölder’s inequality with exponent p − 1 > 1, gives that   t 0

μ(B(ζ, y)) y n−s

 p −1

dy y

p−1  tε



t

0

μ(B(ζ, y)) dy . y n−s y 1+ε

Thus we have that  Ψ p (ζ )  0

1 t 0



μ (B(ζ, y)) μ (B(ζ, t)) p −1

dy dt . y (n−s)+ε+1 t (n−s)( p −1)−ε+1

 Now we check that for any open set G ⊂ S, G Ψ p dσ  Cs, p (G), condition that by Proposition 2.4 gives that the measure Ψ p dσ is a trace measure for Is [L p ].  Let μG be the (s, p)-extremal capacitary measure of G, and w = Is (Is (μG ) p −1 ). p−1)n , Assertion (vi) in Proposition 2.8 gives that since p ≥ 2, for any 1 < δ < ( n−sp δ w ∈ A1 , and for any η ∈ S and y > 0, w δ (B(η, y))  w δ (η). yn

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We fix δ. Since by (iii) in Proposition 2.8, w δ  1 on G, we have that 



Ψ p w δ dσ.

Ψ p dσ  G

G

Let η ∈ B(ζ, t). Then B(ζ, t) ⊂ B(η, 4t), and since 0 < y < t, we have 

μ (B(ζ, y)) μ (B(ζ, t)) p −1 ≤





μ(B(η, 4t)) p −1 dμ(η). B(ζ,y)

Integrating with respect to wδ (ζ )dσ (ζ ) we have 



μ(B(ζ, y))μ(B(ζ, t)) p −1 w δ (ζ )dσ (ζ ) S     μ(B(η, 4t)) p −1 dμ(η)w δ (ζ )dσ (ζ ) S B(ζ,y)   ≤ μ(B(η, 4t)) p −1 w δ (B(η, y))dμ(η). S

Using the above estimates, we deduce that 

  Ψ (ζ ) dσ (ζ )  p

G

1  μ(B(η, 4t))  p  −1  t

S 0

t n−s

0

w δ (B(η, y))dy dt dμ(η). y n−s+ε+1 t 1−ε

The estimate (2.10) gives then that the above integral can be bounded, up to a constant by 1  μ(B(η, 4t))  p  −1  t

 

w δ (η)dy dt dμ(η) y −s+ε+1 t 1−ε

t n−s 0  1 μ(B(η, 4t))  p  −1 dt = w δ (η)t s−ε 1−ε dμ(η) n−s t t S 0      1 μ(B(η, 4t)) p −1 δ dt dμ(η) ≈ = w (η) Ws, p (μ)(η)w δ (η)dμ(η). t n−sp t S 0 S

S 0

 

Now, since Ws, p (μ) is bounded on S, then μ is a trace measure for Is [L p ] ( Theorem 2.7) and we can bound the above integral, up to a constant, by 

δ

S



M

w (η)dμ(η) =

μ({η; w(η) ≥ t})t

δ−1



M

dt 

0

Cs, p ({η; w(η) ≥ t})t δ−1 dt.

0

Since Cs, p ({η; w(η) ≥ t}) 

123

Cs, p (G) , t

Holomorphic potentials and multipliers for Hardy-Sobolev spaces

by (v) in Proposition 2.8, this last integral is bounded, up to a constant, by 

M

Cs, p (G)

t δ−2 dt ≈ Cs, p (G).

0

That finishes the proof of the case p ≥ 2. Let us prove (2), that is, if C2s (μ) is bounded, then it is a multiplier for Hs2 . Once again, we must check that 

1

|R k Vs, p,λ (μ)(ρζ )|(1 − ρ)k−s

0

dρ 1−ρ

p dσ

p

is a trace measure for Hs . It is enough to observe that on one hand that, since n−2s < 1, |C2s (μ)| I2s (μ) Ws,2 (μ) and, the boundedness of C2s (μ), gives the boundedness of Ws,2 (μ). On the other hand, the estimate (3.6) holds for C2s (μ), that is, 

1

|R k C2s (μ)(ρζ )|(1 − ρ)k−s

0

dρ  1−ρ



1

0

μ(B(ζ, δ)) dδ . δ n−s δ

With this observation, the same arguments used to finish the case p ≥ 2 for the potentials Vs, p,λ (μ), prove that if C2s (μ) is bounded, then it is a multiplier for Hs2 . 4 Proof of Theorem 1.2 Proof We begin with the case p < 2 and n − s < 1. We will show that 

Mrad [Us, p,λ (μ E )]  Is [Is [μ E ] p −1 ].

(4.1)

Since μ E is the capacitary extremal measure associated to E, we have that the function  Is [Is (μ E ) p −1 ] is bounded (see Proposition 2.8), and consequently, if (4.1) holds, Us, p,λ (μ E ) is also a bounded function. Theorem 1.1 gives then that the potential p Us, p,λ (μ E ) is a multiplier for Hs . So we are led to show (4.1). Since λ > n − s, we have  |Us, p,λ (μ E )(z)| ≈ 0

 ≤

1

0

Next, since  0

1 μ

1 p  −1

S

1

 

S

μ E (B(ζ, 1 − r )) (1 − r )n−sp μ E (B(ζ, 1 − r )) (1 − r )n−s

 p −1  p −1

(1 − r )λ−n |1 − r zζ |λ

dσ (ζ )

1 |1 −

zζ |n−s

dr 1−r

dσ (ζ )

dr 1−r

< 1, and μ E (B(ζ, 1 − r )) is a decreasing function on r , we have

E (B(ζ, 1 − r )) (1 − r )n−s

 p −1

dr  1−r

 0

1

μ E (B(ζ, (1 − r ))) dr (1 − r )n−s 1−r

 p −1 .

123

C. Cascante et al.

Hence, if we write z = ρη, 0 < ρ < 1, η ∈ S,   |Us, p,λ (μ E )(z)| 

S

1

0

μ E (B(ζ, (1 − r ))) dr (1 − r )n−s 1−r

 Is [Is (μ E )

p  −1

](z)  Is [Is (μ E )

 p −1

p  −1

dσ (ζ ) |1 − zζ |n−s

](η).

In particular, we deduce (4.1). Next we deal with the case p ≥ 2. In this case we will check that Mrad [Vs, p,λ (μ E )](η)  Ws, p [μ E ](η),

(4.2)

which again gives that since μ E is the capacitary extremal function that Vs, p,λ (μ E ) is p a bounded and hence a multiplier for Hs . We have that Vs, p,λ [μ E ](z)  p −2   1  (1 − r )λ+sp−n (1 − r )λ+sp−n dr . ≤ dμ E (ζ ) dμ E (ξ ) λ λ 1 −r S |1 − r zζ | S |1 − r zξ | 0 Now, write z = ρη and fix δ < 1. Since |1 − r zζ | ≈ (1 − r ) + (1 − ρ) + |1 − ηζ |, we have 



dμ E (ζ )

S

dμ E (ζ )

≥

|1 − r zζ |λ

B(η,δ)

|1 − r zζ |λ



μ E (B(η, δ)) . (δ + 1 − r + 1 − ρ)λ

On the other hand, 

dμ E (ξ ) S

|1 − r zξ |λ

  

 S

dμ E (ξ ) 1



dδ

|1−r zξ |<δ

μ E (B(η, δ))

0

δ λ+1

dδ . (δ + 1 − r + 1 − ρ)λ+1

The fact that we are assuming that p ≥ 2 gives that p  − 2 ≤ 0, and, consequently the above estimates give that  |Vs, p,λ [μ E ](ρη)|  0

1 1



μ E (B(η, δ)) p −1

0



(1 − r )(λ+sp−n)( p −1) dδdr .  (δ+1 − r +1 − ρ)λ+1+λ( p −2) (1 − r )

But  0

123

1



1 (1 − r )(λ−(n−sp))( p −1) dr  (n−sp)( p −1)+1 .  (δ + 1 − r )λ( p −1)+1 1 − r δ

Holomorphic potentials and multipliers for Hardy-Sobolev spaces

Plugging this last estimate in the above one, we obtain 1 μ

 Mrad [Vs, p,λ [μ E ]](η)  0

E (B(η, δ)) δ n−sp

 p −1

dδ = Ws, p [μ E ](η), δ

and that ends the proof of this case. Finally, since Mrad [C2s (μ E )]  Ws,2 (μ E ), we also deduce that C2s (μ E ) is a pointwise multiplier for Hs2 . 5 Applications In this section we give some applications of the above results on multipliers for HardySobolev spaces. In the first application we show that there exists a strong solution of the Corona p Theorem for multipliers of Hs for some particular data. Proposition 5.1 Let 1 < p < ∞, 0 < n − sp < 1 and assume in addition that n−s < 1 if p < 2. Let K i , i = 1, . . . , l be compact subsets of S such that li=1 K i = S. p Let Vi i = 1, . . . , l be the potential multipliers for Hs given in Theorem 1.2 associated to the extremal measures of the compact sets K i . We then have that there there exist p multipliers of Hs , gi , i = 1, . . . , l, such that 1=

l 

Vi gi .

i=1 p

That is, there exists a strong solution of the Corona Theorem for multipliers of Hs with data V1 , . . . , Vl .

Proof Since Vi ∈ Hs , for each i = 1, . . . , l, there exists a.e Vi∗ (η) = limr →1 Vi (r η) ∈ H p . On the other hand, by (iii) in Proposition 2.8, for almost every η ∈ K i , ReVi∗ (η)  1. In addition, Proposition 2.9 gives that ReVi∗ (η) ≥ 0. Consequently, p

Re

l 

Vi∗ (η)  1,

i=1

a.e. η ∈ S. Thus, if P(z, η) denotes de Poisson-Szegö kernel, Re

l  i=1

 Vi (z) =

S

P(z, η)Re

l 

Vi∗ (η)dσ (η)  1,

i=1

for any z ∈ B. This estimate gives that V :=

l

1 Vi p Hs ,

∈ H ∞ . Thus, by (3)

i=1

in Theorem 2.10, to prove that V is a multiplier of

it is sufficient to show

123

C. Cascante et al. p

p

that R k V ∈ Mult (Hs → Hs−k ) for some positive integer k > s. Note that   | f R k V |  α·β=k | f (R α1 V1 )β1 · · · (R αl Vl )βl |, where α · β = lj=1 α j β j . Since p p Vi ∈ Mult (Hs ), assertions (5) and (4) in Theorem 2.10 give R αi Vi ∈ Mult (Ht → p p p α β α β Ht−αi ) for any t ≤ s. Thus (R 1 V1 ) 1 · · · (R l Vl ) l ∈ Mult (Hs → Hs−k ). This concludes the proof. Our last application shows that under the same hypothesis than before, for compact sets K ⊂ S of nonisotropic Riesz capacity zero there exists a sequence (m k )k of p p multipliers for Hs which converges in Hs , and such that lim inf ρ→1 |m k (ρ(η))| = ∞ for any η ∈ K . In this sense, we could say that such compact sets are weak exceptional p sets for the multipliers of Hs . Proposition 5.2 Let 1 < p < ∞, 0 < n − sp < 1. Assume in addition that n − s < 1 if p < 2. Let K ⊂ S be a compact set such that Cs, p (K ) = 0. We then have that there p exists a sequence (m k )k of multipliers of Hs , such that p

1. The sequence (m k )k converges to a function F in Hs . 2. There exists C > 0 such that lim inf ρ→1 |m k (ρη)| ≥ Ck, for any η ∈ K . Proof For any k, let G k ⊂ S be an open set satisfying that K ⊂ G k and Cs, p (G k ) ≤ 21k . Let μk be the extremal potential capacity associated to G k and Fk the corresponding holomorphic potential given in (2.11) and (2.12). By Theorem 1.2, Fk is a multiplier p for Hs , and we also have (see [3]), that Fk  Hsp  Cs, p (G k ) ≤

1 . 2k

k Fi . These functions verify the required properFor k ≥ 1, we define m k := i=1 ties, since by Proposition 2.9, for each η ∈ K we have lim inf ReFk (ρη)  Ws, p (μk )(η)  1, ρ→1

which ends the proof. References 1. Adams, D.R.: Weighted nonlinear potential theory. Trans. Amer. Math. Soc. 297, 73–94 (1986) 2. Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. Springer, Berlin (1996) 3. Ahern, P., Cohn, W.: Exceptional sets for Hardy Sobolev functions, p > 1. Indiana Univ. Math. J. 38(2), 417–453 (1989) 4. Beatrous, F.; Burbea, J.: Holomorphic Sobolev spaces on the ball. Dissertationes Mathematicae, vol. 276. Ars Polona (1989) 5. Böe, B.: Construction of multipliers for Bessel potential spaces, unpublished 6. Cascante, C., Ortega, J.M.: Carleson measures for weighted Hardy-Sobolev spaces. Nagoya Math. J. 186, 29–68 (2007) 7. Cascante, C., Ortega, J.M., Verbitsky, I.E.: On L p − L q trace inequalities. J. London Math. Soc. 74(2), 497–511 (2006) 8. Cohn, W.S., Verbitsky, I.E.: Non-linear potential theory on the ball, with applications to exceptional and boundary interpolation sets. Michigan Math. J. 42, 79–97 (1995)

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Holomorphic potentials and multipliers for Hardy-Sobolev spaces 9. Hedberg, L.I., Wolff, ThH: Thin sets in nonlinear potential theory. Ann. Inst. Fourier (Grenoble) 33, 161–187 (1983) 10. Maz’ya, V.G., Shaposhnikova, T.O.: Theory of Sobolev multipliers. With applications to differential and integral operators. Grundlehren der Mathematischen Wissenschaften, vol. 337. Springer, Berlin (2009) 11. Maz’ya, V.G., Verbitsky, I.E.: Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. Ark. Mat. 33, 81–115 (1995) 12. Ortega, J.M., Fàbrega, J.: Holomorphic Triebel-Lizorkin spaces. J. Funct. Anal. 151(1), 177–212 (1997) 13. Ortega, J.M., Fabrega, J.: Multipliers in Hardy-Sobolev spaces. Int. Eq. Op. Th. 55, 535–560 (2006)

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