## Contents (901)

### BibTeX

@MISC{Badr901contents,

author = {N. Badr and Université Claude and Bernard Lyon and F. Bernicot},

title = {Contents},

year = {901}

}

### OpenURL

### Abstract

The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz transforms.

### Citations

594 |
Introduction to Fourier Analysis on Euclidean Spaces
- Stein, Weiss
- 1971
(Show Context)
Citation Context ...‖ 1 q p q ‖f q ‖ 1 q p q ( |f| q∗∗1 q (t) + |∇f| q∗∗1 ) p q(t) + ‖ |∇f| q∗∗ ‖ 1 ) q p q ) + ‖ |∇f| q ‖ 1 q p q = C (‖f‖p + ‖ |∇f| ‖p) = C‖f‖ W 1 p , where we used that for l > 1, ‖f ∗∗ ‖l ∼ ‖f‖l (see =-=[34]-=-, Chapter V: Lemma 3.21 p.191 and Theorem 3.21, p.201). Moreover, from Corollary 3.7, we have W 1 p ⊂ W 1 r + W 1 ∞ for r < p < ∞. Therefore W 1 p is an interpolation space between W 1 r and W 1 ∞ for... |

369 |
Interpolation Spaces. An Introduction
- Bergh
- 1976
(Show Context)
Citation Context ...≤ |f(x) − f(y)| dµ(x) � rQµ(Q) µ(Q) −1/p ‖MHL,p−ǫ(|∇f|)‖ p . Q Finally the L p -boundedness of MHL,p−ǫ concludes the proof. Q 6 ⊓⊔1.3 The K-method of real interpolation The reader can refer to [12], =-=[13]-=- for details on the development of this theory. Here we only recall the essentials to be used in the sequel. Let A0, A1 be two normed vector spaces embedded in a topological Hausdorff vector space V .... |

246 |
R.: Interpolation of Operators
- Bennett, Sharpley
- 1988
(Show Context)
Citation Context ...(Q) ∣ ≤ |f(x) − f(y)| dµ(x) � rQµ(Q) µ(Q) −1/p ‖MHL,p−ǫ(|∇f|)‖ p . Q Finally the L p -boundedness of MHL,p−ǫ concludes the proof. Q 6 ⊓⊔1.3 The K-method of real interpolation The reader can refer to =-=[12]-=-, [13] for details on the development of this theory. Here we only recall the essentials to be used in the sequel. Let A0, A1 be two normed vector spaces embedded in a topological Hausdorff vector spa... |

147 | Sobolev Spaces - Maz’ya - 1985 |

138 |
H p spaces of several variables
- Fefferman, Stein
- 1972
(Show Context)
Citation Context ...he last years, many works were related to the study of specific Hardy spaces defined according to a particular operator (Riesz transforms, Maximal regularity operator, Calderón-Zygmund operators, ... =-=[17, 18, 24, 25, 26, 29, 34]-=-). Mainly one of the most interesting questions in this theory is the interpolation of these spaces with Lebesgue spaces in order to prove boundedness of some operators. Although the theory of Hardy s... |

129 |
Extensions of Hardy spaces and their use in analysis
- Coifman, Weiss
- 1977
(Show Context)
Citation Context ...cular Hardy-Sobolev space. In the study of Hardy spaces (see [17]), we have seen that our abstract Hardy space correspond to the “classical” Hardy space (the one defined by R. Coifman and G. Weiss in =-=[21]-=-), when we choose our operator BQ as the exact oscillation operator. Here we want to study the HardySobolev space defined with a regular version of this particular collection B. For all ball Q, let φQ... |

118 |
Lectures on Analysis on Metric Spaces
- Heinonen
- 2001
(Show Context)
Citation Context ...ak Poincaré inequality (Pqloc) for some 1 ≤ q < ∞. Then, for 1 ≤ p1 < p < p2 ≤ ∞ with p > q0, H 1 p,X is an interpolation space between H1 p1,X and H1 p2,X . 8.2. Weighted Sobolev spaces. We refer to =-=[24]-=-, [29] for the definitions used in this subsection. Let Ω be an open subset of Rn equipped with the Euclidean distance, w ∈ L1,loc(Rn ) with w > 0, dµ = wdx. We assume that µ is q-admissible for some ... |

97 |
Sobolev met Poincaré
- Hajlasz, Koskela
(Show Context)
Citation Context ...covered by the existing references. 2The interested reader may find a wealth of examples of spaces satisfying doubling and Poincaré inequalities –to which our results apply– in [1], [4], [15], [18], =-=[23]-=-. Some comments about the generality of Theorem 1.1- 1.4 are in order. First of all, completeness of the Riemannian manifold is not necessary (see Remark 4.3). Also, our technique can be adapted to mo... |

96 |
Quasiconformal maps in metric spaces with controlled geometry
- Heinonen, Koskela
- 1998
(Show Context)
Citation Context ...ure spaces is an interpolation space between In this section we consider (X, d, µ) a metric-measure space with µ doubling. 7.1. Upper gradients and Poincaré inequality. Definition 7.1 (Upper gradient =-=[26]-=-). Let u : X → R be a Borel function. We say that a Borel function g : X → [0, +∞] is an upper gradient of u if |u(γ(b))−u(γ(a))| ≤ ∫ b a g(γ(t))dt for all 1-Lipschitz curve γ : [a, b] → X 1 . Remark ... |

95 |
Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9
- Cheeger
- 1999
(Show Context)
Citation Context ...Pq) if one can take r1 = ∞. 7.2. Interpolation of the Sobolev spaces H1 p. Before defining the Sobolev spaces H1 p it is convenient to recall the following proposition. Proposition 7.6. (see [22] and =-=[10]-=- Theorem 4.38) Let (X, d, µ) be a complete metricmeasure space, with µ doubling and satisfying a weak Poincaré inequality (Pq) for some 1 < q < ∞. Then there exist an integer N, C ≥ 1 and a linear ope... |

73 |
Isoperimetric and Sobolev inequalities for Carnot-Carathéodory metrics and the existence of minimal surfaces
- Garofalo, Nhieu
(Show Context)
Citation Context ... for the set of bounded metric Lipschitz function. Remark 8.2. For all 1 ≤ p ≤ ∞, H 1 p,X = W 1 p,X (Ω) := {f ∈ Lp(Ω) : |Xf| ∈ Lp(Ω)}, where Xf is defined in the distributional sense (see for example =-=[19]-=- Lemma 7.6). 3 that is relative to the metric ρ of Carnot-Carathéodory. 21 j=1 ) 1 2 .Adapting the same method, we obtain the following interpolation theorem for the H 1 p,X . Theorem 8.3. Consider (... |

61 |
Interpolation of Linear Operators
- Krein, Petunin, et al.
- 1978
(Show Context)
Citation Context ...K ′ (fn, t 1 q ) ≤ K(f − fn, t, W 1 q , W 1 ∞ ) + K′ (fn, t) (∫ )1 t (∫ q t ≤ C + C 0 (∫ t 0 |f − fn| q∗ (s)ds |∇fn| q∗ (s)ds )1 q . + C 0 |∇f − ∇fn| q∗ (s)ds Now we invoke the following theorem from =-=[15]-=- page 67-68 stated there in the Euclidean case. As the proof is the same, we state it in the more general case: Theorem 4.3. Let M be a measured space. Consider a sequence of measurable functions (ψn)... |

49 |
Aspects of Sobolev-type inequalities
- Saloff-Coste
(Show Context)
Citation Context ... M, µ(B(x, 1)) ≥ v, then M satisfies (1.3). Here µ(B(x, 1)) is the Riemannian volume of the open ball B(x, 1). For more general cases where we have (1.3) for some p’s depending on the hypotheses, see =-=[19]-=-. Note that if (1.3) holds for some 1 ≤ p < n then it holds for all p ≤ q < n –see [19], Chapter 3–. We have also non linear versions of Gagliardo-Nirenberg inequalities proved by Rivière-Strzelecki [... |

42 |
Croissance polynomiale et période des fonctions harmoniques
- Guivarc’h
(Show Context)
Citation Context ...ase the CarnotCarathéodory metric ρ is is a distance, and G equipped with the distance ρ is complete and defines the same topology as that of G as a manifold (see [13] page 1148). From the results in =-=[21]-=-, [32], it is known that G satisfies (Dloc). Moreover, if G has polynomial growth it satisfies (D). From the results in [33], [35], G admits a local Poincaré 22inequality (P1loc). If G has polynomial... |

42 |
Saloff-Coste L., Isopérimétrie pour les groupes et les variétés
- Coulhon
- 1993
(Show Context)
Citation Context ...ng a Galois covering manifold of a compact manifold whose deck transformation group has polynomial growth –see [10]–. We can also take the example of a Cayley graph of a finitely generated group –see =-=[7]-=-, [19]–. We get also the following Corollary: Corollary 1.5. Let M be a complete Riemannian manifold satisfying (D) and (P2). Then (1.2) holds for all 2 ≤ p < l < ∞. Note that (P ′ 2 ) is always satis... |

41 |
Analysis and geometry on groups, Cambridge Tracts
- Varopoulos, Coulhon
- 1992
(Show Context)
Citation Context ... ∈ M, for all t > 0 (G) |∇xpt(x, y)| ≤ Then inequality (1.2) holds for all 1 ≤ p < l < ∞. C √ tµ(B(y, √ t)) . Note that a Lie group of polynomial growth satisfies the hypotheses of Corollary 1.4 –see =-=[8]-=-–. Hence it verifies (1.2) for all 1 ≤ p < l < ∞. Another example of a space satysfying the hypotheses of Corollary 1.4 is given by taking a Galois covering manifold of a compact manifold whose deck t... |

37 |
Multipliers on fractional Sobolev spaces
- Strichartz
- 1967
(Show Context)
Citation Context ...dySobolev spaces is still not unified. Before we state our results, let us briefly review the existing literature related to this subject. The Hardy-Sobolev spaces were first studied by R. Strichartz =-=[36]-=- in R n . Related works are [9], [31], [19], [33]. They deal with “classical” Hardy-Sobolev spaces HS 1 on R n , which correspond to the Sobolev version of the Coifman-Weiss Hardy space H 1 CW (Rn ) :... |

37 |
Balls and metrics defined by vector fields
- Nagel, Stein, et al.
- 1985
(Show Context)
Citation Context ...e CarnotCarathéodory metric ρ is is a distance, and G equipped with the distance ρ is complete and defines the same topology as that of G as a manifold (see [13] page 1148). From the results in [21], =-=[32]-=-, it is known that G satisfies (Dloc). Moreover, if G has polynomial growth it satisfies (D). From the results in [33], [35], G admits a local Poincaré 22inequality (P1loc). If G has polynomial growt... |

33 |
Riesz transform on manifolds and heat kernel regularity
- Auscher, Coulhon, et al.
(Show Context)
Citation Context ...+ ∆) 1 2 is bounded from HS 1 (r),ato to L1 for any r ≥ q if q ̸= 1 (resp. r > 1 if q = 1). Consequently, (I + ∆) 1 2 is bounded from W 1,p to L p for any p ∈ [q,2]. Remark 4.7 We refer the reader to =-=[5, 4]-=- for the study of inequality (RRp) for p ∈ (1,2] (which corresponds to the boundedness of ∆ 1 2 from ˙ W 1,p to Lp ) under Poincaré inequality. The new point here is the limit point (RR1). Proof : We ... |

32 |
On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on R n and related estimates, preprint 2004-04, Université de Paris-Sud, Mathématiques
- Auscher
(Show Context)
Citation Context ...∀ξ,ζ ∈ C n , λ|ξ| 2 ≤ Re ( Aξ · ξ ) and |Aξ · ζ| ≤ Λ|ξ||ζ|. We define the second order divergence form operators ˜L(f) := −div(A∇f), L(f) := f + ˜ L(f). We refer the reader to the work of P. Auscher (=-=[3]-=-) and of S. Hofmann and S. Mayboroda ([29]). The two last authors defined a Hardy space H1 L associated to this operator and gave several characterizations using maximal square functions. With a parti... |

30 |
Non-Gaussian aspects of heat kernel behaviour
- Davies
- 1997
(Show Context)
Citation Context ... of ∆ 1 2 is ∫ ∞ 0 kernel ∂tpt verifies ∂tpt(x,y) dt √ t . Under our hypotheses, the partial derivative of the heat |∂tpt(x,y)| ≤ C tµ(B(y, √ t)) e−α d2 (x,y) t (37) for every x, y ∈ M and t > 0 (see =-=[23]-=-, Theorem 4 and [27], Corollary 3.3). Let Q a ball of radius r > 0 and y, z ∈ Q. We therefore have ∫ ∫ ∞ ∣ ∂t(pt(x,y) − pt(x,z)) M\4Q 0 dt ∫ ∫ ∞ √ dt t ∣ dµ(x) ≤ |∂t(pt(x,y) − pt(x,z))| M\4Q 0 dt √ dt... |

27 |
Nonlinear approximation and the space
- Cohen, DeVore, et al.
(Show Context)
Citation Context ...tz spaces 10 4. Proof of Theorem 1.1 and Theorem 1.6 11 5. Another symmetrization inequality 14 6. Proof of Theorem 1.8 14 References 15 1. Introduction Cohen-Meyer-Oru [6], Cohen-Devore-Petrushev-Xu =-=[5]-=-, proved the following GagliardoNirenberg type inequality (1.1) ‖f‖1 n−1 n ∗ ≤ C‖ |∇f| ‖ 1 ‖f‖ 1 n B −(n−1) ∞,∞ for all f ∈ W 1 1 (Rn) (1∗ = n ). The proof of (1.1) is rather involved and based on n−1... |

26 | Duality of Hardy and BMO spaces associated with operators with heat kernel bounds
- Duong, Yan
(Show Context)
Citation Context ...he last years, many works were related to the study of specific Hardy spaces defined according to a particular operator (Riesz transforms, Maximal regularity operator, Calderón-Zygmund operators, ... =-=[17, 18, 24, 25, 26, 29, 34]-=-). Mainly one of the most interesting questions in this theory is the interpolation of these spaces with Lebesgue spaces in order to prove boundedness of some operators. Although the theory of Hardy s... |

24 | Definitions of Sobolev classes on metric spaces
- Franchi, Hajlasz, et al.
(Show Context)
Citation Context ...radius r > 0 we have (Pq) ( ∫ − |f − fB| q dµ B )1 q ( ∫ ≤ Cr − |∇f| q dµ Then Lip(M) ∩ . W 1 p is dense in . W 1 p for q ≤ p < ∞. Proof. The proof of item 2. is implicit in the proof of Theorem 9 in =-=[17]-=-. We obtain for the K-functional of the homogeneous Sobolev spaces the following homogeneous form of Theorem 1.2, weaker in the particular case r = q, but again sufficient for us to interpolate. Theor... |

22 | Weighted norm inequalities, off diagonal estimates and elliptic operators. Part I: General operator theory and weights
- Auscher, Martell
(Show Context)
Citation Context ...′ 0 ∩ L q , ∥ ∥M ♯ B,β ′(T ∗ ∥ f) ∥ L q ≤ c‖f‖L q. (27) 24Now we use a “good lambdas” argument to compare the maximal operators. We use a Sobolevversion of the result of P. Auscher and J.M. Martell: =-=[7]-=-, Theorem 3.1. With its notation, take a function F. We define for all balls Q The assumption (23) shows that By definition of M ♯ B,β ′, we have GQ = B ∗ Q F and HQ = A ∗ Q F. µ(Q) −1/σ ‖HQ‖W −1,σ � ... |

22 | Gaussian upper bounds for the heat kernel on arbitrary manifolds
- Grigor’yan
- 1997
(Show Context)
Citation Context ...ernel ∂tpt verifies ∂tpt(x,y) dt √ t . Under our hypotheses, the partial derivative of the heat |∂tpt(x,y)| ≤ C tµ(B(y, √ t)) e−α d2 (x,y) t (37) for every x, y ∈ M and t > 0 (see [23], Theorem 4 and =-=[27]-=-, Corollary 3.3). Let Q a ball of radius r > 0 and y, z ∈ Q. We therefore have ∫ ∫ ∞ ∣ ∂t(pt(x,y) − pt(x,z)) M\4Q 0 dt ∫ ∫ ∞ √ dt t ∣ dµ(x) ≤ |∂t(pt(x,y) − pt(x,z))| M\4Q 0 dt √ dtdµ(x) t ∫ ∫ ∞ dt ≤ C... |

19 |
The Poincaré inequality is an open ended condition
- Keith, Zhong
(Show Context)
Citation Context ...nown that (Pq) implies (Pp) when p ≥ q (see [28]). Thus if the set of q such that (Pq) holds is not empty, then it is an interval unbounded on the right. A recent result of S. Keith and X. Zhong (see =-=[30]-=-) asserts that this interval is open in [1,+∞[ : Theorem 1.5 Let (X,d,µ) be a complete metric-measure space with µ doubling and admitting a Poincaré inequality (Pq), for some 1 < q < ∞. Then there exi... |

18 |
Riesz transforms for 1 ≤ p ≤ 2
- Coulhon, Duong
- 1999
(Show Context)
Citation Context ...sed that ∫ Lemma 2.1 ). Similarly, we prove that ∫ d(x,y)> √ t e−γ d2 (x,y) s M\4Q ( ∫ ∞ 0 e −α d2 (x,y) t dµ(x) dt √ t ) dt √ t dµ(x) ≤ Cγµ(Q(y, √ t −γ s))e ) 1 tµ(Q(z, √ t)) e−α d2 (x,z) dt t √t s (=-=[22]-=-, dµ(x) ≤ C r . Taking the supremum over all y,z ∈ Q, all balls Q and applying Theorem 0.7, we obtain that T is bounded from ˙ HS 1 r,ato to L1 for r > 1. Corollary 0.8 implies boundedness of T from ˙... |

17 |
Espaces de Sobolev sur les variétés Riemanniennes
- Aubin
- 1976
(Show Context)
Citation Context ...n the tangent space (forgetting the subscript x for simplicity) and ‖ · ‖p for the norm on Lp(M, µ), 1 ≤ p ≤ +∞. Our goal is to prove Theorem 1.2. 3.1. Non-homogeneous Sobolev spaces. Definition 3.1 (=-=[2]-=-). Let M be a C ∞ Riemannian manifold of dimension n. Write E 1 p for the vector space of C ∞ functions ϕ such that ϕ and |∇ϕ| ∈ Lp, 1 ≤ p < ∞. We define the Sobolev space W 1 p as the completion of E... |

17 |
Wheeden: Weighted Sobolev–Poincaré inequalities for Grushin type operators
- Franchi, Gutiérrez, et al.
- 1994
(Show Context)
Citation Context ...d assuming that id : (Ω, ρ) → (Ω, |.|) is an homeomorphism, we obtain our interpolation result. As an example we take vectors fields satisfying a Hörmander condition or vectors fields of Grushin type =-=[16]-=-. 8.3. Lie Groups. In all this subsection, we consider G a connected unimodular Lie group equipped with a Haar measure dµ and a family of left invariant vector fields X1, ..., Xk such that the Xi’s sa... |

16 |
Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces
- Hajlasz
- 2003
(Show Context)
Citation Context ...quality (Pq) if one can take r1 = ∞. 7.2. Interpolation of the Sobolev spaces H1 p. Before defining the Sobolev spaces H1 p it is convenient to recall the following proposition. Proposition 7.6. (see =-=[22]-=- and [10] Theorem 4.38) Let (X, d, µ) be a complete metricmeasure space, with µ doubling and satisfying a weak Poincaré inequality (Pq) for some 1 < q < ∞. Then there exist an integer N, C ≥ 1 and a l... |

15 |
On L p -estimates for square roots of second order elliptic operators
- Auscher
- 2004
(Show Context)
Citation Context ...roof follows from Proposition 3.3 and the Reiteration Theorem (see [12], Theorem 2.4). ⊓⊔ All these results are based on the well adapted “Calderón-Zygmund decomposition”. The first one (described in =-=[2]-=- by P. Auscher) was written for homogeneous Sobolev spaces. We can write an analog result of Proposition 3.1 for homogeneous Sobolev spaces. Then we estimate the functional K (as in [10]) and obtain t... |

15 |
Weighted Sobolev–Poincaré inequalities and pointwise inequalities for a class of degenerate elliptic equations
- Franchi
- 1991
(Show Context)
Citation Context ...sult is not covered by the existing references. 2The interested reader may find a wealth of examples of spaces satisfying doubling and Poincaré inequalities –to which our results apply– in [1], [4], =-=[15]-=-, [18], [23]. Some comments about the generality of Theorem 1.1- 1.4 are in order. First of all, completeness of the Riemannian manifold is not necessary (see Remark 4.3). Also, our technique can be a... |

12 |
Analyse harmonique sur certains espaces homogènes
- Coifman, Weiss
- 1971
(Show Context)
Citation Context ...). (D) Observe that if M satisfies (D) then diam(M) < ∞ ⇔ µ(M) < ∞ (see [1]). Therefore if M is a complete non-compact Riemannian manifold satisfying (D) then µ(M) = ∞. Theorem 1.2 (Maximal theorem) (=-=[20]-=-) Let M be a Riemannian manifold satisfying (D). Denote by MHL the uncentered Hardy-Littlewood maximal function over open balls of M defined by MHLf(x) := sup Q ball x∈Q |f|Q ∫ where fE := − fdµ := E ... |

12 | Non smooth calculus
- Heinonen
- 2007
(Show Context)
Citation Context ...definitions used in this subsection. Let Ω be an open subset of Rn equipped with the Euclidean distance, w ∈ L1,loc(Rn ) with w > 0, dµ = wdx. We assume that µ is q-admissible for some 1 < q < ∞ (see =-=[25]-=- for the definition). This is equivalent to say, (see [23]), that µ is doubling and there exists C > 0 such that for every ball B ⊂ Rn of radius r > 0 and for every function ϕ ∈ C∞ (B), ∫ (Pq) |ϕ − ϕB... |

10 |
Special functions of bounded variation in doubling metric measure spaces. Calculus of variations: topics from the mathematical heritage of
- Ambrosio, Jr, et al.
- 2004
(Show Context)
Citation Context ...e (global) doubling property (D) if there exists a constant C > 0, such that for all x ∈ M, r > 0 we have µ(Q(x,2r)) ≤ Cµ(Q(x,r)). (D) Observe that if M satisfies (D) then diam(M) < ∞ ⇔ µ(M) < ∞ (see =-=[1]-=-). Therefore if M is a complete non-compact Riemannian manifold satisfying (D) then µ(M) = ∞. Theorem 1.2 (Maximal theorem) ([20]) Let M be a Riemannian manifold satisfying (D). Denote by MHL the unce... |

10 |
The solution of the Kato square root problem for second order elliptic operators on R n
- Tchamitchian
(Show Context)
Citation Context ...ction f ∈ W 1,2 (R n ) ‖BQ(f)‖ W 1,2 ≤ c‖f‖ W 1,2. Moreover ∇L −1/2 is bounded from W 1,2 into W 1,2 . 33The W 1,2 boundedness essentially comes from the L 2 -boundedness of the Riesz transform (see =-=[6]-=-). As done in [29], using “Gaffney estimates” for the semigroup, we obtain the following proposition : Proposition 4.9 The operator ∇L −1/2 is continuous from HW 1 F,ato into W 1,1 . The proof is left... |

10 |
Axiomatic Theory of Sobolev Spaces
- Gol’dshtein, Troyanov
(Show Context)
Citation Context ... ∞. We define the Sobolev space W 1 p as the completion of E 1 p for the norm ‖ϕ‖ W 1 p = ‖ϕ‖p + ‖ |∇ϕ| ‖p. We denote W 1 ∞ for the set of all bounded Lipschitz functions on M. Proposition 3.2. ([2], =-=[20]-=-) Let M be a complete Riemannian manifold. Then C∞ 0 and in particular Lip0 is dense in W 1 p for 1 ≤ p < ∞. Definition 3.3 (Poincaré inequality on M). We say that a complete Riemannian manifold M adm... |

10 |
Parabolic Harnack inequality for divergence form second order differential operator
- Saloff-Coste
- 1995
(Show Context)
Citation Context ...pology as that of G as a manifold (see [13] page 1148). From the results in [21], [32], it is known that G satisfies (Dloc). Moreover, if G has polynomial growth it satisfies (D). From the results in =-=[33]-=-, [35], G admits a local Poincaré 22inequality (P1loc). If G has polynomial growth, then it admits a global Poincaré inequality (P1). Interpolation of non-homogeneous Sobolev spaces. We define the no... |

10 |
Sobolev inequalities, truncation method, and John domains, inPapersonAnalysis, Rep
- Hajlasz
- 2001
(Show Context)
Citation Context ...at (1.5) implies the weak q ) l l ‖ |∇f| ‖q ‖f‖ . Consequently the type inequality (q, l), that is µ({|f| > λ}) ≤ ( C λ q 1− l Bα ∞,∞ 7strong type (q, l), that is ‖f‖l ≤ C‖ |∇f| ‖ q l principle (see =-=[11]-=-, [16]). q ‖f‖ q 1− l Bα ∞,∞ , follows by Maz’ya’s truncation □ Proof of Corollary 1.3. Remark that Riemannian manifolds with non-negative Ricci curvature satisfy (D) (with Cd = 2n ) , (P1). They also... |

10 |
and BMO spaces associated to divergence form elliptic operators, in preparation
- Hofmann, Mayboroda, et al.
(Show Context)
Citation Context ...he last years, many works were related to the study of specific Hardy spaces defined according to a particular operator (Riesz transforms, Maximal regularity operator, Calderón-Zygmund operators, ... =-=[15, 16, 24, 25, 26, 29, 34]-=-). Mainly one of the most interesting questions in this theory is the interpolation of these spaces with Lebesgue spaces in order to prove boundedness of some operators. Although the theory of Hardy s... |

9 |
Approximation and imbedding theorems for weighted Sobolev Spaces associated with Lipschitz continuous vector
- Franchi, Serrapioni, et al.
- 1997
(Show Context)
Citation Context ...s not covered by the existing references. 2The interested reader may find a wealth of examples of spaces satisfying doubling and Poincaré inequalities –to which our results apply– in [1], [4], [15], =-=[18]-=-, [23]. Some comments about the generality of Theorem 1.1- 1.4 are in order. First of all, completeness of the Riemannian manifold is not necessary (see Remark 4.3). Also, our technique can be adapted... |

9 |
Modulus and the Poincaré inequality on metric measure spaces
- Keith
(Show Context)
Citation Context ...q) − |f − fB| q )1 ( ∫ q dµ ≤ Cr − |∇f| q )1 q dµ B B 6Remark 2.9. Since C ∞ 0 is dense in W 1 q , if M admits (Pq) for all f ∈ C ∞ 0 then (Pq) holds for all f ∈ W 1 q . In fact, by Theorem 1.3.4 in =-=[13]-=-, M admits (Pq) for all f ∈ ˙ E 1 q . The following recent result of Keith and Zhong [14] improves the exponent of Poincaré inequality: Theorem 2.10. Let (X, d, µ) be a complete metric-measure space w... |

8 |
New Abstract Hardy Spaces
- Bernicot, Zhao
(Show Context)
Citation Context ...he last years, many works were related to the study of specific Hardy spaces defined according to a particular operator (Riesz transforms, Maximal regularity operator, Calderón-Zygmund operators, ... =-=[17, 18, 24, 25, 26, 29, 34]-=-). Mainly one of the most interesting questions in this theory is the interpolation of these spaces with Lebesgue spaces in order to prove boundedness of some operators. Although the theory of Hardy s... |

7 |
Atomic decomposition of H p spaces associated with some Schrödinger operators Indiana Univ
- Dziubański
- 1998
(Show Context)
Citation Context |

7 | Pointwise characterizations of Hardy-Sobolev functions
- Koskela, Saksman
(Show Context)
Citation Context .... Before we state our results, let us briefly review the existing literature related to this subject. The Hardy-Sobolev spaces were first studied by R. Strichartz [36] in R n . Related works are [9], =-=[31]-=-, [19], [33]. They deal with “classical” Hardy-Sobolev spaces HS 1 on R n , which correspond to the Sobolev version of the Coifman-Weiss Hardy space H 1 CW (Rn ) : HS 1 is the set of functions f ∈ H1 ... |

7 |
Fonctions harmoniques sur les groupes de
- Varopoulos
- 1987
(Show Context)
Citation Context ... as that of G as a manifold (see [13] page 1148). From the results in [21], [32], it is known that G satisfies (Dloc). Moreover, if G has polynomial growth it satisfies (D). From the results in [33], =-=[35]-=-, G admits a local Poincaré 22inequality (P1loc). If G has polynomial growth, then it admits a global Poincaré inequality (P1). Interpolation of non-homogeneous Sobolev spaces. We define the non-homo... |

6 | Coulhon T., Riesz transforms on manifolds and Poincaré inequalities, preprint 2004
- Auscher
(Show Context)
Citation Context ...1,2). Then for all p ∈ (q,2], the operator T admits a continuous extension from W 1,p to L p . We apply these last two theorems to the square root of the positive Laplace-Beltrami operator ∆ 1/2 . In =-=[4]-=-, P. Auscher and T. Coulhon proved that under the doubling property (D) and a Poincaré inequality (Pq) for some q ∈ [1,2), (RRp) (which is equivalent to the boundedness of ∆ 1/2 from ˙ W 1,p to L p ) ... |

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Real interpolation of Sobolev spaces
- Badr
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Citation Context ... . 33 The aim of the present work is to define atomic Hardy-Sobolev spaces and interpolate them with Sobolev spaces on Riemannian manifolds. One of the motivations is our Sobolev interpolation result =-=[10]-=-, [11] in different geometric frames, under the doubling property and Poincaré inequalities. After this result, it is interesting to consider a “nice” subspace of W 1,1 – as is the Hardy space for L 1... |

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Hardy-Sobolev spaces and maximal functions
- Miyachi
- 1990
(Show Context)
Citation Context ...state our results, let us briefly review the existing literature related to this subject. The Hardy-Sobolev spaces were first studied by R. Strichartz [36] in R n . Related works are [9], [31], [19], =-=[33]-=-. They deal with “classical” Hardy-Sobolev spaces HS 1 on R n , which correspond to the Sobolev version of the Coifman-Weiss Hardy space H 1 CW (Rn ) : HS 1 is the set of functions f ∈ H1 CW such that... |