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44
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 61 (15 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Hardy spaces of differential forms on Riemannian manifolds
, 2006
"... Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H pboundedness for Riesz transfo ..."
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Cited by 58 (12 self)
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Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H pboundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ∞ functional calculus and Hodge decomposition, are given.
QUASILINEAR AND HESSIAN EQUATIONS OF LANE–EMDEN TYPE
, 2005
"... The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane–Emden type, including the following two model problems: −∆pu = u q + µ, Fk[−u] = u q + µ, u ≥ 0, on R n, or on a bounded domain Ω ⊂ R n. Here ∆p is the pLaplacia ..."
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Cited by 33 (3 self)
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The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane–Emden type, including the following two model problems: −∆pu = u q + µ, Fk[−u] = u q + µ, u ≥ 0, on R n, or on a bounded domain Ω ⊂ R n. Here ∆p is the pLaplacian defined by ∆pu = div (∇u∇u  p−2), and Fk[u] is the kHessian defined as the sum of k × k principal minors of the Hessian matrix D 2 u (k = 1, 2,...,n); µ is a nonnegative measurable function (or measure) on Ω. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data µ ∈ Ls (Ω), s> 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q−p+1) pq for the first equation, and s = n(q−k) 2kq for the second one. Furthermore, a complete characterization of removable singularities is given. Our methods are based on systematic use of Wolff’s potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpeläinen and Mal´y, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasilinear equations and equations of MongeAmpère type.
Hardy spaces and divergence operators on strongly Lipschitz domain
 of R n , J. Funct. Anal
"... Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the nontangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under su ..."
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Cited by 29 (3 self)
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Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the nontangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under suitable assumptions on L, we identify this maximal Hardy space with atomic Hardy spaces, namely with H 1 (R n) if Ω = R n, H 1 r(Ω) under the Dirichlet boundary condition, and H1 z (Ω) under the Neumann boundary condition. In particular, we obtain a new proof of the atomic decomposition for H1 z (Ω). A version for local Hardy spaces is also given. We also present an overview of the theory of Hardy spaces and BMO spaces on Lipschitz domains with proofs.
Topological conjugacy of circle diffeomorphisms
 Erg. Th. Dyn. Syst
, 1997
"... The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy [1]. In [5] one of us gave a new criterion. There is an example satisfying Denjoy’s bounded variation condition rather than [5]’s Zygmund condition and vice versa. ..."
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Cited by 11 (3 self)
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The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy [1]. In [5] one of us gave a new criterion. There is an example satisfying Denjoy’s bounded variation condition rather than [5]’s Zygmund condition and vice versa. This paper will give the third criterion which is implied by either of the above criteria. 1.
On the negative eigenvalues of a class of Schrödinger operators
 AMS Translations, Series
, 1998
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Harmonic measure, L² estimates and the Schwarzian derivative
 J. ANAL. MATH
, 1994
"... We consider several results, each of which uses some type of "L²" estimate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points of a curve in terms of a certain geometric square function. Our next result is an LP estima ..."
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Cited by 5 (5 self)
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We consider several results, each of which uses some type of "L²" estimate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points of a curve in terms of a certain geometric square function. Our next result is an LP estimate relating the derivative of a conformal mapping to its Schwarzian derivative. One consequence of this is an estimate on harmonic measure generalizing Lavrentiev's estimate for rectifiable domains. Finally, we consider L z estimates for Schwarzian derivatives and the question of when a Riemann mapping ~ has log ~ in BMO.
On the fundamental solution of an elliptic equation in nondivergence form
"... Abstract. We consider the existence and asymptotics for the fundamental solution of an elliptic operator in nondivergence form, L(x, ∂x) = aij(x)∂i∂i, for n ≥ 3. We assume that the coefficients have modulus of continuity satisfying the square Dini condition. For fixed y, we construct a solution of ..."
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Cited by 5 (2 self)
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Abstract. We consider the existence and asymptotics for the fundamental solution of an elliptic operator in nondivergence form, L(x, ∂x) = aij(x)∂i∂i, for n ≥ 3. We assume that the coefficients have modulus of continuity satisfying the square Dini condition. For fixed y, we construct a solution of LZy(x) = 0 for 0 < x −y  < ε with explicit leading order term which is O(x−y  2−n e I(x,y) ) as x → y, where I(x, y) is given by an integral and plays an important role for the fundamental solution: if I(x, y) approaches a finite limit as x → y, then we can solve L(x, ∂x)F(x, y) = δ(x − y), and F(x, y) is asymptotic as x → y to the fundamental solution for the constant coefficient operator L(y, ∂x). On the other hand, if I(x, y) → − ∞ as x → y then the solution Zy(x) violates the “extended maximum principle ” of Gilbarg & Serrin [8] and is a distributional solution of L(x, ∂x)Zy(x) = 0 for x − y  < ε although Zy is not even bounded as x → y. 1.
Boundedness of the fractional maximal operator in local Morreytype spaces
 Institute of Mathematics, AS CR
"... Abstract. The problem of the boundedness of the fractional maximal operator Mα, 0 ≤ α < n in local Morreytype spaces is reduced to the problem of the boundedness of the Hardy operator in weighted Lpspaces on the cone of nonnegative nonincreasing functions. This allows obtaining sharp sufficie ..."
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Cited by 4 (2 self)
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Abstract. The problem of the boundedness of the fractional maximal operator Mα, 0 ≤ α < n in local Morreytype spaces is reduced to the problem of the boundedness of the Hardy operator in weighted Lpspaces on the cone of nonnegative nonincreasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. 1.
Uniform bounds and exponential time decay . . .
, 2005
"... We consider the nonlinear VPFP system with a coulombian repulsive interaction potential and a generic confining potential in space dimension d ≥ 3. Using spectral and kinetic methods we prove the existence and uniqueness of a mild solution in weighted spaces and for small charge we find an explicit ..."
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Cited by 4 (1 self)
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We consider the nonlinear VPFP system with a coulombian repulsive interaction potential and a generic confining potential in space dimension d ≥ 3. Using spectral and kinetic methods we prove the existence and uniqueness of a mild solution in weighted spaces and for small charge we find an explicit exponential rate of convergence to the equilibrium in terms of the Witten Laplacian associated to the linear equation.