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117
Brownian motion and harmonic analysis on Sierpinski carpets
 MR MR1701339 (2000i:60083
, 1999
"... Abstract. We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to con ..."
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Cited by 46 (9 self)
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Abstract. We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting. 1
On the Floer homology of cotangent bundles
, 2004
"... This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T ∗ M of a compact orientable manifold M. The first result is a new L ∞ estimate for the solutions of the Floer equation, which allows to deal with a larger and more natural cla ..."
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Cited by 40 (7 self)
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This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T ∗ M of a compact orientable manifold M. The first result is a new L ∞ estimate for the solutions of the Floer equation, which allows to deal with a larger and more natural class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M, in the periodic case, or of the based loop space of M, in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian which is the Legendre transform of a Lagrangian on TM, and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W 1,2 free or based loops on M.
Sobolev inequalities in disguise
 Indiana Univ. Math. J
, 1995
"... We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff argu ..."
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Cited by 38 (4 self)
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We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff arguments. This method has interesting consequences concerning Trudinger type inequalities. 1. Introduction. On R n, the classical Sobolev inequality [27] indicates that, for every smooth enough function f with compact support,
Differentiation And The BalianLow Theorem
 J. Fourier Anal. Appl
, 1995
"... . The BalianLow theorem (BLT) is a key result in timefrequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system fe 2ßimbt g(t \Gamma na)g m;n2Z with ab = 1 forms an orthonormal basis for L 2 (R), then `Z 1 \Gamma1 jt g(t)j 2 dt ' `Z 1 \Gamma1 jfl ..."
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Cited by 34 (18 self)
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. The BalianLow theorem (BLT) is a key result in timefrequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system fe 2ßimbt g(t \Gamma na)g m;n2Z with ab = 1 forms an orthonormal basis for L 2 (R), then `Z 1 \Gamma1 jt g(t)j 2 dt ' `Z 1 \Gamma1 jfl g(fl)j 2 dfl ' = +1: The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that (g 0 ) (fl) = 2ßifl g(fl), the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form fe 2ßibm t g(t \Gamma an )g such that f(an ; bm )g has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems fe 2ßimbt g(t \Gamma na)g that form exact frames, and a new proof of the BLT for exact frame...
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
Marcus M.: Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour
 J. Anal. Math
, 1992
"... Abstract. On a bounded smooth domain Ω ⊂ R N we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to ∂Ω. We derive global a priori bounds of the Keller–Osserman type. Using a Phragmen–Lindelöf alternative for generalize ..."
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Cited by 21 (2 self)
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Abstract. On a bounded smooth domain Ω ⊂ R N we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to ∂Ω. We derive global a priori bounds of the Keller–Osserman type. Using a Phragmen–Lindelöf alternative for generalized sub and superharmonic functions we discuss existence, nonexistence and uniqueness of socalled large solutions, i.e., solution which tend to infinity at ∂Ω. The approach develops the one used by the same authors [2] for a problem with a power nonlinearity instead of the exponential nonlinearity. 1.
On Restrictions And Extensions Of The Besov And TriebelLizorkin Spaces With Respect To Lipschitz Domains
"... The restrictions B s pq and F s pq of the Besov and TriebelLizorkin spaces of tempered distributions B s pq (R n ) and F s pq (R n ) to Lipschitz domains\Omega ae R n are studied. For general values of parameters (s 2 R, p ? 0, q ? 0) a "universal" linear bounded extension operator from ..."
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Cited by 21 (3 self)
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The restrictions B s pq and F s pq of the Besov and TriebelLizorkin spaces of tempered distributions B s pq (R n ) and F s pq (R n ) to Lipschitz domains\Omega ae R n are studied. For general values of parameters (s 2 R, p ? 0, q ? 0) a "universal" linear bounded extension operator from B s pq and F s pq into the corresponding spaces on R n is constructed. The construction is based on a new variant of the Calder'on reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of B s pq and F s pq in terms of their values in\Omega are also obtained. Introduction The purpose of this paper is to construct a linear operator E which extends functions and distributions from a given Lipschitz domain\Omega ae R n to all of R n and possesses the following property: If a distribution f 2 D 0(\Omega\Gamma can be somehow extended to a tempered distribution g on R n which is "regular" in the sense that g 2 X for some function or...
Elliptic regularity and essential selfadjointness of Dirichlet operators on R^n
, 1996
"... We prove a new regularity result for operators of type L = \Delta +B\Deltar+c, on open sets\Omega ae IR n , provided B :\Omega ! IR n , c :\Omega ! IR satisfy mild integrability conditions. As a consequence we prove that L = \Delta + r log ae \Delta r with domain C 1 0 (IR n ) is essen ..."
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Cited by 16 (7 self)
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We prove a new regularity result for operators of type L = \Delta +B\Deltar+c, on open sets\Omega ae IR n , provided B :\Omega ! IR n , c :\Omega ! IR satisfy mild integrability conditions. As a consequence we prove that L = \Delta + r log ae \Delta r with domain C 1 0 (IR n ) is essentially selfadjoint on L 2 (IR n ; aedx), if ae 2 H 1;1 loc (IR n ) and r log ae 2 L fl loc (IR n ; dx) for some fl ? n.
Isoperimetry between exponential and Gaussian
 Electronic J. Prob
"... We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1 ..."
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Cited by 16 (7 self)
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We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1
Generalized Robin Boundary Conditions, RobintoDirichlet Maps, and KreinType Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains
 IN PERSPECTIVES IN PARTIAL DIFFERENTIAL EQUATIONS, HARMONIC ANALYSIS AND APPLICATIONS, D. MITREA AND M. MITREA (EDS.), PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS, AMERICAN MATHEMATICAL SOCIETY
, 2008
"... We study generalized Robin boundary conditions, RobintoDirichlet maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2. We also discuss the case of bounded C 1,rdomains, (1/2) < r < 1. ..."
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Cited by 16 (8 self)
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We study generalized Robin boundary conditions, RobintoDirichlet maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2. We also discuss the case of bounded C 1,rdomains, (1/2) < r < 1.