Results 1  10
of
93
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
 J. FUNCT. ANAL
, 1999
"... We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities. ..."
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Cited by 126 (7 self)
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We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities.
On Talagrand's Deviation Inequalities For Product Measures
, 1996
"... We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. ..."
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Cited by 97 (0 self)
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We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M.
Heat kernel estimates for Dirichlet fractional Laplacian
 J. European Math. Soc
"... In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally ..."
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Cited by 49 (22 self)
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In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C 1,1 open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a nonlocal operator on open sets.
Intrinsic ultracontractivity and conditional gauge for symmetric stable processes on rough domains, preprint
"... It is shown in this paper that the conditional gauge theorem holds for symmetric:stable processes on bounded C1, 1 domains in Rn where 0<:<2 and n2. Two of the major tools used to prove this conditional gauge theorem are logarithmic Sobolev inequality and intrinsic ultracontractivity. 1997 A ..."
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Cited by 46 (18 self)
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It is shown in this paper that the conditional gauge theorem holds for symmetric:stable processes on bounded C1, 1 domains in Rn where 0<:<2 and n2. Two of the major tools used to prove this conditional gauge theorem are logarithmic Sobolev inequality and intrinsic ultracontractivity. 1997 Academic Press 1.
Functional inequalities for empty essential spectrum
 J. Funct. Anal
, 2000
"... In terms of the equivalence of Poincare ́ inequality and the existence of spectral gap, the superPoincare ́ inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corre ..."
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Cited by 44 (13 self)
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In terms of the equivalence of Poincare ́ inequality and the existence of spectral gap, the superPoincare ́ inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator. This inequality recovers known Sobolev and Nash type ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also proved for a more general setting including symmetric jump processes. Moreover, estimates of inequality constants are also presented, which lead to a proof of a result on ultracontractivity suggested recently by D. Stroock. Finally, concentration of reference measures for superPoincare ́ inequalities is studied, the resulting estimates extend previous ones for Poincare ́ and logSobolev inequalities.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Potential theory of special subordinators and subordinate killed stable processes
 J. Theoret. Probab
, 2006
"... In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one ..."
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Cited by 31 (19 self)
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In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a onetoone correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κfat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D.
Optimal Smoothing and Decay Estimates for Viscously Damped Conservation Laws, with Application to the 2D Navier Stokes Equation
, 1994
"... Optimal bounds on the Lp (IR n) smoothing and decay are established for certain viscously damped conservations laws, of which the vorticity formulation of the Navier–Stokes equation on IR 2 is a basic example. From the smoothing bounds, we obtain pointwise bounds that provide optimal control on th ..."
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Cited by 31 (1 self)
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Optimal bounds on the Lp (IR n) smoothing and decay are established for certain viscously damped conservations laws, of which the vorticity formulation of the Navier–Stokes equation on IR 2 is a basic example. From the smoothing bounds, we obtain pointwise bounds that provide optimal control on the spatial decay of solutions. We apply this in a study of the physically important case in which the integral of the initial data (i.e., the total vorticity in the example) vanishes. We show in this case that as the time t increases, the L 1 (IR n) norm of the solution decays to zero in two stages: for large initial data, there is a slow decay period during which the L 1 (IR n) norm falls off with an inverse power of the logarithm of t. Then, once the norm has fallen below a critical value, it decays away to zero with t −1/2. Again, this is optimal, and all of the constants in these estimates are explicitly computable in terms of the initial data.
Twosided heat kernel estimates for censored stablelike processes
, 2008
"... In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrins ..."
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Cited by 30 (19 self)
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In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp twosided estimates for the transition density functions of a large class of censored αstablelike processes in C 1,1 open sets. We further obtain sharp twosided estimates for the Green functions of these censored αstablelike processes in bounded C 1,1 open sets.