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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 23 (6 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
Rényi dimension and gaussian filtering
 New York J. Math
"... Abstract. Consider the partition function S q µ(ɛ) associated in the theory of Rényi dimension to a finite Borel measure μ on Euclidean dspace. This partition function S q µ(ɛ) is the sum of the qth powers of the measure applied to a partition of dspace into dcubes of width ɛ. WefurtherGuérin’s ..."
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Cited by 1 (1 self)
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Abstract. Consider the partition function S q µ(ɛ) associated in the theory of Rényi dimension to a finite Borel measure μ on Euclidean dspace. This partition function S q µ(ɛ) is the sum of the qth powers of the measure applied to a partition of dspace into dcubes of width ɛ. WefurtherGuérin’s investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convolution of μ against an approximate identity of Gaussians. We prove a Lipschitztype estimate on the partition function. This bound on the partition function leads to results regarding the computation of Rényi dimension. It also shows that the partition function is of Oregular variation. We find situations where one can or cannot replace the partition function by a discrete version. We discover that the slopes of the leastsquare best fit linear approximations to the partition function cannot always be used to calculate upper and lower Rényi dimension. Contents
Geometricarithmetic averaging of dyadic weights ∗
, 2009
"... The theory of (Muckenhoupt) weights arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing Ap weights from a measurably varying family of dyadic ..."
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The theory of (Muckenhoupt) weights arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing Ap weights from a measurably varying family of dyadic Ap weights. This averaging process is suggested by the relationship between the Ap weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Hölder (RHp) conditions from families of dyadic RHp weights, and extends to the polydisc as well. 1
INEQUALITIES FOR POISSON INTEGRALS WITH SLOWLY GROWING DIMENSIONAL CONSTANTS
"... Abstract. Let Pt be the Poisson kernel. We study the following L p inequality for the Poisson integral P f(x, t) = (Pt ∗ f)(x) with respect to a Carleson measure µ: P f L p (R n+1 +,dµ) ≤ cp,n κ(µ) 1 p f  L p (R n,dx), where 1 < p < ∞ and κ(µ) is the Carleson norm of µ. It was shown by Ver ..."
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Abstract. Let Pt be the Poisson kernel. We study the following L p inequality for the Poisson integral P f(x, t) = (Pt ∗ f)(x) with respect to a Carleson measure µ: P f L p (R n+1 +,dµ) ≤ cp,n κ(µ) 1 p f  L p (R n,dx), where 1 < p < ∞ and κ(µ) is the Carleson norm of µ. It was shown by Verbitsky [V] that for p> 2 the constant cp,n can be taken to be independent of the dimension n. We show that c2,n = O((log n) 1 2) and that cp,n = O(n 1 p − 1 2) for 1 < p < 2 as n → ∞. We observe that standard proofs of this inequality rely on doubling properties of cubes and lead to a value of cp,n that grows exponentially with n. 1.
1.4 Energy estimates of higher order..................... 20
"... 2.4 Some SIO – type representation formulas and estimates........ 29 2.5 The estimates on ∇ρ and osc ρ...................... 36 2.6 Estimates on div w............................ 38 2.7 Dynamics of the interface and a priori estimates........... 40 2.8 Nondegeneracy of the contact angle............ ..."
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2.4 Some SIO – type representation formulas and estimates........ 29 2.5 The estimates on ∇ρ and osc ρ...................... 36 2.6 Estimates on div w............................ 38 2.7 Dynamics of the interface and a priori estimates........... 40 2.8 Nondegeneracy of the contact angle.................. 42 2.9 Hölder continuity of the ∇u(t) in Ω ± t.................. 42