Results 1  10
of
18
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 ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
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
The law of the iterated logarithm for Kleinian groups
"... Suppose G is an analytically finite, geometrically infinite Kleinian group and that there is a lower bound on the injectivity radius for M = B/G. Then the limit set has positive Hausdorff measure with respect to the gauge function '(t) = t 2 q log 1 t log log log 1 t : If, in addition, the group is ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Suppose G is an analytically finite, geometrically infinite Kleinian group and that there is a lower bound on the injectivity radius for M = B/G. Then the limit set has positive Hausdorff measure with respect to the gauge function '(t) = t 2 q log 1 t log log log 1 t : If, in addition, the group is topological tame and 6= S 2 , then the limit set has nite measure with respect to this gauge. We also obtain corollaries on the size of the conical limit set, the existence and ergodicity of conformal densities, and the differentiability of quasiconformal conjugacies.
On the small ball inequality in three dimensions
 Duke Math. J
"... Abstract. Let hR denote an L ∞ normalized Haar function adapted to a dyadic rectangle R⊂[0, 1] 3. We show that there is a positiveη< 1 2 so that for all integers n, and coefficients α(R) we have ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Abstract. Let hR denote an L ∞ normalized Haar function adapted to a dyadic rectangle R⊂[0, 1] 3. We show that there is a positiveη< 1 2 so that for all integers n, and coefficients α(R) we have
QUASILINEAR AND HESSIAN EQUATIONS OF LANE–EMDEN TYPE
, 2005
"... Abstract. 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 p ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. 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. 1.
On maximal functions for MikhlinHörmander multipliers
 Adv. Math
"... Abstract. Given MikhlinHörmander multipliers mi, i = 1,..., N, with uniform estimates we prove an optimal p log(N + 1) bound in L p for the maximal function sup i F −1 [mi b f]  and related bounds for maximal functions generated by dilations. These improve results in [7]. Given a symbol m satisfy ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Abstract. Given MikhlinHörmander multipliers mi, i = 1,..., N, with uniform estimates we prove an optimal p log(N + 1) bound in L p for the maximal function sup i F −1 [mi b f]  and related bounds for maximal functions generated by dilations. These improve results in [7]. Given a symbol m satisfying 1.
PoincaréSobolev And Isoperimetric Inequalities, Maximal Functions, And HalfSpace Estimates For The Gradient
"... this article). In addition, related to Lecture 2, the paper [FLW] concerning Poincar'e's inequality for vector fields of Hormander type now exists in preprint form. Lecture 1: Fractional maximal functions ([W1], [SWZ]) Consider the fractional maximal function on R ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
this article). In addition, related to Lecture 2, the paper [FLW] concerning Poincar'e's inequality for vector fields of Hormander type now exists in preprint form. Lecture 1: Fractional maximal functions ([W1], [SWZ]) Consider the fractional maximal function on R
Theory of Sobolev multipliers: with applications to differential and integral operators,
, 2010
"... This monumental book, to which for brevity we will be referring to as Sobolev Multipliers, was published by Springer in the prestigious series Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. It is devoted to a broadly developed theory of pointwise mul ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
This monumental book, to which for brevity we will be referring to as Sobolev Multipliers, was published by Springer in the prestigious series Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. It is devoted to a broadly developed theory of pointwise multipliers (or more precisely, multiplication operators) acting in pairs of Sobolev spaces, as well as other spaces of differentiable functions, along with their applications in analysis, partial differential equations, and mathematical physics. To distinguish pointwise multipliers from wellstudied Fourier multipliers and to emphasize the underlying class of function spaces, they are conveniently called Sobolev multipliers. The book under review consists of two parts. Part I presents a general theory of multipliers in pairs of Sobolev spaces and their generalizations, while Part II is concerned with applications to the Schrödinger operator and its relativistic counerpart, as well as to studies of solutions of certain elliptic partial differential equations, both linear and quasilinear, in divergence and nondivergence form, along with related pseudodifferential and integral equations. Questions of optimal regularity of the
Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator
"... Abstract. We give explicit analytic criteria for two problems associated with the Schrödinger operator H = − ∆ + Q on L 2 (R n) where Q ∈ D ′ (R n) is an arbitrary real or complexvalued potential. First, we obtain necessary and sufficient conditions on Q so that the quadratic form 〈Q·, · 〉 has zer ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. We give explicit analytic criteria for two problems associated with the Schrödinger operator H = − ∆ + Q on L 2 (R n) where Q ∈ D ′ (R n) is an arbitrary real or complexvalued potential. First, we obtain necessary and sufficient conditions on Q so that the quadratic form 〈Q·, · 〉 has zero relative bound with respect to the Laplacian. For Q ∈ L 1 loc (Rn), this property can be expressed in the form of the integral inequality: ∣ u(x)  2 Q(x)dx∣ ∣ ≤ ǫ ∇u2 L 2 (R n) + C(ǫ) u2 L 2 (R n) , ∀u ∈ C ∞ 0 (R n), Rn for an arbitrarily small ǫ> 0 and some C(ǫ)> 0. One of the major steps here is the reduction to a similar inequality with nonnegative function ∇(1−∆) −1 Q  2 +(1−∆) −1 Q  in place of Q. This provides a complete solution to the infinitesimal form boundedness problem for the Schrödinger operator, and leads to new broad classes of admissible distributional potentials Q, which extend the usual Lp and Kato classes, as well as those based on the wellknown conditions of Fefferman–Phong and Chang–Wilson–Wolff. Secondly, we characterize Trudinger’s subordination property where C(ǫ) in the above inequality is subject to the condition C(ǫ) ≤ c ǫ−β (β> 0) as ǫ → +0. Such quadratic form inequalities can be understood entirely in the framework of Morrey–Campanato spaces, using mean oscillations of ∇(1−∆) −1 Q and (1−∆) −1 Q on balls or cubes. A version of this condition where ǫ ∈ (0, +∞) is equivalent to the multiplicative inequality:
DECOMPOSITION OF THE DISTRIBUTION ON BMO SPACE
"... In this paper, we characterize − → f so that if the inequality − → ∣ f. (u∇v − v∇u) dx R d ≤ C ‖u ‖. 1 ‖v ‖. H H 1 holds for all u, v ∈ D ( R d) , then − → f can be represented in the form f = ∇g + Div H where g ∈ BMO ( R d) , H is a skewsymmetric matrix field such that H ∈ BMO ( R d) d 2 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
In this paper, we characterize − → f so that if the inequality − → ∣ f. (u∇v − v∇u) dx R d ≤ C ‖u ‖. 1 ‖v ‖. H H 1 holds for all u, v ∈ D ( R d) , then − → f can be represented in the form f = ∇g + Div H where g ∈ BMO ( R d) , H is a skewsymmetric matrix field such that H ∈ BMO ( R d) d 2