Generalized Dirac Operators on Nonsmooth Manifolds and Maxwell's Equations.
Generalized Hardy operators and normalizing measures.
Generalized Hörmander conditions and weighted endpoint estimates
We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u,Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u,v) for the operators to be bounded from to...
Geometric Fourier analysis
In this paper we continue the study of the Fourier transform on , , analyzing the “almost-orthogonality” of the different directions of the space with respect to the Fourier transform. We prove two theorems: the first is related to an angular Littlewood-Paley square function, and we obtain estimates in terms of powers of , where is the number of equal angles considered in . The second is an extension of the Hardy-Littlewood maximal function when one consider cylinders of , , of fixed eccentricity...
Global estimates for singular integrals of the composition of the maximal operator and the Green's operator.
Global maximal estimates for solutions to the Schrödinger equation
Global maximal estimates are considered for solutions to an initial value problem for the Schrödinger equation.
Global mild solutions of the micropolar fluid system in critical spaces
We prove the global in time existence of a small solution for the 3D micropolar fluid system in critical Fourier-Herz spaces by using the Fourier localization method and Littlewood-Paley theory.
Global well-posedness for the 2-D Boussinesq system with temperature-dependent thermal diffusivity
We prove the global well-posedness of the 2-D Boussinesq system with temperature dependent thermal diffusivity and zero viscosity coefficient.
Good λ inequalities for the area integral and the nontangential maximal function
Good-λ inequalities for wavelets of compact support
For a wavelet ψ of compact support, we define a square function and a maximal function NΛ. We then obtain the equivalence of these functions for 0 < p < ∞. We show this equivalence by using good-λ inequalities.
, bmo, blo and Littlewood-Paley -functions with non-doubling measures.
spaces associated with Schrödinger operators with potentials from reverse Hölder classes
Let A = -Δ + V be a Schrödinger operator on , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of if the maximal function belongs to , where is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space admits a special atomic decomposition.
spaces over open subsets of
Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications
Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable...
Hardy spaces, A..., and singular integrals on chord-arc domains
Hardy spaces and the Dirichlet problem on Lipschitz domains.
Our concern in this paper is to describe a class of Hardy spaces Hp(D) for 1 ≤ p < 2 on a Lipschitz domain D ⊂ Rn when n ≥ 3, and a certain smooth counterpart of Hp(D) on Rn-1, by providing an atomic decomposition and a description of their duals.
Hardy's integral inequality for commutators of Hardy operators.
Harmonic analysis and the geometry of subsets of Rn.
This subject has several natural points of view, but we shall start with the one that corresponds to the following question: Is there something like Littlewood-Paley theory which is useful for analyzing the geometry of subsets of Rn, in much the same way that traditional Littlewood-Paley theory is good for analyzing functions and operators?
Harmonic analysis of the space BV.
We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is almost characterized by wavelet expansions in the following sense: if a function f is in BV, its coefficient sequence in a BV normalized wavelet basis satisfies a class of weak-l1 type estimates. These weak estimates can be employed to prove many interesting results. We use them to identify the interpolation spaces between BV and Sobolev...
Higher integrability for maximal oscillatory Fourier integrals.