Functions Of Bounded Mean Oscillation Research Articles

Variational inequalities for the commutators of rough operators with BMO functions

Starting with the relatively simple observation that the variational estimates of the commutators of the standard Calderón-Zygmund operators with the bounded mean oscillation (BMO) functions can be obtained from their weighted variational estimates, we establish similar variational estimates for the commutators of the BMO functions with rough singular integrals, which do not admit any weighted variational estimates. The proof involves several Littlewood-Paley-type inequalities with the commutators as well as Bony decomposition and related para-product estimates.

Characterization of BMO using Wavelets through Triebel-Lizorkin spaces

In the present article it is presented a characterization of all those functions in the space of bounded mean oscillation functions, BMO, in terms of an appropriate wavelet, using an isomorphism between the aforementioned space and the homogeneous space of Triebel-Lizorkin F˙ 0,2∞ . In addition, a new inequality that involves the vector inequality of the maximal function of Hardy-Littlewood is prov

Multilinear Calderón-Zygmund operators and their commutators with BMO functions in variable exponent Morrey spaces

The boundedness of multilinear Calderon-Zygmund operators and their commutators with bounded mean oscillation (BMO) functions in variable exponent Morrey spaces are obtained.

Relation Between BMO and A2 Weight Functions

In this paper, we relate Bounded Mean Oscillation (BMO) function and A2 weight function. We show that logarithm of any A2 function is a BMO function and every BMO function is equal to a constant multiple of the logarithm of an A2 weight function. Moreover, we show that logarithm of any Ap weight function for 1 < p < ∞ is a BMO function.

Weighted Norm Inequalities on Graphs

Let (Γ,μ) be an infinite graph endowed with a reversible Markov kernel p and let P be the corresponding operator. We also consider the associated discrete gradient ∇. We assume that μ is doubling, a uniform lower bound for p(x,y) when p(x,y)>0, and gaussian upper estimates for the iterates of p. Under these conditions (and in some cases assuming further some Poincare inequality) we study the comparability of (I−P)1/2 f and ∇f in Lebesgue spaces with Muckenhoupt weights. Also, we establish weighted norm inequalities for a Littlewood–Paley–Stein square function, its formal adjoint, and commutators of the Riesz transform with bounded mean oscillation functions.

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good- λ inequality with two parameters and the other uses Calderón–Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all L p spaces for 1 < p < ∞ . Pointwise estimates are then replaced by appropriate localized L p – L q estimates. We obtain weighted L p estimates for a range of p that is different from ( 1 , ∞ ) and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.

𝐴_{𝑝} weights for nondoubling measures in 𝑅ⁿ and applications

We study an analogue of the classical theory of A p ( μ ) A_p(\mu ) weights in R n \mathbb {R}^n without assuming that the underlying measure μ \mu is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón- Zygmund operators with bounded mean oscillation functions ( B M O BMO ), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611–635. Finally, we study self–improving properties of Poincaré–B.M.O. type inequalities within this context; more precisely, we show that if f f is a locally integrable function satisfying 1 μ ( Q ) ∫ Q | f − f Q | d μ ≤ a ( Q ) \frac {1}{\mu (Q)}\int _{Q} |f-f_{Q}| d\mu \le a(Q) for all cubes Q Q , then it is possible to deduce a higher L p L^p integrability result for f f , assuming a certain simple geometric condition on the functional a a .

A similarity principle for locally solvable vector fields

This paper establishes a weak similarity principle for the class of locally solvable complex vector fields in the plane. The main tool is a local solvability result in an appropriate space of bounded mean oscillation functions.

Factorization Theorems for Hardy Spaces in Several Variables

The purpose of this paper is to extend to Hardy spaces in several variables certain well known factorization theorems on the unit disk. The extensions will be carried out for various spaces of holomorphic functions on the unit ball of C as well as for Hardy spaces defined by the Riesz systems on R. These results together with their proofs yield new characterizations of the space BMO (Bounded Mean Oscillation) and show a close relationship between BMO functions and certain linear operators on various LI and H2 spaces. The main tools are the result of Fefferman and Stein [8] on the duality between HI and BMO and a new characterization of BMO in terms of boundedness on L2 of the commutator of a singular integral operator with a multiplication operator. We begin by illustrating these ideas in the one dimensional case: Let F be holomorphic in {I z I < 1} and satisfy sup, 5 F(rete) I dO ? 1 (i.e., F is in H'(dO)). It is well known that F = GG2 with G1, G2 holomorphic and sup, I G,(rel0) 1' ! 1 (i.e., G, e H2(dO)). Write F = f + if, G, = gj + ig withf, g1, g, real. Thenf = Im(GG2) = sg1 1 + gi. Thusafunction f is an imaginary (or real) part of an HI function if and only if it can be represented as glg2 + g192 for L2 functions g, and g2. Furthermore,