Vector-valued Inequalities Research Articles

Extrapolation of Weighted norm inequalities for multivariable operators and applications

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderon-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on l2-valued extensions of bounded linear operators is also obtained.

Extrapolation from A∞ weights and applications

We generalize the A p extrapolation theorem of Rubio de Francia to A ∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good- λ inequalities for proving L p norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calderón–Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good- λ inequalities, of Coifman's inequality relating singular integrals and the maximal operator, of the Fefferman–Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt–Wheeden inequality relating the fractional integral operator and the fractional maximal operator.

On the existence of principal values for the cauchy integral on weighted lebesgue spaces for non-doubling measures

Let T be a Calderon-Zygmund operator in a "non-homogeneous" space (X,d,µ), where, in particular, the measure µ may be non- doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil and A. Vol- berg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T?, which is the supremum of the trun- cated operators associated with T. Specifically, for 1 < p < 1, we obtain sucient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L p weighted inequality holds for T?. The main tool to deal with this problem is the theory of vector-valued inequalities for T? and some re- lated operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderon-Zygmund operators in non-homogeneous spaces, developed in our previous pa- per (GM). For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C? to characterize the existence of principal values for functions in weighted L p .

A vector-valued Grothendieck inequality with an application to (p, q)-completely bounded operators

A vector-valued Grothendieck inequality with an application to (p, q)-completely bounded operators

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

Abstract We give a characterization of the weights 𝑢(·) and 𝑣(·) for which the fractional maximal operator 𝑀𝑠 is bounded from the weighted Lebesgue spaces 𝐿𝑝(𝑙𝑟, 𝑣𝑑𝑥) into 𝐿𝑞(𝑙𝑟, 𝑢𝑑𝑥) whenever 0 ≤ 𝑠 &lt; 𝑛, 1 &lt; 𝑝, 𝑟 &lt; ∞, and 1 ≤ 𝑞 &lt; ∞.

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

We give a characterization of the weights u(A) and v(A) for which the fractional maximal operator Ms is bounded from the weighted Lebesgue spaces Lp(lr;vdx) into Lq(lr;udx) whenever 0 i s < n, 1 < p;r < 1, and 1 i q < 1.

Vector Valued Inequalities for Hankel Transforms

The disc multiplier may be seen as a vector valued operator when we consider its projections in terms of the spherical harmonics. We prove the boundedness of this operator, which in this form represents a vector valued Hankel Transform, on the spaces L l q p (rn−1 dr) when 2n/(n + 1) < p, q < 2n/(n − 1).

Spherical Summability and a Vector-Valued Inequality

Spherical Summability and a Vector-Valued Inequality

Vector-Valued Inequalities, Factorization, and Extrapolation for a Family of Rough Operators

We prove very general weighted norm inequalities for rough maximal and singular integral operators whose kernels satisfy some common Fourier transform decay estimates. Examples include homogeneous singular integral operators with kernels which do not necessarily satisfy a Dini condition as well as several operators whose measures are supported on lower dimensional sets, such as the discrete spherical maximal operator and the Hilbert transform along a homogeneous curve. The weights are seen to satisfy analogues of Jones′ factorization theorem and Rubio de Francia′s extrapolation theorem, and so are complete with respect to these important properties. We also obtain the corresponding weighted vector-valued inequalities for these operators and for the maximal operator associated with a starlike set.

Vector-valued inequalities with weights

This paper deals with the following problem: Let T be a given operator. Find conditions on v(x) (resp. u(x)) such that ? |Tf(x)|pu(x) dx = C ? |f(x)|pv(x) dx is satisfied for some u(x) (resp. v(x)). Using vector-valued inequalities the problem is solved for: Carleson's maximal operator of Fourier partial sums, Littlewood-Paley square functions, Hilbert transform of functions valued in U.M.D. Banach spaces and operators in the upper-half plane.

On vector-valued inequalities for Sidon sets and sets of interpolation

Let E be a Sidon subset of the integers and suppose X is a Banach space. Then Pisier has shown that E-spectral polynomials with values in X behave like Rademacher sums with respect to Lp-norms. We consider the situation when X is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if E is a set of interpolation (I0-set). However, for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if X is restricted to be “natural” then the result holds for all Sidon sets. We also consider spaces with plurisubharmonic norms and introduce the class of analytic Sidon sets.

Almost everywhere convergence of Vilenkin-Fourier series of 𝐻¹ functions

In [5] Ladhawala and Pankratz proved that if f f is in dyadic H 1 {H^1} , then any lacunary sequence of partial sums of the Walsh-Fourier series of f f converges a.e. We generalize their theorem to Vilenkin-Fourier series. In obtaining this result, we prove a vector-valued inequality for the partial sums of Vilenkin-Fourier series.

On restriction theorems of maximal-type

In this paper we prove a vector-valued restriction theorem for the Fourier transform. This result allows us to establish certain sharp restriction theorems and a partial generalization of the Riemann-Lebesgue lemma for certain p > 1. 1. Introduction. Vector-valued inequalities for Bochner-Riesz multipliers have been known for some time in two dimensions and for certain exponents in higher dimensions (cf. [2], [3], [4]). In this note our main result is the corresponding vector-valued restriction theorem for the Fourier transform which is valid everywhere the restriction problem is now known to be true. In two dimensions the result is proved by using the techniques of oscillatory integrals while in higher dimensions our inequalities follow trivially from known ones. Finally, using these results we then obtain a lacunary restriction theorem of maximal-type for the Fourier transform.

Weighted and Vector-Valued Inequalities for Potential Operators

Weighted and Vector-Valued Inequalities for Potential Operators

Weighted and vector-valued inequalities for potential operators

In this paper we develop some aspect of a general theory parallel to the Calderón-Zygmund theory for operator valued kernels, where the operators considered map functions defined on ${R^n}$ into functions defined on $R_ + ^{n + 1} = {R^n} \times [0,\infty )$. In particular, we apply the obtained results to get vector-valued inequalities for the Poisson integral and fractional integrals. Some weighted norm inequalities are also considered for fractional integrals.

On systems of vector-valued linear inequalities

where ~~~~~~~ is an indexed set of elements of E and (a,},, , is a corresponding set of real numbers. The set I of indices is of arbitrary cardinality. The system (1) is said to be consistent, if there exists an f~ E satisfying (1). In [2] a consistency condition for the system (1) and a duality theorem for a linear extremum problem with the system (1) as constraints are obtained in terms of the closed convex cone in E x R spanned by the set ((xv, ~,)),EI. We shall deal with systems of linear inequalities of the form

Vector Valued Inequalities for Fourier Series

Vector Valued Inequalities for Fourier Series

Vector-valued inequalities for Fourier series

Denoting by S ∗ {S^\ast } the maximal partial sum operator of Fourier series, we prove that S ∗ ( f 1 , f 2 , … , f k , … ) = ( S ∗ f 1 , S ∗ f 2 , … , S ∗ f k , … ) {S^\ast }({f_1},{f_2}, \ldots ,{f_k}, \ldots ) = ({S^\ast }{f_1},{S^\ast }{f_2}, \ldots ,{S^\ast }{f_k}, \ldots ) is a bounded operator from L p ( l r ) {L^p}({l^r}) to itself, 1 &gt; p , r &gt; ∞ 1 &gt; p,r &gt; \infty . Thus, we extend the theorem of Carleson and Hunt on pointwise convergence of Fourier series to the case of vector valued functions. We give also an application to the rectangular convergence of double Fourier series.