Norm Inequalities For Operators Research Articles

A Covering Theorem with applications

We prove a Covering Theorem that allows us to prove modified norm inequalities for general maximal operators. We will also give applications to convergence of a sequence of linear operators and the differentiation of the integral.

Trace class norm inequalities for localization operators

We give sharp estimates on the norms in the trace class of localization operators in terms of their symbols.

An interpolation approach to Hardy–Littlewood inequalities for norms of operators on sequence spaces

We revisit estimates of Hardy and Littlewood for norms of operators on sequence spaces, and exploit recent advances in interpolation theory to provide relatively simple proofs.

Mixed Norm Inequalities for Directional Operators Associated to Potentials

We study mixed norm inequalities for directional operators which appear applying the method of rotations to homogeneous operators with variable kernel and with the homogeneity of Riesz potentials. The results are sharp for a range of values of the parameter and for all its values when the inequalities are restricted to radial functions.

Two-weight norm inequalities for maximal operators and fractional integrals on non-homogenous spaces

Let μ be a non-negative Borel measure on R d . Fix a real number n, 0 < n ≤ d, and assume that μ is n-dimensional in the following sense: the measure of a cube is smaller than the length of its side raised to the n-th power. CalderOn-Zygmund operators, Hardy and BMO spaces, and some other topics in Harmonic Analysis have been successfully handled in this setting recently, although the measure may be non-doubling. The aim of this paper is to study two-weight norm inequalities for radial fractional maximal functions associated to such p. Namely, we characterize those pairs of weights for which these maximal operators satisfy strong and weak type inequalities. Sawyer and radial Muckenhoupt type conditions are respectively the solutions for these problems. Furthermore, if we strengthen Muckenhoupt conditions by adding a power-bump to the right-hand side weight or even by introducing a certain Orlicz norm, strong type inequalities can be achieved. As a consequence, two-weight norm inequalities for fractional integrals associated to μ are obtained. Finally, for the particular case of the Hardy-Littlewood radial maximal function, we show how, in contrast with the classical situation, radial Muckenhoupt weights may fail to satisfy a reverse Holder's inequality and also strong type inequalities do not necessarily hold for them.

Oscillation inequalities for rectangles

In this paper we extend previously obtained results on LP norm inequalities (1 < p < oo) for square functions, oscillation and variation operators, with Z actions, to the case of Zd actions. The technique involves the use of a result about vector valued maximal functions, due to Fefferman and Stein, to reduce the problem to a situation where we can apply our previous results.

Positive Definite Functions and Operator Inequalities

We construct several examples of positive definite functions, and use the positive definite matrices arising from them to derive several inequalities for norms of operators. 1991 Mathematics Subject Classification 42A82, 47A63, 15A45, 15A60.

New Proofs of Two-Weight Norm Inequalities for the Maximal Operator

Abstract We give a new and simpler proof of Sawyer's theorem characterizing the weights governing the two-weight, strong-type norm inequality for the Hardy-Littlewood maximal operator and the fractional maximal operator. As a further application of our techniques, we give new proofs of two sufficient conditions for such weights due to Wheeden and Sawyer.

Some Norm Inequalities for Operators

AbstractLet Ai , Bi and Xi (i = 1, 2,…,n) be operators on a separable Hilbert space. It is shown that if f and g are nonnegative continuous functions on [0, ∞) which satisfy the relation f(t)g(t) = t for all t in [0, ∞), thenfor every r &gt; 0 and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.

Weighted norm inequalities for geometric fractional maximal operators

For 0 ≤α < ∞ let Tαf denote one of the operators $$M_{\alpha ,0} f(x) = \mathop {\sup }\limits_{I \mathrel\backepsilon x} \left| I \right|^\alpha \exp \left( {\frac{1}{{\left| I \right|}}\int_I {\log \left| f \right|} } \right),M_{\alpha ,0}^* f(x) = \mathop {\lim }\limits_{r \searrow 0} \mathop {\sup }\limits_{I \mathrel\backepsilon x} \left| I \right|^\alpha \left( {\frac{1}{{\left| I \right|}}\int_I {\left| f \right|^r } } \right)^{{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r}} .$$ We characterize the pairs of weights (u, v) for which Tα is a bounded operator from Lp(v) to Lq(u), 0 0 the norm inequalities for α=0 in [4, 16]. As an application we give lower bounds for convolutions ϕ ⋆ f, where ϕ is a radially decreasing function.

Sharp Two-weight, weak-type norm inequalities for singular integral operators

We give a sufficient condition for singular integral operators and, more generally, Calderon-Zygmund operators to satisfy the weak (p, p) inequality u({x ∈ R : |Tf(x)| > t}) ≤ C tp ∫ R n |f |pv dx, 1 0. This conditions is stronger than the Ap condition and is sharp since it fails when δ = 0.

Weighted Norm Inequalities for Fractional Integral Operators With Rough Kernel

AbstractGiven function Ω on ℝn , we define the fractional maximal operator and the fractional integral operator by and respectively, where 0 &lt; α &lt; n. In this paper we study the weighted norm inequalities of MΩα and TΩα for appropriate α, s and A(p, q) weights in the case that Ω∈ Ls(Sn-1)(s&gt; 1), homogeneous of degree zero.

Inequalities for norms of some integral operators

Let (A(a)u)(x) := ∫ a 0 (1 − xt)−1u(t) dt, 0 < a < 1 . Properties of the operators A(a) as a → 1 are studied. It is proved that A := A(1) is a bounded, positive self-adjoint operator in H = L2[0, 1] , ||A|| π , while A : C(0, 1) → C(0, 1) is unbounded. Mathematics subject classification (1991): 35R30.

Weighted Lorentz norm inequalities for general maximal operators associated with certain families of Borel measures

Given a certain family ℱ of positive Borel measures and γ ∈ [0, 1), we define a general onesided maximal operatorand we study weighted inequalities inLp,qspaces for these operators. Our results contain, as particular cases, the characterisation of weighted Lorentz norm inequalities for some well-known one-sided maximal operators such as the one-sided Hardy–Littlewood maximal operator associated with a general measure, the one-sided fractional maximal operatorand the maximal operatorassociated with the Cesèro-α averages.

Weighted norm inequalities for the geometric maximal operator

We consider two closely related but distinct operators, $$ \align M_0f(x)&= \sup_{I\ni x}\exp\left(\frac{1}{|I|}\int_I\log|f|\,dy\right) \quad\text{and}\\ M_0^*f(x) &= \lim_{r\rightarrow0} \sup_{I\ni x}\left(\frac{1}{|I|}\int_I|f|^r\,dy\right)^{1/r}. \endalign $$ We give sufficient conditions for the two operators to be equal and show that these conditions are sharp. We also prove two-weight, weighted norm inequalities for both operators using our earlier results about weighted norm inequalities for the minimal operator: $$ \text{\mgran{m}} f(x) = \inf_{I \ni x} \frac{1}{|I|}\int_I |f|\,dy. $$ This extends the work of X. Shi; H. Wei, S. Xianliang and S. Qiyu; X. Yin and B. Muckenhoupt; and C. Sbordone and I. Wik.

Weighted norm inequalities for integral operators

Weighted norm inequalities for integral operators

A Characterization of two Weight Norm Inequalities for One-Sided Operators of Fractional Type

AbstractIn this paper we give a characterization of the pairs of weights (ω, v) such that T maps Lp(v) into Lq(ω),where T is a general one-sided operator that includes as a particular case theWeyl fractional integral. As an applicationwe solve the following problem: given a weight v, when is there a nontrivial weight ω such that T maps Lp(v) into Lq(ω)?

A new proof of the two weight norm inequality for the one-sided fractional maximal operator

We give a new proof of the two weight norm inequality for the one-sided, fractional maximal operator, $M_\alpha ^+$, simplifying the original proof of Martín-Reyes and de la Torre.

Log-majorizations and norm inequalities for exponential operators

Concise but self-contained reviews are given on theories of majorization and symmetrically normed ideals, including the proofs of the Lidskii-Wielandt and the Gelfand-Naimark theorems. Based on these reviews, we discuss logarithmic majorizations and norm

Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.